If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.
Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the evenvalued terms.
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?
A palindromic number reads the same both ways. The largest palindrome made from the product of two 2digit numbers is 9009 = 91 × 99.
Find the largest palindrome made from the product of two 3digit numbers.
2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.
What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?
The sum of the squares of the first ten natural numbers is,
$$1^2 + 2^2 + ... + 10^2 = 385$$The square of the sum of the first ten natural numbers is,
$$(1 + 2 + ... + 10)^2 = 55^2 = 3025$$Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is $3025  385 = 2640$.
Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
What is the 10 001st prime number?
The four adjacent digits in the 1000digit number that have the greatest product are 9 × 9 × 8 × 9 = 5832.
73167176531330624919225119674426574742355349194934
96983520312774506326239578318016984801869478851843
85861560789112949495459501737958331952853208805511
12540698747158523863050715693290963295227443043557
66896648950445244523161731856403098711121722383113
62229893423380308135336276614282806444486645238749
30358907296290491560440772390713810515859307960866
70172427121883998797908792274921901699720888093776
65727333001053367881220235421809751254540594752243
52584907711670556013604839586446706324415722155397
53697817977846174064955149290862569321978468622482
83972241375657056057490261407972968652414535100474
82166370484403199890008895243450658541227588666881
16427171479924442928230863465674813919123162824586
17866458359124566529476545682848912883142607690042
24219022671055626321111109370544217506941658960408
07198403850962455444362981230987879927244284909188
84580156166097919133875499200524063689912560717606
05886116467109405077541002256983155200055935729725
71636269561882670428252483600823257530420752963450
Find the thirteen adjacent digits in the 1000digit number that have the greatest product. What is the value of this product?
A Pythagorean triplet is a set of three natural numbers, a < b < c, for which,
For example, 3^{2} + 4^{2} = 9 + 16 = 25 = 5^{2}.
There exists exactly one Pythagorean triplet for which a + b + c = 1000.
Find the product abc.
The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million.
In the 20×20 grid below, four numbers along a diagonal line have been marked in red.
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91
22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80
24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50
32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70
67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21
24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72
21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95
78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92
16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57
86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58
19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40
04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66
88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69
04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36
20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16
20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48
The product of these numbers is 26 × 63 × 78 × 14 = 1788696.
What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20×20 grid?
The sequence of triangle numbers is generated by adding the natural numbers. So the 7^{th} triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
Let us list the factors of the first seven triangle numbers:
1: 1
3: 1,3
6: 1,2,3,6
10: 1,2,5,10
15: 1,3,5,15
21: 1,3,7,21
28: 1,2,4,7,14,28
We can see that 28 is the first triangle number to have over five divisors.
What is the value of the first triangle number to have over five hundred divisors?
Work out the first ten digits of the sum of the following onehundred 50digit numbers.
46376937677490009712648124896970078050417018260538
74324986199524741059474233309513058123726617309629
91942213363574161572522430563301811072406154908250
23067588207539346171171980310421047513778063246676
89261670696623633820136378418383684178734361726757
28112879812849979408065481931592621691275889832738
44274228917432520321923589422876796487670272189318
47451445736001306439091167216856844588711603153276
70386486105843025439939619828917593665686757934951
62176457141856560629502157223196586755079324193331
64906352462741904929101432445813822663347944758178
92575867718337217661963751590579239728245598838407
58203565325359399008402633568948830189458628227828
80181199384826282014278194139940567587151170094390
35398664372827112653829987240784473053190104293586
86515506006295864861532075273371959191420517255829
71693888707715466499115593487603532921714970056938
54370070576826684624621495650076471787294438377604
53282654108756828443191190634694037855217779295145
36123272525000296071075082563815656710885258350721
45876576172410976447339110607218265236877223636045
17423706905851860660448207621209813287860733969412
81142660418086830619328460811191061556940512689692
51934325451728388641918047049293215058642563049483
62467221648435076201727918039944693004732956340691
15732444386908125794514089057706229429197107928209
55037687525678773091862540744969844508330393682126
18336384825330154686196124348767681297534375946515
80386287592878490201521685554828717201219257766954
78182833757993103614740356856449095527097864797581
16726320100436897842553539920931837441497806860984
48403098129077791799088218795327364475675590848030
87086987551392711854517078544161852424320693150332
59959406895756536782107074926966537676326235447210
69793950679652694742597709739166693763042633987085
41052684708299085211399427365734116182760315001271
65378607361501080857009149939512557028198746004375
35829035317434717326932123578154982629742552737307
94953759765105305946966067683156574377167401875275
88902802571733229619176668713819931811048770190271
25267680276078003013678680992525463401061632866526
36270218540497705585629946580636237993140746255962
24074486908231174977792365466257246923322810917141
91430288197103288597806669760892938638285025333403
34413065578016127815921815005561868836468420090470
23053081172816430487623791969842487255036638784583
11487696932154902810424020138335124462181441773470
63783299490636259666498587618221225225512486764533
67720186971698544312419572409913959008952310058822
95548255300263520781532296796249481641953868218774
76085327132285723110424803456124867697064507995236
37774242535411291684276865538926205024910326572967
23701913275725675285653248258265463092207058596522
29798860272258331913126375147341994889534765745501
18495701454879288984856827726077713721403798879715
38298203783031473527721580348144513491373226651381
34829543829199918180278916522431027392251122869539
40957953066405232632538044100059654939159879593635
29746152185502371307642255121183693803580388584903
41698116222072977186158236678424689157993532961922
62467957194401269043877107275048102390895523597457
23189706772547915061505504953922979530901129967519
86188088225875314529584099251203829009407770775672
11306739708304724483816533873502340845647058077308
82959174767140363198008187129011875491310547126581
97623331044818386269515456334926366572897563400500
42846280183517070527831839425882145521227251250327
55121603546981200581762165212827652751691296897789
32238195734329339946437501907836945765883352399886
75506164965184775180738168837861091527357929701337
62177842752192623401942399639168044983993173312731
32924185707147349566916674687634660915035914677504
99518671430235219628894890102423325116913619626622
73267460800591547471830798392868535206946944540724
76841822524674417161514036427982273348055556214818
97142617910342598647204516893989422179826088076852
87783646182799346313767754307809363333018982642090
10848802521674670883215120185883543223812876952786
71329612474782464538636993009049310363619763878039
62184073572399794223406235393808339651327408011116
66627891981488087797941876876144230030984490851411
60661826293682836764744779239180335110989069790714
85786944089552990653640447425576083659976645795096
66024396409905389607120198219976047599490197230297
64913982680032973156037120041377903785566085089252
16730939319872750275468906903707539413042652315011
94809377245048795150954100921645863754710598436791
78639167021187492431995700641917969777599028300699
15368713711936614952811305876380278410754449733078
40789923115535562561142322423255033685442488917353
44889911501440648020369068063960672322193204149535
41503128880339536053299340368006977710650566631954
81234880673210146739058568557934581403627822703280
82616570773948327592232845941706525094512325230608
22918802058777319719839450180888072429661980811197
77158542502016545090413245809786882778948721859617
72107838435069186155435662884062257473692284509516
20849603980134001723930671666823555245252804609722
53503534226472524250874054075591789781264330331690
The following iterative sequence is defined for the set of positive integers:
n → n/2 (n is even)
n → 3n + 1 (n is odd)
Using the rule above and starting with 13, we generate the following sequence:
It can be seen that this sequence (starting at 13 and finishing at 1) contains 10 terms. Although it has not been proved yet (Collatz Problem), it is thought that all starting numbers finish at 1.
Which starting number, under one million, produces the longest chain?
NOTE: Once the chain starts the terms are allowed to go above one million.
Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner.
How many such routes are there through a 20×20 grid?
2^{15} = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.
What is the sum of the digits of the number 2^{1000}?
If the numbers 1 to 5 are written out in words: one, two, three, four, five, then there are 3 + 3 + 5 + 4 + 4 = 19 letters used in total.
If all the numbers from 1 to 1000 (one thousand) inclusive were written out in words, how many letters would be used?
NOTE: Do not count spaces or hyphens. For example, 342 (three hundred and fortytwo) contains 23 letters and 115 (one hundred and fifteen) contains 20 letters. The use of "and" when writing out numbers is in compliance with British usage.
By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.
3
7 4
2 4 6
8 5 9 3
That is, 3 + 7 + 4 + 9 = 23.
Find the maximum total from top to bottom of the triangle below:
75
95 64
17 47 82
18 35 87 10
20 04 82 47 65
19 01 23 75 03 34
88 02 77 73 07 63 67
99 65 04 28 06 16 70 92
41 41 26 56 83 40 80 70 33
41 48 72 33 47 32 37 16 94 29
53 71 44 65 25 43 91 52 97 51 14
70 11 33 28 77 73 17 78 39 68 17 57
91 71 52 38 17 14 91 43 58 50 27 29 48
63 66 04 68 89 53 67 30 73 16 69 87 40 31
04 62 98 27 23 09 70 98 73 93 38 53 60 04 23
NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing onehundred rows; it cannot be solved by brute force, and requires a clever method! ;o)
You are given the following information, but you may prefer to do some research for yourself.
 1 Jan 1900 was a Monday.
 Thirty days has September,
April, June and November.
All the rest have thirtyone,
Saving February alone,
Which has twentyeight, rain or shine.
And on leap years, twentynine.  A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400.
How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?
n! means n × (n − 1) × ... × 3 × 2 × 1
For example, 10! = 10 × 9 × ... × 3 × 2 × 1 = 3628800,
and the sum of the digits in the number 10! is 3 + 6 + 2 + 8 + 8 + 0 + 0 = 27.
Find the sum of the digits in the number 100!
Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are called amicable numbers.
For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.
Evaluate the sum of all the amicable numbers under 10000.
Using names.txt (right click and 'Save Link/Target As...'), a 46K text file containing over fivethousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position in the list to obtain a name score.
For example, when the list is sorted into alphabetical order, COLIN, which is worth 3 + 15 + 12 + 9 + 14 = 53, is the 938th name in the list. So, COLIN would obtain a score of 938 × 53 = 49714.
What is the total of all the name scores in the file?
A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.
A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.
As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.
Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.
A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:
012 021 102 120 201 210
What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?
The Fibonacci sequence is defined by the recurrence relation:
F_{n} = F_{n−1} + F_{n−2}, where F_{1} = 1 and F_{2} = 1.
Hence the first 12 terms will be:
F_{1} = 1
F_{2} = 1
F_{3} = 2
F_{4} = 3
F_{5} = 5
F_{6} = 8
F_{7} = 13
F_{8} = 21
F_{9} = 34
F_{10} = 55
F_{11} = 89
F_{12} = 144
The 12th term, F_{12}, is the first term to contain three digits.
What is the index of the first term in the Fibonacci sequence to contain 1000 digits?
A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:
^{1}/_{2} = 0.5 ^{1}/_{3} = 0.(3) ^{1}/_{4} = 0.25 ^{1}/_{5} = 0.2 ^{1}/_{6} = 0.1(6) ^{1}/_{7} = 0.(142857) ^{1}/_{8} = 0.125 ^{1}/_{9} = 0.(1) ^{1}/_{10} = 0.1
Where 0.1(6) means 0.166666..., and has a 1digit recurring cycle. It can be seen that ^{1}/_{7} has a 6digit recurring cycle.
Find the value of d < 1000 for which ^{1}/_{d} contains the longest recurring cycle in its decimal fraction part.
Euler discovered the remarkable quadratic formula:
$n^2 + n + 41$
It turns out that the formula will produce 40 primes for the consecutive integer values $0 \le n \le 39$. However, when $n = 40, 40^2 + 40 + 41 = 40(40 + 1) + 41$ is divisible by 41, and certainly when $n = 41, 41^2 + 41 + 41$ is clearly divisible by 41.
The incredible formula $n^2  79n + 1601$ was discovered, which produces 80 primes for the consecutive values $0 \le n \le 79$. The product of the coefficients, −79 and 1601, is −126479.
Considering quadratics of the form:
$n^2 + an + b$, where $a < 1000$ and $b \le 1000$where $n$ is the modulus/absolute value of $n$
e.g. $11 = 11$ and $4 = 4$
Find the product of the coefficients, $a$ and $b$, for the quadratic expression that produces the maximum number of primes for consecutive values of $n$, starting with $n = 0$.
Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:
21 22 23 24 25
20 7 8 9 10
19 6 1 2 11
18 5 4 3 12
17 16 15 14 13
It can be verified that the sum of the numbers on the diagonals is 101.
What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?
Consider all integer combinations of a^{b} for 2 ≤ a ≤ 5 and 2 ≤ b ≤ 5:
2^{2}=4, 2^{3}=8, 2^{4}=16, 2^{5}=32
3^{2}=9, 3^{3}=27, 3^{4}=81, 3^{5}=243
4^{2}=16, 4^{3}=64, 4^{4}=256, 4^{5}=1024
5^{2}=25, 5^{3}=125, 5^{4}=625, 5^{5}=3125
If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:
4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125
How many distinct terms are in the sequence generated by a^{b} for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?
Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:
1634 = 1^{4} + 6^{4} + 3^{4} + 4^{4}
8208 = 8^{4} + 2^{4} + 0^{4} + 8^{4}
9474 = 9^{4} + 4^{4} + 7^{4} + 4^{4}
As 1 = 1^{4} is not a sum it is not included.
The sum of these numbers is 1634 + 8208 + 9474 = 19316.
Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.
In the United Kingdom the currency is made up of pound (£) and pence (p). There are eight coins in general circulation:
1p, 2p, 5p, 10p, 20p, 50p, £1 (100p), and £2 (200p).
It is possible to make £2 in the following way:
1×£1 + 1×50p + 2×20p + 1×5p + 1×2p + 3×1p
How many different ways can £2 be made using any number of coins?
We shall say that an ndigit number is pandigital if it makes use of all the digits 1 to n exactly once; for example, the 5digit number, 15234, is 1 through 5 pandigital.
The product 7254 is unusual, as the identity, 39 × 186 = 7254, containing multiplicand, multiplier, and product is 1 through 9 pandigital.
Find the sum of all products whose multiplicand/multiplier/product identity can be written as a 1 through 9 pandigital.
The fraction ^{49}/_{98} is a curious fraction, as an inexperienced mathematician in attempting to simplify it may incorrectly believe that ^{49}/_{98} = ^{4}/_{8}, which is correct, is obtained by cancelling the 9s.
We shall consider fractions like, ^{30}/_{50} = ^{3}/_{5}, to be trivial examples.
There are exactly four nontrivial examples of this type of fraction, less than one in value, and containing two digits in the numerator and denominator.
If the product of these four fractions is given in its lowest common terms, find the value of the denominator.
145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.
Find the sum of all numbers which are equal to the sum of the factorial of their digits.
Note: as 1! = 1 and 2! = 2 are not sums they are not included.
The number, 197, is called a circular prime because all rotations of the digits: 197, 971, and 719, are themselves prime.
There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, and 97.
How many circular primes are there below one million?
The decimal number, 585 = 1001001001_{2} (binary), is palindromic in both bases.
Find the sum of all numbers, less than one million, which are palindromic in base 10 and base 2.
(Please note that the palindromic number, in either base, may not include leading zeros.)
The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.
Find the sum of the only eleven primes that are both truncatable from left to right and right to left.
NOTE: 2, 3, 5, and 7 are not considered to be truncatable primes.
Take the number 192 and multiply it by each of 1, 2, and 3:
192 × 1 = 192
192 × 2 = 384
192 × 3 = 576
By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3)
The same can be achieved by starting with 9 and multiplying by 1, 2, 3, 4, and 5, giving the pandigital, 918273645, which is the concatenated product of 9 and (1,2,3,4,5).
What is the largest 1 to 9 pandigital 9digit number that can be formed as the concatenated product of an integer with (1,2, ... , n) where n > 1?
If p is the perimeter of a right angle triangle with integral length sides, {a,b,c}, there are exactly three solutions for p = 120.
{20,48,52}, {24,45,51}, {30,40,50}
For which value of p ≤ 1000, is the number of solutions maximised?
An irrational decimal fraction is created by concatenating the positive integers:
0.123456789101112131415161718192021...
It can be seen that the 12^{th} digit of the fractional part is 1.
If d_{n} represents the n^{th} digit of the fractional part, find the value of the following expression.
d_{1} × d_{10} × d_{100} × d_{1000} × d_{10000} × d_{100000} × d_{1000000}
We shall say that an ndigit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4digit pandigital and is also prime.
What is the largest ndigit pandigital prime that exists?
The n^{th} term of the sequence of triangle numbers is given by, t_{n} = ½n(n+1); so the first ten triangle numbers are:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is 19 + 11 + 25 = 55 = t_{10}. If the word value is a triangle number then we shall call the word a triangle word.
Using words.txt (right click and 'Save Link/Target As...'), a 16K text file containing nearly twothousand common English words, how many are triangle words?
The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting substring divisibility property.
Let d_{1} be the 1^{st} digit, d_{2} be the 2^{nd} digit, and so on. In this way, we note the following:
 d_{2}d_{3}d_{4}=406 is divisible by 2
 d_{3}d_{4}d_{5}=063 is divisible by 3
 d_{4}d_{5}d_{6}=635 is divisible by 5
 d_{5}d_{6}d_{7}=357 is divisible by 7
 d_{6}d_{7}d_{8}=572 is divisible by 11
 d_{7}d_{8}d_{9}=728 is divisible by 13
 d_{8}d_{9}d_{10}=289 is divisible by 17
Find the sum of all 0 to 9 pandigital numbers with this property.
Pentagonal numbers are generated by the formula, P_{n}=n(3n−1)/2. The first ten pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ...
It can be seen that P_{4} + P_{7} = 22 + 70 = 92 = P_{8}. However, their difference, 70 − 22 = 48, is not pentagonal.
Find the pair of pentagonal numbers, P_{j} and P_{k}, for which their sum and difference are pentagonal and D = P_{k} − P_{j} is minimised; what is the value of D?
Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
Triangle  T_{n}=n(n+1)/2  1, 3, 6, 10, 15, ...  
Pentagonal  P_{n}=n(3n−1)/2  1, 5, 12, 22, 35, ...  
Hexagonal  H_{n}=n(2n−1)  1, 6, 15, 28, 45, ... 
It can be verified that T_{285} = P_{165} = H_{143} = 40755.
Find the next triangle number that is also pentagonal and hexagonal.
It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
9 = 7 + 2×1^{2}
15 = 7 + 2×2^{2}
21 = 3 + 2×3^{2}
25 = 7 + 2×3^{2}
27 = 19 + 2×2^{2}
33 = 31 + 2×1^{2}
It turns out that the conjecture was false.
What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
The first two consecutive numbers to have two distinct prime factors are:
14 = 2 × 7
15 = 3 × 5
The first three consecutive numbers to have three distinct prime factors are:
644 = 2² × 7 × 23
645 = 3 × 5 × 43
646 = 2 × 17 × 19.
Find the first four consecutive integers to have four distinct prime factors each. What is the first of these numbers?
The series, 1^{1} + 2^{2} + 3^{3} + ... + 10^{10} = 10405071317.
Find the last ten digits of the series, 1^{1} + 2^{2} + 3^{3} + ... + 1000^{1000}.
The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4digit numbers are permutations of one another.
There are no arithmetic sequences made up of three 1, 2, or 3digit primes, exhibiting this property, but there is one other 4digit increasing sequence.
What 12digit number do you form by concatenating the three terms in this sequence?
The prime 41, can be written as the sum of six consecutive primes:
This is the longest sum of consecutive primes that adds to a prime below onehundred.
The longest sum of consecutive primes below onethousand that adds to a prime, contains 21 terms, and is equal to 953.
Which prime, below onemillion, can be written as the sum of the most consecutive primes?
By replacing the 1^{st} digit of the 2digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.
By replacing the 3^{rd} and 4^{th} digits of 56**3 with the same digit, this 5digit number is the first example having seven primes among the ten generated numbers, yielding the family: 56003, 56113, 56333, 56443, 56663, 56773, and 56993. Consequently 56003, being the first member of this family, is the smallest prime with this property.
Find the smallest prime which, by replacing part of the number (not necessarily adjacent digits) with the same digit, is part of an eight prime value family.
It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different order.
Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits.
There are exactly ten ways of selecting three from five, 12345:
123, 124, 125, 134, 135, 145, 234, 235, 245, and 345
In combinatorics, we use the notation, $\displaystyle \binom 5 3 = 10$.
In general, $\displaystyle \binom n r = \dfrac{n!}{r!(nr)!}$, where $r \le n$, $n! = n \times (n1) \times ... \times 3 \times 2 \times 1$, and $0! = 1$.
It is not until $n = 23$, that a value exceeds onemillion: $\displaystyle \binom {23} {10} = 1144066$.
How many, not necessarily distinct, values of $\displaystyle \binom n r$ for $1 \le n \le 100$, are greater than onemillion?
In the card game poker, a hand consists of five cards and are ranked, from lowest to highest, in the following way:
 High Card: Highest value card.
 One Pair: Two cards of the same value.
 Two Pairs: Two different pairs.
 Three of a Kind: Three cards of the same value.
 Straight: All cards are consecutive values.
 Flush: All cards of the same suit.
 Full House: Three of a kind and a pair.
 Four of a Kind: Four cards of the same value.
 Straight Flush: All cards are consecutive values of same suit.
 Royal Flush: Ten, Jack, Queen, King, Ace, in same suit.
The cards are valued in the order:
2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace.
If two players have the same ranked hands then the rank made up of the highest value wins; for example, a pair of eights beats a pair of fives (see example 1 below). But if two ranks tie, for example, both players have a pair of queens, then highest cards in each hand are compared (see example 4 below); if the highest cards tie then the next highest cards are compared, and so on.
Consider the following five hands dealt to two players:
Hand  Player 1  Player 2  Winner  
1  5H 5C 6S 7S KD Pair of Fives  2C 3S 8S 8D TD Pair of Eights  Player 2  
2  5D 8C 9S JS AC Highest card Ace  2C 5C 7D 8S QH Highest card Queen  Player 1  
3  2D 9C AS AH AC Three Aces  3D 6D 7D TD QD Flush with Diamonds  Player 2  
4  4D 6S 9H QH QC Pair of Queens Highest card Nine  3D 6D 7H QD QS Pair of Queens Highest card Seven  Player 1  
5  2H 2D 4C 4D 4S Full House With Three Fours  3C 3D 3S 9S 9D Full House with Three Threes  Player 1 
The file, poker.txt, contains onethousand random hands dealt to two players. Each line of the file contains ten cards (separated by a single space): the first five are Player 1's cards and the last five are Player 2's cards. You can assume that all hands are valid (no invalid characters or repeated cards), each player's hand is in no specific order, and in each hand there is a clear winner.
How many hands does Player 1 win?
If we take 47, reverse and add, 47 + 74 = 121, which is palindromic.
Not all numbers produce palindromes so quickly. For example,
349 + 943 = 1292,
1292 + 2921 = 4213
4213 + 3124 = 7337
That is, 349 took three iterations to arrive at a palindrome.
Although no one has proved it yet, it is thought that some numbers, like 196, never produce a palindrome. A number that never forms a palindrome through the reverse and add process is called a Lychrel number. Due to the theoretical nature of these numbers, and for the purpose of this problem, we shall assume that a number is Lychrel until proven otherwise. In addition you are given that for every number below tenthousand, it will either (i) become a palindrome in less than fifty iterations, or, (ii) no one, with all the computing power that exists, has managed so far to map it to a palindrome. In fact, 10677 is the first number to be shown to require over fifty iterations before producing a palindrome: 4668731596684224866951378664 (53 iterations, 28digits).
Surprisingly, there are palindromic numbers that are themselves Lychrel numbers; the first example is 4994.
How many Lychrel numbers are there below tenthousand?
NOTE: Wording was modified slightly on 24 April 2007 to emphasise the theoretical nature of Lychrel numbers.
A googol (10^{100}) is a massive number: one followed by onehundred zeros; 100^{100} is almost unimaginably large: one followed by twohundred zeros. Despite their size, the sum of the digits in each number is only 1.
Considering natural numbers of the form, a^{b}, where a, b < 100, what is the maximum digital sum?
It is possible to show that the square root of two can be expressed as an infinite continued fraction.
$\sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}}$
By expanding this for the first four iterations, we get:
$1 + \frac 1 2 = \frac 32 = 1.5$
$1 + \frac 1 {2 + \frac 1 2} = \frac 7 5 = 1.4$
$1 + \frac 1 {2 + \frac 1 {2+\frac 1 2}} = \frac {17}{12} = 1.41666 \dots$
$1 + \frac 1 {2 + \frac 1 {2+\frac 1 {2+\frac 1 2}}} = \frac {41}{29} = 1.41379 \dots$
The next three expansions are $\frac {99}{70}$, $\frac {239}{169}$, and $\frac {577}{408}$, but the eighth expansion, $\frac {1393}{985}$, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.
In the first onethousand expansions, how many fractions contain a numerator with more digits than the denominator?
Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed.
37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18 5 4 3 12 29
40 19 6 1 2 11 28
41 20 7 8 9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49
It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 ≈ 62%.
If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%?
Each character on a computer is assigned a unique code and the preferred standard is ASCII (American Standard Code for Information Interchange). For example, uppercase A = 65, asterisk (*) = 42, and lowercase k = 107.
A modern encryption method is to take a text file, convert the bytes to ASCII, then XOR each byte with a given value, taken from a secret key. The advantage with the XOR function is that using the same encryption key on the cipher text, restores the plain text; for example, 65 XOR 42 = 107, then 107 XOR 42 = 65.
For unbreakable encryption, the key is the same length as the plain text message, and the key is made up of random bytes. The user would keep the encrypted message and the encryption key in different locations, and without both "halves", it is impossible to decrypt the message.
Unfortunately, this method is impractical for most users, so the modified method is to use a password as a key. If the password is shorter than the message, which is likely, the key is repeated cyclically throughout the message. The balance for this method is using a sufficiently long password key for security, but short enough to be memorable.
Your task has been made easy, as the encryption key consists of three lower case characters. Using p059_cipher.txt (right click and 'Save Link/Target As...'), a file containing the encrypted ASCII codes, and the knowledge that the plain text must contain common English words, decrypt the message and find the sum of the ASCII values in the original text.
The primes 3, 7, 109, and 673, are quite remarkable. By taking any two primes and concatenating them in any order the result will always be prime. For example, taking 7 and 109, both 7109 and 1097 are prime. The sum of these four primes, 792, represents the lowest sum for a set of four primes with this property.
Find the lowest sum for a set of five primes for which any two primes concatenate to produce another prime.
Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
Triangle  P_{3,n}=n(n+1)/2  1, 3, 6, 10, 15, ...  
Square  P_{4,n}=n^{2}  1, 4, 9, 16, 25, ...  
Pentagonal  P_{5,n}=n(3n−1)/2  1, 5, 12, 22, 35, ...  
Hexagonal  P_{6,n}=n(2n−1)  1, 6, 15, 28, 45, ...  
Heptagonal  P_{7,n}=n(5n−3)/2  1, 7, 18, 34, 55, ...  
Octagonal  P_{8,n}=n(3n−2)  1, 8, 21, 40, 65, ... 
The ordered set of three 4digit numbers: 8128, 2882, 8281, has three interesting properties.
 The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
 Each polygonal type: triangle (P_{3,127}=8128), square (P_{4,91}=8281), and pentagonal (P_{5,44}=2882), is represented by a different number in the set.
 This is the only set of 4digit numbers with this property.
Find the sum of the only ordered set of six cyclic 4digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.
The cube, 41063625 (345^{3}), can be permuted to produce two other cubes: 56623104 (384^{3}) and 66430125 (405^{3}). In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.
Find the smallest cube for which exactly five permutations of its digits are cube.
The 5digit number, 16807=7^{5}, is also a fifth power. Similarly, the 9digit number, 134217728=8^{9}, is a ninth power.
How many ndigit positive integers exist which are also an nth power?
All square roots are periodic when written as continued fractions and can be written in the form:
$\displaystyle \quad \quad \sqrt{N}=a_0+\frac 1 {a_1+\frac 1 {a_2+ \frac 1 {a3+ \dots}}}$For example, let us consider $\sqrt{23}:$
$\quad \quad \sqrt{23}=4+\sqrt{23}4=4+\frac 1 {\frac 1 {\sqrt{23}4}}=4+\frac 1 {1+\frac{\sqrt{23}3}7}$If we continue we would get the following expansion:
$\displaystyle \quad \quad \sqrt{23}=4+\frac 1 {1+\frac 1 {3+ \frac 1 {1+\frac 1 {8+ \dots}}}}$The process can be summarised as follows:
$\quad \quad a_0=4, \frac 1 {\sqrt{23}4}=\frac {\sqrt{23}+4} 7=1+\frac {\sqrt{23}3} 7$
$\quad \quad a_1=1, \frac 7 {\sqrt{23}3}=\frac {7(\sqrt{23}+3)} {14}=3+\frac {\sqrt{23}3} 2$
$\quad \quad a_2=3, \frac 2 {\sqrt{23}3}=\frac {2(\sqrt{23}+3)} {14}=1+\frac {\sqrt{23}4} 7$
$\quad \quad a_3=1, \frac 7 {\sqrt{23}4}=\frac {7(\sqrt{23}+4)} 7=8+\sqrt{23}4$
$\quad \quad a_4=8, \frac 1 {\sqrt{23}4}=\frac {\sqrt{23}+4} 7=1+\frac {\sqrt{23}3} 7$
$\quad \quad a_5=1, \frac 7 {\sqrt{23}3}=\frac {7 (\sqrt{23}+3)} {14}=3+\frac {\sqrt{23}3} 2$
$\quad \quad a_6=3, \frac 2 {\sqrt{23}3}=\frac {2(\sqrt{23}+3)} {14}=1+\frac {\sqrt{23}4} 7$
$\quad \quad a_7=1, \frac 7 {\sqrt{23}4}=\frac {7(\sqrt{23}+4)} {7}=8+\sqrt{23}4$
It can be seen that the sequence is repeating. For conciseness, we use the notation $\sqrt{23}=[4;(1,3,1,8)]$, to indicate that the block (1,3,1,8) repeats indefinitely.
The first ten continued fraction representations of (irrational) square roots are:
$\quad \quad \sqrt{2}=[1;(2)]$, period=$1$
$\quad \quad \sqrt{3}=[1;(1,2)]$, period=$2$
$\quad \quad \sqrt{5}=[2;(4)]$, period=$1$
$\quad \quad \sqrt{6}=[2;(2,4)]$, period=$2$
$\quad \quad \sqrt{7}=[2;(1,1,1,4)]$, period=$4$
$\quad \quad \sqrt{8}=[2;(1,4)]$, period=$2$
$\quad \quad \sqrt{10}=[3;(6)]$, period=$1$
$\quad \quad \sqrt{11}=[3;(3,6)]$, period=$2$
$\quad \quad \sqrt{12}=[3;(2,6)]$, period=$2$
$\quad \quad \sqrt{13}=[3;(1,1,1,1,6)]$, period=$5$
Exactly four continued fractions, for $N \le 13$, have an odd period.
How many continued fractions for $N \le 10\,000$ have an odd period?
The square root of 2 can be written as an infinite continued fraction.
$\sqrt{2} = 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + ...}}}}$
The infinite continued fraction can be written, $\sqrt{2} = [1; (2)]$, $(2)$ indicates that 2 repeats ad infinitum. In a similar way, $\sqrt{23} = [4; (1, 3, 1, 8)]$.
It turns out that the sequence of partial values of continued fractions for square roots provide the best rational approximations. Let us consider the convergents for $\sqrt{2}$.
$ 1 + \dfrac{1}{2} = \dfrac{3}{2}\\ 1 + \dfrac{1}{2 + \dfrac{1}{2}} = \dfrac{7}{5}\\ 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2}}} = \dfrac{17}{12}\\ 1 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2}}}} = \dfrac{41}{29} $
Hence the sequence of the first ten convergents for $\sqrt{2}$ are:
$1, \dfrac{3}{2}, \dfrac{7}{5}, \dfrac{17}{12}, \dfrac{41}{29}, \dfrac{99}{70}, \dfrac{239}{169}, \dfrac{577}{408}, \dfrac{1393}{985}, \dfrac{3363}{2378}, ...$
What is most surprising is that the important mathematical constant,
$e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ... , 1, 2k, 1, ...]$.
The first ten terms in the sequence of convergents for e are:
$2, 3, \dfrac{8}{3}, \dfrac{11}{4}, \dfrac{19}{7}, \dfrac{87}{32}, \dfrac{106}{39}, \dfrac{193}{71}, \dfrac{1264}{465}, \dfrac{1457}{536}, ...$
The sum of digits in the numerator of the 10^{th} convergent is $1 + 4 + 5 + 7 = 17$.
Find the sum of digits in the numerator of the 100^{th} convergent of the continued fraction for $e$.
Consider quadratic Diophantine equations of the form:
x^{2} – Dy^{2} = 1
For example, when D=13, the minimal solution in x is 649^{2} – 13×180^{2} = 1.
It can be assumed that there are no solutions in positive integers when D is square.
By finding minimal solutions in x for D = {2, 3, 5, 6, 7}, we obtain the following:
3^{2} – 2×2^{2} = 1
2^{2} – 3×1^{2} = 1
9^{2} – 5×4^{2} = 1
5^{2} – 6×2^{2} = 1
8^{2} – 7×3^{2} = 1
Hence, by considering minimal solutions in x for D ≤ 7, the largest x is obtained when D=5.
Find the value of D ≤ 1000 in minimal solutions of x for which the largest value of x is obtained.
By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.
3
7 4
2 4 6
8 5 9 3
That is, 3 + 7 + 4 + 9 = 23.
Find the maximum total from top to bottom in triangle.txt (right click and 'Save Link/Target As...'), a 15K text file containing a triangle with onehundred rows.
NOTE: This is a much more difficult version of Problem 18. It is not possible to try every route to solve this problem, as there are 2^{99} altogether! If you could check one trillion (10^{12}) routes every second it would take over twenty billion years to check them all. There is an efficient algorithm to solve it. ;o)
Consider the following "magic" 3gon ring, filled with the numbers 1 to 6, and each line adding to nine.
Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described by the set: 4,3,2; 6,2,1; 5,1,3.
It is possible to complete the ring with four different totals: 9, 10, 11, and 12. There are eight solutions in total.
Total  Solution Set 
9  4,2,3; 5,3,1; 6,1,2 
9  4,3,2; 6,2,1; 5,1,3 
10  2,3,5; 4,5,1; 6,1,3 
10  2,5,3; 6,3,1; 4,1,5 
11  1,4,6; 3,6,2; 5,2,4 
11  1,6,4; 5,4,2; 3,2,6 
12  1,5,6; 2,6,4; 3,4,5 
12  1,6,5; 3,5,4; 2,4,6 
By concatenating each group it is possible to form 9digit strings; the maximum string for a 3gon ring is 432621513.
Using the numbers 1 to 10, and depending on arrangements, it is possible to form 16 and 17digit strings. What is the maximum 16digit string for a "magic" 5gon ring?
Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6.
n  Relatively Prime  φ(n)  n/φ(n) 
2  1  1  2 
3  1,2  2  1.5 
4  1,3  2  2 
5  1,2,3,4  4  1.25 
6  1,5  2  3 
7  1,2,3,4,5,6  6  1.1666... 
8  1,3,5,7  4  2 
9  1,2,4,5,7,8  6  1.5 
10  1,3,7,9  4  2.5 
It can be seen that n=6 produces a maximum n/φ(n) for n ≤ 10.
Find the value of n ≤ 1,000,000 for which n/φ(n) is a maximum.
Euler's Totient function, φ(n) [sometimes called the phi function], is used to determine the number of positive numbers less than or equal to n which are relatively prime to n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6.
The number 1 is considered to be relatively prime to every positive number, so φ(1)=1.
Interestingly, φ(87109)=79180, and it can be seen that 87109 is a permutation of 79180.
Find the value of n, 1 < n < 10^{7}, for which φ(n) is a permutation of n and the ratio n/φ(n) produces a minimum.
Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we get:
1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8
It can be seen that 2/5 is the fraction immediately to the left of 3/7.
By listing the set of reduced proper fractions for d ≤ 1,000,000 in ascending order of size, find the numerator of the fraction immediately to the left of 3/7.
Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we get:
1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8
It can be seen that there are 21 elements in this set.
How many elements would be contained in the set of reduced proper fractions for d ≤ 1,000,000?
Consider the fraction, n/d, where n and d are positive integers. If n<d and HCF(n,d)=1, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we get:
1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8
It can be seen that there are 3 fractions between 1/3 and 1/2.
How many fractions lie between 1/3 and 1/2 in the sorted set of reduced proper fractions for d ≤ 12,000?
The number 145 is well known for the property that the sum of the factorial of its digits is equal to 145:
1! + 4! + 5! = 1 + 24 + 120 = 145
Perhaps less well known is 169, in that it produces the longest chain of numbers that link back to 169; it turns out that there are only three such loops that exist:
169 → 363601 → 1454 → 169
871 → 45361 → 871
872 → 45362 → 872
It is not difficult to prove that EVERY starting number will eventually get stuck in a loop. For example,
69 → 363600 → 1454 → 169 → 363601 (→ 1454)
78 → 45360 → 871 → 45361 (→ 871)
540 → 145 (→ 145)
Starting with 69 produces a chain of five nonrepeating terms, but the longest nonrepeating chain with a starting number below one million is sixty terms.
How many chains, with a starting number below one million, contain exactly sixty nonrepeating terms?
It turns out that 12 cm is the smallest length of wire that can be bent to form an integer sided right angle triangle in exactly one way, but there are many more examples.
12 cm: (3,4,5)
24 cm: (6,8,10)
30 cm: (5,12,13)
36 cm: (9,12,15)
40 cm: (8,15,17)
48 cm: (12,16,20)
In contrast, some lengths of wire, like 20 cm, cannot be bent to form an integer sided right angle triangle, and other lengths allow more than one solution to be found; for example, using 120 cm it is possible to form exactly three different integer sided right angle triangles.
120 cm: (30,40,50), (20,48,52), (24,45,51)
Given that L is the length of the wire, for how many values of L ≤ 1,500,000 can exactly one integer sided right angle triangle be formed?
It is possible to write five as a sum in exactly six different ways:
4 + 1
3 + 2
3 + 1 + 1
2 + 2 + 1
2 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
How many different ways can one hundred be written as a sum of at least two positive integers?
It is possible to write ten as the sum of primes in exactly five different ways:
7 + 3
5 + 5
5 + 3 + 2
3 + 3 + 2 + 2
2 + 2 + 2 + 2 + 2
What is the first value which can be written as the sum of primes in over five thousand different ways?
Let p(n) represent the number of different ways in which n coins can be separated into piles. For example, five coins can be separated into piles in exactly seven different ways, so p(5)=7.
OOOOO 
OOOO O 
OOO OO 
OOO O O 
OO OO O 
OO O O O 
O O O O O 
Find the least value of n for which p(n) is divisible by one million.
A common security method used for online banking is to ask the user for three random characters from a passcode. For example, if the passcode was 531278, they may ask for the 2nd, 3rd, and 5th characters; the expected reply would be: 317.
The text file, keylog.txt, contains fifty successful login attempts.
Given that the three characters are always asked for in order, analyse the file so as to determine the shortest possible secret passcode of unknown length.
It is well known that if the square root of a natural number is not an integer, then it is irrational. The decimal expansion of such square roots is infinite without any repeating pattern at all.
The square root of two is 1.41421356237309504880..., and the digital sum of the first one hundred decimal digits is 475.
For the first one hundred natural numbers, find the total of the digital sums of the first one hundred decimal digits for all the irrational square roots.
In the 5 by 5 matrix below, the minimal path sum from the top left to the bottom right, by only moving to the right and down, is indicated in bold red and is equal to 2427.
Find the minimal path sum from the top left to the bottom right by only moving right and down in matrix.txt (right click and "Save Link/Target As..."), a 31K text file containing an 80 by 80 matrix.
NOTE: This problem is a more challenging version of Problem 81.
The minimal path sum in the 5 by 5 matrix below, by starting in any cell in the left column and finishing in any cell in the right column, and only moving up, down, and right, is indicated in red and bold; the sum is equal to 994.
Find the minimal path sum from the left column to the right column in matrix.txt (right click and "Save Link/Target As..."), a 31K text file containing an 80 by 80 matrix.
NOTE: This problem is a significantly more challenging version of Problem 81.
In the 5 by 5 matrix below, the minimal path sum from the top left to the bottom right, by moving left, right, up, and down, is indicated in bold red and is equal to 2297.
Find the minimal path sum from the top left to the bottom right by moving left, right, up, and down in matrix.txt (right click and "Save Link/Target As..."), a 31K text file containing an 80 by 80 matrix.
In the game, Monopoly, the standard board is set up in the following way:
GO  A1  CC1  A2  T1  R1  B1  CH1  B2  B3  JAIL 
H2  C1  
T2  U1  
H1  C2  
CH3  C3  
R4  R2  
G3  D1  
CC3  CC2  
G2  D2  
G1  D3  
G2J  F3  U2  F2  F1  R3  E3  E2  CH2  E1  FP 
A player starts on the GO square and adds the scores on two 6sided dice to determine the number of squares they advance in a clockwise direction. Without any further rules we would expect to visit each square with equal probability: 2.5%. However, landing on G2J (Go To Jail), CC (community chest), and CH (chance) changes this distribution.
In addition to G2J, and one card from each of CC and CH, that orders the player to go directly to jail, if a player rolls three consecutive doubles, they do not advance the result of their 3rd roll. Instead they proceed directly to jail.
At the beginning of the game, the CC and CH cards are shuffled. When a player lands on CC or CH they take a card from the top of the respective pile and, after following the instructions, it is returned to the bottom of the pile. There are sixteen cards in each pile, but for the purpose of this problem we are only concerned with cards that order a movement; any instruction not concerned with movement will be ignored and the player will remain on the CC/CH square.
 Community Chest (2/16 cards):
 Advance to GO
 Go to JAIL
 Chance (10/16 cards):
 Advance to GO
 Go to JAIL
 Go to C1
 Go to E3
 Go to H2
 Go to R1
 Go to next R (railway company)
 Go to next R
 Go to next U (utility company)
 Go back 3 squares.
The heart of this problem concerns the likelihood of visiting a particular square. That is, the probability of finishing at that square after a roll. For this reason it should be clear that, with the exception of G2J for which the probability of finishing on it is zero, the CH squares will have the lowest probabilities, as 5/8 request a movement to another square, and it is the final square that the player finishes at on each roll that we are interested in. We shall make no distinction between "Just Visiting" and being sent to JAIL, and we shall also ignore the rule about requiring a double to "get out of jail", assuming that they pay to get out on their next turn.
By starting at GO and numbering the squares sequentially from 00 to 39 we can concatenate these twodigit numbers to produce strings that correspond with sets of squares.
Statistically it can be shown that the three most popular squares, in order, are JAIL (6.24%) = Square 10, E3 (3.18%) = Square 24, and GO (3.09%) = Square 00. So these three most popular squares can be listed with the sixdigit modal string: 102400.
If, instead of using two 6sided dice, two 4sided dice are used, find the sixdigit modal string.
By counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles:
Although there exists no rectangular grid that contains exactly two million rectangles, find the area of the grid with the nearest solution.
A spider, S, sits in one corner of a cuboid room, measuring 6 by 5 by 3, and a fly, F, sits in the opposite corner. By travelling on the surfaces of the room the shortest "straight line" distance from S to F is 10 and the path is shown on the diagram.
However, there are up to three "shortest" path candidates for any given cuboid and the shortest route doesn't always have integer length.
It can be shown that there are exactly 2060 distinct cuboids, ignoring rotations, with integer dimensions, up to a maximum size of M by M by M, for which the shortest route has integer length when M = 100. This is the least value of M for which the number of solutions first exceeds two thousand; the number of solutions when M = 99 is 1975.
Find the least value of M such that the number of solutions first exceeds one million.
The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is 28. In fact, there are exactly four numbers below fifty that can be expressed in such a way:
28 = 2^{2} + 2^{3} + 2^{4}
33 = 3^{2} + 2^{3} + 2^{4}
49 = 5^{2} + 2^{3} + 2^{4}
47 = 2^{2} + 3^{3} + 2^{4}
How many numbers below fifty million can be expressed as the sum of a prime square, prime cube, and prime fourth power?
A natural number, N, that can be written as the sum and product of a given set of at least two natural numbers, {a_{1}, a_{2}, ... , a_{k}} is called a productsum number: N = a_{1} + a_{2} + ... + a_{k} = a_{1} × a_{2} × ... × a_{k}.
For example, 6 = 1 + 2 + 3 = 1 × 2 × 3.
For a given set of size, k, we shall call the smallest N with this property a minimal productsum number. The minimal productsum numbers for sets of size, k = 2, 3, 4, 5, and 6 are as follows.
k=2: 4 = 2 × 2 = 2 + 2
k=3: 6 = 1 × 2 × 3 = 1 + 2 + 3
k=4: 8 = 1 × 1 × 2 × 4 = 1 + 1 + 2 + 4
k=5: 8 = 1 × 1 × 2 × 2 × 2 = 1 + 1 + 2 + 2 + 2
k=6: 12 = 1 × 1 × 1 × 1 × 2 × 6 = 1 + 1 + 1 + 1 + 2 + 6
Hence for 2≤k≤6, the sum of all the minimal productsum numbers is 4+6+8+12 = 30; note that 8 is only counted once in the sum.
In fact, as the complete set of minimal productsum numbers for 2≤k≤12 is {4, 6, 8, 12, 15, 16}, the sum is 61.
What is the sum of all the minimal productsum numbers for 2≤k≤12000?
For a number written in Roman numerals to be considered valid there are basic rules which must be followed. Even though the rules allow some numbers to be expressed in more than one way there is always a "best" way of writing a particular number.
For example, it would appear that there are at least six ways of writing the number sixteen:
IIIIIIIIIIIIIIII
VIIIIIIIIIII
VVIIIIII
XIIIIII
VVVI
XVI
However, according to the rules only XIIIIII and XVI are valid, and the last example is considered to be the most efficient, as it uses the least number of numerals.
The 11K text file, roman.txt (right click and 'Save Link/Target As...'), contains one thousand numbers written in valid, but not necessarily minimal, Roman numerals; see About... Roman Numerals for the definitive rules for this problem.
Find the number of characters saved by writing each of these in their minimal form.
Note: You can assume that all the Roman numerals in the file contain no more than four consecutive identical units.
Each of the six faces on a cube has a different digit (0 to 9) written on it; the same is done to a second cube. By placing the two cubes sidebyside in different positions we can form a variety of 2digit numbers.
For example, the square number 64 could be formed:
In fact, by carefully choosing the digits on both cubes it is possible to display all of the square numbers below onehundred: 01, 04, 09, 16, 25, 36, 49, 64, and 81.
For example, one way this can be achieved is by placing {0, 5, 6, 7, 8, 9} on one cube and {1, 2, 3, 4, 8, 9} on the other cube.
However, for this problem we shall allow the 6 or 9 to be turned upsidedown so that an arrangement like {0, 5, 6, 7, 8, 9} and {1, 2, 3, 4, 6, 7} allows for all nine square numbers to be displayed; otherwise it would be impossible to obtain 09.
In determining a distinct arrangement we are interested in the digits on each cube, not the order.
{1, 2, 3, 4, 5, 6} is equivalent to {3, 6, 4, 1, 2, 5}
{1, 2, 3, 4, 5, 6} is distinct from {1, 2, 3, 4, 5, 9}
But because we are allowing 6 and 9 to be reversed, the two distinct sets in the last example both represent the extended set {1, 2, 3, 4, 5, 6, 9} for the purpose of forming 2digit numbers.
How many distinct arrangements of the two cubes allow for all of the square numbers to be displayed?
The points P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) are plotted at integer coordinates and are joined to the origin, O(0,0), to form ΔOPQ.
There are exactly fourteen triangles containing a right angle that can be formed when each coordinate lies between 0 and 2 inclusive; that is,
0 ≤ x_{1}, y_{1}, x_{2}, y_{2} ≤ 2.
Given that 0 ≤ x_{1}, y_{1}, x_{2}, y_{2} ≤ 50, how many right triangles can be formed?
A number chain is created by continuously adding the square of the digits in a number to form a new number until it has been seen before.
For example,
44 → 32 → 13 → 10 → 1 → 1
85 → 89 → 145 → 42 → 20 → 4 → 16 → 37 → 58 → 89
Therefore any chain that arrives at 1 or 89 will become stuck in an endless loop. What is most amazing is that EVERY starting number will eventually arrive at 1 or 89.
How many starting numbers below ten million will arrive at 89?
By using each of the digits from the set, {1, 2, 3, 4}, exactly once, and making use of the four arithmetic operations (+, −, *, /) and brackets/parentheses, it is possible to form different positive integer targets.
For example,
8 = (4 * (1 + 3)) / 2
14 = 4 * (3 + 1 / 2)
19 = 4 * (2 + 3) − 1
36 = 3 * 4 * (2 + 1)
Note that concatenations of the digits, like 12 + 34, are not allowed.
Using the set, {1, 2, 3, 4}, it is possible to obtain thirtyone different target numbers of which 36 is the maximum, and each of the numbers 1 to 28 can be obtained before encountering the first nonexpressible number.
Find the set of four distinct digits, a < b < c < d, for which the longest set of consecutive positive integers, 1 to n, can be obtained, giving your answer as a string: abcd.
It is easily proved that no equilateral triangle exists with integral length sides and integral area. However, the almost equilateral triangle 556 has an area of 12 square units.
We shall define an almost equilateral triangle to be a triangle for which two sides are equal and the third differs by no more than one unit.
Find the sum of the perimeters of all almost equilateral triangles with integral side lengths and area and whose perimeters do not exceed one billion (1,000,000,000).
The proper divisors of a number are all the divisors excluding the number itself. For example, the proper divisors of 28 are 1, 2, 4, 7, and 14. As the sum of these divisors is equal to 28, we call it a perfect number.
Interestingly the sum of the proper divisors of 220 is 284 and the sum of the proper divisors of 284 is 220, forming a chain of two numbers. For this reason, 220 and 284 are called an amicable pair.
Perhaps less well known are longer chains. For example, starting with 12496, we form a chain of five numbers:
12496 → 14288 → 15472 → 14536 → 14264 (→ 12496 → ...)
Since this chain returns to its starting point, it is called an amicable chain.
Find the smallest member of the longest amicable chain with no element exceeding one million.
Su Doku (Japanese meaning number place) is the name given to a popular puzzle concept. Its origin is unclear, but credit must be attributed to Leonhard Euler who invented a similar, and much more difficult, puzzle idea called Latin Squares. The objective of Su Doku puzzles, however, is to replace the blanks (or zeros) in a 9 by 9 grid in such that each row, column, and 3 by 3 box contains each of the digits 1 to 9. Below is an example of a typical starting puzzle grid and its solution grid.


A well constructed Su Doku puzzle has a unique solution and can be solved by logic, although it may be necessary to employ "guess and test" methods in order to eliminate options (there is much contested opinion over this). The complexity of the search determines the difficulty of the puzzle; the example above is considered easy because it can be solved by straight forward direct deduction.
The 6K text file, sudoku.txt (right click and 'Save Link/Target As...'), contains fifty different Su Doku puzzles ranging in difficulty, but all with unique solutions (the first puzzle in the file is the example above).
By solving all fifty puzzles find the sum of the 3digit numbers found in the top left corner of each solution grid; for example, 483 is the 3digit number found in the top left corner of the solution grid above.
The first known prime found to exceed one million digits was discovered in 1999, and is a Mersenne prime of the form 2^{6972593}−1; it contains exactly 2,098,960 digits. Subsequently other Mersenne primes, of the form 2^{p}−1, have been found which contain more digits.
However, in 2004 there was found a massive nonMersenne prime which contains 2,357,207 digits: 28433×2^{7830457}+1.
Find the last ten digits of this prime number.
By replacing each of the letters in the word CARE with 1, 2, 9, and 6 respectively, we form a square number: 1296 = 36^{2}. What is remarkable is that, by using the same digital substitutions, the anagram, RACE, also forms a square number: 9216 = 96^{2}. We shall call CARE (and RACE) a square anagram word pair and specify further that leading zeroes are not permitted, neither may a different letter have the same digital value as another letter.
Using words.txt (right click and 'Save Link/Target As...'), a 16K text file containing nearly twothousand common English words, find all the square anagram word pairs (a palindromic word is NOT considered to be an anagram of itself).
What is the largest square number formed by any member of such a pair?
NOTE: All anagrams formed must be contained in the given text file.
Comparing two numbers written in index form like 2^{11} and 3^{7} is not difficult, as any calculator would confirm that 2^{11} = 2048 < 3^{7} = 2187.
However, confirming that 632382^{518061} > 519432^{525806} would be much more difficult, as both numbers contain over three million digits.
Using base_exp.txt (right click and 'Save Link/Target As...'), a 22K text file containing one thousand lines with a base/exponent pair on each line, determine which line number has the greatest numerical value.
NOTE: The first two lines in the file represent the numbers in the example given above.
If a box contains twentyone coloured discs, composed of fifteen blue discs and six red discs, and two discs were taken at random, it can be seen that the probability of taking two blue discs, P(BB) = (15/21)×(14/20) = 1/2.
The next such arrangement, for which there is exactly 50% chance of taking two blue discs at random, is a box containing eightyfive blue discs and thirtyfive red discs.
By finding the first arrangement to contain over 10^{12} = 1,000,000,000,000 discs in total, determine the number of blue discs that the box would contain.
If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next term, as there are infinitely many polynomial functions that can model the sequence.
As an example, let us consider the sequence of cube numbers. This is defined by the generating function,
u_{n} = n^{3}: 1, 8, 27, 64, 125, 216, ...
Suppose we were only given the first two terms of this sequence. Working on the principle that "simple is best" we should assume a linear relationship and predict the next term to be 15 (common difference 7). Even if we were presented with the first three terms, by the same principle of simplicity, a quadratic relationship should be assumed.
We shall define OP(k, n) to be the n^{th} term of the optimum polynomial generating function for the first k terms of a sequence. It should be clear that OP(k, n) will accurately generate the terms of the sequence for n ≤ k, and potentially the first incorrect term (FIT) will be OP(k, k+1); in which case we shall call it a bad OP (BOP).
As a basis, if we were only given the first term of sequence, it would be most sensible to assume constancy; that is, for n ≥ 2, OP(1, n) = u_{1}.
Hence we obtain the following OPs for the cubic sequence:
OP(1, n) = 1  1, 1, 1, 1, ... 
OP(2, n) = 7n−6  1, 8, 15, ... 
OP(3, n) = 6n^{2}−11n+6  1, 8, 27, 58, ... 
OP(4, n) = n^{3}  1, 8, 27, 64, 125, ... 
Clearly no BOPs exist for k ≥ 4.
By considering the sum of FITs generated by the BOPs (indicated in red above), we obtain 1 + 15 + 58 = 74.
Consider the following tenth degree polynomial generating function:
u_{n} = 1 − n + n^{2} − n^{3} + n^{4} − n^{5} + n^{6} − n^{7} + n^{8} − n^{9} + n^{10}
Find the sum of FITs for the BOPs.
Three distinct points are plotted at random on a Cartesian plane, for which 1000 ≤ x, y ≤ 1000, such that a triangle is formed.
Consider the following two triangles:
A(340,495), B(153,910), C(835,947)
X(175,41), Y(421,714), Z(574,645)
It can be verified that triangle ABC contains the origin, whereas triangle XYZ does not.
Using triangles.txt (right click and 'Save Link/Target As...'), a 27K text file containing the coordinates of one thousand "random" triangles, find the number of triangles for which the interior contains the origin.
NOTE: The first two examples in the file represent the triangles in the example given above.
Let S(A) represent the sum of elements in set A of size n. We shall call it a special sum set if for any two nonempty disjoint subsets, B and C, the following properties are true:
 S(B) ≠ S(C); that is, sums of subsets cannot be equal.
 If B contains more elements than C then S(B) > S(C).
If S(A) is minimised for a given n, we shall call it an optimum special sum set. The first five optimum special sum sets are given below.
n = 1: {1}
n = 2: {1, 2}
n = 3: {2, 3, 4}
n = 4: {3, 5, 6, 7}
n = 5: {6, 9, 11, 12, 13}
It seems that for a given optimum set, A = {a_{1}, a_{2}, ... , a_{n}}, the next optimum set is of the form B = {b, a_{1}+b, a_{2}+b, ... ,a_{n}+b}, where b is the "middle" element on the previous row.
By applying this "rule" we would expect the optimum set for n = 6 to be A = {11, 17, 20, 22, 23, 24}, with S(A) = 117. However, this is not the optimum set, as we have merely applied an algorithm to provide a near optimum set. The optimum set for n = 6 is A = {11, 18, 19, 20, 22, 25}, with S(A) = 115 and corresponding set string: 111819202225.
Given that A is an optimum special sum set for n = 7, find its set string.
NOTE: This problem is related to Problem 105 and Problem 106.
The Fibonacci sequence is defined by the recurrence relation:
F_{n} = F_{n−1} + F_{n−2}, where F_{1} = 1 and F_{2} = 1.
It turns out that F_{541}, which contains 113 digits, is the first Fibonacci number for which the last nine digits are 19 pandigital (contain all the digits 1 to 9, but not necessarily in order). And F_{2749}, which contains 575 digits, is the first Fibonacci number for which the first nine digits are 19 pandigital.
Given that F_{k} is the first Fibonacci number for which the first nine digits AND the last nine digits are 19 pandigital, find k.
Let S(A) represent the sum of elements in set A of size n. We shall call it a special sum set if for any two nonempty disjoint subsets, B and C, the following properties are true:
 S(B) ≠ S(C); that is, sums of subsets cannot be equal.
 If B contains more elements than C then S(B) > S(C).
For example, {81, 88, 75, 42, 87, 84, 86, 65} is not a special sum set because 65 + 87 + 88 = 75 + 81 + 84, whereas {157, 150, 164, 119, 79, 159, 161, 139, 158} satisfies both rules for all possible subset pair combinations and S(A) = 1286.
Using sets.txt (right click and "Save Link/Target As..."), a 4K text file with onehundred sets containing seven to twelve elements (the two examples given above are the first two sets in the file), identify all the special sum sets, A_{1}, A_{2}, ..., A_{k}, and find the value of S(A_{1}) + S(A_{2}) + ... + S(A_{k}).
NOTE: This problem is related to Problem 103 and Problem 106.
Let S(A) represent the sum of elements in set A of size n. We shall call it a special sum set if for any two nonempty disjoint subsets, B and C, the following properties are true:
 S(B) ≠ S(C); that is, sums of subsets cannot be equal.
 If B contains more elements than C then S(B) > S(C).
For this problem we shall assume that a given set contains n strictly increasing elements and it already satisfies the second rule.
Surprisingly, out of the 25 possible subset pairs that can be obtained from a set for which n = 4, only 1 of these pairs need to be tested for equality (first rule). Similarly, when n = 7, only 70 out of the 966 subset pairs need to be tested.
For n = 12, how many of the 261625 subset pairs that can be obtained need to be tested for equality?
NOTE: This problem is related to Problem 103 and Problem 105.
The following undirected network consists of seven vertices and twelve edges with a total weight of 243.
The same network can be represented by the matrix below.
A  B  C  D  E  F  G  
A    16  12  21       
B  16      17  20     
C  12      28    31   
D  21  17  28    18  19  23 
E    20    18      11 
F      31  19      27 
G        23  11  27   
However, it is possible to optimise the network by removing some edges and still ensure that all points on the network remain connected. The network which achieves the maximum saving is shown below. It has a weight of 93, representing a saving of 243 − 93 = 150 from the original network.
Using network.txt (right click and 'Save Link/Target As...'), a 6K text file containing a network with forty vertices, and given in matrix form, find the maximum saving which can be achieved by removing redundant edges whilst ensuring that the network remains connected.
In the following equation x, y, and n are positive integers.
$$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$For n = 4 there are exactly three distinct solutions:
$$\begin{align} \dfrac{1}{5} + \dfrac{1}{20} &= \dfrac{1}{4}\\ \dfrac{1}{6} + \dfrac{1}{12} &= \dfrac{1}{4}\\ \dfrac{1}{8} + \dfrac{1}{8} &= \dfrac{1}{4} \end{align} $$What is the least value of n for which the number of distinct solutions exceeds onethousand?
NOTE: This problem is an easier version of Problem 110; it is strongly advised that you solve this one first.
In the game of darts a player throws three darts at a target board which is split into twenty equal sized sections numbered one to twenty.
The score of a dart is determined by the number of the region that the dart lands in. A dart landing outside the red/green outer ring scores zero. The black and cream regions inside this ring represent single scores. However, the red/green outer ring and middle ring score double and treble scores respectively.
At the centre of the board are two concentric circles called the bull region, or bullseye. The outer bull is worth 25 points and the inner bull is a double, worth 50 points.
There are many variations of rules but in the most popular game the players will begin with a score 301 or 501 and the first player to reduce their running total to zero is a winner. However, it is normal to play a "doubles out" system, which means that the player must land a double (including the double bullseye at the centre of the board) on their final dart to win; any other dart that would reduce their running total to one or lower means the score for that set of three darts is "bust".
When a player is able to finish on their current score it is called a "checkout" and the highest checkout is 170: T20 T20 D25 (two treble 20s and double bull).
There are exactly eleven distinct ways to checkout on a score of 6:
D3  
D1  D2  
S2  D2  
D2  D1  
S4  D1  
S1  S1  D2 
S1  T1  D1 
S1  S3  D1 
D1  D1  D1 
D1  S2  D1 
S2  S2  D1 
Note that D1 D2 is considered different to D2 D1 as they finish on different doubles. However, the combination S1 T1 D1 is considered the same as T1 S1 D1.
In addition we shall not include misses in considering combinations; for example, D3 is the same as 0 D3 and 0 0 D3.
Incredibly there are 42336 distinct ways of checking out in total.
How many distinct ways can a player checkout with a score less than 100?
In the following equation x, y, and n are positive integers.
1 x 
+  1 y 
=  1 n 
It can be verified that when n = 1260 there are 113 distinct solutions and this is the least value of n for which the total number of distinct solutions exceeds one hundred.
What is the least value of n for which the number of distinct solutions exceeds four million?
NOTE: This problem is a much more difficult version of Problem 108 and as it is well beyond the limitations of a brute force approach it requires a clever implementation.
Considering 4digit primes containing repeated digits it is clear that they cannot all be the same: 1111 is divisible by 11, 2222 is divisible by 22, and so on. But there are nine 4digit primes containing three ones:
1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111
We shall say that M(n, d) represents the maximum number of repeated digits for an ndigit prime where d is the repeated digit, N(n, d) represents the number of such primes, and S(n, d) represents the sum of these primes.
So M(4, 1) = 3 is the maximum number of repeated digits for a 4digit prime where one is the repeated digit, there are N(4, 1) = 9 such primes, and the sum of these primes is S(4, 1) = 22275. It turns out that for d = 0, it is only possible to have M(4, 0) = 2 repeated digits, but there are N(4, 0) = 13 such cases.
In the same way we obtain the following results for 4digit primes.
Digit, d  M(4, d)  N(4, d)  S(4, d) 
0  2  13  67061 
1  3  9  22275 
2  3  1  2221 
3  3  12  46214 
4  3  2  8888 
5  3  1  5557 
6  3  1  6661 
7  3  9  57863 
8  3  1  8887 
9  3  7  48073 
For d = 0 to 9, the sum of all S(4, d) is 273700.
Find the sum of all S(10, d).
Working from lefttoright if no digit is exceeded by the digit to its left it is called an increasing number; for example, 134468.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420.
We shall call a positive integer that is neither increasing nor decreasing a "bouncy" number; for example, 155349.
Clearly there cannot be any bouncy numbers below onehundred, but just over half of the numbers below onethousand (525) are bouncy. In fact, the least number for which the proportion of bouncy numbers first reaches 50% is 538.
Surprisingly, bouncy numbers become more and more common and by the time we reach 21780 the proportion of bouncy numbers is equal to 90%.
Find the least number for which the proportion of bouncy numbers is exactly 99%.
Working from lefttoright if no digit is exceeded by the digit to its left it is called an increasing number; for example, 134468.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420.
We shall call a positive integer that is neither increasing nor decreasing a "bouncy" number; for example, 155349.
As n increases, the proportion of bouncy numbers below n increases such that there are only 12951 numbers below onemillion that are not bouncy and only 277032 nonbouncy numbers below 10^{10}.
How many numbers below a googol (10^{100}) are not bouncy?
A row measuring seven units in length has red blocks with a minimum length of three units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one grey square. There are exactly seventeen ways of doing this.
How many ways can a row measuring fifty units in length be filled?
NOTE: Although the example above does not lend itself to the possibility, in general it is permitted to mix block sizes. For example, on a row measuring eight units in length you could use red (3), grey (1), and red (4).
NOTE: This is a more difficult version of Problem 114.
A row measuring n units in length has red blocks with a minimum length of m units placed on it, such that any two red blocks (which are allowed to be different lengths) are separated by at least one black square.
Let the fillcount function, F(m, n), represent the number of ways that a row can be filled.
For example, F(3, 29) = 673135 and F(3, 30) = 1089155.
That is, for m = 3, it can be seen that n = 30 is the smallest value for which the fillcount function first exceeds one million.
In the same way, for m = 10, it can be verified that F(10, 56) = 880711 and F(10, 57) = 1148904, so n = 57 is the least value for which the fillcount function first exceeds one million.
For m = 50, find the least value of n for which the fillcount function first exceeds one million.
A row of five grey square tiles is to have a number of its tiles replaced with coloured oblong tiles chosen from red (length two), green (length three), or blue (length four).
If red tiles are chosen there are exactly seven ways this can be done.
If green tiles are chosen there are three ways.
And if blue tiles are chosen there are two ways.
Assuming that colours cannot be mixed there are 7 + 3 + 2 = 12 ways of replacing the grey tiles in a row measuring five units in length.
How many different ways can the grey tiles in a row measuring fifty units in length be replaced if colours cannot be mixed and at least one coloured tile must be used?
NOTE: This is related to Problem 117.
Using a combination of grey square tiles and oblong tiles chosen from: red tiles (measuring two units), green tiles (measuring three units), and blue tiles (measuring four units), it is possible to tile a row measuring five units in length in exactly fifteen different ways.
How many ways can a row measuring fifty units in length be tiled?
NOTE: This is related to Problem 116.
Using all of the digits 1 through 9 and concatenating them freely to form decimal integers, different sets can be formed. Interestingly with the set {2,5,47,89,631}, all of the elements belonging to it are prime.
How many distinct sets containing each of the digits one through nine exactly once contain only prime elements?
The number 512 is interesting because it is equal to the sum of its digits raised to some power: 5 + 1 + 2 = 8, and 8^{3} = 512. Another example of a number with this property is 614656 = 28^{4}.
We shall define a_{n} to be the nth term of this sequence and insist that a number must contain at least two digits to have a sum.
You are given that a_{2} = 512 and a_{10} = 614656.
Find a_{30}.
Let r be the remainder when (a−1)^{n} + (a+1)^{n} is divided by a^{2}.
For example, if a = 7 and n = 3, then r = 42: 6^{3} + 8^{3} = 728 ≡ 42 mod 49. And as n varies, so too will r, but for a = 7 it turns out that r_{max} = 42.
For 3 ≤ a ≤ 1000, find ∑ r_{max}.
A bag contains one red disc and one blue disc. In a game of chance a player takes a disc at random and its colour is noted. After each turn the disc is returned to the bag, an extra red disc is added, and another disc is taken at random.
The player pays £1 to play and wins if they have taken more blue discs than red discs at the end of the game.
If the game is played for four turns, the probability of a player winning is exactly 11/120, and so the maximum prize fund the banker should allocate for winning in this game would be £10 before they would expect to incur a loss. Note that any payout will be a whole number of pounds and also includes the original £1 paid to play the game, so in the example given the player actually wins £9.
Find the maximum prize fund that should be allocated to a single game in which fifteen turns are played.
The most naive way of computing n^{15} requires fourteen multiplications:
n × n × ... × n = n^{15}
But using a "binary" method you can compute it in six multiplications:
n × n = n^{2}
n^{2} × n^{2} = n^{4}
n^{4} × n^{4} = n^{8}
n^{8} × n^{4} = n^{12}
n^{12} × n^{2} = n^{14}
n^{14} × n = n^{15}
However it is yet possible to compute it in only five multiplications:
n × n = n^{2}
n^{2} × n = n^{3}
n^{3} × n^{3} = n^{6}
n^{6} × n^{6} = n^{12}
n^{12} × n^{3} = n^{15}
We shall define m(k) to be the minimum number of multiplications to compute n^{k}; for example m(15) = 5.
For 1 ≤ k ≤ 200, find ∑ m(k).
Let p_{n} be the nth prime: 2, 3, 5, 7, 11, ..., and let r be the remainder when (p_{n}−1)^{n} + (p_{n}+1)^{n} is divided by p_{n}^{2}.
For example, when n = 3, p_{3} = 5, and 4^{3} + 6^{3} = 280 ≡ 5 mod 25.
The least value of n for which the remainder first exceeds 10^{9} is 7037.
Find the least value of n for which the remainder first exceeds 10^{10}.
The radical of n, rad(n), is the product of the distinct prime factors of n. For example, 504 = 2^{3} × 3^{2} × 7, so rad(504) = 2 × 3 × 7 = 42.
If we calculate rad(n) for 1 ≤ n ≤ 10, then sort them on rad(n), and sorting on n if the radical values are equal, we get:
Unsorted 
Sorted 

n 
rad(n) 
n 
rad(n) 
k 

1  1 
1  1  1 

2  2 
2  2  2 

3  3 
4  2  3 

4  2 
8  2  4 

5  5 
3  3  5 

6  6 
9  3  6 

7  7 
5  5  7 

8  2 
6  6  8 

9  3 
7  7  9 

10  10 
10  10  10 
Let E(k) be the kth element in the sorted n column; for example, E(4) = 8 and E(6) = 9.
If rad(n) is sorted for 1 ≤ n ≤ 100000, find E(10000).
The palindromic number 595 is interesting because it can be written as the sum of consecutive squares: 6^{2} + 7^{2} + 8^{2} + 9^{2} + 10^{2} + 11^{2} + 12^{2}.
There are exactly eleven palindromes below onethousand that can be written as consecutive square sums, and the sum of these palindromes is 4164. Note that 1 = 0^{2} + 1^{2} has not been included as this problem is concerned with the squares of positive integers.
Find the sum of all the numbers less than 10^{8} that are both palindromic and can be written as the sum of consecutive squares.
The minimum number of cubes to cover every visible face on a cuboid measuring 3 x 2 x 1 is twentytwo.
If we then add a second layer to this solid it would require fortysix cubes to cover every visible face, the third layer would require seventyeight cubes, and the fourth layer would require onehundred and eighteen cubes to cover every visible face.
However, the first layer on a cuboid measuring 5 x 1 x 1 also requires twentytwo cubes; similarly the first layer on cuboids measuring 5 x 3 x 1, 7 x 2 x 1, and 11 x 1 x 1 all contain fortysix cubes.
We shall define C(n) to represent the number of cuboids that contain n cubes in one of its layers. So C(22) = 2, C(46) = 4, C(78) = 5, and C(118) = 8.
It turns out that 154 is the least value of n for which C(n) = 10.
Find the least value of n for which C(n) = 1000.
The radical of n, rad(n), is the product of distinct prime factors of n. For example, 504 = 2^{3} × 3^{2} × 7, so rad(504) = 2 × 3 × 7 = 42.
We shall define the triplet of positive integers (a, b, c) to be an abchit if:
 GCD(a, b) = GCD(a, c) = GCD(b, c) = 1
 a < b
 a + b = c
 rad(abc) < c
For example, (5, 27, 32) is an abchit, because:
 GCD(5, 27) = GCD(5, 32) = GCD(27, 32) = 1
 5 < 27
 5 + 27 = 32
 rad(4320) = 30 < 32
It turns out that abchits are quite rare and there are only thirtyone abchits for c < 1000, with ∑c = 12523.
Find ∑c for c < 120000.
A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting at "12 o'clock" and numbering the tiles 2 to 7 in an anticlockwise direction.
New rings are added in the same fashion, with the next rings being numbered 8 to 19, 20 to 37, 38 to 61, and so on. The diagram below shows the first three rings.
By finding the difference between tile n and each of its six neighbours we shall define PD(n) to be the number of those differences which are prime.
For example, working clockwise around tile 8 the differences are 12, 29, 11, 6, 1, and 13. So PD(8) = 3.
In the same way, the differences around tile 17 are 1, 17, 16, 1, 11, and 10, hence PD(17) = 2.
It can be shown that the maximum value of PD(n) is 3.
If all of the tiles for which PD(n) = 3 are listed in ascending order to form a sequence, the 10th tile would be 271.
Find the 2000th tile in this sequence.
A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k; for example, R(6) = 111111.
Given that n is a positive integer and GCD(n, 10) = 1, it can be shown that there always exists a value, k, for which R(k) is divisible by n, and let A(n) be the least such value of k; for example, A(7) = 6 and A(41) = 5.
The least value of n for which A(n) first exceeds ten is 17.
Find the least value of n for which A(n) first exceeds onemillion.
A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k; for example, R(6) = 111111.
Given that n is a positive integer and GCD(n, 10) = 1, it can be shown that there always exists a value, k, for which R(k) is divisible by n, and let A(n) be the least such value of k; for example, A(7) = 6 and A(41) = 5.
You are given that for all primes, p > 5, that p − 1 is divisible by A(p). For example, when p = 41, A(41) = 5, and 40 is divisible by 5.
However, there are rare composite values for which this is also true; the first five examples being 91, 259, 451, 481, and 703.
Find the sum of the first twentyfive composite values of n for which
GCD(n, 10) = 1 and n − 1 is divisible by A(n).
There are some prime values, p, for which there exists a positive integer, n, such that the expression n^{3} + n^{2}p is a perfect cube.
For example, when p = 19, 8^{3} + 8^{2}×19 = 12^{3}.
What is perhaps most surprising is that for each prime with this property the value of n is unique, and there are only four such primes below onehundred.
How many primes below one million have this remarkable property?
A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k.
For example, R(10) = 1111111111 = 11×41×271×9091, and the sum of these prime factors is 9414.
Find the sum of the first forty prime factors of R(10^{9}).
A number consisting entirely of ones is called a repunit. We shall define R(k) to be a repunit of length k; for example, R(6) = 111111.
Let us consider repunits of the form R(10^{n}).
Although R(10), R(100), or R(1000) are not divisible by 17, R(10000) is divisible by 17. Yet there is no value of n for which R(10^{n}) will divide by 19. In fact, it is remarkable that 11, 17, 41, and 73 are the only four primes below onehundred that can be a factor of R(10^{n}).
Find the sum of all the primes below onehundred thousand that will never be a factor of R(10^{n}).
Consider the consecutive primes p_{1} = 19 and p_{2} = 23. It can be verified that 1219 is the smallest number such that the last digits are formed by p_{1} whilst also being divisible by p_{2}.
In fact, with the exception of p_{1} = 3 and p_{2} = 5, for every pair of consecutive primes, p_{2} > p_{1}, there exist values of n for which the last digits are formed by p_{1} and n is divisible by p_{2}. Let S be the smallest of these values of n.
Find ∑ S for every pair of consecutive primes with 5 ≤ p_{1} ≤ 1000000.
Given the positive integers, x, y, and z, are consecutive terms of an arithmetic progression, the least value of the positive integer, n, for which the equation, x^{2} − y^{2} − z^{2} = n, has exactly two solutions is n = 27:
34^{2} − 27^{2} − 20^{2} = 12^{2} − 9^{2} − 6^{2} = 27
It turns out that n = 1155 is the least value which has exactly ten solutions.
How many values of n less than one million have exactly ten distinct solutions?
The positive integers, x, y, and z, are consecutive terms of an arithmetic progression. Given that n is a positive integer, the equation, x^{2} − y^{2} − z^{2} = n, has exactly one solution when n = 20:
13^{2} − 10^{2} − 7^{2} = 20
In fact there are twentyfive values of n below one hundred for which the equation has a unique solution.
How many values of n less than fifty million have exactly one solution?
Consider the infinite polynomial series $A_F(x) = x F_1 + x^2 F_2 + x^3 F_3 + \dots$, where $F_k$ is the $k$th term in the Fibonacci sequence: $1, 1, 2, 3, 5, 8, \dots$; that is, $F_k = F_{k1} + F_{k2}$, $F_1 = 1$ and $F_2 = 1$.
For this problem we shall be interested in values of $x$ for which $A_F(x)$ is a positive integer.
Surprisingly  $\begin{align*} A_F(\tfrac{1}{2}) &= (\tfrac{1}{2})\times 1 + (\tfrac{1}{2})^2\times 1 + (\tfrac{1}{2})^3\times 2 + (\tfrac{1}{2})^4\times 3 + (\tfrac{1}{2})^5\times 5 + \cdots \\ &= \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{2}{8} + \tfrac{3}{16} + \tfrac{5}{32} + \cdots \\ &= 2 \end{align*}$ 
The corresponding values of x for the first five natural numbers are shown below.
$x$  $A_F(x)$ 

$\sqrt{2}1$  1 
$\tfrac{1}{2}$  2 
$\frac{\sqrt{13}2}{3}$  3 
$\frac{\sqrt{89}5}{8}$  4 
$\frac{\sqrt{34}3}{5}$  5 
We shall call $A_F(x)$ a golden nugget if $x$ is rational, because they become increasingly rarer; for example, the 10th golden nugget is 74049690.
Find the 15th golden nugget.
Consider the isosceles triangle with base length, $b = 16$, and legs, $L = 17$.
By using the Pythagorean theorem it can be seen that the height of the triangle, $h = \sqrt{17^2  8^2} = 15$, which is one less than the base length.
With $b = 272$ and $L = 305$, we get $h = 273$, which is one more than the base length, and this is the second smallest isosceles triangle with the property that $h = b \pm 1$.
Find $\sum L$ for the twelve smallest isosceles triangles for which $h = b \pm 1$ and $b$, $L$ are positive integers.
Let (a, b, c) represent the three sides of a right angle triangle with integral length sides. It is possible to place four such triangles together to form a square with length c.
For example, (3, 4, 5) triangles can be placed together to form a 5 by 5 square with a 1 by 1 hole in the middle and it can be seen that the 5 by 5 square can be tiled with twentyfive 1 by 1 squares.
However, if (5, 12, 13) triangles were used then the hole would measure 7 by 7 and these could not be used to tile the 13 by 13 square.
Given that the perimeter of the right triangle is less than onehundred million, how many Pythagorean triangles would allow such a tiling to take place?
Consider the infinite polynomial series $A_G(x) = x G_1 + x^2 G_2 + x^3 G_3 + \cdots$, where $G_k$ is the $k$th term of the second order recurrence relation $G_k = G_{k1} + G_{k2}$, $G_1 = 1$ and $G_2 = 4$; that is, $1, 4, 5, 9, 14, 23, \dots$ .
For this problem we shall be concerned with values of $x$ for which $A_G(x)$ is a positive integer.
The corresponding values of $x$ for the first five natural numbers are shown below.
$x$  $A_G(x)$ 

$\frac{\sqrt{5}1}{4}$  1 
$\tfrac{2}{5}$  2 
$\frac{\sqrt{22}2}{6}$  3 
$\frac{\sqrt{137}5}{14}$  4 
$\tfrac{1}{2}$  5 
We shall call $A_G(x)$ a golden nugget if $x$ is rational, because they become increasingly rarer; for example, the 20th golden nugget is 211345365.
Find the sum of the first thirty golden nuggets.
A positive integer, n, is divided by d and the quotient and remainder are q and r respectively. In addition d, q, and r are consecutive positive integer terms in a geometric sequence, but not necessarily in that order.
For example, 58 divided by 6 has quotient 9 and remainder 4. It can also be seen that 4, 6, 9 are consecutive terms in a geometric sequence (common ratio 3/2).
We will call such numbers, n, progressive.
Some progressive numbers, such as 9 and 10404 = 102^{2}, happen to also be perfect squares.
The sum of all progressive perfect squares below one hundred thousand is 124657.
Find the sum of all progressive perfect squares below one trillion (10^{12}).
Find the smallest x + y + z with integers x > y > z > 0 such that x + y, x − y, x + z, x − z, y + z, y − z are all perfect squares.
Let ABC be a triangle with all interior angles being less than 120 degrees. Let X be any point inside the triangle and let XA = p, XC = q, and XB = r.
Fermat challenged Torricelli to find the position of X such that p + q + r was minimised.
Torricelli was able to prove that if equilateral triangles AOB, BNC and AMC are constructed on each side of triangle ABC, the circumscribed circles of AOB, BNC, and AMC will intersect at a single point, T, inside the triangle. Moreover he proved that T, called the Torricelli/Fermat point, minimises p + q + r. Even more remarkable, it can be shown that when the sum is minimised, AN = BM = CO = p + q + r and that AN, BM and CO also intersect at T.
If the sum is minimised and a, b, c, p, q and r are all positive integers we shall call triangle ABC a Torricelli triangle. For example, a = 399, b = 455, c = 511 is an example of a Torricelli triangle, with p + q + r = 784.
Find the sum of all distinct values of p + q + r ≤ 120000 for Torricelli triangles.
In laser physics, a "white cell" is a mirror system that acts as a delay line for the laser beam. The beam enters the cell, bounces around on the mirrors, and eventually works its way back out.
The specific white cell we will be considering is an ellipse with the equation 4x^{2} + y^{2} = 100
The section corresponding to −0.01 ≤ x ≤ +0.01 at the top is missing, allowing the light to enter and exit through the hole.
The light beam in this problem starts at the point (0.0,10.1) just outside the white cell, and the beam first impacts the mirror at (1.4,9.6).
Each time the laser beam hits the surface of the ellipse, it follows the usual law of reflection "angle of incidence equals angle of reflection." That is, both the incident and reflected beams make the same angle with the normal line at the point of incidence.
In the figure on the left, the red line shows the first two points of contact between the laser beam and the wall of the white cell; the blue line shows the line tangent to the ellipse at the point of incidence of the first bounce.
The slope m of the tangent line at any point (x,y) of the given ellipse is: m = −4x/y
The normal line is perpendicular to this tangent line at the point of incidence.
The animation on the right shows the first 10 reflections of the beam.
How many times does the beam hit the internal surface of the white cell before exiting?
Some positive integers n have the property that the sum [ n + reverse(n) ] consists entirely of odd (decimal) digits. For instance, 36 + 63 = 99 and 409 + 904 = 1313. We will call such numbers reversible; so 36, 63, 409, and 904 are reversible. Leading zeroes are not allowed in either n or reverse(n).
There are 120 reversible numbers below onethousand.
How many reversible numbers are there below onebillion (10^{9})?
The smallest positive integer n for which the numbers n^{2}+1, n^{2}+3, n^{2}+7, n^{2}+9, n^{2}+13, and n^{2}+27 are consecutive primes is 10. The sum of all such integers n below onemillion is 1242490.
What is the sum of all such integers n below 150 million?
In a 3x2 crosshatched grid, a total of 37 different rectangles could be situated within that grid as indicated in the sketch.
There are 5 grids smaller than 3x2, vertical and horizontal dimensions being important, i.e. 1x1, 2x1, 3x1, 1x2 and 2x2. If each of them is crosshatched, the following number of different rectangles could be situated within those smaller grids:
1x1  1 
2x1  4 
3x1  8 
1x2  4 
2x2  18 
Adding those to the 37 of the 3x2 grid, a total of 72 different rectangles could be situated within 3x2 and smaller grids.
How many different rectangles could be situated within 47x43 and smaller grids?
We can easily verify that none of the entries in the first seven rows of Pascal's triangle are divisible by 7:
1  
1  1  
1  2  1  
1  3  3  1  
1  4  6  4  1  
1  5  10  10  5  1  
1  6  15  20  15  6  1 
However, if we check the first one hundred rows, we will find that only 2361 of the 5050 entries are not divisible by 7.
Find the number of entries which are not divisible by 7 in the first one billion (10^{9}) rows of Pascal's triangle.
Looking at the table below, it is easy to verify that the maximum possible sum of adjacent numbers in any direction (horizontal, vertical, diagonal or antidiagonal) is 16 (= 8 + 7 + 1).
−2  5  3  2 
9  −6  5  1 
3  2  7  3 
−1  8  −4  8 
Now, let us repeat the search, but on a much larger scale:
First, generate four million pseudorandom numbers using a specific form of what is known as a "Lagged Fibonacci Generator":
For 1 ≤ k ≤ 55, s_{k} = [100003 − 200003k + 300007k^{3}] (modulo 1000000) − 500000.
For 56 ≤ k ≤ 4000000, s_{k} = [s_{k−24} + s_{k−55} + 1000000] (modulo 1000000) − 500000.
Thus, s_{10} = −393027 and s_{100} = 86613.
The terms of s are then arranged in a 2000×2000 table, using the first 2000 numbers to fill the first row (sequentially), the next 2000 numbers to fill the second row, and so on.
Finally, find the greatest sum of (any number of) adjacent entries in any direction (horizontal, vertical, diagonal or antidiagonal).
In a triangular array of positive and negative integers, we wish to find a subtriangle such that the sum of the numbers it contains is the smallest possible.
In the example below, it can be easily verified that the marked triangle satisfies this condition having a sum of −42.
We wish to make such a triangular array with one thousand rows, so we generate 500500 pseudorandom numbers s_{k} in the range ±2^{19}, using a type of random number generator (known as a Linear Congruential Generator) as follows:
t := 0
for k = 1 up to k = 500500:
t := (615949*t + 797807) modulo 2^{20}
s_{k} := t−2^{19}
Thus: s_{1} = 273519, s_{2} = −153582, s_{3} = 450905 etc
Our triangular array is then formed using the pseudorandom numbers thus:
s_{2} s_{3}
s_{4} s_{5} s_{6}
s_{7} s_{8} s_{9} s_{10}
...
Subtriangles can start at any element of the array and extend down as far as we like (takingin the two elements directly below it from the next row, the three elements directly below from the row after that, and so on).
The "sum of a subtriangle" is defined as the sum of all the elements it contains.
Find the smallest possible subtriangle sum.
A printing shop runs 16 batches (jobs) every week and each batch requires a sheet of special colourproofing paper of size A5.
Every Monday morning, the foreman opens a new envelope, containing a large sheet of the special paper with size A1.
He proceeds to cut it in half, thus getting two sheets of size A2. Then he cuts one of them in half to get two sheets of size A3 and so on until he obtains the A5size sheet needed for the first batch of the week.
All the unused sheets are placed back in the envelope.
At the beginning of each subsequent batch, he takes from the envelope one sheet of paper at random. If it is of size A5, he uses it. If it is larger, he repeats the 'cutinhalf' procedure until he has what he needs and any remaining sheets are always placed back in the envelope.
Excluding the first and last batch of the week, find the expected number of times (during each week) that the foreman finds a single sheet of paper in the envelope.
Give your answer rounded to six decimal places using the format x.xxxxxx .
There are several ways to write the number 1/2 as a sum of inverse squares using distinct integers.
For instance, the numbers {2,3,4,5,7,12,15,20,28,35} can be used:
$$\begin{align}\dfrac{1}{2} &= \dfrac{1}{2^2} + \dfrac{1}{3^2} + \dfrac{1}{4^2} + \dfrac{1}{5^2} +\\ &\quad \dfrac{1}{7^2} + \dfrac{1}{12^2} + \dfrac{1}{15^2} + \dfrac{1}{20^2} +\\ &\quad \dfrac{1}{28^2} + \dfrac{1}{35^2}\end{align}$$
In fact, only using integers between 2 and 45 inclusive, there are exactly three ways to do it, the remaining two being: {2,3,4,6,7,9,10,20,28,35,36,45} and {2,3,4,6,7,9,12,15,28,30,35,36,45}.
How many ways are there to write the number 1/2 as a sum of inverse squares using distinct integers between 2 and 80 inclusive?
As we all know the equation x^{2}=1 has no solutions for real x.
If we however introduce the imaginary number i this equation has two solutions: x=i and x=i.
If we go a step further the equation (x3)^{2}=4 has two complex solutions: x=3+2i and x=32i.
x=3+2i and x=32i are called each others' complex conjugate.
Numbers of the form a+bi are called complex numbers.
In general a+bi and a−bi are each other's complex conjugate.
A Gaussian Integer is a complex number a+bi such that both a and b are integers.
The regular integers are also Gaussian integers (with b=0).
To distinguish them from Gaussian integers with b ≠ 0 we call such integers "rational integers."
A Gaussian integer is called a divisor of a rational integer n if the result is also a Gaussian integer.
If for example we divide 5 by 1+2i we can simplify $\dfrac{5}{1 + 2i}$ in the following manner:
Multiply numerator and denominator by the complex conjugate of 1+2i: 1−2i.
The result is $\dfrac{5}{1 + 2i} = \dfrac{5}{1 + 2i}\dfrac{1  2i}{1  2i} = \dfrac{5(1  2i)}{1  (2i)^2} = \dfrac{5(1  2i)}{1  (4)} = \dfrac{5(1  2i)}{5} = 1  2i$.
So 1+2i is a divisor of 5.
Note that 1+i is not a divisor of 5 because $\dfrac{5}{1 + i} = \dfrac{5}{2}  \dfrac{5}{2}i$.
Note also that if the Gaussian Integer (a+bi) is a divisor of a rational integer n, then its complex conjugate (a−bi) is also a divisor of n.
In fact, 5 has six divisors such that the real part is positive: {1, 1 + 2i, 1 − 2i, 2 + i, 2 − i, 5}.
The following is a table of all of the divisors for the first five positive rational integers:
n  Gaussian integer divisors with positive real part  Sum s(n) of these divisors 
1  1  1 
2  1, 1+i, 1i, 2  5 
3  1, 3  4 
4  1, 1+i, 1i, 2, 2+2i, 22i,4  13 
5  1, 1+2i, 12i, 2+i, 2i, 5  12 
For divisors with positive real parts, then, we have: $\sum \limits_{n = 1}^{5} {s(n)} = 35$.
For $\sum \limits_{n = 1}^{10^5} {s(n)} = 17924657155$.
What is $\sum \limits_{n = 1}^{10^8} {s(n)}$?
A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next lower level.
Then, we calculate the number of paths leading from the apex to each position:
A path starts at the apex and progresses downwards to any of the three spheres directly below the current position.
Consequently, the number of paths to reach a certain position is the sum of the numbers immediately above it (depending on the position, there are up to three numbers above it).
The result is Pascal's pyramid and the numbers at each level n are the coefficients of the trinomial expansion (x + y + z)^{n}.
How many coefficients in the expansion of (x + y + z)^{200000} are multiples of 10^{12}?
An electric circuit uses exclusively identical capacitors of the same value C.
The capacitors can be connected in series or in parallel to form subunits, which can then be connected in series or in parallel with other capacitors or other subunits to form larger subunits, and so on up to a final circuit.
Using this simple procedure and up to n identical capacitors, we can make circuits having a range of different total capacitances. For example, using up to n=3 capacitors of 60 F each, we can obtain the following 7 distinct total capacitance values:
If we denote by D(n) the number of distinct total capacitance values we can obtain when using up to n equalvalued capacitors and the simple procedure described above, we have: D(1)=1, D(2)=3, D(3)=7 ...
Find D(18).
Reminder : When connecting capacitors C_{1}, C_{2} etc in parallel, the total capacitance is C_{T} = C_{1} + C_{2} +...,
whereas when connecting them in series, the overall capacitance is given by: $\dfrac{1}{C_T} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + ...$
Starting from zero the natural numbers are written down in base 10 like this:
0 1 2 3 4 5 6 7 8 9 10 11 12....
Consider the digit d=1. After we write down each number n, we will update the number of ones that have occurred and call this number f(n,1). The first values for f(n,1), then, are as follows:
n  f(n,1) 
0  0 
1  1 
2  1 
3  1 
4  1 
5  1 
6  1 
7  1 
8  1 
9  1 
10  2 
11  4 
12  5 
Note that f(n,1) never equals 3.
So the first two solutions of the equation f(n,1)=n are n=0 and n=1. The next solution is n=199981.
In the same manner the function f(n,d) gives the total number of digits d that have been written down after the number n has been written.
In fact, for every digit d ≠ 0, 0 is the first solution of the equation f(n,d)=n.
Let s(d) be the sum of all the solutions for which f(n,d)=n.
You are given that s(1)=22786974071.
Find ∑ s(d) for 1 ≤ d ≤ 9.
Note: if, for some n, f(n,d)=n for more than one value of d this value of n is counted again for every value of d for which f(n,d)=n.
Consider the diophantine equation ^{1}/_{a}+^{1}/_{b}= ^{p}/_{10n} with a, b, p, n positive integers and a ≤ b.
For n=1 this equation has 20 solutions that are listed below:
^{1}/_{1}+^{1}/_{1}=^{20}/_{10}  ^{1}/_{1}+^{1}/_{2}=^{15}/_{10}  ^{1}/_{1}+^{1}/_{5}=^{12}/_{10}  ^{1}/_{1}+^{1}/_{10}=^{11}/_{10}  ^{1}/_{2}+^{1}/_{2}=^{10}/_{10} 
^{1}/_{2}+^{1}/_{5}=^{7}/_{10}  ^{1}/_{2}+^{1}/_{10}=^{6}/_{10}  ^{1}/_{3}+^{1}/_{6}=^{5}/_{10}  ^{1}/_{3}+^{1}/_{15}=^{4}/_{10}  ^{1}/_{4}+^{1}/_{4}=^{5}/_{10} 
^{1}/_{4}+^{1}/_{20}=^{3}/_{10}  ^{1}/_{5}+^{1}/_{5}=^{4}/_{10}  ^{1}/_{5}+^{1}/_{10}=^{3}/_{10}  ^{1}/_{6}+^{1}/_{30}=^{2}/_{10}  ^{1}/_{10}+^{1}/_{10}=^{2}/_{10} 
^{1}/_{11}+^{1}/_{110}=^{1}/_{10}  ^{1}/_{12}+^{1}/_{60}=^{1}/_{10}  ^{1}/_{14}+^{1}/_{35}=^{1}/_{10}  ^{1}/_{15}+^{1}/_{30}=^{1}/_{10}  ^{1}/_{20}+^{1}/_{20}=^{1}/_{10} 
How many solutions has this equation for 1 ≤ n ≤ 9?
Taking three different letters from the 26 letters of the alphabet, character strings of length three can be formed.
Examples are 'abc', 'hat' and 'zyx'.
When we study these three examples we see that for 'abc' two characters come lexicographically after its neighbour to the left.
For 'hat' there is exactly one character that comes lexicographically after its neighbour to the left. For 'zyx' there are zero characters that come lexicographically after its neighbour to the left.
In all there are 10400 strings of length 3 for which exactly one character comes lexicographically after its neighbour to the left.
We now consider strings of n ≤ 26 different characters from the alphabet.
For every n, p(n) is the number of strings of length n for which exactly one character comes lexicographically after its neighbour to the left.
What is the maximum value of p(n)?
A composite number can be factored many different ways. For instance, not including multiplication by one, 24 can be factored in 7 distinct ways:
24 = 2x3x4
24 = 2x2x6
24 = 4x6
24 = 3x8
24 = 2x12
24 = 24
Recall that the digital root of a number, in base 10, is found by adding together the digits of that number, and repeating that process until a number is arrived at that is less than 10. Thus the digital root of 467 is 8.
We shall call a Digital Root Sum (DRS) the sum of the digital roots of the individual factors of our number.
The chart below demonstrates all of the DRS values for 24.
Factorisation  Digital Root Sum 

2x2x2x3 
9 
2x3x4 
9 
2x2x6 
10 
4x6 
10 
3x8 
11 
2x12 
5 
24 
6 
The maximum Digital Root Sum of 24 is 11.
The function mdrs(n) gives the maximum Digital Root Sum of n. So mdrs(24)=11.
Find ∑ mdrs(n) for 1 < n < 1,000,000.
For any N, let f(N) be the last five digits before the trailing zeroes in N!.
For example,
9! = 362880 so f(9)=36288
10! = 3628800 so f(10)=36288
20! = 2432902008176640000 so f(20)=17664
Find f(1,000,000,000,000)
A triomino is a shape consisting of three squares joined via the edges. There are two basic forms:
If all possible orientations are taken into account there are six:
Any n by m grid for which nxm is divisible by 3 can be tiled with triominoes.
If we consider tilings that can be obtained by reflection or rotation from another tiling as different there are 41 ways a 2 by 9 grid can be tiled with triominoes:
In how many ways can a 9 by 12 grid be tiled in this way by triominoes?
In the hexadecimal number system numbers are represented using 16 different digits:
The hexadecimal number AF when written in the decimal number system equals 10x16+15=175.
In the 3digit hexadecimal numbers 10A, 1A0, A10, and A01 the digits 0,1 and A are all present.
Like numbers written in base ten we write hexadecimal numbers without leading zeroes.
How many hexadecimal numbers containing at most sixteen hexadecimal digits exist with all of the digits 0,1, and A present at least once?
Give your answer as a hexadecimal number.
(A,B,C,D,E and F in upper case, without any leading or trailing code that marks the number as hexadecimal and without leading zeroes , e.g. 1A3F and not: 1a3f and not 0x1a3f and not $1A3F and not #1A3F and not 0000001A3F)
Consider an equilateral triangle in which straight lines are drawn from each vertex to the middle of the opposite side, such as in the size 1 triangle in the sketch below.
Sixteen triangles of either different shape or size or orientation or location can now be observed in that triangle. Using size 1 triangles as building blocks, larger triangles can be formed, such as the size 2 triangle in the above sketch. Onehundred and four triangles of either different shape or size or orientation or location can now be observed in that size 2 triangle.
It can be observed that the size 2 triangle contains 4 size 1 triangle building blocks. A size 3 triangle would contain 9 size 1 triangle building blocks and a size n triangle would thus contain n^{2} size 1 triangle building blocks.
If we denote T(n) as the number of triangles present in a triangle of size n, then
T(1) = 16
T(2) = 104
Find T(36).
How many 20 digit numbers n (without any leading zero) exist such that no three consecutive digits of n have a sum greater than 9?
A segment is uniquely defined by its two endpoints.
By considering two line segments in plane geometry there are three possibilities:
the segments have zero points, one point, or infinitely many points in common.
Moreover when two segments have exactly one point in common it might be the case that that common point is an endpoint of either one of the segments or of both. If a common point of two segments is not an endpoint of either of the segments it is an interior point of both segments.
We will call a common point T of two segments L_{1} and L_{2} a true intersection point of L_{1} and L_{2} if T is the only common point of L_{1} and L_{2} and T is an interior point of both segments.
Consider the three segments L_{1}, L_{2}, and L_{3}:
L_{1}: (27, 44) to (12, 32)
L_{2}: (46, 53) to (17, 62)
L_{3}: (46, 70) to (22, 40)
It can be verified that line segments L_{2} and L_{3} have a true intersection point. We note that as the one of the end points of L_{3}: (22,40) lies on L_{1} this is not considered to be a true point of intersection. L_{1} and L_{2} have no common point. So among the three line segments, we find one true intersection point.
Now let us do the same for 5000 line segments. To this end, we generate 20000 numbers using the socalled "Blum Blum Shub" pseudorandom number generator.
s_{0} = 290797
s_{n+1} = s_{n}×s_{n} (modulo 50515093)
t_{n} = s_{n} (modulo 500)
To create each line segment, we use four consecutive numbers t_{n}. That is, the first line segment is given by:
(t_{1}, t_{2}) to (t_{3}, t_{4})
The first four numbers computed according to the above generator should be: 27, 144, 12 and 232. The first segment would thus be (27,144) to (12,232).
How many distinct true intersection points are found among the 5000 line segments?
A 4x4 grid is filled with digits d, 0 ≤ d ≤ 9.
It can be seen that in the grid
6 3 3 0
5 0 4 3
0 7 1 4
1 2 4 5
the sum of each row and each column has the value 12. Moreover the sum of each diagonal is also 12.
In how many ways can you fill a 4x4 grid with the digits d, 0 ≤ d ≤ 9 so that each row, each column, and both diagonals have the same sum?
For two positive integers a and b, the Ulam sequence U(a,b) is defined by U(a,b)_{1} = a, U(a,b)_{2} = b and for k > 2, U(a,b)_{k} is the smallest integer greater than U(a,b)_{(k1)} which can be written in exactly one way as the sum of two distinct previous members of U(a,b).
For example, the sequence U(1,2) begins with
1, 2, 3 = 1 + 2, 4 = 1 + 3, 6 = 2 + 4, 8 = 2 + 6, 11 = 3 + 8;
5 does not belong to it because 5 = 1 + 4 = 2 + 3 has two representations as the sum of two previous members, likewise 7 = 1 + 6 = 3 + 4.
Find ∑ U(2,2n+1)_{k} for 2 ≤ n ≤10, where k = 10^{11}.
Consider the number 142857. We can rightrotate this number by moving the last digit (7) to the front of it, giving us 714285.
It can be verified that 714285=5×142857.
This demonstrates an unusual property of 142857: it is a divisor of its rightrotation.
Find the last 5 digits of the sum of all integers n, 10 < n < 10^{100}, that have this property.
Define f(0)=1 and f(n) to be the number of different ways n can be expressed as a sum of integer powers of 2 using each power no more than twice.
For example, f(10)=5 since there are five different ways to express 10:
1 + 1 + 8
1 + 1 + 4 + 4
1 + 1 + 2 + 2 + 4
2 + 4 + 4
2 + 8
What is f(10^{25})?
Take the number 6 and multiply it by each of 1273 and 9854:
6 × 1273 = 7638
6 × 9854 = 59124
By concatenating these products we get the 1 to 9 pandigital 763859124. We will call 763859124 the "concatenated product of 6 and (1273,9854)". Notice too, that the concatenation of the input numbers, 612739854, is also 1 to 9 pandigital.
The same can be done for 0 to 9 pandigital numbers.
What is the largest 0 to 9 pandigital 10digit concatenated product of an integer with two or more other integers, such that the concatenation of the input numbers is also a 0 to 9 pandigital 10digit number?
For a positive integer n, let f(n) be the sum of the squares of the digits (in base 10) of n, e.g.
f(3) = 3^{2} = 9,
f(25) = 2^{2} + 5^{2} = 4 + 25 = 29,
f(442) = 4^{2} + 4^{2} + 2^{2} = 16 + 16 + 4 = 36
Find the last nine digits of the sum of all n, 0 < n < 10^{20}, such that f(n) is a perfect square.
How many 18digit numbers n (without leading zeros) are there such that no digit occurs more than three times in n?
We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry. For example, using exactly thirtytwo square tiles we can form two different square laminae:
With onehundred tiles, and not necessarily using all of the tiles at one time, it is possible to form fortyone different square laminae.
Using up to one million tiles how many different square laminae can be formed?
We shall define a square lamina to be a square outline with a square "hole" so that the shape possesses vertical and horizontal symmetry.
Given eight tiles it is possible to form a lamina in only one way: 3x3 square with a 1x1 hole in the middle. However, using thirtytwo tiles it is possible to form two distinct laminae.
If t represents the number of tiles used, we shall say that t = 8 is type L(1) and t = 32 is type L(2).
Let N(n) be the number of t ≤ 1000000 such that t is type L(n); for example, N(15) = 832.
What is ∑ N(n) for 1 ≤ n ≤ 10?
For example, f(10)=5 since there are five different ways to express 10:
10 = 8+2 = 8+1+1 = 4+4+2 = 4+2+2+1+1 = 4+4+1+1
It can be shown that for every fraction p/q (p>0, q>0) there exists at least one integer n such that
f(n)/f(n1)=p/q.
For instance, the smallest n for which f(n)/f(n1)=13/17 is 241.
The binary expansion of 241 is 11110001.
Reading this binary number from the most significant bit to the least significant bit there are 4 one's, 3 zeroes and 1 one. We shall call the string 4,3,1 the Shortened Binary Expansion of 241.
Find the Shortened Binary Expansion of the smallest n for which
f(n)/f(n1)=123456789/987654321.
Give your answer as comma separated integers, without any whitespaces.
The four rightangled triangles with sides (9,12,15), (12,16,20), (5,12,13) and (12,35,37) all have one of the shorter sides (catheti) equal to 12. It can be shown that no other integer sided rightangled triangle exists with one of the catheti equal to 12.
Find the smallest integer that can be the length of a cathetus of exactly 47547 different integer sided rightangled triangles.
Let ABCD be a convex quadrilateral, with diagonals AC and BD. At each vertex the diagonal makes an angle with each of the two sides, creating eight corner angles.
For example, at vertex A, the two angles are CAD, CAB.
We call such a quadrilateral for which all eight corner angles have integer values when measured in degrees an "integer angled quadrilateral". An example of an integer angled quadrilateral is a square, where all eight corner angles are 45°. Another example is given by DAC = 20°, BAC = 60°, ABD = 50°, CBD = 30°, BCA = 40°, DCA = 30°, CDB = 80°, ADB = 50°.
What is the total number of nonsimilar integer angled quadrilaterals?
Note: In your calculations you may assume that a calculated angle is integral if it is within a tolerance of 10^{9} of an integer value.
It can be seen that each pair of consecutive digits of 45656 has a difference of one.
A number for which every pair of consecutive digits has a difference of one is called a step number.
A pandigital number contains every decimal digit from 0 to 9 at least once.
How many pandigital step numbers less than 10^{40} are there?
Find the number of integers 1 < n < 10^{7}, for which n and n + 1 have the same number of positive divisors. For example, 14 has the positive divisors 1, 2, 7, 14 while 15 has 1, 3, 5, 15.
For any integer n, consider the three functions
f_{1,n}(x,y,z) = x^{n+1} + y^{n+1} − z^{n+1}
f_{2,n}(x,y,z) = (xy + yz + zx)*(x^{n1} + y^{n1} − z^{n1})
f_{3,n}(x,y,z) = xyz*(x^{n2} + y^{n2} − z^{n2})
and their combination
f_{n}(x,y,z) = f_{1,n}(x,y,z) + f_{2,n}(x,y,z) − f_{3,n}(x,y,z)
We call (x,y,z) a golden triple of order k if x, y, and z are all rational numbers of the form a / b with
0 < a < b ≤ k and there is (at least) one integer n, so that f_{n}(x,y,z) = 0.
Let s(x,y,z) = x + y + z.
Let t = u / v be the sum of all distinct s(x,y,z) for all golden triples (x,y,z) of order 35.
All the s(x,y,z) and t must be in reduced form.
Find u + v.
Having three black objects B and one white object W they can be grouped in 7 ways like this:
(BBBW)  (B,BBW)  (B,B,BW)  (B,B,B,W)  (B,BB,W)  (BBB,W)  (BB,BW) 
In how many ways can sixty black objects B and forty white objects W be thus grouped?
The RSA encryption is based on the following procedure:
Generate two distinct primes p and q.
Compute n=pq and φ=(p1)(q1).
Find an integer e, 1<e<φ, such that gcd(e,φ)=1.
A message in this system is a number in the interval [0,n1].
A text to be encrypted is then somehow converted to messages (numbers in the interval [0,n1]).
To encrypt the text, for each message, m, c=m^{e} mod n is calculated.
To decrypt the text, the following procedure is needed: calculate d such that ed=1 mod φ, then for each encrypted message, c, calculate m=c^{d} mod n.
There exist values of e and m such that m^{e} mod n=m.
We call messages m for which m^{e} mod n=m unconcealed messages.
An issue when choosing e is that there should not be too many unconcealed messages.
For instance, let p=19 and q=37.
Then n=19*37=703 and φ=18*36=648.
If we choose e=181, then, although gcd(181,648)=1 it turns out that all possible messages
m (0≤m≤n1) are unconcealed when calculating m^{e} mod n.
For any valid choice of e there exist some unconcealed messages.
It's important that the number of unconcealed messages is at a minimum.
Choose p=1009 and q=3643.
Find the sum of all values of e, 1<e<φ(1009,3643) and gcd(e,φ)=1, so that the number of unconcealed messages for this value of e is at a minimum.
Let N be a positive integer and let N be split into k equal parts, r = N/k, so that N = r + r + ... + r.
Let P be the product of these parts, P = r × r × ... × r = r^{k}.
For example, if 11 is split into five equal parts, 11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2, then P = 2.2^{5} = 51.53632.
Let M(N) = P_{max} for a given value of N.
It turns out that the maximum for N = 11 is found by splitting eleven into four equal parts which leads to P_{max} = (11/4)^{4}; that is, M(11) = 14641/256 = 57.19140625, which is a terminating decimal.
However, for N = 8 the maximum is achieved by splitting it into three equal parts, so M(8) = 512/27, which is a nonterminating decimal.
Let D(N) = N if M(N) is a nonterminating decimal and D(N) = N if M(N) is a terminating decimal.
For example, ∑ D(N) for 5 ≤ N ≤ 100 is 2438.
Find ∑ D(N) for 5 ≤ N ≤ 10000.
Consider the set I_{r} of points (x,y) with integer coordinates in the interior of the circle with radius r, centered at the origin, i.e. x^{2} + y^{2} < r^{2}.
For a radius of 2, I_{2} contains the nine points (0,0), (1,0), (1,1), (0,1), (1,1), (1,0), (1,1), (0,1) and (1,1). There are eight triangles having all three vertices in I_{2} which contain the origin in the interior. Two of them are shown below, the others are obtained from these by rotation.
For a radius of 3, there are 360 triangles containing the origin in the interior and having all vertices in I_{3} and for I_{5} the number is 10600.
How many triangles are there containing the origin in the interior and having all three vertices in I_{105}?
The game Number Mind is a variant of the well known game Master Mind.
Instead of coloured pegs, you have to guess a secret sequence of digits. After each guess you're only told in how many places you've guessed the correct digit. So, if the sequence was 1234 and you guessed 2036, you'd be told that you have one correct digit; however, you would NOT be told that you also have another digit in the wrong place.
For instance, given the following guesses for a 5digit secret sequence,
90342 ;2 correct
70794 ;0 correct
39458 ;2 correct
34109 ;1 correct
51545 ;2 correct
12531 ;1 correct
The correct sequence 39542 is unique.
Based on the following guesses,
5616185650518293 ;2 correct
3847439647293047 ;1 correct
5855462940810587 ;3 correct
9742855507068353 ;3 correct
4296849643607543 ;3 correct
3174248439465858 ;1 correct
4513559094146117 ;2 correct
7890971548908067 ;3 correct
8157356344118483 ;1 correct
2615250744386899 ;2 correct
8690095851526254 ;3 correct
6375711915077050 ;1 correct
6913859173121360 ;1 correct
6442889055042768 ;2 correct
2321386104303845 ;0 correct
2326509471271448 ;2 correct
5251583379644322 ;2 correct
1748270476758276 ;3 correct
4895722652190306 ;1 correct
3041631117224635 ;3 correct
1841236454324589 ;3 correct
2659862637316867 ;2 correct
Find the unique 16digit secret sequence.
Here are the records from a busy telephone system with one million users:
RecNr  Caller  Called 

1  200007  100053 
2  600183  500439 
3  600863  701497 
...  ...  ... 
The telephone number of the caller and the called number in record n are Caller(n) = S_{2n1} and Called(n) = S_{2n} where S_{1,2,3,...} come from the "Lagged Fibonacci Generator":
For 1 ≤ k ≤ 55, S_{k} = [100003  200003k + 300007k^{3}] (modulo 1000000)
For 56 ≤ k, S_{k} = [S_{k24} + S_{k55}] (modulo 1000000)
If Caller(n) = Called(n) then the user is assumed to have misdialled and the call fails; otherwise the call is successful.
From the start of the records, we say that any pair of users X and Y are friends if X calls Y or viceversa. Similarly, X is a friend of a friend of Z if X is a friend of Y and Y is a friend of Z; and so on for longer chains.
The Prime Minister's phone number is 524287. After how many successful calls, not counting misdials, will 99% of the users (including the PM) be a friend, or a friend of a friend etc., of the Prime Minister?
A composite is a number containing at least two prime factors. For example, 15 = 3 × 5; 9 = 3 × 3; 12 = 2 × 2 × 3.
There are ten composites below thirty containing precisely two, not necessarily distinct, prime factors: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26.
How many composite integers, n < 10^{8}, have precisely two, not necessarily distinct, prime factors?
The hyperexponentiation or tetration of a number a by a positive integer b, denoted by a↑↑b or ^{b}a, is recursively defined by:
a↑↑1 = a,
a↑↑(k+1) = a^{(a↑↑k)}.
Thus we have e.g. 3↑↑2 = 3^{3} = 27, hence 3↑↑3 = 3^{27} = 7625597484987 and 3↑↑4 is roughly 10^{3.6383346400240996*10^12}.
Find the last 8 digits of 1777↑↑1855.
Consider the following configuration of 64 triangles:
We wish to colour the interior of each triangle with one of three colours: red, green or blue, so that no two neighbouring triangles have the same colour. Such a colouring shall be called valid. Here, two triangles are said to be neighbouring if they share an edge.
Note: if they only share a vertex, then they are not neighbours.
For example, here is a valid colouring of the above grid:
A colouring C' which is obtained from a colouring C by rotation or reflection is considered distinct from C unless the two are identical.
How many distinct valid colourings are there for the above configuration?
Let S_{m} = (x_{1}, x_{2}, ... , x_{m}) be the mtuple of positive real numbers with x_{1} + x_{2} + ... + x_{m} = m for which P_{m} = x_{1} * x_{2}^{2} * ... * x_{m}^{m} is maximised.
For example, it can be verified that [P_{10}] = 4112 ([ ] is the integer part function).
Find Σ[P_{m}] for 2 ≤ m ≤ 15.
A particular school offers cash rewards to children with good attendance and punctuality. If they are absent for three consecutive days or late on more than one occasion then they forfeit their prize.
During an nday period a trinary string is formed for each child consisting of L's (late), O's (on time), and A's (absent).
Although there are eightyone trinary strings for a 4day period that can be formed, exactly fortythree strings would lead to a prize:
OOOO OOOA OOOL OOAO OOAA OOAL OOLO OOLA OAOO OAOA
OAOL OAAO OAAL OALO OALA OLOO OLOA OLAO OLAA AOOO
AOOA AOOL AOAO AOAA AOAL AOLO AOLA AAOO AAOA AAOL
AALO AALA ALOO ALOA ALAO ALAA LOOO LOOA LOAO LOAA
LAOO LAOA LAAO
How many "prize" strings exist over a 30day period?
Let $x$ be a real number.
A best approximation to $x$ for the denominator bound $d$ is a rational number $\frac r s $ in reduced form, with $s \le d$, such that any rational number which is closer to $x$ than $\frac r s$ has a denominator larger than $d$:
For example, the best approximation to $\sqrt {13}$ for the denominator bound 20 is $\frac {18} 5$ and the best approximation to $\sqrt {13}$ for the denominator bound 30 is $\frac {101}{28}$.
Find the sum of all denominators of the best approximations to $\sqrt n$ for the denominator bound $10^{12}$, where $n$ is not a perfect square and $ 1 < n \le 100000$.
A positive integer n is called squarefree, if no square of a prime divides n, thus 1, 2, 3, 5, 6, 7, 10, 11 are squarefree, but not 4, 8, 9, 12.
How many squarefree numbers are there below 2^{50}?
Consider graphs built with the units A: and B: , where the units are glued along the vertical edges as in the graph .
A configuration of type (a,b,c) is a graph thus built of a units A and b units B, where the graph's vertices are coloured using up to c colours, so that no two adjacent vertices have the same colour.
The compound graph above is an example of a configuration of type (2,2,6), in fact of type (2,2,c) for all c ≥ 4.
Let N(a,b,c) be the number of configurations of type (a,b,c).
For example, N(1,0,3) = 24, N(0,2,4) = 92928 and N(2,2,3) = 20736.
Find the last 8 digits of N(25,75,1984).
Let's call an integer sided triangle with exactly one angle of 60 degrees a 60degree triangle.
Let r be the radius of the inscribed circle of such a 60degree triangle.
There are 1234 60degree triangles for which r ≤ 100.
Let T(n) be the number of 60degree triangles for which r ≤ n, so
T(100) = 1234, T(1000) = 22767, and T(10000) = 359912.
Find T(1053779).
Build a triangle from all positive integers in the following way:
1
2 3
4 5 6
7 8 9 10
11 12 13 14 15
16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45
46 47 48 49 50 51 52 53 54 55
56 57 58 59 60 61 62 63 64 65 66
. . .
Each positive integer has up to eight neighbours in the triangle.
A set of three primes is called a prime triplet if one of the three primes has the other two as neighbours in the triangle.
For example, in the second row, the prime numbers 2 and 3 are elements of some prime triplet.
If row 8 is considered, it contains two primes which are elements of some prime triplet, i.e. 29 and 31.
If row 9 is considered, it contains only one prime which is an element of some prime triplet: 37.
Define S(n) as the sum of the primes in row n which are elements of any prime triplet.
Then S(8)=60 and S(9)=37.
You are given that S(10000)=950007619.
Find S(5678027) + S(7208785).
Given is the function f(x) = ⌊2^{30.403243784x2}⌋ × 10^{9} ( ⌊ ⌋ is the floorfunction),
the sequence u_{n} is defined by u_{0} = 1 and u_{n+1} = f(u_{n}).
Find u_{n} + u_{n+1} for n = 10^{12}.
Give your answer with 9 digits after the decimal point.
A best approximation to a real number $x$ for the denominator bound $d$ is a rational number $\frac r s$ (in reduced form) with $s \le d$, so that any rational number $\frac p q$ which is closer to $x$ than $\frac r s$ has $q > d$.
Usually the best approximation to a real number is uniquely determined for all denominator bounds. However, there are some exceptions, e.g. $\frac 9 {40}$ has the two best approximations $\frac 1 4$ and $\frac 1 5$ for the denominator bound $6$. We shall call a real number $x$ ambiguous, if there is at least one denominator bound for which $x$ possesses two best approximations. Clearly, an ambiguous number is necessarily rational.
How many ambiguous numbers $x=\frac p q, 0 < x < \frac 1 {100}$, are there whose denominator $q$ does not exceed $10^8$?
Three circles of equal radius are placed inside a larger circle such that each pair of circles is tangent to one another and the inner circles do not overlap. There are four uncovered "gaps" which are to be filled iteratively with more tangent circles.
At each iteration, a maximally sized circle is placed in each gap, which creates more gaps for the next iteration. After 3 iterations (pictured), there are 108 gaps and the fraction of the area which is not covered by circles is 0.06790342, rounded to eight decimal places.
What fraction of the area is not covered by circles after 10 iterations?
Give your answer rounded to eight decimal places using the format x.xxxxxxxx .
We shall define a sqube to be a number of the form, p^{2}q^{3}, where p and q are distinct primes.
For example, 200 = 5^{2}2^{3} or 120072949 = 23^{2}61^{3}.
The first five squbes are 72, 108, 200, 392, and 500.
Interestingly, 200 is also the first number for which you cannot change any single digit to make a prime; we shall call such numbers, primeproof. The next primeproof sqube which contains the contiguous substring "200" is 1992008.
Find the 200th primeproof sqube containing the contiguous substring "200".
For any set A of numbers, let sum(A) be the sum of the elements of A.
Consider the set B = {1,3,6,8,10,11}.
There are 20 subsets of B containing three elements, and their sums are:
sum({1,3,6}) = 10,
sum({1,3,8}) = 12,
sum({1,3,10}) = 14,
sum({1,3,11}) = 15,
sum({1,6,8}) = 15,
sum({1,6,10}) = 17,
sum({1,6,11}) = 18,
sum({1,8,10}) = 19,
sum({1,8,11}) = 20,
sum({1,10,11}) = 22,
sum({3,6,8}) = 17,
sum({3,6,10}) = 19,
sum({3,6,11}) = 20,
sum({3,8,10}) = 21,
sum({3,8,11}) = 22,
sum({3,10,11}) = 24,
sum({6,8,10}) = 24,
sum({6,8,11}) = 25,
sum({6,10,11}) = 27,
sum({8,10,11}) = 29.
Some of these sums occur more than once, others are unique.
For a set A, let U(A,k) be the set of unique sums of kelement subsets of A, in our example we find U(B,3) = {10,12,14,18,21,25,27,29} and sum(U(B,3)) = 156.
Now consider the 100element set S = {1^{2}, 2^{2}, ... , 100^{2}}.
S has 100891344545564193334812497256 50element subsets.
Determine the sum of all integers which are the sum of exactly one of the 50element subsets of S, i.e. find sum(U(S,50)).
Three mirrors are arranged in the shape of an equilateral triangle, with their reflective surfaces pointing inwards. There is an infinitesimal gap at each vertex of the triangle through which a laser beam may pass.
Label the vertices A, B and C. There are 2 ways in which a laser beam may enter vertex C, bounce off 11 surfaces, then exit through the same vertex: one way is shown below; the other is the reverse of that.
There are 80840 ways in which a laser beam may enter vertex C, bounce off 1000001 surfaces, then exit through the same vertex.
In how many ways can a laser beam enter at vertex C, bounce off 12017639147 surfaces, then exit through the same vertex?
The binomial coefficients $\displaystyle \binom n k$ can be arranged in triangular form, Pascal's triangle, like this:
1  
1  1  
1  2  1  
1  3  3  1  
1  4  6  4  1  
1  5  10  10  5  1  
1  6  15  20  15  6  1  
1  7  21  35  35  21  7  1 
It can be seen that the first eight rows of Pascal's triangle contain twelve distinct numbers: 1, 2, 3, 4, 5, 6, 7, 10, 15, 20, 21 and 35.
A positive integer n is called squarefree if no square of a prime divides n. Of the twelve distinct numbers in the first eight rows of Pascal's triangle, all except 4 and 20 are squarefree. The sum of the distinct squarefree numbers in the first eight rows is 105.
Find the sum of the distinct squarefree numbers in the first 51 rows of Pascal's triangle.
A Hamming number is a positive number which has no prime factor larger than 5.
So the first few Hamming numbers are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15.
There are 1105 Hamming numbers not exceeding 10^{8}.
We will call a positive number a generalised Hamming number of type n, if it has no prime factor larger than n.
Hence the Hamming numbers are the generalised Hamming numbers of type 5.
How many generalised Hamming numbers of type 100 are there which don't exceed 10^{9}?
Peter has nine foursided (pyramidal) dice, each with faces numbered 1, 2, 3, 4.
Colin has six sixsided (cubic) dice, each with faces numbered 1, 2, 3, 4, 5, 6.
Peter and Colin roll their dice and compare totals: the highest total wins. The result is a draw if the totals are equal.
What is the probability that Pyramidal Pete beats Cubic Colin? Give your answer rounded to seven decimal places in the form 0.abcdefg
Find the unique positive integer whose square has the form 1_2_3_4_5_6_7_8_9_0,
where each “_” is a single digit.
For some positive integers k, there exists an integer partition of the form 4^{t} = 2^{t} + k,
where 4^{t}, 2^{t}, and k are all positive integers and t is a real number.
The first two such partitions are 4^{1} = 2^{1} + 2 and 4^{1.5849625...} = 2^{1.5849625...} + 6.
Partitions where t is also an integer are called perfect.
For any m ≥ 1 let P(m) be the proportion of such partitions that are perfect with k ≤ m.
Thus P(6) = 1/2.
In the following table are listed some values of P(m)
P(5) = 1/1
P(10) = 1/2
P(15) = 2/3
P(20) = 1/2
P(25) = 1/2
P(30) = 2/5
...
P(180) = 1/4
P(185) = 3/13
Find the smallest m for which P(m) < 1/12345
A robot moves in a series of onefifth circular arcs (72°), with a free choice of a clockwise or an anticlockwise arc for each step, but no turning on the spot.
One of 70932 possible closed paths of 25 arcs starting northward is
Given that the robot starts facing North, how many journeys of 70 arcs in length can it take that return it, after the final arc, to its starting position?
(Any arc may be traversed multiple times.)
A kinput binary truth table is a map from k input bits (binary digits, 0 [false] or 1 [true]) to 1 output bit. For example, the 2input binary truth tables for the logical AND and XOR functions are:
x  y  x AND y 

0  0  0 
0  1  0 
1  0  0 
1  1  1 
x  y  x XOR y 

0  0  0 
0  1  1 
1  0  1 
1  1  0 
How many 6input binary truth tables, τ, satisfy the formula
for all 6bit inputs (a, b, c, d, e, f)?
Let O be the point (0,0) and C the point (r/4,r/4).
Let N(r) be the number of points B in S(r), so that the triangle OBC has an obtuse angle, i.e. the largest angle α satisfies 90°<α<180°.
So, for example, N(4)=24 and N(8)=100.
What is N(1,000,000,000)?
For a positive integer n, let σ_{2}(n) be the sum of the squares of its divisors. For example,
Find the sum of all n, 0 < n < 64,000,000 such that σ_{2}(n) is a perfect square.
An axisaligned cuboid, specified by parameters { (x_{0},y_{0},z_{0}), (dx,dy,dz) }, consists of all points (X,Y,Z) such that x_{0} ≤ X ≤ x_{0}+dx, y_{0} ≤ Y ≤ y_{0}+dy and z_{0} ≤ Z ≤ z_{0}+dz. The volume of the cuboid is the product, dx × dy × dz. The combined volume of a collection of cuboids is the volume of their union and will be less than the sum of the individual volumes if any cuboids overlap.
Let C_{1},...,C_{50000} be a collection of 50000 axisaligned cuboids such that C_{n} has parameters
x_{0} = S_{6n5} modulo 10000
y_{0} = S_{6n4} modulo 10000
z_{0} = S_{6n3} modulo 10000
dx = 1 + (S_{6n2} modulo 399)
dy = 1 + (S_{6n1} modulo 399)
dz = 1 + (S_{6n} modulo 399)
where S_{1},...,S_{300000} come from the "Lagged Fibonacci Generator":
For 1 ≤ k ≤ 55, S_{k} = [100003  200003k + 300007k^{3}] (modulo 1000000)
For 56 ≤ k, S_{k} = [S_{k24} + S_{k55}] (modulo 1000000)
Thus, C_{1} has parameters {(7,53,183),(94,369,56)}, C_{2} has parameters {(2383,3563,5079),(42,212,344)}, and so on.
The combined volume of the first 100 cuboids, C_{1},...,C_{100}, is 723581599.
What is the combined volume of all 50000 cuboids, C_{1},...,C_{50000} ?
A 30×30 grid of squares contains 900 fleas, initially one flea per square.
When a bell is rung, each flea jumps to an adjacent square at random (usually 4 possibilities, except for fleas on the edge of the grid or at the corners).
What is the expected number of unoccupied squares after 50 rings of the bell? Give your answer rounded to six decimal places.
Let φ be Euler's totient function, i.e. for a natural number n, φ(n) is the number of k, 1 ≤ k ≤ n, for which gcd(k,n) = 1.
By iterating φ, each positive integer generates a decreasing chain of numbers ending in 1.
E.g. if we start with 5 the sequence 5,4,2,1 is generated.
Here is a listing of all chains with length 4:
7,6,2,1
8,4,2,1
9,6,2,1
10,4,2,1
12,4,2,1
14,6,2,1
18,6,2,1
Only two of these chains start with a prime, their sum is 12.
What is the sum of all primes less than 40000000 which generate a chain of length 25?
Consider the problem of building a wall out of 2×1 and 3×1 bricks (horizontal×vertical dimensions) such that, for extra strength, the gaps between horizontallyadjacent bricks never line up in consecutive layers, i.e. never form a "running crack".
For example, the following 9×3 wall is not acceptable due to the running crack shown in red:
There are eight ways of forming a crackfree 9×3 wall, written W(9,3) = 8.
Calculate W(32,10).
Consider numbers t(n) of the form t(n) = 2n^{2}1 with n > 1.
The first such numbers are 7, 17, 31, 49, 71, 97, 127 and 161.
It turns out that only 49 = 7*7 and 161 = 7*23 are not prime.
For n ≤ 10000 there are 2202 numbers t(n) that are prime.
How many numbers t(n) are prime for n ≤ 50,000,000 ?
A positive integer with k (decimal) digits is called balanced if its first ⌈^{k}/_{2}⌉ digits sum to the same value as its last ⌈^{k}/_{2}⌉ digits, where ⌈x⌉, pronounced ceiling of x, is the smallest integer ≥ x, thus ⌈π⌉ = 4 and ⌈5⌉ = 5.
So, for example, all palindromes are balanced, as is 13722.
Let T(n) be the sum of all balanced numbers less than 10^{n}.
Thus: T(1) = 45, T(2) = 540 and T(5) = 334795890.
Find T(47) mod 3^{15}
Consider the right angled triangle with sides a=7, b=24 and c=25.
The area of this triangle is 84, which is divisible by the perfect numbers 6 and 28.
Moreover it is a primitive right angled triangle as gcd(a,b)=1 and gcd(b,c)=1.
Also c is a perfect square.
We will call a right angled triangle perfect if
it is a primitive right angled triangle
its hypotenuse is a perfect square
We will call a right angled triangle superperfect if
it is a perfect right angled triangle and
its area is a multiple of the perfect numbers 6 and 28.
How many perfect rightangled triangles with c≤10^{16} exist that are not superperfect?
Let A and B be bit strings (sequences of 0's and 1's).
If A is equal to the leftmost length(A) bits of B, then A is said to be a prefix of B.
For example, 00110 is a prefix of 001101001, but not of 00111 or 100110.
A prefixfree code of size n is a collection of n distinct bit strings such that no string is a prefix of any other. For example, this is a prefixfree code of size 6:
0000, 0001, 001, 01, 10, 11
Now suppose that it costs one penny to transmit a '0' bit, but four pence to transmit a '1'.
Then the total cost of the prefixfree code shown above is 35 pence, which happens to be the cheapest possible for the skewed pricing scheme in question.
In short, we write Cost(6) = 35.
What is Cost(10^{9}) ?
Let D_{0} be the twoletter string "Fa". For n≥1, derive D_{n} from D_{n1} by the stringrewriting rules:
"a" → "aRbFR"
"b" → "LFaLb"
Thus, D_{0} = "Fa", D_{1} = "FaRbFR", D_{2} = "FaRbFRRLFaLbFR", and so on.
These strings can be interpreted as instructions to a computer graphics program, with "F" meaning "draw forward one unit", "L" meaning "turn left 90 degrees", "R" meaning "turn right 90 degrees", and "a" and "b" being ignored. The initial position of the computer cursor is (0,0), pointing up towards (0,1).
Then D_{n} is an exotic drawing known as the Heighway Dragon of order n. For example, D_{10} is shown below; counting each "F" as one step, the highlighted spot at (18,16) is the position reached after 500 steps.
What is the position of the cursor after 10^{12} steps in D_{50} ?
Give your answer in the form x,y with no spaces.
We shall call a positive integer A an "Alexandrian integer", if there exist integers p, q, r such that:
A = p · q · r and 

= 

+ 

+ 

For example, 630 is an Alexandrian integer (p = 5, q = −7, r = −18). In fact, 630 is the 6^{th} Alexandrian integer, the first 6 Alexandrian integers being: 6, 42, 120, 156, 420 and 630.
Find the 150000^{th} Alexandrian integer.
What is the length of the shortest pipe, of internal radius 50mm, that can fully contain 21 balls of radii 30mm, 31mm, ..., 50mm?
Give your answer in micrometres (10^{6} m) rounded to the nearest integer.
Let us call an integer sided triangle with sides a ≤ b ≤ c barely acute if the sides satisfy
a^{2} + b^{2} = c^{2} + 1.
How many barely acute triangles are there with perimeter ≤ 25,000,000?
Let us call an integer sided triangle with sides a ≤ b ≤ c barely obtuse if the sides satisfy
a^{2} + b^{2} = c^{2}  1.
How many barely obtuse triangles are there with perimeter ≤ 75,000,000?
The sequence 1, 1, 1, 3, 5, 9, 17, 31, 57, 105, 193, 355, 653, 1201 ...
is defined by T_{1} = T_{2} = T_{3} = 1 and T_{n} = T_{n1} + T_{n2} + T_{n3}.
It can be shown that 27 does not divide any terms of this sequence.
In fact, 27 is the first odd number with this property.
Find the 124^{th} odd number that does not divide any terms of the above sequence.
The blancmange curve is the set of points $(x, y)$ such that $0 \le x \le 1$ and $y = \sum \limits_{n = 0}^{\infty} {\dfrac{s(2^n x)}{2^n}}$, where $s(x)$ is the distance from $x$ to the nearest integer.
The area under the blancmange curve is equal to ½, shown in pink in the diagram below.
Let C be the circle with centre $\left ( \frac{1}{4}, \frac{1}{2} \right )$ and radius $\frac{1}{4}$, shown in black in the diagram.
What area under the blancmange curve is enclosed by C?
Give your answer rounded to eight decimal places in the form 0.abcdefgh
"The Chase" is a game played with two dice and an even number of players.
The players sit around a table; the game begins with two opposite players having one die each. On each turn, the two players with a die roll it.
If a player rolls a 1, he passes the die to his neighbour on the left; if he rolls a 6, he passes the die to his neighbour on the right; otherwise, he keeps the die for the next turn.
The game ends when one player has both dice after they have been rolled and passed; that player has then lost.
In a game with 100 players, what is the expected number of turns the game lasts?
Give your answer rounded to ten significant digits.
Let S_{n} be the regular nsided polygon – or shape – whose vertices v_{k} (k = 1,2,…,n) have coordinates:
x_{k} = cos( ^{2k1}/_{n} ×180° )  
y_{k} = sin( ^{2k1}/_{n} ×180° ) 
Each S_{n} is to be interpreted as a filled shape consisting of all points on the perimeter and in the interior.
The Minkowski sum, S+T, of two shapes S and T is the result of adding every point in S to every point in T, where point addition is performed coordinatewise: (u, v) + (x, y) = (u+x, v+y).
For example, the sum of S_{3} and S_{4} is the sixsided shape shown in pink below:
How many sides does S_{1864} + S_{1865} + … + S_{1909} have?
Consider the number 3600. It is very special, because
3600 = 20^{2} + 2×40^{2}
3600 = 30^{2} + 3×30^{2}
3600 = 45^{2} + 7×15^{2}
Similarly, we find that 88201 = 99^{2} + 280^{2} = 287^{2} + 2×54^{2} = 283^{2} + 3×52^{2} = 197^{2} + 7×84^{2}.
In 1747, Euler proved which numbers are representable as a sum of two squares. We are interested in the numbers n which admit representations of all of the following four types:
n = a_{2}^{2} + 2 b_{2}^{2}
n = a_{3}^{2} + 3 b_{3}^{2}
n = a_{7}^{2} + 7 b_{7}^{2},
where the a_{k} and b_{k} are positive integers.
There are 75373 such numbers that do not exceed 10^{7}.
How many such numbers are there that do not exceed 2×10^{9}?
For any two strings of digits, A and B, we define F_{A,B} to be the sequence (A,B,AB,BAB,ABBAB,...) in which each term is the concatenation of the previous two.
Further, we define D_{A,B}(n) to be the n^{th} digit in the first term of F_{A,B} that contains at least n digits.
Example:
Let A=1415926535, B=8979323846. We wish to find D_{A,B}(35), say.
The first few terms of F_{A,B} are:
1415926535
8979323846
14159265358979323846
897932384614159265358979323846
14159265358979323846897932384614159265358979323846
Then D_{A,B}(35) is the 35^{th} digit in the fifth term, which is 9.
Now we use for A the first 100 digits of π behind the decimal point:
14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679
and for B the next hundred digits:
82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196 .
Find ∑_{n = 0,1,...,17} 10^{n}× D_{A,B}((127+19n)×7^{n}) .
The binomial coefficient $\displaystyle \binom {10} 3 = 120$.
$120 = 2^3 \times 3 \times 5 = 2 \times 2 \times 2 \times 3 \times 5$, and $2 + 2 + 2 + 3 + 5 = 14$.
So the sum of the terms in the prime factorisation of $\displaystyle \binom {10} 3$ is $14$.
Find the sum of the terms in the prime factorisation of $\displaystyle \binom {20\,000\,000} {15\,000\,000}$.
Two players share an unbiased coin and take it in turns to play "The Race". On Player 1's turn, he tosses the coin once: if it comes up Heads, he scores one point; if it comes up Tails, he scores nothing. On Player 2's turn, she chooses a positive integer T and tosses the coin T times: if it comes up all Heads, she scores 2^{T1} points; otherwise, she scores nothing. Player 1 goes first. The winner is the first to 100 or more points.
On each turn Player 2 selects the number, T, of coin tosses that maximises the probability of her winning.
What is the probability that Player 2 wins?
Give your answer rounded to eight decimal places in the form 0.abcdefgh .
Let f(N) be the number of points with integer coordinates that are on a circle passing through (0,0), (N,0),(0,N), and (N,N).
It can be shown that f(10000) = 36.
What is the sum of all positive integers N ≤ 10^{11} such that f(N) = 420 ?
For an integer n ≥ 4, we define the lower prime square root of n, denoted by lps(n), as the largest prime ≤ √n and the upper prime square root of n, ups(n), as the smallest prime ≥ √n.
So, for example, lps(4) = 2 = ups(4), lps(1000) = 31, ups(1000) = 37.
Let us call an integer n ≥ 4 semidivisible, if one of lps(n) and ups(n) divides n, but not both.
The sum of the semidivisible numbers not exceeding 15 is 30, the numbers are 8, 10 and 12.
15 is not semidivisible because it is a multiple of both lps(15) = 3 and ups(15) = 5.
As a further example, the sum of the 92 semidivisible numbers up to 1000 is 34825.
What is the sum of all semidivisible numbers not exceeding 999966663333 ?
Given is the arithmeticgeometric sequence u(k) = (9003k)r^{k1}.
Let s(n) = Σ_{k=1...n}u(k).
Find the value of r for which s(5000) = 600,000,000,000.
Give your answer rounded to 12 places behind the decimal point.
Suppliers 'A' and 'B' provided the following numbers of products for the luxury hamper market:
Product  'A'  'B' 

Beluga Caviar  5248  640 
Christmas Cake  1312  1888 
Gammon Joint  2624  3776 
Vintage Port  5760  3776 
Champagne Truffles  3936  5664 
Although the suppliers try very hard to ship their goods in perfect condition, there is inevitably some spoilage  i.e. products gone bad.
The suppliers compare their performance using two types of statistic:
 The five perproduct spoilage rates for each supplier are equal to the number of products gone bad divided by the number of products supplied, for each of the five products in turn.
 The overall spoilage rate for each supplier is equal to the total number of products gone bad divided by the total number of products provided by that supplier.
To their surprise, the suppliers found that each of the five perproduct spoilage rates was worse (higher) for 'B' than for 'A' by the same factor (ratio of spoilage rates), m>1; and yet, paradoxically, the overall spoilage rate was worse for 'A' than for 'B', also by a factor of m.
There are thirtyfive m>1 for which this surprising result could have occurred, the smallest of which is 1476/1475.
What's the largest possible value of m?
Give your answer as a fraction reduced to its lowest terms, in the form u/v.
Let T(n) be the number of tours over a 4 × n playing board such that:
 The tour starts in the top left corner.
 The tour consists of moves that are up, down, left, or right one square.
 The tour visits each square exactly once.
 The tour ends in the bottom left corner.
The diagram shows one tour over a 4 × 10 board:
T(10) is 2329. What is T(10^{12}) modulo 10^{8}?
Create a sequence of numbers using the "Blum Blum Shub" pseudorandom number generator:
s_{0}  =  14025256 
s_{n+1}  =  s_{n}^{2} mod 20300713 
Concatenate these numbers s_{0}s_{1}s_{2}… to create a string w of infinite length.
Then, w = 14025256741014958470038053646…
For a positive integer k, if no substring of w exists with a sum of digits equal to k, p(k) is defined to be zero. If at least one substring of w exists with a sum of digits equal to k, we define p(k) = z, where z is the starting position of the earliest such substring.
For instance:
The substrings 1, 14, 1402, …
with respective sums of digits equal to 1, 5, 7, …
start at position 1, hence p(1) = p(5) = p(7) = … = 1.
The substrings 4, 402, 4025, …
with respective sums of digits equal to 4, 6, 11, …
start at position 2, hence p(4) = p(6) = p(11) = … = 2.
The substrings 02, 0252, …
with respective sums of digits equal to 2, 9, …
start at position 3, hence p(2) = p(9) = … = 3.
Note that substring 025 starting at position 3, has a sum of digits equal to 7, but there was an earlier substring (starting at position 1) with a sum of digits equal to 7, so p(7) = 1, not 3.
We can verify that, for 0 < k ≤ 10^{3}, ∑ p(k) = 4742.
Find ∑ p(k), for 0 < k ≤ 2×10^{15}.
A set of disks numbered 1 through 100 are placed in a line in random order.
What is the probability that we have a partial derangement such that exactly 22 prime number discs are found away from their natural positions?
(Any number of nonprime disks may also be found in or out of their natural positions.)
Give your answer rounded to 12 places behind the decimal point in the form 0.abcdefghijkl.
There are 1111 ways in which five 6sided dice (sides numbered 1 to 6) can be rolled so that the top three sum to 15. Some examples are:
D_{1},D_{2},D_{3},D_{4},D_{5} = 4,3,6,3,5
D_{1},D_{2},D_{3},D_{4},D_{5} = 4,3,3,5,6
D_{1},D_{2},D_{3},D_{4},D_{5} = 3,3,3,6,6
D_{1},D_{2},D_{3},D_{4},D_{5} = 6,6,3,3,3
In how many ways can twenty 12sided dice (sides numbered 1 to 12) be rolled so that the top ten sum to 70?
For a positive integer n, let σ(n) be the sum of all divisors of n, so e.g. σ(6) = 1 + 2 + 3 + 6 = 12.
A perfect number, as you probably know, is a number with σ(n) = 2n.
Let us define the perfection quotient of a positive integer as  p(n)  =  σ(n) n 
. 
Find the sum of all positive integers n ≤ 10^{18} for which p(n) has the form k + ^{1}⁄_{2}, where k is an integer.
Given the set {1,2,...,n}, we define f(n,k) as the number of its kelement subsets with an odd sum of elements. For example, f(5,3) = 4, since the set {1,2,3,4,5} has four 3element subsets having an odd sum of elements, i.e.: {1,2,4}, {1,3,5}, {2,3,4} and {2,4,5}.
When all three values n, k and f(n,k) are odd, we say that they make
an oddtriplet [n,k,f(n,k)].
There are exactly five oddtriplets with n ≤ 10, namely:
[1,1,f(1,1) = 1], [5,1,f(5,1) = 3], [5,5,f(5,5) = 1], [9,1,f(9,1) = 5] and [9,9,f(9,9) = 1].
How many oddtriplets are there with n ≤ 10^{12} ?
A positive fraction whose numerator is less than its denominator is called a proper fraction.
For any denominator, d, there will be d−1 proper fractions; for example, with d = 12:
^{1}/_{12} , ^{2}/_{12} , ^{3}/_{12} , ^{4}/_{12} , ^{5}/_{12} , ^{6}/_{12} , ^{7}/_{12} , ^{8}/_{12} , ^{9}/_{12} , ^{10}/_{12} , ^{11}/_{12} .
We shall call a fraction that cannot be cancelled down a resilient fraction.
Furthermore we shall define the resilience of a denominator, R(d), to be the ratio of its proper fractions that are resilient; for example, R(12) = ^{4}/_{11} .
In fact, d = 12 is the smallest denominator having a resilience R(d) < ^{4}/_{10} .
Find the smallest denominator d, having a resilience R(d) < ^{15499}/_{94744} .
You probably know the game Fifteen Puzzle. Here, instead of numbered tiles, we have seven red tiles and eight blue tiles.
A move is denoted by the uppercase initial of the direction (Left, Right, Up, Down) in which the tile is slid, e.g. starting from configuration (S), by the sequence LULUR we reach the configuration (E):
(S)  , (E) 
For each path, its checksum is calculated by (pseudocode):
checksum = (checksum × 243 + m_{1}) mod 100 000 007
checksum = (checksum × 243 + m_{2}) mod 100 000 007
…
checksum = (checksum × 243 + m_{n}) mod 100 000 007
L  76 
R  82 
U  85 
D  68 
For the sequence LULUR given above, the checksum would be 19761398.
Now, starting from configuration (S), find all shortest ways to reach configuration (T).
(S)  , (T) 
What is the sum of all checksums for the paths having the minimal length?
We shall call a fraction that cannot be cancelled down a resilient fraction.
Furthermore we shall define the resilience of a denominator, R(d), to be the ratio of its proper fractions that are resilient; for example, R(12) = ^{4}⁄_{11}.
The resilience of a number d > 1 is then  φ(d) d − 1 
, where φ is Euler's totient function. 
We further define the coresilience of a number n > 1 as C(n)  =  n − φ(n) n − 1  . 
The coresilience of a prime p is C(p)  =  1 p − 1  . 
Find the sum of all composite integers 1 < n ≤ 2×10^{11}, for which C(n) is a unit fraction.
A definition for an ellipse is:
Given a circle c with centre M and radius r and a point G such that d(G,M)<r, the locus of the points that are equidistant from c and G form an ellipse.
Given are the points M(2000,1500) and G(8000,1500).
Given is also the circle c with centre M and radius 15000.
The locus of the points that are equidistant from G and c form an ellipse e.
From a point P outside e the two tangents t_{1} and t_{2} to the ellipse are drawn.
Let the points where t_{1} and t_{2} touch the ellipse be R and S.
For how many lattice points P is angle RPS greater than 45 degrees?
Consider the region constrained by 1 ≤ x and 0 ≤ y ≤ ^{1}/_{x}.
Let S_{1} be the largest square that can fit under the curve.
Let S_{2} be the largest square that fits in the remaining area, and so on.
Let the index of S_{n} be the pair (left, below) indicating the number of squares to the left of S_{n} and the number of squares below S_{n}.
The diagram shows some such squares labelled by number.
S_{2} has one square to its left and none below, so the index of S_{2} is (1,0).
It can be seen that the index of S_{32} is (1,1) as is the index of S_{50}.
50 is the largest n for which the index of S_{n} is (1,1).
What is the largest n for which the index of S_{n} is (3,3)?
The first number n for which φ(n)=13! is 6227180929.
Find the 150,000^{th} such number.
Let S = {2, 3, 5, ..., 4999} be the set of prime numbers less than 5000.
Find the number of subsets of S, the sum of whose elements is a prime number.
Enter the rightmost 16 digits as your answer.
Find the number of nonempty subsets of {1^{1}, 2^{2}, 3^{3},..., 250250^{250250}}, the sum of whose elements is divisible by 250. Enter the rightmost 16 digits as your answer.
A triplet of positive integers (a,b,c) is called a Cardano Triplet if it satisfies the condition:
$$\sqrt[3]{a + b \sqrt{c}} + \sqrt[3]{a  b \sqrt{c}} = 1$$For example, (2,1,5) is a Cardano Triplet.
There exist 149 Cardano Triplets for which a+b+c ≤ 1000.
Find how many Cardano Triplets exist such that a+b+c ≤ 110,000,000.
Given a set of points on a plane, we define a convex hole to be a convex polygon having as vertices any of the given points and not containing any of the given points in its interior (in addition to the vertices, other given points may lie on the perimeter of the polygon).
As an example, the image below shows a set of twenty points and a few such convex holes. The convex hole shown as a red heptagon has an area equal to 1049694.5 square units, which is the highest possible area for a convex hole on the given set of points.
For our example, we used the first 20 points (T_{2k−1}, T_{2k}), for k = 1,2,…,20, produced with the pseudorandom number generator:
S_{0}  =_{ }  290797_{ } 
S_{n+1}  =_{ }  S_{n}^{2} mod 50515093 
T_{n}  =_{ }  ( S_{n} mod 2000 ) − 1000^{ } 
i.e. (527, 144), (−488, 732), (−454, −947), …
What is the maximum area for a convex hole on the set containing the first 500 points in the pseudorandom sequence?
Specify your answer including one digit after the decimal point.
A small child has a “number caterpillar” consisting of forty jigsaw pieces, each with one number on it, which, when connected together in a line, reveal the numbers 1 to 40 in order.
Every night, the child's father has to pick up the pieces of the caterpillar that have been scattered across the play room. He picks up the pieces at random and places them in the correct order.
As the caterpillar is built up in this way, it forms distinct segments that gradually merge together.
The number of segments starts at zero (no pieces placed), generally increases up to about eleven or twelve, then tends to drop again before finishing at a single segment (all pieces placed).
For example:
Piece Placed  Segments So Far 

12  1 
4  2 
29  3 
6  4 
34  5 
5  4 
35  4 
…  … 
Let M be the maximum number of segments encountered during a random tidyup of the caterpillar.
For a caterpillar of ten pieces, the number of possibilities for each M is
M  Possibilities 

1  512 
2  250912 
3  1815264 
4  1418112 
5  144000 
so the most likely value of M is 3 and the average value is ^{385643}⁄_{113400} = 3.400732, rounded to six decimal places.
The most likely value of M for a fortypiece caterpillar is 11; but what is the average value of M?
Give your answer rounded to six decimal places.
Define f(n) as the sum of the factorials of the digits of n. For example, f(342) = 3! + 4! + 2! = 32.
Define sf(n) as the sum of the digits of f(n). So sf(342) = 3 + 2 = 5.
Define g(i) to be the smallest positive integer n such that sf(n) = i. Though sf(342) is 5, sf(25) is also 5, and it can be verified that g(5) is 25.
Define sg(i) as the sum of the digits of g(i). So sg(5) = 2 + 5 = 7.
Further, it can be verified that g(20) is 267 and ∑ sg(i) for 1 ≤ i ≤ 20 is 156.
What is ∑ sg(i) for 1 ≤ i ≤ 150?
We define the roundedsquareroot of a positive integer n as the square root of n rounded to the nearest integer.
The following procedure (essentially Heron's method adapted to integer arithmetic) finds the roundedsquareroot of n:
Let d be the number of digits of the number n.
If d is odd, set $x_0 = 2 \times 10^{(d1)/2}$.
If d is even, set $x_0 = 7 \times 10^{(d2)/2}$.
Repeat:
until $x_{k+1} = x_k$.
As an example, let us find the roundedsquareroot of n = 4321.
n has 4 digits, so $x_0 = 7 \times 10^{(42)/2} = 70$.
$$x_1 = \Biggl\lfloor{\dfrac{70 + \lceil{4321 / 70}\rceil}{2}}\Biggr\rfloor = 66\\
x_2 = \Biggl\lfloor{\dfrac{66 + \lceil{4321 / 66}\rceil}{2}}\Biggr\rfloor = 66$$
Since $x_2 = x_1$, we stop here.
So, after just two iterations, we have found that the roundedsquareroot of 4321 is 66 (the actual square root is 65.7343137…).
The number of iterations required when using this method is surprisingly low.
For example, we can find the roundedsquareroot of a 5digit integer (10,000 ≤ n ≤ 99,999) with an average of 3.2102888889 iterations (the average value was rounded to 10 decimal places).
Using the procedure described above, what is the average number of iterations required to find the roundedsquareroot of a 14digit number (10^{13} ≤ n < 10^{14})?
Give your answer rounded to 10 decimal places.
Note: The symbols $\lfloor x \rfloor$ and $\lceil x \rceil$ represent the floor function and ceiling function respectively.
Tatami are rectangular mats, used to completely cover the floor of a room, without overlap.
Assuming that the only type of available tatami has dimensions 1×2, there are obviously some limitations for the shape and size of the rooms that can be covered.
For this problem, we consider only rectangular rooms with integer dimensions a, b and even size s = a·b.
We use the term 'size' to denote the floor surface area of the room, and — without loss of generality — we add the condition a ≤ b.
There is one rule to follow when laying out tatami: there must be no points where corners of four different mats meet.
For example, consider the two arrangements below for a 4×4 room:
The arrangement on the left is acceptable, whereas the one on the right is not: a red "X" in the middle, marks the point where four tatami meet.
Because of this rule, certain evensized rooms cannot be covered with tatami: we call them tatamifree rooms.
Further, we define T(s) as the number of tatamifree rooms of size s.
The smallest tatamifree room has size s = 70 and dimensions 7×10.
All the other rooms of size s = 70 can be covered with tatami; they are: 1×70, 2×35 and 5×14.
Hence, T(70) = 1.
Similarly, we can verify that T(1320) = 5 because there are exactly 5 tatamifree rooms of size s = 1320:
20×66, 22×60, 24×55, 30×44 and 33×40.
In fact, s = 1320 is the smallest roomsize s for which T(s) = 5.
Find the smallest roomsize s for which T(s) = 200.
Given is an integer sided triangle ABC with sides a ≤ b ≤ c.
(AB = c, BC = a and AC = b).
The angular bisectors of the triangle intersect the sides at points E, F and G (see picture below).
The segments EF, EG and FG partition the triangle ABC into four smaller triangles: AEG, BFE, CGF and EFG.
It can be proven that for each of these four triangles the ratio area(ABC)/area(subtriangle) is rational.
However, there exist triangles for which some or all of these ratios are integral.
How many triangles ABC with perimeter≤100,000,000 exist so that the ratio area(ABC)/area(AEG) is integral?
A sequence is defined as:
 g_{k} = 1, for 0 ≤ k ≤ 1999
 g_{k} = g_{k2000} + g_{k1999}, for k ≥ 2000.
Find g_{k} mod 20092010 for k = 10^{18}.
A positive integer will be called reachable if it can result from an arithmetic expression obeying the following rules:
 Uses the digits 1 through 9, in that order and exactly once each.
 Any successive digits can be concatenated (for example, using the digits 2, 3 and 4 we obtain the number 234).
 Only the four usual binary arithmetic operations (addition, subtraction, multiplication and division) are allowed.
 Each operation can be used any number of times, or not at all.
 Unary minus is not allowed.
 Any number of (possibly nested) parentheses may be used to define the order of operations.
For example, 42 is reachable, since (1/23) * ((4*5)6) * (789) = 42.
What is the sum of all positive reachable integers?
A game is played with three piles of stones and two players.
At her turn, a player removes one or more stones from the piles. However, if she takes stones from more than one pile, she must remove the same number of stones from each of the selected piles.
In other words, the player chooses some N>0 and removes:
 N stones from any single pile; or
 N stones from each of any two piles (2N total); or
 N stones from each of the three piles (3N total).
A winning configuration is one where the first player can force a win.
For example, (0,0,13), (0,11,11) and (5,5,5) are winning configurations because the first player can immediately remove all stones.
A losing configuration is one where the second player can force a win, no matter what the first player does.
For example, (0,1,2) and (1,3,3) are losing configurations: any legal move leaves a winning configuration for the second player.
Consider all losing configurations (x_{i},y_{i},z_{i}) where x_{i} ≤ y_{i} ≤ z_{i} ≤ 100.
We can verify that ∑ (x_{i}+y_{i}+z_{i}) = 173895 for these.
Find ∑ (x_{i}+y_{i}+z_{i}) where (x_{i},y_{i},z_{i}) ranges over the losing configurations
with x_{i} ≤ y_{i} ≤ z_{i} ≤ 1000.
Let us call a positive integer k a squarepivot, if there is a pair of integers m > 0 and n ≥ k, such that the sum of the (m+1) consecutive squares up to k equals the sum of the m consecutive squares from (n+1) on:
Some small squarepivots are
 4: 3^{2} + 4^{2} = 5^{2}
 21: 20^{2} + 21^{2} = 29^{2}
 24: 21^{2} + 22^{2} + 23^{2} + 24^{2} = 25^{2} + 26^{2} + 27^{2}
 110: 108^{2} + 109^{2} + 110^{2} = 133^{2} + 134^{2}
Find the sum of all distinct squarepivots ≤ 10^{10}.
The following equation represents the continuous topography of a mountainous region, giving the elevation h at any point (x,y):
A mosquito intends to fly from A(200,200) to B(1400,1400), without leaving the area given by 0 ≤ x, y ≤ 1600.
Because of the intervening mountains, it first rises straight up to a point A', having elevation f. Then, while remaining at the same elevation f, it flies around any obstacles until it arrives at a point B' directly above B.
First, determine f_{min} which is the minimum constant elevation allowing such a trip from A to B, while remaining in the specified area.
Then, find the length of the shortest path between A' and B', while flying at that constant elevation f_{min}.
Give that length as your answer, rounded to three decimal places.
Note: For convenience, the elevation function shown above is repeated below, in a form suitable for most programming languages:
h=( 50000.005*(x*x+y*y+x*y)+12.5*(x+y) ) * exp( abs(0.000001*(x*x+y*y)0.0015*(x+y)+0.7) )
Consider the number 6. The divisors of 6 are: 1,2,3 and 6.
Every number from 1 up to and including 6 can be written as a sum of distinct divisors of 6:
1=1, 2=2, 3=1+2, 4=1+3, 5=2+3, 6=6.
A number n is called a practical number if every number from 1 up to and including n can be expressed as a sum of distinct divisors of n.
A pair of consecutive prime numbers with a difference of six is called a sexy pair (since "sex" is the Latin word for "six"). The first sexy pair is (23, 29).
We may occasionally find a triplepair, which means three consecutive sexy prime pairs, such that the second member of each pair is the first member of the next pair.
We shall call a number n such that :
 (n9, n3), (n3,n+3), (n+3, n+9) form a triplepair, and
 the numbers n8, n4, n, n+4 and n+8 are all practical,
Find the sum of the first four engineers’ paradises.
Consider all the triangles having:
 All their vertices on lattice points.
 Circumcentre at the origin O.
 Orthocentre at the point H(5, 0).
There are nine such triangles having a perimeter ≤ 50.
Listed and shown in ascending order of their perimeter, they are:
A(4, 3), B(5, 0), C(4, 3) A(4, 3), B(5, 0), C(4, 3) A(3, 4), B(5, 0), C(3, 4) A(3, 4), B(5, 0), C(3, 4) A(0, 5), B(5, 0), C(0, 5) A(1, 8), B(8, 1), C(4, 7) A(8, 1), B(1, 8), C(4, 7) A(2, 9), B(9, 2), C(6, 7) A(9, 2), B(2, 9), C(6, 7) 
The sum of their perimeters, rounded to four decimal places, is 291.0089.
Find all such triangles with a perimeter ≤ 10^{5}.
Enter as your answer the sum of their perimeters rounded to four decimal places.
2^{N} binary digits can be placed in a circle so that all the Ndigit clockwise subsequences are distinct.
For N=3, two such circular arrangements are possible, ignoring rotations:
For the first arrangement, the 3digit subsequences, in clockwise order, are:
000, 001, 010, 101, 011, 111, 110 and 100.
Each circular arrangement can be encoded as a number by concatenating the binary digits starting with the subsequence of all zeros as the most significant bits and proceeding clockwise. The two arrangements for N=3 are thus represented as 23 and 29:
Calling S(N) the sum of the unique numeric representations, we can see that S(3) = 23 + 29 = 52.
Find S(5).
The divisors of 12 are: 1,2,3,4,6 and 12.
The largest divisor of 12 that does not exceed the square root of 12 is 3.
We shall call the largest divisor of an integer n that does not exceed the square root of n the pseudo square root (PSR) of n.
It can be seen that PSR(3102)=47.
Let p be the product of the primes below 190.
Find PSR(p) mod 10^{16}.
You are given a unique investment opportunity.
Starting with £1 of capital, you can choose a fixed proportion, f, of your capital to bet on a fair coin toss repeatedly for 1000 tosses.
Your return is double your bet for heads and you lose your bet for tails.
For example, if f = 1/4, for the first toss you bet £0.25, and if heads comes up you win £0.5 and so then have £1.5. You then bet £0.375 and if the second toss is tails, you have £1.125.
Choosing f to maximize your chances of having at least £1,000,000,000 after 1,000 flips, what is the chance that you become a billionaire?
All computations are assumed to be exact (no rounding), but give your answer rounded to 12 digits behind the decimal point in the form 0.abcdefghijkl.
It can be verified that there are 23 positive integers less than 1000 that are divisible by at least four distinct primes less than 100.
Find how many positive integers less than 10^{16} are divisible by at least four distinct primes less than 100.
A root or zero of a polynomial P(x) is a solution to the equation P(x) = 0.
Define P_{n} as the polynomial whose coefficients are the digits of n.
For example, P_{5703}(x) = 5x^{3} + 7x^{2} + 3.
We can see that:
 P_{n}(0) is the last digit of n,
 P_{n}(1) is the sum of the digits of n,
 P_{n}(10) is n itself.
Define Z(k) as the number of positive integers, n, not exceeding k for which the polynomial P_{n} has at least one integer root.
It can be verified that Z(100 000) is 14696.
What is Z(10^{16})?
A square piece of paper with integer dimensions N×N is placed with a corner at the origin and two of its sides along the x and yaxes. Then, we cut it up respecting the following rules:
 We only make straight cuts between two points lying on different sides of the square, and having integer coordinates.
 Two cuts cannot cross, but several cuts can meet at the same border point.
 Proceed until no more legal cuts can be made.
Counting any reflections or rotations as distinct, we call C(N) the number of ways to cut an N×N square. For example, C(1) = 2 and C(2) = 30 (shown below).
What is C(30) mod 10^{8} ?
For a positive number n, define S(n) as the sum of the integers x, for which 1<x<n and
x^{3}≡1 mod n.
When n=91, there are 8 possible values for x, namely : 9, 16, 22, 29, 53, 74, 79, 81.
Thus, S(91)=9+16+22+29+53+74+79+81=363.
Find S(13082761331670030).
For a positive number n, define C(n) as the number of the integers x, for which 1<x<n and
x^{3}≡1 mod n.
When n=91, there are 8 possible values for x, namely : 9, 16, 22, 29, 53, 74, 79, 81.
Thus, C(91)=8.
Find the sum of the positive numbers n≤10^{11} for which C(n)=242.
Consider equations of the form: a^{2} + b^{2} = N, 0 ≤ a ≤ b, a, b and N integer.
For N=65 there are two solutions:
a=1, b=8 and a=4, b=7.
We call S(N) the sum of the values of a of all solutions of a^{2} + b^{2} = N, 0 ≤ a ≤ b, a, b and N integer.
Thus S(65) = 1 + 4 = 5.
Find ∑ S(N), for all squarefree N only divisible by primes of the form 4k+1 with 4k+1 < 150.
For each integer p > 1 coprime to 10 there is a positive divisibility multiplier m < p which preserves divisibility by p for the following function on any positive integer, n:
f(n) = (all but the last digit of n) + (the last digit of n) * m
That is, if m is the divisibility multiplier for p, then f(n) is divisible by p if and only if n is divisible by p.
(When n is much larger than p, f(n) will be less than n and repeated application of f provides a multiplicative divisibility test for p.)
For example, the divisibility multiplier for 113 is 34.
f(76275) = 7627 + 5 * 34 = 7797 : 76275 and 7797 are both divisible by 113
f(12345) = 1234 + 5 * 34 = 1404 : 12345 and 1404 are both not divisible by 113
The sum of the divisibility multipliers for the primes that are coprime to 10 and less than 1000 is 39517. What is the sum of the divisibility multipliers for the primes that are coprime to 10 and less than 10^{7}?
Let us define a balanced sculpture of order n as follows:
 A polyomino made up of n+1 tiles known as the blocks (n tiles)
and the plinth (remaining tile);  the plinth has its centre at position (x = 0, y = 0);
 the blocks have ycoordinates greater than zero (so the plinth is the unique lowest tile);
 the centre of mass of all the blocks, combined, has xcoordinate equal to zero.
When counting the sculptures, any arrangements which are simply reflections about the yaxis, are not counted as distinct. For example, the 18 balanced sculptures of order 6 are shown below; note that each pair of mirror images (about the yaxis) is counted as one sculpture:
There are 964 balanced sculptures of order 10 and 360505 of order 15.
How many balanced sculptures are there of order 18?
Consider the triangles with integer sides a, b and c with a ≤ b ≤ c.
An integer sided triangle (a,b,c) is called primitive if gcd(a,b,c)=1.
How many primitive integer sided triangles exist with a perimeter not exceeding 10 000 000?
A modified Collatz sequence of integers is obtained from a starting value $a_1$ in the following way:
$a_{n+1} = \, \,\, \frac {a_n} 3 \quad$ if $a_n$ is divisible by $3$. We shall denote this as a large downward step, "D".
$a_{n+1} = \frac {4 a_n+2} 3 \, \,$ if $a_n$ divided by $3$ gives a remainder of $1$. We shall denote this as an upward step, "U".
$a_{n+1} = \frac {2 a_n 1} 3 \, \,$ if $a_n$ divided by $3$ gives a remainder of $2$. We shall denote this as a small downward step, "d".
The sequence terminates when some $a_n = 1$.
Given any integer, we can list out the sequence of steps.
For instance if $a_1=231$, then the sequence $\{a_n\}=\{231,77,51,17,11,7,10,14,9,3,1\}$ corresponds to the steps "DdDddUUdDD".
Of course, there are other sequences that begin with that same sequence "DdDddUUdDD....".
For instance, if $a_1=1004064$, then the sequence is DdDddUUdDDDdUDUUUdDdUUDDDUdDD.
In fact, $1004064$ is the smallest possible $a_1 > 10^6$ that begins with the sequence DdDddUUdDD.
What is the smallest $a_1 > 10^{15}$ that begins with the sequence "UDDDUdddDDUDDddDdDddDDUDDdUUDd"?
Given the values of integers $1 < a_1 < a_2 < \dots < a_n$, consider the linear combination
$q_1 a_1+q_2 a_2 + \dots + q_n a_n=b$, using only integer values $q_k \ge 0$.
Note that for a given set of $a_k$, it may be that not all values of $b$ are possible.
For instance, if $a_1=5$ and $a_2=7$, there are no $q_1 \ge 0$ and $q_2 \ge 0$ such that $b$ could be
$1, 2, 3, 4, 6, 8, 9, 11, 13, 16, 18$ or $23$.
In fact, $23$ is the largest impossible value of $b$ for $a_1=5$ and $a_2=7$.
We therefore call $f(5, 7) = 23$.
Similarly, it can be shown that $f(6, 10, 15)=29$ and $f(14, 22, 77) = 195$.
Find $\displaystyle \sum f( p\, q,p \, r, q \, r)$, where $p$, $q$ and $r$ are prime numbers and $p < q < r < 5000$.
How many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter that does not exceed 10^{8}?
A laborious ant walks randomly on a 5x5 grid. The walk starts from the central square. At each step, the ant moves to an adjacent square at random, without leaving the grid; thus there are 2, 3 or 4 possible moves at each step depending on the ant's position.
At the start of the walk, a seed is placed on each square of the lower row. When the ant isn't carrying a seed and reaches a square of the lower row containing a seed, it will start to carry the seed. The ant will drop the seed on the first empty square of the upper row it eventually reaches.
What's the expected number of steps until all seeds have been dropped in the top row?
Give your answer rounded to 6 decimal places.
You are given a pizza (perfect circle) that has been cut into m·n equal pieces and you want to have exactly one topping on each slice.
Let f(m,n) denote the number of ways you can have toppings on the pizza with m different toppings (m ≥ 2), using each topping on exactly n slices (n ≥ 1).
Reflections are considered distinct, rotations are not.
Thus, for instance, f(2,1) = 1, f(2,2) = f(3,1) = 2 and f(3,2) = 16.
f(3,2) is shown below:
Find the sum of all f(m,n) such that f(m,n) ≤ 10^{15}.
For nonnegative integers $m$, $n$, the Ackermann function $A(m,n)$ is defined as follows: $$ A(m,n) = \cases{ n+1 &$\htmltext{ if }m=0$\cr A(m1,1) &$\htmltext{ if }m>0 \htmltext{ and } n=0$\cr A(m1,A(m,n1)) &$\htmltext{ if }m>0 \htmltext{ and } n>0$\cr }$$
For example $A(1,0) = 2$, $A(2,2) = 7$ and $A(3,4) = 125$.
Find $\displaystyle\sum_{n=0}^6 A(n,n)$ and give your answer mod $14^8$.
Consider the triangle with sides 6, 8 and 10. It can be seen that the perimeter and the area are both equal to 24.
So the area/perimeter ratio is equal to 1.
Consider also the triangle with sides 13, 14 and 15. The perimeter equals 42 while the area is equal to 84.
So for this triangle the area/perimeter ratio is equal to 2.
Find the sum of the perimeters of all integer sided triangles for which the area/perimeter ratios are equal to positive integers not exceeding 1000.
The 3digit number 376 in the decimal numbering system is an example of numbers with the special property that its square ends with the same digits: 376^{2} = 141376. Let's call a number with this property a steady square.
Steady squares can also be observed in other numbering systems. In the base 14 numbering system, the 3digit number c37 is also a steady square: c37^{2} = aa0c37, and the sum of its digits is c+3+7=18 in the same numbering system. The letters a, b, c and d are used for the 10, 11, 12 and 13 digits respectively, in a manner similar to the hexadecimal numbering system.
For 1 ≤ n ≤ 9, the sum of the digits of all the ndigit steady squares in the base 14 numbering system is 2d8 (582 decimal). Steady squares with leading 0's are not allowed.
Find the sum of the digits of all the ndigit steady squares in the base 14 numbering system for
1 ≤ n ≤ 10000 (decimal) and give your answer in the base 14 system using lower case letters where necessary.
Albert chooses a positive integer k, then two real numbers a, b are randomly chosen in the interval [0,1] with uniform distribution.
The square root of the sum (k·a+1)^{2} + (k·b+1)^{2} is then computed and rounded to the nearest integer. If the result is equal to k, he scores k points; otherwise he scores nothing.
For example, if k = 6, a = 0.2 and b = 0.85, then (k·a+1)^{2} + (k·b+1)^{2} = 42.05.
The square root of 42.05 is 6.484... and when rounded to the nearest integer, it becomes 6.
This is equal to k, so he scores 6 points.
It can be shown that if he plays 10 turns with k = 1, k = 2, ..., k = 10, the expected value of his total score, rounded to five decimal places, is 10.20914.
If he plays 10^{5} turns with k = 1, k = 2, k = 3, ..., k = 10^{5}, what is the expected value of his total score, rounded to five decimal places?
Barbara is a mathematician and a basketball player. She has found that the probability of scoring a point when shooting from a distance x is exactly (1  ^{x}/_{q}), where q is a real constant greater than 50.
During each practice run, she takes shots from distances x = 1, x = 2, ..., x = 50 and, according to her records, she has precisely a 2 % chance to score a total of exactly 20 points.
Find q and give your answer rounded to 10 decimal places.
The quadtree encoding allows us to describe a 2^{N}×2^{N} black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this:
 the first bit deals with the complete 2^{N}×2^{N} region;
 "0" denotes a split:
the current 2^{n}×2^{n} region is divided into 4 subregions of dimension 2^{n1}×2^{n1},
the next bits contains the description of the top left, top right, bottom left and bottom right subregions  in that order;  "10" indicates that the current region contains only black pixels;
 "11" indicates that the current region contains only white pixels.
Consider the following 4×4 image (colored marks denote places where a split can occur):
This image can be described by several sequences, for example :
"001010101001011111011010101010", of length 30, or
"0100101111101110", of length 16, which is the minimal sequence for this image.
For a positive integer N, define D_{N} as the 2^{N}×2^{N} image with the following coloring scheme:
 the pixel with coordinates x = 0, y = 0 corresponds to the bottom left pixel,
 if (x  2^{N1})^{2} + (y  2^{N1})^{2} ≤ 2^{2N2} then the pixel is black,
 otherwise the pixel is white.
What is the length of the minimal sequence describing D_{24} ?
For any prime p the number N(p,q) is defined by
N(p,q) = ∑_{n=0 to q} T_{n}*p^{n}
with T_{n} generated by the following random number generator:
S_{0} = 290797
S_{n+1} = S_{n}^{2} mod 50515093
T_{n} = S_{n} mod p
Let Nfac(p,q) be the factorial of N(p,q).
Let NF(p,q) be the number of factors p in Nfac(p,q).
You are given that NF(3,10000) mod 3^{20}=624955285.
Find NF(61,10^{7}) mod 61^{10}
Let C(x,y) be a circle passing through the points (x, y), (x, y+1), (x+1, y) and (x+1, y+1).
For positive integers m and n, let E(m,n) be a configuration which consists of the m·n circles:
{ C(x,y): 0 ≤ x < m, 0 ≤ y < n, x and y are integers }
An Eulerian cycle on E(m,n) is a closed path that passes through each arc exactly once.
Many such paths are possible on E(m,n), but we are only interested in those which are not selfcrossing:
A noncrossing path just touches itself at lattice points, but it never crosses itself.
The image below shows E(3,3) and an example of an Eulerian noncrossing path.
Let L(m,n) be the number of Eulerian noncrossing paths on E(m,n).
For example, L(1,2) = 2, L(2,2) = 37 and L(3,3) = 104290.
Find L(6,10) mod 10^{10}.
How many integers 0 ≤ n < 10^{18} have the property that the sum of the digits of n equals the sum of digits of 137n?
A prime number $p$ is called a Panaitopol prime if $p = \dfrac{x^4  y^4}{x^3 + y^3}$ for some positive integers $x$ and $y$.
Find how many Panaitopol primes are less than 5×10^{15}.
We shall define a pythagorean polygon to be a convex polygon with the following properties:
 there are at least three vertices,
 no three vertices are aligned,
 each vertex has integer coordinates,
 each edge has integer length.
For a given integer n, define P(n) as the number of distinct pythagorean polygons for which the perimeter is ≤ n.
Pythagorean polygons should be considered distinct as long as none is a translation of another.
You are given that P(4) = 1, P(30) = 3655 and P(60) = 891045.
Find P(120).
An even positive integer N will be called admissible, if it is a power of 2 or its distinct prime factors are consecutive primes.
The first twelve admissible numbers are 2,4,6,8,12,16,18,24,30,32,36,48.
If N is admissible, the smallest integer M > 1 such that N+M is prime, will be called the pseudoFortunate number for N.
For example, N=630 is admissible since it is even and its distinct prime factors are the consecutive primes 2,3,5 and 7.
The next prime number after 631 is 641; hence, the pseudoFortunate number for 630 is M=11.
It can also be seen that the pseudoFortunate number for 16 is 3.
Find the sum of all distinct pseudoFortunate numbers for admissible numbers N less than 10^{9}.
For a positive integer k, define d(k) as the sum of the digits of k in its usual decimal representation. Thus d(42) = 4+2 = 6.
For a positive integer n, define S(n) as the number of positive integers k < 10^{n} with the following properties :
 k is divisible by 23 and
 d(k) = 23.
Find S(11^{12}) and give your answer mod 10^{9}.
We call the convex area enclosed by two circles a lenticular hole if:
 The centres of both circles are on lattice points.
 The two circles intersect at two distinct lattice points.
 The interior of the convex area enclosed by both circles does not contain any lattice points.
Consider the circles:
C_{0}: x^{2}+y^{2}=25
C_{1}: (x+4)^{2}+(y4)^{2}=1
C_{2}: (x12)^{2}+(y4)^{2}=65
The circles C_{0}, C_{1} and C_{2} are drawn in the picture below.
C_{0} and C_{1} form a lenticular hole, as well as C_{0} and C_{2}.
We call an ordered pair of positive real numbers (r_{1}, r_{2}) a lenticular pair if there exist two circles with radii r_{1} and r_{2} that form a lenticular hole. We can verify that (1, 5) and (5, √65) are the lenticular pairs of the example above.
Let L(N) be the number of distinct lenticular pairs (r_{1}, r_{2}) for which 0 < r_{1} ≤ r_{2} ≤ N.
We can verify that L(10) = 30 and L(100) = 3442.
Find L(100 000).
Given is an integer sided triangle ABC with BC ≤ AC ≤ AB.
k is the angular bisector of angle ACB.
m is the tangent at C to the circumscribed circle of ABC.
n is a line parallel to m through B.
The intersection of n and k is called E.
How many triangles ABC with a perimeter not exceeding 100 000 exist such that BE has integral length?
Each new term in the Fibonacci sequence is generated by adding the previous two terms.
Starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.
Every positive integer can be uniquely written as a sum of nonconsecutive terms of the Fibonacci sequence. For example, 100 = 3 + 8 + 89.
Such a sum is called the Zeckendorf representation of the number.
For any integer n>0, let z(n) be the number of terms in the Zeckendorf representation of n.
Thus, z(5) = 1, z(14) = 2, z(100) = 3 etc.
Also, for 0<n<10^{6}, ∑ z(n) = 7894453.
Find ∑ z(n) for 0<n<10^{17}.
Larry and Robin play a memory game involving of a sequence of random numbers between 1 and 10, inclusive, that are called out one at a time. Each player can remember up to 5 previous numbers. When the called number is in a player's memory, that player is awarded a point. If it's not, the player adds the called number to his memory, removing another number if his memory is full.
Both players start with empty memories. Both players always add new missed numbers to their memory but use a different strategy in deciding which number to remove:
Larry's strategy is to remove the number that hasn't been called in the longest time.
Robin's strategy is to remove the number that's been in the memory the longest time.
Example game:
Turn  Called number 
Larry's memory 
Larry's score 
Robin's memory 
Robin's score 

1  1  1  0  1  0 
2  2  1,2  0  1,2  0 
3  4  1,2,4  0  1,2,4  0 
4  6  1,2,4,6  0  1,2,4,6  0 
5  1  1,2,4,6  1  1,2,4,6  1 
6  8  1,2,4,6,8  1  1,2,4,6,8  1 
7  10  1,4,6,8,10  1  2,4,6,8,10  1 
8  2  1,2,6,8,10  1  2,4,6,8,10  2 
9  4  1,2,4,8,10  1  2,4,6,8,10  3 
10  1  1,2,4,8,10  2  1,4,6,8,10  3 
Denoting Larry's score by L and Robin's score by R, what is the expected value of LR after 50 turns? Give your answer rounded to eight decimal places using the format x.xxxxxxxx .
Four points with integer coordinates are selected:
A(a, 0), B(b, 0), C(0, c) and D(0, d),
with 0 < a < b and 0 < c < d.
Point P, also with integer coordinates, is chosen on the line AC so that the three triangles ABP, CDP and BDP are all similar.
It is easy to prove that the three triangles can be similar, only if a=c.
So, given that a=c, we are looking for triplets (a,b,d) such that at least one point P (with integer coordinates) exists on AC, making the three triangles ABP, CDP and BDP all similar.
For example, if (a,b,d)=(2,3,4), it can be easily verified that point P(1,1) satisfies the above condition. Note that the triplets (2,3,4) and (2,4,3) are considered as distinct, although point P(1,1) is common for both.
If b+d < 100, there are 92 distinct triplets (a,b,d) such that point P exists.
If b+d < 100 000, there are 320471 distinct triplets (a,b,d) such that point P exists.
If b+d < 100 000 000, how many distinct triplets (a,b,d) are there such that point P exists?
In a very simplified form, we can consider proteins as strings consisting of hydrophobic (H) and polar (P) elements, e.g. HHPPHHHPHHPH.
For this problem, the orientation of a protein is important; e.g. HPP is considered distinct from PPH. Thus, there are 2^{n} distinct proteins consisting of n elements.
When one encounters these strings in nature, they are always folded in such a way that the number of HH contact points is as large as possible, since this is energetically advantageous.
As a result, the Helements tend to accumulate in the inner part, with the Pelements on the outside.
Natural proteins are folded in three dimensions of course, but we will only consider protein folding in two dimensions.
The figure below shows two possible ways that our example protein could be folded (HH contact points are shown with red dots).
The folding on the left has only six HH contact points, thus it would never occur naturally.
On the other hand, the folding on the right has nine HH contact points, which is optimal for this string.
Assuming that H and P elements are equally likely to occur in any position along the string, the average number of HH contact points in an optimal folding of a random protein string of length 8 turns out to be 850 / 2^{8}=3.3203125.
What is the average number of HH contact points in an optimal folding of a random protein string of length 15?
Give your answer using as many decimal places as necessary for an exact result.
Nim is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any heap until no stones remain.
We'll consider the threeheap normalplay version of Nim, which works as follows:
 At the start of the game there are three heaps of stones.
 On his turn the player removes any positive number of stones from any single heap.
 The first player unable to move (because no stones remain) loses.
If (n_{1},n_{2},n_{3}) indicates a Nim position consisting of heaps of size n_{1}, n_{2} and n_{3} then there is a simple function X(n_{1},n_{2},n_{3}) — that you may look up or attempt to deduce for yourself — that returns:
 zero if, with perfect strategy, the player about to move will eventually lose; or
 nonzero if, with perfect strategy, the player about to move will eventually win.
For example X(1,2,3) = 0 because, no matter what the current player does, his opponent can respond with a move that leaves two heaps of equal size, at which point every move by the current player can be mirrored by his opponent until no stones remain; so the current player loses. To illustrate:
 current player moves to (1,2,1)
 opponent moves to (1,0,1)
 current player moves to (0,0,1)
 opponent moves to (0,0,0), and so wins.
For how many positive integers n ≤ 2^{30} does X(n,2n,3n) = 0 ?
A positive integer n is powerful if p^{2} is a divisor of n for every prime factor p in n.
A positive integer n is a perfect power if n can be expressed as a power of another positive integer.
A positive integer n is an Achilles number if n is powerful but not a perfect power. For example, 864 and 1800 are Achilles numbers: 864 = 2^{5}·3^{3} and 1800 = 2^{3}·3^{2}·5^{2}.
We shall call a positive integer S a Strong Achilles number if both S and φ(S) are Achilles numbers.^{1}
For example, 864 is a Strong Achilles number: φ(864) = 288 = 2^{5}·3^{2}. However, 1800 isn't a Strong Achilles number because: φ(1800) = 480 = 2^{5}·3^{1}·5^{1}.
There are 7 Strong Achilles numbers below 10^{4} and 656 below 10^{8}.
How many Strong Achilles numbers are there below 10^{18}?
^{1} φ denotes Euler's totient function.
For a positive integer n, define f(n) as the least positive multiple of n that, written in base 10, uses only digits ≤ 2.
Thus f(2)=2, f(3)=12, f(7)=21, f(42)=210, f(89)=1121222.
Also, $\sum \limits_{n = 1}^{100} {\dfrac{f(n)}{n}} = 11363107$.
Find $\sum \limits_{n=1}^{10000} {\dfrac{f(n)}{n}}$.
For any positive integer n the function next_prime(n) returns the smallest prime p
such that p>n.
The sequence a(n) is defined by:
a(1)=next_prime(10^{14}) and a(n)=next_prime(a(n1)) for n>1.
The fibonacci sequence f(n) is defined by: f(0)=0, f(1)=1 and f(n)=f(n1)+f(n2) for n>1.
The sequence b(n) is defined as f(a(n)).
Find ∑ b(n) for 1≤n≤100 000. Give your answer mod 1234567891011.
Let's call S the (infinite) string that is made by concatenating the consecutive positive integers (starting from 1) written down in base 10.
Thus, S = 1234567891011121314151617181920212223242...
It's easy to see that any number will show up an infinite number of times in S.
Let's call f(n) the starting position of the n^{th} occurrence of n in S.
For example, f(1)=1, f(5)=81, f(12)=271 and f(7780)=111111365.
Find ∑ f(3^{k}) for 1≤k≤13.
The following game is a classic example of Combinatorial Game Theory:
Two players start with a strip of n white squares and they take alternate turns.
On each turn, a player picks two contiguous white squares and paints them black.
The first player who cannot make a move loses.
 If n = 1, there are no valid moves, so the first player loses automatically.
 If n = 2, there is only one valid move, after which the second player loses.
 If n = 3, there are two valid moves, but both leave a situation where the second player loses.
 If n = 4, there are three valid moves for the first player; she can win the game by painting the two middle squares.
 If n = 5, there are four valid moves for the first player (shown below in red); but no matter what she does, the second player (blue) wins.
So, for 1 ≤ n ≤ 5, there are 3 values of n for which the first player can force a win.
Similarly, for 1 ≤ n ≤ 50, there are 40 values of n for which the first player can force a win.
For 1 ≤ n ≤ 1 000 000, how many values of n are there for which the first player can force a win?
k defects are randomly distributed amongst n integratedcircuit chips produced by a factory (any number of defects may be found on a chip and each defect is independent of the other defects).
Let p(k,n) represent the probability that there is a chip with at least 3 defects.
For instance p(3,7) ≈ 0.0204081633.
Find p(20 000, 1 000 000) and give your answer rounded to 10 decimal places in the form 0.abcdefghij
A program written in the programming language Fractran consists of a list of fractions.
The internal state of the Fractran Virtual Machine is a positive integer, which is initially set to a seed value. Each iteration of a Fractran program multiplies the state integer by the first fraction in the list which will leave it an integer.
For example, one of the Fractran programs that John Horton Conway wrote for primegeneration consists of the following 14 fractions:

, 

, 

, 

, 

, 

, 

, 

, 

, 

, 

, 

, 

, 

. 
Starting with the seed integer 2, successive iterations of the program produce the sequence:
15, 825, 725, 1925, 2275, 425, ..., 68, 4, 30, ..., 136, 8, 60, ..., 544, 32, 240, ...
The powers of 2 that appear in this sequence are 2^{2}, 2^{3}, 2^{5}, ...
It can be shown that all the powers of 2 in this sequence have prime exponents and that all the primes appear as exponents of powers of 2, in proper order!
If someone uses the above Fractran program to solve Project Euler Problem 7 (find the 10001^{st} prime), how many iterations would be needed until the program produces 2^{10001st prime} ?
In the classic "Crossing Ladders" problem, we are given the lengths x and y of two ladders resting on the opposite walls of a narrow, level street. We are also given the height h above the street where the two ladders cross and we are asked to find the width of the street (w).
Here, we are only concerned with instances where all four variables are positive integers.
For example, if x = 70, y = 119 and h = 30, we can calculate that w = 56.
In fact, for integer values x, y, h and 0 < x < y < 200, there are only five triplets (x,y,h) producing integer solutions for w:
(70, 119, 30), (74, 182, 21), (87, 105, 35), (100, 116, 35) and (119, 175, 40).
For integer values x, y, h and 0 < x < y < 1 000 000, how many triplets (x,y,h) produce integer solutions for w?
Alice and Bob play the game Nim Square.
Nim Square is just like ordinary threeheap normal play Nim, but the players may only remove a square number of stones from a heap.
The number of stones in the three heaps is represented by the ordered triple (a,b,c).
If 0≤a≤b≤c≤29 then the number of losing positions for the next player is 1160.
Find the number of losing positions for the next player if 0≤a≤b≤c≤100 000.
ABCD is a convex, integer sided quadrilateral with 1 ≤ AB < BC < CD < AD.
BD has integer length. O is the midpoint of BD. AO has integer length.
We'll call ABCD a biclinic integral quadrilateral if AO = CO ≤ BO = DO.
For example, the following quadrilateral is a biclinic integral quadrilateral:
AB = 19, BC = 29, CD = 37, AD = 43, BD = 48 and AO = CO = 23.
Let B(N) be the number of distinct biclinic integral quadrilaterals ABCD that satisfy AB^{2}+BC^{2}+CD^{2}+AD^{2} ≤ N.
We can verify that B(10 000) = 49 and B(1 000 000) = 38239.
Find B(10 000 000 000).
 A Sierpiński graph of order1 (S_{1}) is an equilateral triangle.
 S_{n+1} is obtained from S_{n} by positioning three copies of S_{n} so that every pair of copies has one common corner.
Let C(n) be the number of cycles that pass exactly once through all the vertices of S_{n}.
For example, C(3) = 8 because eight such cycles can be drawn on S_{3}, as shown below:
It can also be verified that :
C(1) = C(2) = 1
C(5) = 71328803586048
C(10 000) mod 10^{8} = 37652224
C(10 000) mod 13^{8} = 617720485
Find C(C(C(10 000))) mod 13^{8}.
In a sliding game a counter may slide horizontally or vertically into an empty space. The objective of the game is to move the red counter from the top left corner of a grid to the bottom right corner; the space always starts in the bottom right corner. For example, the following sequence of pictures show how the game can be completed in five moves on a 2 by 2 grid.
Let S(m,n) represent the minimum number of moves to complete the game on an m by n grid. For example, it can be verified that S(5,4) = 25.
There are exactly 5482 grids for which S(m,n) = p^{2}, where p < 100 is prime.
How many grids does S(m,n) = p^{2}, where p < 10^{6} is prime?
The moon has been opened up, and land can be obtained for free, but there is a catch. You have to build a wall around the land that you stake out, and building a wall on the moon is expensive. Every country has been allotted a 500 m by 500 m square area, but they will possess only that area which they wall in. 251001 posts have been placed in a rectangular grid with 1 meter spacing. The wall must be a closed series of straight lines, each line running from post to post.
The bigger countries of course have built a 2000 m wall enclosing the entire 250 000 m^{2} area. The Duchy of Grand Fenwick, has a tighter budget, and has asked you (their Royal Programmer) to compute what shape would get best maximum enclosedarea/walllength ratio.
You have done some preliminary calculations on a sheet of paper.
For a 2000 meter wall enclosing the 250 000 m^{2} area the
enclosedarea/walllength ratio is 125.
Although not allowed , but to get an idea if this is anything better: if you place a circle inside the square area touching the four sides the area will be equal to π*250^{2} m^{2} and the perimeter will be π*500 m, so the enclosedarea/walllength ratio will also be 125.
However, if you cut off from the square four triangles with sides 75 m, 75 m and 75√2 m the total area becomes 238750 m^{2} and the perimeter becomes 1400+300√2 m. So this gives an enclosedarea/walllength ratio of 130.87, which is significantly better.
Find the maximum enclosedarea/walllength ratio.
Give your answer rounded to 8 places behind the decimal point in the form abc.defghijk.
Sam and Max are asked to transform two digital clocks into two "digital root" clocks.
A digital root clock is a digital clock that calculates digital roots step by step.
When a clock is fed a number, it will show it and then it will start the calculation, showing all the intermediate values until it gets to the result.
For example, if the clock is fed the number 137, it will show: "137" → "11" → "2" and then it will go black, waiting for the next number.
Every digital number consists of some light segments: three horizontal (top, middle, bottom) and four vertical (topleft, topright, bottomleft, bottomright).
Number "1" is made of vertical topright and bottomright, number "4" is made by middle horizontal and vertical topleft, topright and bottomright. Number "8" lights them all.
The clocks consume energy only when segments are turned on/off.
To turn on a "2" will cost 5 transitions, while a "7" will cost only 4 transitions.
Sam and Max built two different clocks.
Sam's clock is fed e.g. number 137: the clock shows "137", then the panel is turned off, then the next number ("11") is turned on, then the panel is turned off again and finally the last number ("2") is turned on and, after some time, off.
For the example, with number 137, Sam's clock requires:
"137"  :  (2 + 5 + 4) × 2 = 22 transitions ("137" on/off). 
"11"  :  (2 + 2) × 2 = 8 transitions ("11" on/off). 
"2"  :  (5) × 2 = 10 transitions ("2" on/off). 
Max's clock works differently. Instead of turning off the whole panel, it is smart enough to turn off only those segments that won't be needed for the next number.
For number 137, Max's clock requires:
"137" 
: 
2 + 5 + 4 = 11 transitions ("137" on) 7 transitions (to turn off the segments that are not needed for number "11"). 
"11" 
: 
0 transitions (number "11" is already turned on correctly) 3 transitions (to turn off the first "1" and the bottom part of the second "1"; the top part is common with number "2"). 
"2" 
: 
4 transitions (to turn on the remaining segments in order to get a "2") 5 transitions (to turn off number "2"). 
Of course, Max's clock consumes less power than Sam's one.
The two clocks are fed all the prime numbers between A = 10^{7} and B = 2×10^{7}.
Find the difference between the total number of transitions needed by Sam's clock and that needed by Max's one.
Let p = p_{1} p_{2} p_{3} ... be an infinite sequence of random digits, selected from {0,1,2,3,4,5,6,7,8,9} with equal probability.
It can be seen that p corresponds to the real number 0.p_{1} p_{2} p_{3} ....
It can also be seen that choosing a random real number from the interval [0,1) is equivalent to choosing an infinite sequence of random digits selected from {0,1,2,3,4,5,6,7,8,9} with equal probability.
For any positive integer n with d decimal digits, let k be the smallest index such that
p_{k, }p_{k+1}, ...p_{k+d1} are the decimal digits of n, in the same order.
Also, let g(n) be the expected value of k; it can be proven that g(n) is always finite and, interestingly, always an integer number.
For example, if n = 535, then
for p = 31415926535897...., we get k = 9
for p = 355287143650049560000490848764084685354..., we get k = 36
etc and we find that g(535) = 1008.
Given that $\sum \limits_{n = 2}^{999} {g \left ( \left \lfloor \dfrac{10^6}{n} \right \rfloor \right )} = 27280188$, find $\sum \limits_{n = 2}^{999999} {g \left ( \left \lfloor \dfrac{10^{16}}{n} \right \rfloor \right )}$.
Note: $\lfloor x \rfloor$ represents the floor function.A firecracker explodes at a height of 100 m above level ground. It breaks into a large number of very small fragments, which move in every direction; all of them have the same initial velocity of 20 m/s.
We assume that the fragments move without air resistance, in a uniform gravitational field with g=9.81 m/s^{2}.
Find the volume (in m^{3}) of the region through which the fragments move before reaching the ground. Give your answer rounded to four decimal places.
Consider the real number $\sqrt 2 + \sqrt 3$.
When we calculate the even powers of $\sqrt 2 + \sqrt 3$
we get:
$(\sqrt 2 + \sqrt 3)^2 = 9.898979485566356 \dots $
$(\sqrt 2 + \sqrt 3)^4 = 97.98979485566356 \dots $
$(\sqrt 2 + \sqrt 3)^6 = 969.998969071069263 \dots $
$(\sqrt 2 + \sqrt 3)^8 = 9601.99989585502907 \dots $
$(\sqrt 2 + \sqrt 3)^{10} = 95049.999989479221 \dots $
$(\sqrt 2 + \sqrt 3)^{12} = 940897.9999989371855 \dots $
$(\sqrt 2 + \sqrt 3)^{14} = 9313929.99999989263 \dots $
$(\sqrt 2 + \sqrt 3)^{16} = 92198401.99999998915 \dots $
It looks as if the number of consecutive nines at the beginning of the fractional part of these powers is nondecreasing.
In fact it can be proven that the fractional part of $(\sqrt 2 + \sqrt 3)^{2 n}$ approaches $1$ for large $n$.
Consider all real numbers of the form $\sqrt p + \sqrt q$ with $p$ and $q$ positive integers and $p < q$, such that the fractional part of $(\sqrt p + \sqrt q)^{ 2 n}$ approaches $1$ for large $n$.
Let $C(p,q,n)$ be the number of consecutive nines at the beginning of the fractional part of $(\sqrt p + \sqrt q)^{ 2 n}$.
Let $N(p,q)$ be the minimal value of $n$ such that $C(p,q,n) \ge 2011$.
Find $\displaystyle \sum N(p,q) \,\, \text{ for } p+q \le 2011$.
Let x_{1}, x_{2},..., x_{n} be a sequence of length n such that:
 x_{1} = 2
 for all 1 < i ≤ n : x_{i1} < x_{i}
 for all i and j with 1 ≤ i, j ≤ n : (x_{i})^{ j} < (x_{j} + 1)^{i}
There are only five such sequences of length 2, namely:
{2,4}, {2,5}, {2,6}, {2,7} and {2,8}.
There are 293 such sequences of length 5; three examples are given below:
{2,5,11,25,55}, {2,6,14,36,88}, {2,8,22,64,181}.
Let t(n) denote the number of such sequences of length n.
You are given that t(10) = 86195 and t(20) = 5227991891.
Find t(10^{10}) and give your answer modulo 10^{9}.
Let N(i) be the smallest integer n such that n! is divisible by (i!)^{1234567890}
Let S(u)=∑ N(i) for 10 ≤ i ≤ u.
S(1000)=614538266565663.
Find S(1 000 000) mod 10^{18}.
A horizontal row comprising of 2n + 1 squares has n red counters placed at one end and n blue counters at the other end, being separated by a single empty square in the centre. For example, when n = 3.
A counter can move from one square to the next (slide) or can jump over another counter (hop) as long as the square next to that counter is unoccupied.
Let M(n) represent the minimum number of moves/actions to completely reverse the positions of the coloured counters; that is, move all the red counters to the right and all the blue counters to the left.
It can be verified M(3) = 15, which also happens to be a triangle number.
If we create a sequence based on the values of n for which M(n) is a triangle number then the first five terms would be:
1, 3, 10, 22, and 63, and their sum would be 99.
Find the sum of the first forty terms of this sequence.
Let T(m, n) be the number of the binomial coefficients ^{i}C_{n} that are divisible by 10 for n ≤ i < m(i, m and n are positive integers).
You are given that T(10^{9}, 10^{7}10) = 989697000.
Find T(10^{18}, 10^{12}10).
Let y_{0}, y_{1}, y_{2},... be a sequence of random unsigned 32 bit integers
(i.e. 0 ≤ y_{i} < 2^{32}, every value equally likely).
For the sequence x_{i} the following recursion is given:
 x_{0} = 0 and
 x_{i} = x_{i1} y_{i1}, for i > 0. (  is the bitwiseOR operator)
It can be seen that eventually there will be an index N such that x_{i} = 2^{32} 1 (a bitpattern of all ones) for all i ≥ N.
Find the expected value of N.
Give your answer rounded to 10 digits after the decimal point.
Let f(n) represent the number of ways one can fill a 3×3×n tower with blocks of 2×1×1.
You're allowed to rotate the blocks in any way you like; however, rotations, reflections etc of the tower itself are counted as distinct.
For example (with q = 100000007) :
f(2) = 229,
f(4) = 117805,
f(10) mod q = 96149360,
f(10^{3}) mod q = 24806056,
f(10^{6}) mod q = 30808124.
Find f(10^{10000}) mod 100000007.
A game is played with two piles of stones and two players. At her turn, a player removes a number of stones from the larger pile. The number of stones she removes must be a positive multiple of the number of stones in the smaller pile.
E.g., let the ordered pair(6,14) describe a configuration with 6 stones in the smaller pile and 14 stones in the larger pile, then the first player can remove 6 or 12 stones from the larger pile.
The player taking all the stones from a pile wins the game.
A winning configuration is one where the first player can force a win. For example, (1,5), (2,6) and (3,12) are winning configurations because the first player can immediately remove all stones in the second pile.
A losing configuration is one where the second player can force a win, no matter what the first player does. For example, (2,3) and (3,4) are losing configurations: any legal move leaves a winning configuration for the second player.
Define S(N) as the sum of (x_{i}+y_{i}) for all losing configurations (x_{i},y_{i}), 0 < x_{i} < y_{i} ≤ N. We can verify that S(10) = 211 and S(10^{4}) = 230312207313.
Find S(10^{16}) mod 7^{10}.
Let $a_n$ be a sequence recursively defined by:$\quad a_1=1,\quad\displaystyle a_n=\biggl(\sum_{k=1}^{n1}k\cdot a_k\biggr)\bmod n$.
So the first 10 elements of $a_n$ are: 1,1,0,3,0,3,5,4,1,9.
Let f(N,M) represent the number of pairs (p,q) such that:
$$ \def\htmltext#1{\style{fontfamily:inherit;}{\text{#1}}} 1\le p\le q\le N \quad\htmltext{and}\quad\biggl(\sum_{i=p}^qa_i\biggr)\bmod M=0 $$
It can be seen that f(10,10)=4 with the pairs (3,3), (5,5), (7,9) and (9,10).
You are also given that f(10^{4},10^{3})=97158.
Find f(10^{12},10^{6}).
A series of three rooms are connected to each other by automatic doors.
Each door is operated by a security card. Once you enter a room the door automatically closes and that security card cannot be used again. A machine at the start will dispense an unlimited number of cards, but each room (including the starting room) contains scanners and if they detect that you are holding more than three security cards or if they detect an unattended security card on the floor, then all the doors will become permanently locked. However, each room contains a box where you may safely store any number of security cards for use at a later stage.
If you simply tried to travel through the rooms one at a time then as you entered room 3 you would have used all three cards and would be trapped in that room forever!
However, if you make use of the storage boxes, then escape is possible. For example, you could enter room 1 using your first card, place one card in the storage box, and use your third card to exit the room back to the start. Then after collecting three more cards from the dispensing machine you could use one to enter room 1 and collect the card you placed in the box a moment ago. You now have three cards again and will be able to travel through the remaining three doors. This method allows you to travel through all three rooms using six security cards in total.
It is possible to travel through six rooms using a total of 123 security cards while carrying a maximum of 3 cards.
Let C be the maximum number of cards which can be carried at any time.
Let R be the number of rooms to travel through.
Let M(C,R) be the minimum number of cards required from the dispensing machine to travel through R rooms carrying up to a maximum of C cards at any time.
For example, M(3,6)=123 and M(4,6)=23.
And, ∑ M(C,6)=146 for 3 ≤ C ≤ 4.
You are given that ∑ M(C,10)=10382 for 3 ≤ C ≤ 10.
Find ∑ M(C,30) for 3 ≤ C ≤ 40.
We are trying to find a hidden number selected from the set of integers {1, 2, ..., n} by asking questions.
Each number (question) we ask, has a cost equal to the number asked and we get one of three possible answers:
 "Your guess is lower than the hidden number", or
 "Yes, that's it!", or
 "Your guess is higher than the hidden number".
Given the value of n, an optimal strategy minimizes the total cost (i.e. the sum of all the questions asked) for the worst possible case. E.g.
If n=3, the best we can do is obviously to ask the number "2". The answer will immediately lead us to find the hidden number (at a total cost = 2).
If n=8, we might decide to use a "binary search" type of strategy: Our first question would be "4" and if the hidden number is higher than 4 we will need one or two additional questions.
Let our second question be "6". If the hidden number is still higher than 6, we will need a third question in order to discriminate between 7 and 8.
Thus, our third question will be "7" and the total cost for this worstcase scenario will be 4+6+7=17.
We can improve considerably the worstcase cost for n=8, by asking "5" as our first question.
If we are told that the hidden number is higher than 5, our second question will be "7", then we'll know for certain what the hidden number is (for a total cost of 5+7=12).
If we are told that the hidden number is lower than 5, our second question will be "3" and if the hidden number is lower than 3 our third question will be "1", giving a total cost of 5+3+1=9.
Since 12>9, the worstcase cost for this strategy is 12. That's better than what we achieved previously with the "binary search" strategy; it is also better than or equal to any other strategy.
So, in fact, we have just described an optimal strategy for n=8.
Let C(n) be the worstcase cost achieved by an optimal strategy for n, as described above.
Thus C(1) = 0, C(2) = 1, C(3) = 2 and C(8) = 12.
Similarly, C(100) = 400 and $\sum \limits_{n = 1}^{100} {C(n)} = 17575$.
Find $\sum \limits_{n = 1}^{200000} {C(n)}$ .
Susan has a prime frog.
Her frog is jumping around over 500 squares numbered 1 to 500.
He can only jump one square to the left or to the right, with equal probability, and he cannot jump outside the range [1;500].
(if it lands at either end, it automatically jumps to the only available square on the next move.)
When he is on a square with a prime number on it, he croaks 'P' (PRIME) with probability 2/3 or 'N' (NOT PRIME) with probability 1/3 just before jumping to the next square.
When he is on a square with a number on it that is not a prime he croaks 'P' with probability 1/3 or 'N' with probability 2/3 just before jumping to the next square.
Given that the frog's starting position is random with the same probability for every square, and given that she listens to his first 15 croaks, what is the probability that she hears the sequence PPPPNNPPPNPPNPN?
Give your answer as a fraction p/q in reduced form.For example,
a(0) = 

+ 

+ 

+ ... = e − 1 
a(1) = 

+ 

+ 

+ ... = 2e − 3 
a(2) = 

+ 

+ 

+ ... = 

e − 6 
It can be shown that a(n) is of the form 

for integers A(n) and B(n). 
For example a(10) = 

. 
Find A(10^{9}) + B(10^{9}) and give your answer mod 77 777 777.
N×N disks are placed on a square game board. Each disk has a black side and white side.
At each turn, you may choose a disk and flip all the disks in the same row and the same column as this disk: thus 2×N1 disks are flipped. The game ends when all disks show their white side. The following example shows a game on a 5×5 board.
It can be proven that 3 is the minimal number of turns to finish this game.
The bottom left disk on the N×N board has coordinates (0,0);
the bottom right disk has coordinates (N1,0) and the top left disk has coordinates (0,N1).
Let C_{N} be the following configuration of a board with N×N disks:
A disk at (x,y) satisfying $N  1 \le \sqrt{x^2 + y^2} \lt N$, shows its black side; otherwise, it shows its white side. C_{5} is shown above.
Let T(N) be the minimal number of turns to finish a game starting from configuration C_{N} or 0 if configuration C_{N} is unsolvable.
We have shown that T(5)=3. You are also given that T(10)=29 and T(1 000)=395253.
Find $\sum \limits_{i = 3}^{31} {T(2^i  i)}$.
A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices.
Let C(r) be the sphere with the centre (0,0,0) and radius r.
Let Z(r) be the set of points on the surface of C(r) with integer coordinates.
Let T(r) be the set of spherical triangles with vertices in Z(r).
Degenerate spherical triangles, formed by three points on the same great arc, are not included in T(r).
Let A(r) be the area of the smallest spherical triangle in T(r).
For example A(14) is 3.294040 rounded to six decimal places.
Find $\sum \limits_{r = 1}^{50} {A(r)}$. Give your answer rounded to six decimal places.
All positive integers can be partitioned in such a way that each and every term of the partition can be expressed as 2^{i}x3^{j}, where i,j ≥ 0.
Let's consider only those such partitions where none of the terms can divide any of the other terms.
For example, the partition of 17 = 2 + 6 + 9 = (2^{1}x3^{0} + 2^{1}x3^{1} + 2^{0}x3^{2}) would not be valid since 2 can divide 6. Neither would the partition 17 = 16 + 1 = (2^{4}x3^{0} + 2^{0}x3^{0}) since 1 can divide 16. The only valid partition of 17 would be 8 + 9 = (2^{3}x3^{0} + 2^{0}x3^{2}).
Many integers have more than one valid partition, the first being 11 having the following two partitions.
11 = 2 + 9 = (2^{1}x3^{0} + 2^{0}x3^{2})
11 = 8 + 3 = (2^{3}x3^{0} + 2^{0}x3^{1})
Let's define P(n) as the number of valid partitions of n. For example, P(11) = 2.
Let's consider only the prime integers q which would have a single valid partition such as P(17).
The sum of the primes q <100 such that P(q)=1 equals 233.
Find the sum of the primes q <1000000 such that P(q)=1.
In Plato's heaven, there exist an infinite number of bowls in a straight line.
Each bowl either contains some or none of a finite number of beans.
A child plays a game, which allows only one kind of move: removing two beans from any bowl, and putting one in each of the two adjacent bowls.
The game ends when each bowl contains either one or no beans.
For example, consider two adjacent bowls containing 2 and 3 beans respectively, all other bowls being empty. The following eight moves will finish the game:
You are given the following sequences:
$\def\htmltext#1{\style{fontfamily:inherit;}{\text{#1}}}$ $ \begin{align} \qquad t_0 &= 123456,\cr \qquad t_i &= \cases{ \;\;\frac{t_{i1}}{2},&$\htmltext{if }t_{i1}\htmltext{ is even}$\cr \left\lfloor\frac{t_{i1}}{2}\right\rfloor\oplus 926252,&$\htmltext{if }t_{i1}\htmltext{ is odd}$\cr}\cr &\qquad\htmltext{where }\lfloor x\rfloor\htmltext{ is the floor function }\cr &\qquad\!\htmltext{and }\oplus\htmltext{is the bitwise XOR operator.}\cr \qquad b_i &= (t_i\bmod2^{11}) + 1.\cr \end{align} $
The first two terms of the last sequence are $b_1 = 289$ and $b_2 = 145$.
If we start with $b_1$ and $b_2$ beans in two adjacent bowls, $3419100$ moves would be required to finish the game.
Consider now $1500$ adjacent bowls containing $b_1, b_2, \ldots, b_{1500}$ beans respectively, all other bowls being empty. Find how many moves it takes before the game ends.
Whenever Peter feels bored, he places some bowls, containing one bean each, in a circle. After this, he takes all the beans out of a certain bowl and drops them one by one in the bowls going clockwise. He repeats this, starting from the bowl he dropped the last bean in, until the initial situation appears again. For example with 5 bowls he acts as follows:
So with 5 bowls it takes Peter 15 moves to return to the initial situation.
Let M(x) represent the number of moves required to return to the initial situation, starting with x bowls. Thus, M(5) = 15. It can also be verified that M(100) = 10920.
Find M(2^{k}+1). Give your answer modulo 7^{9}.
A train is used to transport four carriages in the order: ABCD. However, sometimes when the train arrives to collect the carriages they are not in the correct order.
To rearrange the carriages they are all shunted on to a large rotating turntable. After the carriages are uncoupled at a specific point the train moves off the turntable pulling the carriages still attached with it. The remaining carriages are rotated 180 degrees. All of the carriages are then rejoined and this process is repeated as often as necessary in order to obtain the least number of uses of the turntable.
Some arrangements, such as ADCB, can be solved easily: the carriages are separated between A and D, and after DCB are rotated the correct order has been achieved.
However, Simple Simon, the train driver, is not known for his efficiency, so he always solves the problem by initially getting carriage A in the correct place, then carriage B, and so on.
Using four carriages, the worst possible arrangements for Simon, which we shall call maximix arrangements, are DACB and DBAC; each requiring him five rotations (although, using the most efficient approach, they could be solved using just three rotations). The process he uses for DACB is shown below.
It can be verified that there are 24 maximix arrangements for six carriages, of which the tenth lexicographic maximix arrangement is DFAECB.
Find the 2011^{th} lexicographic maximix arrangement for eleven carriages.
Let {a_{1}, a_{2},..., a_{n}} be an integer sequence of length n such that:
 a_{1} = 6
 for all 1 ≤ i < n : φ(a_{i}) < φ(a_{i+1}) < a_{i} < a_{i+1}^{1}
Let S(N) be the number of such sequences with a_{n} ≤ N.
For example, S(10) = 4: {6}, {6, 8}, {6, 8, 9} and {6, 10}.
We can verify that S(100) = 482073668 and S(10 000) mod 10^{8} = 73808307.
Find S(20 000 000) mod 10^{8}.
^{1} φ denotes Euler's totient function.
A rectangular sheet of grid paper with integer dimensions w × h is given. Its grid spacing is 1.
When we cut the sheet along the grid lines into two pieces and rearrange those pieces without overlap, we can make new rectangles with different dimensions.
For example, from a sheet with dimensions 9 × 4 , we can make rectangles with dimensions 18 × 2, 12 × 3 and 6 × 6 by cutting and rearranging as below:
Similarly, from a sheet with dimensions 9 × 8 , we can make rectangles with dimensions 18 × 4 and 12 × 6 .
For a pair w and h, let F(w,h) be the number of distinct rectangles that can be made from a sheet with dimensions w × h .
For example, F(2,1) = 0, F(2,2) = 1, F(9,4) = 3 and F(9,8) = 2.
Note that rectangles congruent to the initial one are not counted in F(w,h).
Note also that rectangles with dimensions w × h and dimensions h × w are not considered distinct.
For an integer N, let G(N) be the sum of F(w,h) for all pairs w and h which satisfy 0 < h ≤ w ≤ N.
We can verify that G(10) = 55, G(10^{3}) = 971745 and G(10^{5}) = 9992617687.
Find G(10^{12}). Give your answer modulo 10^{8}.
"And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of the river were level meadows. And on one side of the river he saw a flock of white sheep, and on the other a flock of black sheep. And whenever one of the white sheep bleated, one of the black sheep would cross over and become white; and when one of the black sheep bleated, one of the white sheep would cross over and become black."
en.wikisource.org
Initially each flock consists of n sheep. Each sheep (regardless of colour) is equally likely to be the next sheep to bleat. After a sheep has bleated and a sheep from the other flock has crossed over, Peredur may remove a number of white sheep in order to maximize the expected final number of black sheep. Let E(n) be the expected final number of black sheep if Peredur uses an optimal strategy.
You are given that E(5) = 6.871346 rounded to 6 places behind the decimal point.
Find E(10 000) and give your answer rounded to 6 places behind the decimal point.
For fixed integers a, b, c, define the crazy function F(n) as follows:
F(n) = n  c for all n > b
F(n) = F(a + F(a + F(a + F(a + n)))) for all n ≤ b.
Also, define $S(a, b, c) = \sum \limits_{n = 0}^{b} {F(n)}$.
For example, if a = 50, b = 2000 and c = 40, then F(0) = 3240 and F(2000) = 2040.
Also, S(50, 2000, 40) = 5204240.
Find the last 9 digits of S(21^{7}, 7^{21}, 12^{7}).
The Golomb's selfdescribing sequence $(G(n))$ is the only nondecreasing sequence of natural numbers such that $n$ appears exactly $G(n)$ times in the sequence. The values of $G(n)$ for the first few $n$ are
You are given that $G(10^3) = 86$, $G(10^6) = 6137$.
You are also given that $\sum G(n^3) = 153506976$ for $1 \le n \lt 10^3$.
Find $\sum G(n^3)$ for $1 \le n \lt 10^6$.
Consider the number 50.
50^{2} = 2500 = 2^{2} × 5^{4}, so φ(2500) = 2 × 4 × 5^{3} = 8 × 5^{3} = 2^{3} × 5^{3}. ^{1}
So 2500 is a square and φ(2500) is a cube.
Find the sum of all numbers n, 1 < n < 10^{10} such that φ(n^{2}) is a cube.
^{1} φ denotes Euler's totient function.
For any positive integer k, a finite sequence a_{i} of fractions x_{i}/y_{i} is defined by:
a_{1} = 1/k and
a_{i} = (x_{i1}+1)/(y_{i1}1) reduced to lowest terms for i>1.
When a_{i} reaches some integer n, the sequence stops. (That is, when y_{i}=1.)
Define f(k) = n.
For example, for k = 20:
1/20 → 2/19 → 3/18 = 1/6 → 2/5 → 3/4 → 4/3 → 5/2 → 6/1 = 6
So f(20) = 6.
Also f(1) = 1, f(2) = 2, f(3) = 1 and ∑ f(k^{3}) = 118937 for 1 ≤ k ≤ 100.
Find ∑ f(k^{3}) for 1 ≤ k ≤ 2×10^{6}.
One variant of N.G. de Bruijn's silver dollar game can be described as follows:
On a strip of squares a number of coins are placed, at most one coin per square. Only one coin, called the silver dollar, has any value. Two players take turns making moves. At each turn a player must make either a regular or a special move.
A regular move consists of selecting one coin and moving it one or more squares to the left. The coin cannot move out of the strip or jump on or over another coin.
Alternatively, the player can choose to make the special move of pocketing the leftmost coin rather than making a regular move. If no regular moves are possible, the player is forced to pocket the leftmost coin.
The winner is the player who pockets the silver dollar.
A winning configuration is an arrangement of coins on the strip where the first player can force a win no matter what the second player does.
Let W(n,c) be the number of winning configurations for a strip of n squares, c worthless coins and one silver dollar.
You are given that W(10,2) = 324 and W(100,10) = 1514704946113500.
Find W(1 000 000, 100) modulo the semiprime 1000 036 000 099 (= 1 000 003 · 1 000 033).
We define the Matrix Sum of a matrix as the maximum sum of matrix elements with each element being the only one in his row and column. For example, the Matrix Sum of the matrix below equals 3315 ( = 863 + 383 + 343 + 959 + 767):
7 53 183 439 863
497 383 563 79 973
287 63 343 169 583
627 343 773 959 943
767 473 103 699 303
Find the Matrix Sum of:
7 53 183 439 863 497 383 563 79 973 287 63 343 169 583
627 343 773 959 943 767 473 103 699 303 957 703 583 639 913
447 283 463 29 23 487 463 993 119 883 327 493 423 159 743
217 623 3 399 853 407 103 983 89 463 290 516 212 462 350
960 376 682 962 300 780 486 502 912 800 250 346 172 812 350
870 456 192 162 593 473 915 45 989 873 823 965 425 329 803
973 965 905 919 133 673 665 235 509 613 673 815 165 992 326
322 148 972 962 286 255 941 541 265 323 925 281 601 95 973
445 721 11 525 473 65 511 164 138 672 18 428 154 448 848
414 456 310 312 798 104 566 520 302 248 694 976 430 392 198
184 829 373 181 631 101 969 613 840 740 778 458 284 760 390
821 461 843 513 17 901 711 993 293 157 274 94 192 156 574
34 124 4 878 450 476 712 914 838 669 875 299 823 329 699
815 559 813 459 522 788 168 586 966 232 308 833 251 631 107
813 883 451 509 615 77 281 613 459 205 380 274 302 35 805
The number 7 is special, because 7 is 111 written in base 2, and 11 written in base 6
(i.e. 7_{10} = 11_{6} = 111_{2}). In other words, 7 is a repunit in at least two bases b > 1.
We shall call a positive integer with this property a strong repunit. It can be verified that there are 8 strong repunits below 50: {1,7,13,15,21,31,40,43}.
Furthermore, the sum of all strong repunits below 1000 equals 15864.
The largest integer ≤ 100 that is only divisible by both the primes 2 and 3 is 96, as 96=32*3=2^{5}*3. For two distinct primes p and q let M(p,q,N) be the largest positive integer ≤N only divisible by both p and q and M(p,q,N)=0 if such a positive integer does not exist.
E.g. M(2,3,100)=96.
M(3,5,100)=75 and not 90 because 90 is divisible by 2 ,3 and 5.
Also M(2,73,100)=0 because there does not exist a positive integer ≤ 100 that is divisible by both 2 and 73.
Let S(N) be the sum of all distinct M(p,q,N). S(100)=2262.
Find S(10 000 000).
Many numbers can be expressed as the sum of a square and a cube. Some of them in more than one way.
Consider the palindromic numbers that can be expressed as the sum of a square and a cube, both greater than 1, in exactly 4 different ways.
For example, 5229225 is a palindromic number and it can be expressed in exactly 4 different ways:
2285^{2} + 20^{3}
2223^{2} + 66^{3}
1810^{2} + 125^{3}
1197^{2} + 156^{3}
Find the sum of the five smallest such palindromic numbers.
An ant moves on a regular grid of squares that are coloured either black or white.
The ant is always oriented in one of the cardinal directions (left, right, up or down) and moves from square to adjacent square according to the following rules:
 if it is on a black square, it flips the colour of the square to white, rotates 90 degrees counterclockwise and moves forward one square.
 if it is on a white square, it flips the colour of the square to black, rotates 90 degrees clockwise and moves forward one square.
Starting with a grid that is entirely white, how many squares are black after 10^{18} moves of the ant?
A list of size n is a sequence of n natural numbers.
Examples are (2,4,6), (2,6,4), (10,6,15,6), and (11).
The greatest common divisor, or gcd, of a list is the largest natural number that divides all entries of the list.
Examples: gcd(2,6,4) = 2, gcd(10,6,15,6) = 1 and gcd(11) = 11.
The least common multiple, or lcm, of a list is the smallest natural number divisible by each entry of the list.
Examples: lcm(2,6,4) = 12, lcm(10,6,15,6) = 30 and lcm(11) = 11.
Let f(G, L, N) be the number of lists of size N with gcd ≥ G and lcm ≤ L. For example:
f(10, 100, 1) = 91.
f(10, 100, 2) = 327.
f(10, 100, 3) = 1135.
f(10, 100, 1000) mod 101^{4} = 3286053.
Find f(10^{6}, 10^{12}, 10^{18}) mod 101^{4}.
A hexagonal orchard of order n is a triangular lattice made up of points within a regular hexagon with side n. The following is an example of a hexagonal orchard of order 5:
Highlighted in green are the points which are hidden from the center by a point closer to it. It can be seen that for a hexagonal orchard of order 5, 30 points are hidden from the center.
Let H(n) be the number of points hidden from the center in a hexagonal orchard of order n.
H(5) = 30. H(10) = 138. H(1 000) = 1177848.
Find H(100 000 000).
Each one of the 25 sheep in a flock must be tested for a rare virus, known to affect 2% of the sheep population. An accurate and extremely sensitive PCR test exists for blood samples, producing a clear positive / negative result, but it is very timeconsuming and expensive.
Because of the high cost, the vetincharge suggests that instead of performing 25 separate tests, the following procedure can be used instead:
The sheep are split into 5 groups of 5 sheep in each group.
For each group, the 5 samples are mixed together and a single test is performed. Then,
 If the result is negative, all the sheep in that group are deemed to be virusfree.
 If the result is positive, 5 additional tests will be performed (a separate test for each animal) to determine the affected individual(s).
Since the probability of infection for any specific animal is only 0.02, the first test (on the pooled samples) for each group will be:
 Negative (and no more tests needed) with probability 0.98^{5} = 0.9039207968.
 Positive (5 additional tests needed) with probability 1  0.9039207968 = 0.0960792032.
Thus, the expected number of tests for each group is 1 + 0.0960792032 × 5 = 1.480396016.
Consequently, all 5 groups can be screened using an average of only 1.480396016 × 5 = 7.40198008 tests, which represents a huge saving of more than 70% !
Although the scheme we have just described seems to be very efficient, it can still be improved considerably (always assuming that the test is sufficiently sensitive and that there are no adverse effects caused by mixing different samples). E.g.:
 We may start by running a test on a mixture of all the 25 samples. It can be verified that in about 60.35% of the cases this test will be negative, thus no more tests will be needed. Further testing will only be required for the remaining 39.65% of the cases.
 If we know that at least one animal in a group of 5 is infected and the first 4 individual tests come out negative, there is no need to run a test on the fifth animal (we know that it must be infected).
 We can try a different number of groups / different number of animals in each group, adjusting those numbers at each level so that the total expected number of tests will be minimised.
To simplify the very wide range of possibilities, there is one restriction we place when devising the most costefficient testing scheme: whenever we start with a mixed sample, all the sheep contributing to that sample must be fully screened (i.e. a verdict of infected / virusfree must be reached for all of them) before we start examining any other animals.
For the current example, it turns out that the most costefficient testing scheme (we'll call it the optimal strategy) requires an average of just 4.155452 tests!
Using the optimal strategy, let T(s,p) represent the average number of tests needed to screen a flock of s sheep for a virus having probability p to be present in any individual.
Thus, rounded to six decimal places, T(25, 0.02) = 4.155452 and T(25, 0.10) = 12.702124.
Find ∑ T(10000, p) for p=0.01, 0.02, 0.03, ... 0.50.
Give your answer rounded to six decimal places.
A moon could be described by the sphere $C(r)$ with centre $(0,0,0)$ and radius $r$.
There are stations on the moon at the points on the surface of $C(r)$ with integer coordinates. The station at $(0,0,r)$ is called North Pole station, the station at $(0,0,r)$ is called South Pole station.
All stations are connected with each other via the shortest road on the great arc through the stations. A journey between two stations is risky. If d is the length of the road between two stations, $\left(\frac{d}{\pi r}\right)^2$ is a measure for the risk of the journey (let us call it the risk of the road). If the journey includes more than two stations, the risk of the journey is the sum of risks of the used roads.
A direct journey from the North Pole station to the South Pole station has the length $\pi r$ and risk 1. The journey from the North Pole station to the South Pole station via $(0,r,0)$ has the same length, but a smaller risk:
\[ \left(\frac{\frac{1}{2}\pi r}{\pi r}\right)^2+\left(\frac{\frac{1}{2}\pi r}{\pi r}\right)^2=0.5 \]The minimal risk of a journey from the North Pole station to the South Pole station on $C(r)$ is $M(r)$.
You are given that $M(7)=0.1784943998$ rounded to 10 digits behind the decimal point.
Find $\displaystyle{\sum_{n=1}^{15}M(2^n1)}$.
Give your answer rounded to 10 digits behind the decimal point in the form a.bcdefghijk.
Consider a honey bee's honeycomb where each cell is a perfect regular hexagon with side length $1$.
One particular cell is occupied by the queen bee.
For a positive real number $L$, let $\text{B}(L)$ count the cells with distance $L$ from the queen bee cell (all distances are measured from centre to centre); you may assume that the honeycomb is large enough to accommodate for any distance we wish to consider.
For example, $\text{B}(\sqrt 3)=6$, $\text{B}(\sqrt {21}) = 12$ and $\text{B}(111\,111\,111) = 54$.
Find the number of $L \le 5 \times 10^{11}$ such that $\text{B}(L) = 450$.
Define Co(n) to be the maximal possible sum of a set of mutually coprime elements from {1, 2, ..., n}.
For example Co(10) is 30 and hits that maximum on the subset {1, 5, 7, 8, 9}.
You are given that Co(30) = 193 and Co(100) = 1356.
Find Co(200000).
Let a_{n} be the largest real root of a polynomial g(x) = x^{3}  2^{n}·x^{2} + n.
For example, a_{2} = 3.86619826...
Find the last eight digits of $\sum \limits_{i = 1}^{30} {\left \lfloor a_i^{987654321} \right \rfloor}$.
Note: $\lfloor a \rfloor$ represents the floor function.
Consider the divisors of 30: 1,2,3,5,6,10,15,30.
It can be seen that for every divisor d of 30, d+30/d is prime.
Find the sum of all positive integers n not exceeding 100 000 000
such that
for every divisor d of n, d+n/d is prime.
A cyclic number with n digits has a very interesting property:
When it is multiplied by 1, 2, 3, 4, ... n, all the products have exactly the same digits, in the same order, but rotated in a circular fashion!
The smallest cyclic number is the 6digit number 142857 :
142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142
The next cyclic number is 0588235294117647 with 16 digits :
0588235294117647 × 1 = 0588235294117647
0588235294117647 × 2 = 1176470588235294
0588235294117647 × 3 = 1764705882352941
...
0588235294117647 × 16 = 9411764705882352
Note that for cyclic numbers, leading zeros are important.
There is only one cyclic number for which, the eleven leftmost digits are 00000000137 and the five rightmost digits are 56789 (i.e., it has the form 00000000137...56789 with an unknown number of digits in the middle). Find the sum of all its digits.
An infinite number of people (numbered 1, 2, 3, etc.) are lined up to get a room at Hilbert's newest infinite hotel. The hotel contains an infinite number of floors (numbered 1, 2, 3, etc.), and each floor contains an infinite number of rooms (numbered 1, 2, 3, etc.).
Initially the hotel is empty. Hilbert declares a rule on how the n^{th} person is assigned a room: person n gets the first vacant room in the lowest numbered floor satisfying either of the following:
 the floor is empty
 the floor is not empty, and if the latest person taking a room in that floor is person m, then m + n is a perfect square
Person 1 gets room 1 in floor 1 since floor 1 is empty.
Person 2 does not get room 2 in floor 1 since 1 + 2 = 3 is not a perfect square.
Person 2 instead gets room 1 in floor 2 since floor 2 is empty.
Person 3 gets room 2 in floor 1 since 1 + 3 = 4 is a perfect square.
Eventually, every person in the line gets a room in the hotel.
Define P(f, r) to be n if person n occupies room r in floor f, and 0 if no person occupies the room. Here are a few examples:
P(1, 1) = 1
P(1, 2) = 3
P(2, 1) = 2
P(10, 20) = 440
P(25, 75) = 4863
P(99, 100) = 19454
Find the sum of all P(f, r) for all positive f and r such that f × r = 71328803586048 and give the last 8 digits as your answer.
Given two points (x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{2}) in three dimensional space, the Manhattan distance between those points is defined as
x_{1}x_{2}+y_{1}y_{2}+z_{1}z_{2}.
Let C(r) be a sphere with radius r and center in the origin O(0,0,0).
Let I(r) be the set of all points with integer coordinates on the surface of C(r).
Let S(r) be the sum of the Manhattan distances of all elements of I(r) to the origin O.
E.g. S(45)=34518.
Find S(10^{10}).
The ThueMorse sequence {T_{n}} is a binary sequence satisfying:
 T_{0} = 0
 T_{2n} = T_{n}
 T_{2n+1} = 1  T_{n}
The first several terms of {T_{n}} are given as follows:
01101001100101101001011001101001....
We define {A_{n}} as the sorted sequence of integers such that the binary expression of each element appears as a subsequence in {T_{n}}.
For example, the decimal number 18 is expressed as 10010 in binary. 10010 appears in {T_{n}} (T_{8} to T_{12}), so 18 is an element of {A_{n}}.
The decimal number 14 is expressed as 1110 in binary. 1110 never appears in {T_{n}}, so 14 is not an element of {A_{n}}.
The first several terms of A_{n} are given as follows:
n  0  1  2  3  4  5  6  7  8  9  10  11  12  … 
A_{n}  0  1  2  3  4  5  6  9  10  11  12  13  18  … 
We can also verify that A_{100} = 3251 and A_{1000} = 80852364498.
Find the last 9 digits of $\sum \limits_{k = 1}^{18} {A_{10^k}}$.
Consider the number 54.
54 can be factored in 7 distinct ways into one or more factors larger than 1:
54, 2×27, 3×18, 6×9, 3×3×6, 2×3×9 and 2×3×3×3.
If we require that the factors are all squarefree only two ways remain: 3×3×6 and 2×3×3×3.
Let's call Fsf(n) the number of ways n can be factored into one or more squarefree factors larger than 1, so Fsf(54)=2.
Let S(n) be ∑ Fsf(k) for k=2 to n.
S(100)=193.
Find S(10 000 000 000).
A cubic Bézier curve is defined by four points: P_{0}, P_{1}, P_{2} and P_{3}.
The curve is constructed as follows:
On the segments P_{0}P_{1}, P_{1}P_{2} and P_{2}P_{3} the points Q_{0},Q_{1} and Q_{2} are drawn such that
P_{0}Q_{0} / P_{0}P_{1} = P_{1}Q_{1} / P_{1}P_{2} = P_{2}Q_{2} / P_{2}P_{3} = t (t in [0,1]).
On the segments Q_{0}Q_{1} and Q_{1}Q_{2} the points R_{0} and R_{1} are drawn such that
Q_{0}R_{0} / Q_{0}Q_{1} = Q_{1}R_{1} / Q_{1}Q_{2} = t for the same value of t.
On the segment R_{0}R_{1} the point B is drawn such that R_{0}B / R_{0}R_{1} = t for the same value of t.
The Bézier curve defined by the points P_{0}, P_{1}, P_{2}, P_{3} is the locus of B as Q_{0} takes all possible positions on the segment P_{0}P_{1}.
(Please note that for all points the value of t is the same.)
At this (external) web address you will find an applet that allows you to drag the points P_{0}, P_{1}, P_{2} and P_{3} to see what the Bézier curve (green curve) defined by those points looks like. You can also drag the point Q_{0} along the segment P_{0}P_{1}.
From the construction it is clear that the Bézier curve will be tangent to the segments P_{0}P_{1} in P_{0} and P_{2}P_{3} in P_{3}.
A cubic Bézier curve with P_{0}=(1,0), P_{1}=(1,v), P_{2}=(v,1) and P_{3}=(0,1) is used to approximate a quarter circle.
The value v > 0 is chosen such that the area enclosed by the lines OP_{0}, OP_{3} and the curve is equal to ^{π}/_{4} (the area of the quarter circle).
That is, if L is the length of the curve, calculate 100 ×  L − π/2 π/2 
There are N seats in a row. N people come after each other to fill the seats according to the following rules:
 If there is any seat whose adjacent seat(s) are not occupied take such a seat.
 If there is no such seat and there is any seat for which only one adjacent seat is occupied take such a seat.
 Otherwise take one of the remaining available seats.
The following figure shows T(4)=8.
We can verify that T(10) = 61632 and T(1 000) mod 100 000 007 = 47255094.
Find T(1 000 000) mod 100 000 007.
The binomial coefficient $\displaystyle{\binom{10^{18}}{10^9}}$ is a number with more than 9 billion ($9\times 10^9$) digits.
Let $M(n,k,m)$ denote the binomial coefficient $\displaystyle{\binom{n}{k}}$ modulo $m$.
Calculate $\displaystyle{\sum M(10^{18},10^9,p\cdot q\cdot r)}$ for $1000\lt p\lt q\lt r\lt 5000$ and $p$,$q$,$r$ prime.
Two players, Anton and Bernhard, are playing the following game.
There is one pile of n stones.
The first player may remove any positive number of stones, but not the whole pile.
Thereafter, each player may remove at most twice the number of stones his opponent took on the previous move.
The player who removes the last stone wins.
E.g. n=5
If the first player takes anything more than one stone the next player will be able to take all remaining stones.
If the first player takes one stone, leaving four, his opponent will take also one stone, leaving three stones.
The first player cannot take all three because he may take at most 2x1=2 stones. So let's say he takes also one stone, leaving 2. The second player can take the two remaining stones and wins.
So 5 is a losing position for the first player.
For some winning positions there is more than one possible move for the first player.
E.g. when n=17 the first player can remove one or four stones.
Let M(n) be the maximum number of stones the first player can take from a winning position at his first turn and M(n)=0 for any other position.
∑ M(n) for n≤100 is 728.
Find ∑ M(n) for n≤10^{18}. Give your answer modulo 10^{8}.
Bozo sort, not to be confused with the slightly less efficient bogo sort, consists out of checking if the input sequence is sorted and if not swapping randomly two elements. This is repeated until eventually the sequence is sorted.
If we consider all permutations of the first 4 natural numbers as input the expectation value of the number of swaps, averaged over all 4! input sequences is 24.75.
The already sorted sequence takes 0 steps.
In this problem we consider the following variant on bozo sort.
If the sequence is not in order we pick three elements at random and shuffle these three elements randomly.
All 3!=6 permutations of those three elements are equally likely.
The already sorted sequence will take 0 steps.
If we consider all permutations of the first 4 natural numbers as input the expectation value of the number of shuffles, averaged over all 4! input sequences is 27.5.
Consider as input sequences the permutations of the first 11 natural numbers.
Averaged over all 11! input sequences, what is the expected number of shuffles this sorting algorithm will perform?
Give your answer rounded to the nearest integer.
The harmonic series $1 + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + ...$ is well known to be divergent.
If we however omit from this series every term where the denominator has a 9 in it, the series remarkably enough converges to approximately 22.9206766193.
This modified harmonic series is called the Kempner series.
Let us now consider another modified harmonic series by omitting from the harmonic series every term where the denominator has 3 or more equal consecutive digits.
One can verify that out of the first 1200 terms of the harmonic series, only 20 terms will be omitted.
These 20 omitted terms are:
$$\dfrac{1}{111}, \dfrac{1}{222}, \dfrac{1}{333}, \dfrac{1}{444}, \dfrac{1}{555}, \dfrac{1}{666}, \dfrac{1}{777}, \dfrac{1}{888}, \dfrac{1}{999}, \dfrac{1}{1000}, \dfrac{1}{1110}, \\\ \dfrac{1}{1111}, \dfrac{1}{1112}, \dfrac{1}{1113}, \dfrac{1}{1114}, \dfrac{1}{1115}, \dfrac{1}{1116}, \dfrac{1}{1117}, \dfrac{1}{1118}, \dfrac{1}{1119}$$
This series converges as well.
Find the value the series converges to.
Give your answer rounded to 10 digits behind the decimal point.
In a standard 52 card deck of playing cards, a set of 4 cards is a Badugi if it contains 4 cards with no pairs and no two cards of the same suit.
Let f(n) be the number of ways to choose n cards with a 4 card subset that is a Badugi. For example, there are 2598960 ways to choose five cards from a standard 52 card deck, of which 514800 contain a 4 card subset that is a Badugi, so f(5) = 514800.
Find ∑ f(n) for 4 ≤ n ≤ 13.
Let us define a geometric triangle as an integer sided triangle with sides a ≤ b ≤ c so that its sides form a geometric progression, i.e. b^{2} = a · c .
An example of such a geometric triangle is the triangle with sides a = 144, b = 156 and c = 169.
There are 861805 geometric triangles with perimeter ≤ 10^{6} .
How many geometric triangles exist with perimeter ≤ 2.5·10^{13} ?
Oregon licence plates consist of three letters followed by a three digit number (each digit can be from [0..9]).
While driving to work Seth plays the following game:
Whenever the numbers of two licence plates seen on his trip add to 1000 that's a win.
E.g. MIC012 and HAN988 is a win and RYU500 and SET500 too. (as long as he sees them in the same trip).
Find the expected number of plates he needs to see for a win.
Give your answer rounded to 8 decimal places behind the decimal point.
Note: We assume that each licence plate seen is equally likely to have any three digit number on it.
Let $R(M, N)$ be the number of lattice points $(x, y)$ which satisfy $M\!\lt\!x\!\le\!N$, $M\!\lt\!y\!\le\!N$ and $\large\left\lfloor\!\frac{y^2}{x^2}\!\right\rfloor$ is odd.
We can verify that $R(0, 100) = 3019$ and $R(100, 10000) = 29750422$.
Find $R(2\cdot10^6, 10^9)$.
Note: $\lfloor x\rfloor$ represents the floor function.
Every triangle has a circumscribed circle that goes through the three vertices. Consider all integer sided triangles for which the radius of the circumscribed circle is integral as well.
Let S(n) be the sum of the radii of the circumscribed circles of all such triangles for which the radius does not exceed n.
S(100)=4950 and S(1200)=1653605.
Find S(10^{7}).
An integer partition of a number n is a way of writing n as a sum of positive integers.
Partitions that differ only in the order of their summands are considered the same. A partition of n into distinct parts is a partition of n in which every part occurs at most once.
The partitions of 5 into distinct parts are:
5, 4+1 and 3+2.
Let f(n) be the maximum product of the parts of any such partition of n into distinct parts and let m(n) be the number of elements of any such partition of n with that product.
So f(5)=6 and m(5)=2.
For n=10 the partition with the largest product is 10=2+3+5, which gives f(10)=30 and m(10)=3.
And their product, f(10)·m(10) = 30·3 = 90
It can be verified that
∑ f(n)·m(n) for 1 ≤ n ≤ 100 = 1683550844462.
Find ∑ f(n)·m(n) for 1 ≤ n ≤ 10^{14}.
Give your answer modulo 982451653, the 50 millionth prime.
Let $S_n$ be an integer sequence produced with the following pseudorandom number generator:
\[ \begin{equation} \begin{split} S_0 & = 290797 \\ S_{n+1} & = S_n^2 \bmod 50515093 \end{split} \end{equation} \]
Let $A(i, j)$ be the minimum of the numbers $S_i, S_{i+1}, \ldots, S_j$ for $i\le j$.
Let $M(N) = \sum A(i, j)$ for $1 \le i \le j \le N$.
We can verify that $M(10) = 432256955$ and $M(10\,000) = 3264567774119$.
Find $M(2\,000\,000\,000)$.
Consider the following set of dice with nonstandard pips:
Die A: 1 4 4 4 4 4
Die B: 2 2 2 5 5 5
Die C: 3 3 3 3 3 6
A game is played by two players picking a die in turn and rolling it. The player who rolls the highest value wins.
If the first player picks die A and the second player picks die B we get
P(second player wins) = ^{7}/_{12} > ^{1}/_{2}
If the first player picks die B and the second player picks die C we get
P(second player wins) = ^{7}/_{12} > ^{1}/_{2}
If the first player picks die C and the second player picks die A we get
P(second player wins) = ^{25}/_{36} > ^{1}/_{2}
So whatever die the first player picks, the second player can pick another die and have a larger than 50% chance of winning.
A set of dice having this property is called a nontransitive set of dice.
We wish to investigate how many sets of nontransitive dice exist. We will assume the following conditions:
 There are three sixsided dice with each side having between 1 and N pips, inclusive.
 Dice with the same set of pips are equal, regardless of which side on the die the pips are located.
 The same pip value may appear on multiple dice; if both players roll the same value neither player wins.
 The sets of dice {A,B,C}, {B,C,A} and {C,A,B} are the same set.
For N = 7 we find there are 9780 such sets.
How many are there for N = 30 ?
There are 16 positive integers that do not have a zero in their digits and that have a digital sum equal to 5, namely:
5, 14, 23, 32, 41, 113, 122, 131, 212, 221, 311, 1112, 1121, 1211, 2111 and 11111.
Their sum is 17891.
Let f(n) be the sum of all positive integers that do not have a zero in their digits and have a digital sum equal to n.
Find $\displaystyle \sum_{i=1}^{17} f(13^i)$.
Give the last 9 digits as your answer.
Let T(n) be the n^{th} triangle number, so T(n) = 

. 
Let dT(n) be the number of divisors of T(n).
E.g.:
T(7) = 28 and dT(7) = 6.
Let Tr(n) be the number of triples (i, j, k) such that 1 ≤ i < j < k ≤ n and dT(i) > dT(j) > dT(k).
Tr(20) = 14, Tr(100) = 5772 and Tr(1000) = 11174776.
Find Tr(60 000 000).
Give the last 18 digits of your answer.
Let f(n) be the number of couples (x,y) with x and y positive integers, x ≤ y and the least common multiple of x and y equal to n.
Let g be the summatory function of f, i.e.: g(n) = ∑ f(i) for 1 ≤ i ≤ n.
You are given that g(10^{6}) = 37429395.
Find g(10^{12}).
An m×n maze is an m×n rectangular grid with walls placed between grid cells such that there is exactly one path from the topleft square to any other square.
The following are examples of a 9×12 maze and a 15×20 maze:
Let C(m,n) be the number of distinct m×n mazes. Mazes which can be formed by rotation and reflection from another maze are considered distinct.
It can be verified that C(1,1) = 1, C(2,2) = 4, C(3,4) = 2415, and C(9,12) = 2.5720e46 (in scientific notation rounded to 5 significant digits).
Find C(100,500) and write your answer in scientific notation rounded to 5 significant digits.
When giving your answer, use a lowercase e to separate mantissa and exponent. E.g. if the answer is 1234567891011 then the answer format would be 1.2346e12.
For a prime p let S(p) = (∑ (pk)!) mod(p) for 1 ≤ k ≤ 5.
For example, if p=7,
(71)! + (72)! + (73)! + (74)! + (75)! = 6! + 5! + 4! + 3! + 2! = 720+120+24+6+2 = 872.
As 872 mod(7) = 4, S(7) = 4.
It can be verified that ∑ S(p) = 480 for 5 ≤ p < 100.
Find ∑ S(p) for 5 ≤ p < 10^{8}.
A polygon is a flat shape consisting of straight line segments that are joined to form a closed chain or circuit. A polygon consists of at least three sides and does not selfintersect.
A set S of positive numbers is said to generate a polygon P if:
 no two sides of P are the same length,
 the length of every side of P is in S, and
 S contains no other value.
For example:
The set {3, 4, 5} generates a polygon with sides 3, 4, and 5 (a triangle).
The set {6, 9, 11, 24} generates a polygon with sides 6, 9, 11, and 24 (a quadrilateral).
The sets {1, 2, 3} and {2, 3, 4, 9} do not generate any polygon at all.
Consider the sequence s, defined as follows:
 s_{1} = 1, s_{2} = 2, s_{3} = 3
 s_{n} = s_{n1} + s_{n3} for n > 3.
Let U_{n} be the set {s_{1}, s_{2}, ..., s_{n}}. For example, U_{10} = {1, 2, 3, 4, 6, 9, 13, 19, 28, 41}.
Let f(n) be the number of subsets of U_{n} which generate at least one polygon.
For example, f(5) = 7, f(10) = 501 and f(25) = 18635853.
Find the last 9 digits of f(10^{18}).
Let f_{5}(n) be the largest integer x for which 5^{x} divides n.
For example, f_{5}(625000) = 7.
Let T_{5}(n) be the number of integers i which satisfy f_{5}((2·i1)!) < 2·f_{5}(i!) and 1 ≤ i ≤ n.
It can be verified that T_{5}(10^{3}) = 68 and T_{5}(10^{9}) = 2408210.
Find T_{5}(10^{18}).
Define the sequence a(n) as the number of adjacent pairs of ones in the binary expansion of n (possibly overlapping).
E.g.: a(5) = a(101_{2}) = 0, a(6) = a(110_{2}) = 1, a(7) = a(111_{2}) = 2
Define the sequence b(n) = (1)^{a(n)}.
This sequence is called the RudinShapiro sequence.
Also consider the summatory sequence of b(n): $s(n) = \sum \limits_{i = 0}^{n} {b(i)}$.
The first couple of values of these sequences are:
n 0 1 2 3 4 5 6 7
a(n) 0 0 0 1 0 0 1 2
b(n) 1 1 1 1 1 1 1 1
s(n) 1 2 3 2 3 4 3 4
The sequence s(n) has the remarkable property that all elements are positive and every positive integer k occurs exactly k times.
Define g(t,c), with 1 ≤ c ≤ t, as the index in s(n) for which t occurs for the c'th time in s(n).
E.g.: g(3,3) = 6, g(4,2) = 7 and g(54321,12345) = 1220847710.
Let F(n) be the fibonacci sequence defined by:
F(0)=F(1)=1 and
F(n)=F(n1)+F(n2) for n>1.
Define GF(t)=g(F(t),F(t1)).
Find $\sum$ GF(t) for 2≤t≤45.
For any triangle T in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside T.
For a given n, consider triangles T such that:
 the vertices of T have integer coordinates with absolute value ≤ n, and
 the foci^{1} of the largestarea ellipse inside T are (√13,0) and (√13,0).
Let A(n) be the sum of the areas of all such triangles.
For example, if n = 8, there are two such triangles. Their vertices are (4,3),(4,3),(8,0) and (4,3),(4,3),(8,0), and the area of each triangle is 36. Thus A(8) = 36 + 36 = 72.
It can be verified that A(10) = 252, A(100) = 34632 and A(1000) = 3529008.
Find A(1 000 000 000).
^{1}The foci (plural of focus) of an ellipse are two points A and B such that for every point P on the boundary of the ellipse, AP + PB is constant.
Let n be an integer and S(n) be the set of factors of n.
A subset A of S(n) is called an antichain of S(n) if A contains only one element or if none of the elements of A divides any of the other elements of A.
For example: S(30) = {1, 2, 3, 5, 6, 10, 15, 30}
{2, 5, 6} is not an antichain of S(30).
{2, 3, 5} is an antichain of S(30).
Let N(n) be the maximum length of an antichain of S(n).
Find ∑ N(n) for 1 ≤ n ≤ 10^{8}
A Harshad or Niven number is a number that is divisible by the sum of its digits.
201 is a Harshad number because it is divisible by 3 (the sum of its digits.)
When we truncate the last digit from 201, we get 20, which is a Harshad number.
When we truncate the last digit from 20, we get 2, which is also a Harshad number.
Let's call a Harshad number that, while recursively truncating the last digit, always results in a Harshad number a right truncatable Harshad number.
Also:
201/3=67 which is prime.
Let's call a Harshad number that, when divided by the sum of its digits, results in a prime a strong Harshad number.
Now take the number 2011 which is prime.
When we truncate the last digit from it we get 201, a strong Harshad number that is also right truncatable.
Let's call such primes strong, right truncatable Harshad primes.
You are given that the sum of the strong, right truncatable Harshad primes less than 10000 is 90619.
Find the sum of the strong, right truncatable Harshad primes less than 10^{14}.
Consider all lattice points (a,b,c) with 0 ≤ a,b,c ≤ N.
From the origin O(0,0,0) all lines are drawn to the other lattice points.
Let D(N) be the number of distinct such lines.
You are given that D(1 000 000) = 831909254469114121.
Find D(10^{10}). Give as your answer the first nine digits followed by the last nine digits.
An unbiased single 4sided die is thrown and its value, T, is noted.
T unbiased 6sided dice are thrown and their scores are added together. The sum, C, is noted.
C unbiased 8sided dice are thrown and their scores are added together. The sum, O, is noted.
O unbiased 12sided dice are thrown and their scores are added together. The sum, D, is noted.
D unbiased 20sided dice are thrown and their scores are added together. The sum, I, is noted.
Find the variance of I, and give your answer rounded to 4 decimal places.
Consider the triangle with sides $\sqrt 5$, $\sqrt {65}$ and $\sqrt {68}$. It can be shown that this triangle has area $9$.
$S(n)$ is the sum of the areas of all triangles with sides $\sqrt{1+b^2}$, $\sqrt {1+c^2}$ and $\sqrt{b^2+c^2}\,$ (for positive integers $b$ and $c$) that have an integral area not exceeding $n$.
The example triangle has $b=2$ and $c=8$.
$S(10^6)=18018206$.
Find $S(10^{10})$.
Let s_{k} be the number of 1’s when writing the numbers from 0 to k in binary.
For example, writing 0 to 5 in binary, we have 0, 1, 10, 11, 100, 101. There are seven 1’s, so s_{5} = 7.
The sequence S = {s_{k} : k ≥ 0} starts {0, 1, 2, 4, 5, 7, 9, 12, ...}.
A game is played by two players. Before the game starts, a number n is chosen. A counter c starts at 0. At each turn, the player chooses a number from 1 to n (inclusive) and increases c by that number. The resulting value of c must be a member of S. If there are no more valid moves, the player loses.
For example:
Let n = 5. c starts at 0.
Player 1 chooses 4, so c becomes 0 + 4 = 4.
Player 2 chooses 5, so c becomes 4 + 5 = 9.
Player 1 chooses 3, so c becomes 9 + 3 = 12.
etc.
Note that c must always belong to S, and each player can increase c by at most n.
Let M(n) be the highest number the first player can choose at her first turn to force a win, and M(n) = 0 if there is no such move. For example, M(2) = 2, M(7) = 1 and M(20) = 4.
Given ∑ (M(n))^{3} = 8150 for 1 ≤ n ≤ 20.
Find ∑ (M(n))^{3} for 1 ≤ n ≤ 1000.
A rectilinear grid is an orthogonal grid where the spacing between the gridlines does not have to be equidistant.
An example of such grid is logarithmic graph paper.
Consider rectilinear grids in the Cartesian coordinate system with the following properties:
 The gridlines are parallel to the axes of the Cartesian coordinate system.
 There are N+2 vertical and N+2 horizontal gridlines. Hence there are (N+1) x (N+1) rectangular cells.
 The equations of the two outer vertical gridlines are x = 1 and x = 1.
 The equations of the two outer horizontal gridlines are y = 1 and y = 1.
 The grid cells are colored red if they overlap with the unit circle, black otherwise.
E.g. here is a picture of the solution for N = 10:
The area occupied by the red cells for N = 10 rounded to 10 digits behind the decimal point is 3.3469640797.
Find the positions for N = 400.
Give as your answer the area occupied by the red cells rounded to 10 digits behind the decimal point.
An n×n grid of squares contains n^{2} ants, one ant per square.
All ants decide to move simultaneously to an adjacent square (usually 4 possibilities, except for ants on the edge of the grid or at the corners).
We define f(n) to be the number of ways this can happen without any ants ending on the same square and without any two ants crossing the same edge between two squares.
You are given that f(4) = 88.
Find f(10).
Jeff eats a pie in an unusual way.
The pie is circular. He starts with slicing an initial cut in the pie along a radius.
While there is at least a given fraction F of pie left, he performs the following procedure:
 He makes two slices from the pie centre to any point of what is remaining of the pie border, any point on the remaining pie border equally likely. This will divide the remaining pie into three pieces.
 Going counterclockwise from the initial cut, he takes the first two pie pieces and eats them.
When less than a fraction F of pie remains, he does not repeat this procedure. Instead, he eats all of the remaining pie.
For x ≥ 1, let E(x) be the expected number of times Jeff repeats the procedure above with F = ^{1}/_{x}.
It can be verified that E(1) = 1, E(2) ≈ 1.2676536759, and E(7.5) ≈ 2.1215732071.
Find E(40) rounded to 10 decimal places behind the decimal point.
The Pythagorean tree is a fractal generated by the following procedure:
Start with a unit square. Then, calling one of the sides its base (in the animation, the bottom side is the base):
 Attach a right triangle to the side opposite the base, with the hypotenuse coinciding with that side and with the sides in a 345 ratio. Note that the smaller side of the triangle must be on the 'right' side with respect to the base (see animation).
 Attach a square to each leg of the right triangle, with one of its sides coinciding with that leg.
 Repeat this procedure for both squares, considering as their bases the sides touching the triangle.
It can be shown that there exists at least one rectangle, whose sides are parallel to the largest square of the Pythagorean tree, which encloses the Pythagorean tree completely.
Find the smallest area possible for such a bounding rectangle, and give your answer rounded to 10 decimal places.
For any positive integer n, the nth weak Goodstein sequence {g_{1}, g_{2}, g_{3}, ...} is defined as:
 g_{1} = n
 for k > 1, g_{k} is obtained by writing g_{k1} in base k, interpreting it as a base k + 1 number, and subtracting 1.
For example, the 6th weak Goodstein sequence is {6, 11, 17, 25, ...}:
 g_{1} = 6.
 g_{2} = 11 since 6 = 110_{2}, 110_{3} = 12, and 12  1 = 11.
 g_{3} = 17 since 11 = 102_{3}, 102_{4} = 18, and 18  1 = 17.
 g_{4} = 25 since 17 = 101_{4}, 101_{5} = 26, and 26  1 = 25.
It can be shown that every weak Goodstein sequence terminates.
Let G(n) be the number of nonzero elements in the nth weak Goodstein sequence.
It can be verified that G(2) = 3, G(4) = 21 and G(6) = 381.
It can also be verified that ∑ G(n) = 2517 for 1 ≤ n < 8.
Find the last 9 digits of ∑ G(n) for 1 ≤ n < 16.
On the parabola y = x^{2}/k, three points A(a, a^{2}/k), B(b, b^{2}/k) and C(c, c^{2}/k) are chosen.
Let F(K, X) be the number of the integer quadruplets (k, a, b, c) such that at least one angle of the triangle ABC is 45degree, with 1 ≤ k ≤ K and X ≤ a < b < c ≤ X.
For example, F(1, 10) = 41 and F(10, 100) = 12492.
Find F(10^{6}, 10^{9}).
Inside a rope of length n, n1 points are placed with distance 1 from each other and from the endpoints. Among these points, we choose m1 points at random and cut the rope at these points to create m segments.
Let E(n, m) be the expected length of the secondshortest segment. For example, E(3, 2) = 2 and E(8, 3) = 16/7. Note that if multiple segments have the same shortest length the length of the secondshortest segment is defined as the same as the shortest length.
Find E(10^{7}, 100). Give your answer rounded to 5 decimal places behind the decimal point.
The first 15 fibonacci numbers are:
1,1,2,3,5,8,13,21,34,55,89,144,233,377,610.
It can be seen that 8 and 144 are not squarefree: 8 is divisible by 4 and 144 is divisible by 4 and by 9.
So the first 13 squarefree fibonacci numbers are:
1,1,2,3,5,13,21,34,55,89,233,377 and 610.
The 200th squarefree fibonacci number is:
971183874599339129547649988289594072811608739584170445.
The last sixteen digits of this number are: 1608739584170445 and in scientific notation this number can be written as 9.7e53.
Find the 100 000 000th squarefree fibonacci number.
Give as your answer its last sixteen digits followed by a comma followed by the number in scientific notation (rounded to one digit after the decimal point).
For the 200th squarefree number the answer would have been: 1608739584170445,9.7e53
Note:
For this problem, assume that for every prime p, the first fibonacci number divisible by p is not divisible by p^{2} (this is part of Wall's conjecture). This has been verified for primes ≤ 3·10^{15}, but has not been proven in general.
If it happens that the conjecture is false, then the accepted answer to this problem isn't guaranteed to be the 100 000 000th squarefree fibonacci number, rather it represents only a lower bound for that number.
A Fibonacci tree is a binary tree recursively defined as:
 T(0) is the empty tree.
 T(1) is the binary tree with only one node.
 T(k) consists of a root node that has T(k1) and T(k2) as children.
On such a tree two players play a takeaway game. On each turn a player selects a node and removes that node along with the subtree rooted at that node.
The player who is forced to take the root node of the entire tree loses.
Here are the winning moves of the first player on the first turn for T(k) from k=1 to k=6.
Let f(k) be the number of winning moves of the first player (i.e. the moves for which the second player has no winning strategy) on the first turn of the game when this game is played on T(k).For example, f(5) = 1 and f(10) = 17.
Find f(10000). Give the last 18 digits of your answer.
The divisors of 6 are 1,2,3 and 6.
The sum of the squares of these numbers is 1+4+9+36=50.
Let sigma2(n) represent the sum of the squares of the divisors of n. Thus sigma2(6)=50.
Let SIGMA2 represent the summatory function of sigma2, that is SIGMA2(n)=∑ sigma2(i) for i=1 to n.The first 6 values of SIGMA2 are: 1,6,16,37,63 and 113.
Find SIGMA2(10^{15}) modulo 10^{9}.
It can be shown that the polynomial n^{4} + 4n^{3} + 2n^{2} + 5n is a multiple of 6 for every integer n. It can also be shown that 6 is the largest integer satisfying this property.
Define M(a, b, c) as the maximum m such that n^{4} + an^{3} + bn^{2} + cn is a multiple of m for all integers n. For example, M(4, 2, 5) = 6.
Also, define S(N) as the sum of M(a, b, c) for all 0 < a, b, c ≤ N.
We can verify that S(10) = 1972 and S(10000) = 2024258331114.
Let F_{k} be the Fibonacci sequence:
F_{0} = 0, F_{1} = 1 and
F_{k} = F_{k1} + F_{k2} for k ≥ 2.
Find the last 9 digits of ∑ S(F_{k}) for 2 ≤ k ≤ 1234567890123.
For integers a and b, we define D(a, b) as the domain enclosed by the parabola y = x^{2} and the line y = a·x + b:
D(a, b) = { (x, y)  x^{2} ≤ y ≤ a·x + b }.
L(a, b) is defined as the number of lattice points contained in D(a, b).
For example, L(1, 2) = 8 and L(2, 1) = 1.
We also define S(N) as the sum of L(a, b) for all the pairs (a, b) such that the area of D(a, b) is a rational number and a,b ≤ N.
We can verify that S(5) = 344 and S(100) = 26709528.
Find S(10^{12}). Give your answer mod 10^{8}.
E_{a} is an ellipse with an equation of the form x^{2} + 4y^{2} = 4a^{2}.
E_{a}' is the rotated image of E_{a} by θ degrees counterclockwise around the origin O(0, 0) for 0° < θ < 90°.
b is the distance to the origin of the two intersection points closest to the origin and c is the distance of the two other intersection points.
We call an ordered triplet (a, b, c) a canonical ellipsoidal triplet if a, b and c are positive integers.
For example, (209, 247, 286) is a canonical ellipsoidal triplet.
Let C(N) be the number of distinct canonical ellipsoidal triplets (a, b, c) for a ≤ N.
It can be verified that C(10^{3}) = 7, C(10^{4}) = 106 and C(10^{6}) = 11845.
Find C(10^{17}).
We wish to tile a rectangle whose length is twice its width.
Let T(0) be the tiling consisting of a single rectangle.
For n > 0, let T(n) be obtained from T(n1) by replacing all tiles in the following manner:
The following animation demonstrates the tilings T(n) for n from 0 to 5:
Let f(n) be the number of points where four tiles meet in T(n).
For example, f(1) = 0, f(4) = 82 and f(10^{9}) mod 17^{7} = 126897180.
Find f(10^{k}) for k = 10^{18}, give your answer modulo 17^{7}.
We are trying to find a hidden number selected from the set of integers {1, 2, ..., n} by asking questions.
Each number (question) we ask, we get one of three possible answers:
 "Your guess is lower than the hidden number" (and you incur a cost of a), or
 "Your guess is higher than the hidden number" (and you incur a cost of b), or
 "Yes, that's it!" (and the game ends).
Given the value of n, a, and b, an optimal strategy minimizes the total cost for the worst possible case.
For example, if n = 5, a = 2, and b = 3, then we may begin by asking "2" as our first question.
If we are told that 2 is higher than the hidden number (for a cost of b=3), then we are sure that "1" is the hidden number (for a total cost of 3).
If we are told that 2 is lower than the hidden number (for a cost of a=2), then our next question will be "4".
If we are told that 4 is higher than the hidden number (for a cost of b=3), then we are sure that "3" is the hidden number (for a total cost of 2+3=5).
If we are told that 4 is lower than the hidden number (for a cost of a=2), then we are sure that "5" is the hidden number (for a total cost of 2+2=4).
Thus, the worstcase cost achieved by this strategy is 5. It can also be shown that this is the lowest worstcase cost that can be achieved.
So, in fact, we have just described an optimal strategy for the given values of n, a, and b.
Let $C(n, a, b)$ be the worstcase cost achieved by an optimal strategy for the given values of n, a and b.
Here are a few examples:
$C(5, 2, 3) = 5$
$C(500, \sqrt 2, \sqrt 3) = 13.22073197\dots$
$C(20000, 5, 7) = 82$
$C(2000000, \sqrt 5, \sqrt 7) = 49.63755955\dots$
Let $F_k$ be the Fibonacci numbers: $F_k=F_{k1}+F_{k2}$ with base cases $F_1=F_2= 1$.
Find $\displaystyle \sum \limits_{k = 1}^{30} {C \left (10^{12}, \sqrt{k}, \sqrt{F_k} \right )}$, and give your answer rounded to 8 decimal places behind the decimal point.
If we calculate a^{2} mod 6 for 0 ≤ a ≤ 5 we get: 0,1,4,3,4,1.
The largest value of a such that a^{2} ≡ a mod 6 is 4.
Let's call M(n) the largest value of a < n such that a^{2} ≡ a (mod n).
So M(6) = 4.
Find ∑ M(n) for 1 ≤ n ≤ 10^{7}.
Let's call a lattice point (x, y) inadmissible if x, y and x + y are all positive perfect squares.
For example, (9, 16) is inadmissible, while (0, 4), (3, 1) and (9, 4) are not.
Consider a path from point (x_{1}, y_{1}) to point (x_{2}, y_{2}) using only unit steps north or east.
Let's call such a path admissible if none of its intermediate points are inadmissible.
Let P(n) be the number of admissible paths from (0, 0) to (n, n).
It can be verified that P(5) = 252, P(16) = 596994440 and P(1000) mod 1 000 000 007 = 341920854.
Find P(10 000 000) mod 1 000 000 007.
Let n be a positive integer. Consider nim positions where:
 There are n nonempty piles.
 Each pile has size less than 2^{n}.
 No two piles have the same size.
Let W(n) be the number of winning nim positions satisfying the above conditions (a position is winning if the first player has a winning strategy). For example, W(1) = 1, W(2) = 6, W(3) = 168, W(5) = 19764360 and W(100) mod 1 000 000 007 = 384777056.
Find W(10 000 000) mod 1 000 000 007.
Let C be the circle with radius r, x^{2} + y^{2} = r^{2}. We choose two points P(a, b) and Q(a, c) so that the line passing through P and Q is tangent to C.
For example, the quadruplet (r, a, b, c) = (2, 6, 2, 7) satisfies this property.
Let F(R, X) be the number of the integer quadruplets (r, a, b, c) with this property, and with 0 < r ≤ R and 0 < a ≤ X.
We can verify that F(1, 5) = 10, F(2, 10) = 52 and F(10, 100) = 3384.
Find F(10^{8}, 10^{9}) + F(10^{9}, 10^{8}).
Let n be a positive integer. Suppose there are stations at the coordinates (x, y) = (2^{i} mod n, 3^{i} mod n) for 0 ≤ i ≤ 2n. We will consider stations with the same coordinates as the same station.
We wish to form a path from (0, 0) to (n, n) such that the x and y coordinates never decrease.
Let S(n) be the maximum number of stations such a path can pass through.
For example, if n = 22, there are 11 distinct stations, and a valid path can pass through at most 5 stations. Therefore, S(22) = 5. The case is illustrated below, with an example of an optimal path:
It can also be verified that S(123) = 14 and S(10000) = 48.
Find ∑ S(k^{5}) for 1 ≤ k ≤ 30.
For integers m, n (0 ≤ n < m), let L(m, n) be an m×m grid with the topright n×n grid removed.
For example, L(5, 3) looks like this:
We want to number each cell of L(m, n) with consecutive integers 1, 2, 3, ... such that the number in every cell is smaller than the number below it and to the left of it.
For example, here are two valid numberings of L(5, 3):
Let LC(m, n) be the number of valid numberings of L(m, n).
It can be verified that LC(3, 0) = 42, LC(5, 3) = 250250, LC(6, 3) = 406029023400 and LC(10, 5) mod 76543217 = 61251715.
Find LC(10000, 5000) mod 76543217.
We say that a ddigit positive number (no leading zeros) is a onechild number if exactly one of its substrings is divisible by d.
For example, 5671 is a 4digit onechild number. Among all its substrings 5, 6, 7, 1, 56, 67, 71, 567, 671 and 5671, only 56 is divisible by 4.
Similarly, 104 is a 3digit onechild number because only 0 is divisible by 3.
1132451 is a 7digit onechild number because only 245 is divisible by 7.
Let F(N) be the number of the onechild numbers less than N.
We can verify that F(10) = 9, F(10^{3}) = 389 and F(10^{7}) = 277674.
Find F(10^{19}).
6174 is a remarkable number; if we sort its digits in increasing order and subtract that number from the number you get when you sort the digits in decreasing order, we get 76411467=6174.
Even more remarkable is that if we start from any 4 digit number and repeat this process of sorting and subtracting, we'll eventually end up with 6174 or immediately with 0 if all digits are equal.
This also works with numbers that have less than 4 digits if we pad the number with leading zeroes until we have 4 digits.
E.g. let's start with the number 0837:
87300378=8352
85322358=6174
6174 is called the Kaprekar constant. The process of sorting and subtracting and repeating this until either 0 or the Kaprekar constant is reached is called the Kaprekar routine.
We can consider the Kaprekar routine for other bases and number of digits.
Unfortunately, it is not guaranteed a Kaprekar constant exists in all cases; either the routine can end up in a cycle for some input numbers or the constant the routine arrives at can be different for different input numbers.
However, it can be shown that for 5 digits and a base b = 6t+3≠9, a Kaprekar constant exists.
E.g. base 15: (10,4,14,9,5)_{15}
base 21: (14,6,20,13,7)_{21}
Define C_{b} to be the Kaprekar constant in base b for 5 digits. Define the function sb(i) to be
 0 if i = C_{b} or if i written in base b consists of 5 identical digits
 the number of iterations it takes the Kaprekar routine in base b to arrive at C_{b}, otherwise
Define S(b) as the sum of sb(i) for 0 < i < b^{5}.
E.g. S(15) = 5274369
S(111) = 400668930299
Find the sum of S(6k+3) for 2 ≤ k ≤ 300.
Give the last 18 digits as your answer.
A set of lattice points S is called a titanic set if there exists a line passing through exactly two points in S.
An example of a titanic set is S = {(0, 0), (0, 1), (0, 2), (1, 1), (2, 0), (1, 0)}, where the line passing through (0, 1) and (2, 0) does not pass through any other point in S.
On the other hand, the set {(0, 0), (1, 1), (2, 2), (4, 4)} is not a titanic set since the line passing through any two points in the set also passes through the other two.
For any positive integer N, let T(N) be the number of titanic sets S whose every point (x, y) satisfies 0 ≤ x, y ≤ N. It can be verified that T(1) = 11, T(2) = 494, T(4) = 33554178, T(111) mod 10^{8} = 13500401 and T(10^{5}) mod 10^{8} = 63259062.
Find T(10^{11}) mod 10^{8}.
A row of n squares contains a frog in the leftmost square. By successive jumps the frog goes to the rightmost square and then back to the leftmost square. On the outward trip he jumps one, two or three squares to the right, and on the homeward trip he jumps to the left in a similar manner. He cannot jump outside the squares. He repeats the roundtrip travel m times.
Let F(m, n) be the number of the ways the frog can travel so that at most one square remains unvisited.
For example, F(1, 3) = 4, F(1, 4) = 15, F(1, 5) = 46, F(2, 3) = 16 and F(2, 100) mod 10^{9} = 429619151.
Find the last 9 digits of F(10, 10^{12}).
A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:
^{1}/_{2} = 0.5 ^{1}/_{3} = 0.(3) ^{1}/_{4} = 0.25 ^{1}/_{5} = 0.2 ^{1}/_{6} = 0.1(6) ^{1}/_{7} = 0.(142857) ^{1}/_{8} = 0.125 ^{1}/_{9} = 0.(1) ^{1}/_{10} = 0.1
Where 0.1(6) means 0.166666..., and has a 1digit recurring cycle. It can be seen that ^{1}/_{7} has a 6digit recurring cycle.
Unit fractions whose denominator has no other prime factors than 2 and/or 5 are not considered to have a recurring cycle.
We define the length of the recurring cycle of those unit fractions as 0.
Let L(n) denote the length of the recurring cycle of 1/n. You are given that ∑ L(n) for 3 ≤ n ≤ 1 000 000 equals 55535191115.
Find ∑ L(n) for 3 ≤ n ≤ 100 000 000.
Let n be a positive integer. An integer triple (a, b, c) is called a factorisation triple of n if:
 1 ≤ a ≤ b ≤ c
 a·b·c = n.
Define f(n) to be a + b + c for the factorisation triple (a, b, c) of n which minimises c / a. One can show that this triple is unique.
For example, f(165) = 19, f(100100) = 142 and f(20!) = 4034872.
Find f(43!).
The look and say sequence goes 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...
The sequence starts with 1 and all other members are obtained by describing the previous member in terms of consecutive digits.
It helps to do this out loud:
1 is 'one one' → 11
11 is 'two ones' → 21
21 is 'one two and one one' → 1211
1211 is 'one one, one two and two ones' → 111221
111221 is 'three ones, two twos and one one' → 312211
...
Define A(n), B(n) and C(n) as the number of ones, twos and threes in the n'th element of the sequence respectively.
One can verify that A(40) = 31254, B(40) = 20259 and C(40) = 11625.
Find A(n), B(n) and C(n) for n = 10^{12}.
Give your answer modulo 2^{30} and separate your values for A, B and C by a comma.
E.g. for n = 40 the answer would be 31254,20259,11625
A positive integer matrix is a matrix whose elements are all positive integers.
Some positive integer matrices can be expressed as a square of a positive integer matrix in two different ways. Here is an example:
We define F(N) as the number of the 2x2 positive integer matrices which have a trace less than N and which can be expressed as a square of a positive integer matrix in two different ways.
We can verify that F(50) = 7 and F(1000) = 1019.
Find F(10^{7}).
Numbers of the form n^{15}+1 are composite for every integer n > 1.
For positive integers n and m let s(n,m) be defined as the sum of the distinct prime factors of n^{15}+1 not exceeding m.
So s(2,10) = 3 and s(2,1000) = 3+11+331 = 345.
Also 10^{15}+1 = 7×11×13×211×241×2161×9091.
So s(10,100) = 31 and s(10,1000) = 483.
Find ∑ s(n,10^{8}) for 1 ≤ n ≤ 10^{11}.
Let H be the hyperbola defined by the equation 12x^{2} + 7xy  12y^{2} = 625.
Next, define X as the point (7, 1). It can be seen that X is in H.
Now we define a sequence of points in H, {P_{i} : i ≥ 1}, as:
 P_{1} = (13, 61/4).
 P_{2} = (43/6, 4).
 For i > 2, P_{i} is the unique point in H that is different from P_{i1} and such that line P_{i}P_{i1} is parallel to line P_{i2}X. It can be shown that P_{i} is welldefined, and that its coordinates are always rational.
You are given that P_{3} = (19/2, 229/24), P_{4} = (1267/144, 37/12) and P_{7} = (17194218091/143327232, 274748766781/1719926784).
Find P_{n} for n = 11^{14} in the following format:
If P_{n} = (a/b, c/d) where the fractions are in lowest terms and the denominators are positive, then the answer is (a + b + c + d) mod 1 000 000 007.
For n = 7, the answer would have been: 806236837.
Let n be a positive integer.
A 6sided die is thrown n times. Let c be the number of pairs of consecutive throws that give the same value.
For example, if n = 7 and the values of the die throws are (1,1,5,6,6,6,3), then the following pairs of consecutive throws give the same value:
(1,1,5,6,6,6,3)
(1,1,5,6,6,6,3)
(1,1,5,6,6,6,3)
Therefore, c = 3 for (1,1,5,6,6,6,3).
Define C(n) as the number of outcomes of throwing a 6sided die n times such that c does not exceed π(n).^{1}
For example, C(3) = 216, C(4) = 1290, C(11) = 361912500 and C(24) = 4727547363281250000.
Define S(L) as ∑ C(n) for 1 ≤ n ≤ L.
For example, S(50) mod 1 000 000 007 = 832833871.
Find S(50 000 000) mod 1 000 000 007.
^{1} π denotes the primecounting function, i.e. π(n) is the number of primes ≤ n.
The above is an example of a cryptic kakuro (also known as cross sums, or even sums cross) puzzle, with its final solution on the right. (The common rules of kakuro puzzles can be found easily on numerous internet sites. Other related information can also be currently found at krazydad.com whose author has provided the puzzle data for this challenge.)
The downloadable text file (kakuro200.txt) contains the description of 200 such puzzles, a mix of 5x5 and 6x6 types. The first puzzle in the file is the above example which is coded as follows:
6,X,X,(vCC),(vI),X,X,X,(hH),B,O,(vCA),(vJE),X,(hFE,vD),O,O,O,O,(hA),O,I,(hJC,vB),O,O,(hJC),H,O,O,O,X,X,X,(hJE),O,O,X
The first character is a numerical digit indicating the size of the information grid. It would be either a 6 (for a 5x5 kakuro puzzle) or a 7 (for a 6x6 puzzle) followed by a comma (,). The extra top line and left column are needed to insert information.
The content of each cell is then described and followed by a comma, going left to right and starting with the top line.
X = Gray cell, not required to be filled by a digit.
O (upper case letter)= White empty cell to be filled by a digit.
A = Or any one of the upper case letters from A to J to be replaced by its equivalent digit in the solved puzzle.
( ) = Location of the encrypted sums. Horizontal sums are preceded by a lower case "h" and vertical sums are preceded by a lower case "v". Those are followed by one or two upper case letters depending if the sum is a single digit or double digit one. For double digit sums, the first letter would be for the "tens" and the second one for the "units". When the cell must contain information for both a horizontal and a vertical sum, the first one is always for the horizontal sum and the two are separated by a comma within the same set of brackets, ex.: (hFE,vD). Each set of brackets is also immediately followed by a comma.
The description of the last cell is followed by a Carriage Return/Line Feed (CRLF) instead of a comma.
The required answer to each puzzle is based on the value of each letter necessary to arrive at the solution and according to the alphabetical order. As indicated under the example puzzle, its answer would be 8426039571. At least 9 out of the 10 encrypting letters are always part of the problem description. When only 9 are given, the missing one must be assigned the remaining digit.
You are given that the sum of the answers for the first 10 puzzles in the file is 64414157580.
Find the sum of the answers for the 200 puzzles.
Two positive numbers A and B are said to be connected (denoted by "A ↔ B") if one of these conditions holds:
(1) A and B have the same length and differ in exactly one digit; for example, 123 ↔ 173.
(2) Adding one digit to the left of A (or B) makes B (or A); for example, 23 ↔ 223 and 123 ↔ 23.
We call a prime P a 2's relative if there exists a chain of connected primes between 2 and P and no prime in the chain exceeds P.
For example, 127 is a 2's relative. One of the possible chains is shown below:
2 ↔ 3 ↔ 13 ↔ 113 ↔ 103 ↔ 107 ↔ 127
However, 11 and 103 are not 2's relatives.
Let F(N) be the sum of the primes ≤ N which are not 2's relatives.
We can verify that F(10^{3}) = 431 and F(10^{4}) = 78728.
Find F(10^{7}).
Consider an infinite row of boxes. Some of the boxes contain a ball. For example, an initial configuration of 2 consecutive occupied boxes followed by 2 empty boxes, 2 occupied boxes, 1 empty box, and 2 occupied boxes can be denoted by the sequence (2, 2, 2, 1, 2), in which the number of consecutive occupied and empty boxes appear alternately.
A turn consists of moving each ball exactly once according to the following rule: Transfer the leftmost ball which has not been moved to the nearest empty box to its right.
After one turn the sequence (2, 2, 2, 1, 2) becomes (2, 2, 1, 2, 3) as can be seen below; note that we begin the new sequence starting at the first occupied box.
A system like this is called a BoxBall System or BBS for short.
It can be shown that after a sufficient number of turns, the system evolves to a state where the consecutive numbers of occupied boxes is invariant. In the example below, the consecutive numbers of occupied boxes evolves to [1, 2, 3]; we shall call this the final state.
We define the sequence {t_{i}}:
 s_{0} = 290797
 s_{k+1} = s_{k}^{2} mod 50515093
 t_{k} = (s_{k} mod 64) + 1
Starting from the initial configuration (t_{0}, t_{1}, …, t_{10}), the final state becomes [1, 3, 10, 24, 51, 75].
Starting from the initial configuration (t_{0}, t_{1}, …, t_{10 000 000}), find the final state.
Give as your answer the sum of the squares of the elements of the final state. For example, if the final state is [1, 2, 3] then 14 ( = 1^{2} + 2^{2} + 3^{2}) is your answer.
A sequence of integers S = {s_{i}} is called an nsequence if it has n elements and each element s_{i} satisfies 1 ≤ s_{i} ≤ n. Thus there are n^{n} distinct nsequences in total. For example, the sequence S = {1, 5, 5, 10, 7, 7, 7, 2, 3, 7} is a 10sequence.
For any sequence S, let L(S) be the length of the longest contiguous subsequence of S with the same value. For example, for the given sequence S above, L(S) = 3, because of the three consecutive 7's.
Let f(n) = ∑ L(S) for all nsequences S.
For example, f(3) = 45, f(7) = 1403689 and f(11) = 481496895121.
Find f(7 500 000) mod 1 000 000 009.
Let a, b and c be positive numbers.
Let W, X, Y, Z be four collinear points where WX = a, XY = b, YZ = c and WZ = a + b + c.
Let C_{in} be the circle having the diameter XY.
Let C_{out} be the circle having the diameter WZ.
The triplet (a, b, c) is called a necklace triplet if you can place k ≥ 3 distinct circles C_{1}, C_{2}, ..., C_{k} such that:
 C_{i} has no common interior points with any C_{j} for 1 ≤ i, j ≤ k and i ≠ j,
 C_{i} is tangent to both C_{in} and C_{out} for 1 ≤ i ≤ k,
 C_{i} is tangent to C_{i+1} for 1 ≤ i < k, and
 C_{k} is tangent to C_{1}.
For example, (5, 5, 5) and (4, 3, 21) are necklace triplets, while it can be shown that (2, 2, 5) is not.
Let T(n) be the number of necklace triplets (a, b, c) such that a, b and c are positive integers, and b ≤ n. For example, T(1) = 9, T(20) = 732 and T(3000) = 438106.
Find T(1 000 000 000).
A unitary divisor d of a number n is a divisor of n that has the property gcd(d, n/d) = 1.
The unitary divisors of 4! = 24 are 1, 3, 8 and 24.
The sum of their squares is 1^{2} + 3^{2} + 8^{2} + 24^{2} = 650.
Let S(n) represent the sum of the squares of the unitary divisors of n. Thus S(4!)=650.
Find S(100 000 000!) modulo 1 000 000 009.
N disks are placed in a row, indexed 1 to N from left to right.
Each disk has a black side and white side. Initially all disks show their white side.
At each turn, two, not necessarily distinct, integers A and B between 1 and N (inclusive) are chosen uniformly at random.
All disks with an index from A to B (inclusive) are flipped.
The following example shows the case N = 8. At the first turn A = 5 and B = 2, and at the second turn A = 4 and B = 6.
Let E(N, M) be the expected number of disks that show their white side after M turns.
We can verify that E(3, 1) = 10/9, E(3, 2) = 5/3, E(10, 4) ≈ 5.157 and E(100, 10) ≈ 51.893.
Find E(10^{10}, 4000).
Give your answer rounded to 2 decimal places behind the decimal point.
Fred the farmer arranges to have a new storage silo installed on his farm and having an obsession for all things square he is absolutely devastated when he discovers that it is circular. Quentin, the representative from the company that installed the silo, explains that they only manufacture cylindrical silos, but he points out that it is resting on a square base. Fred is not amused and insists that it is removed from his property.
Quick thinking Quentin explains that when granular materials are delivered from above a conical slope is formed and the natural angle made with the horizontal is called the angle of repose. For example if the angle of repose, $\alpha = 30$ degrees, and grain is delivered at the centre of the silo then a perfect cone will form towards the top of the cylinder. In the case of this silo, which has a diameter of 6m, the amount of space wasted would be approximately 32.648388556 m^{3}. However, if grain is delivered at a point on the top which has a horizontal distance of $x$ metres from the centre then a cone with a strangely curved and sloping base is formed. He shows Fred a picture.
We shall let the amount of space wasted in cubic metres be given by $V(x)$. If $x = 1.114785284$, which happens to have three squared decimal places, then the amount of space wasted, $V(1.114785284) \approx 36$. Given the range of possible solutions to this problem there is exactly one other option: $V(2.511167869) \approx 49$. It would be like knowing that the square is king of the silo, sitting in splendid glory on top of your grain.
Fred's eyes light up with delight at this elegant resolution, but on closer inspection of Quentin's drawings and calculations his happiness turns to despondency once more. Fred points out to Quentin that it's the radius of the silo that is 6 metres, not the diameter, and the angle of repose for his grain is 40 degrees. However, if Quentin can find a set of solutions for this particular silo then he will be more than happy to keep it.
If Quick thinking Quentin is to satisfy frustratingly fussy Fred the farmer's appetite for all things square then determine the values of $x$ for all possible square space wastage options and calculate $\sum x$ correct to 9 decimal places.
Let S(n,m) = ∑φ(n × i) for 1 ≤ i ≤ m. (φ is Euler's totient function)
You are given that S(510510,10^{6} )= 45480596821125120.
Find S(510510,10^{11}).
Give the last 9 digits of your answer.
Let E(x_{0}, y_{0}) be the number of steps it takes to determine the greatest common divisor of x_{0} and y_{0} with Euclid's algorithm. More formally:
x_{1} = y_{0}, y_{1} = x_{0} mod y_{0}
x_{n} = y_{n1}, y_{n} = x_{n1} mod y_{n1}
E(x_{0}, y_{0}) is the smallest n such that y_{n} = 0.
We have E(1,1) = 1, E(10,6) = 3 and E(6,10) = 4.
Define S(N) as the sum of E(x,y) for 1 ≤ x,y ≤ N.
We have S(1) = 1, S(10) = 221 and S(100) = 39826.
Find S(5·10^{6}).
Recall that a graph is a collection of vertices and edges connecting the vertices, and that two vertices connected by an edge are called adjacent.
Graphs can be embedded in Euclidean space by associating each vertex with a point in the Euclidean space.
A flexible graph is an embedding of a graph where it is possible to move one or more vertices continuously so that the distance between at least two nonadjacent vertices is altered while the distances between each pair of adjacent vertices is kept constant.
A rigid graph is an embedding of a graph which is not flexible.
Informally, a graph is rigid if by replacing the vertices with fully rotating hinges and the edges with rods that are unbending and inelastic, no parts of the graph can be moved independently from the rest of the graph.
The grid graphs embedded in the Euclidean plane are not rigid, as the following animation demonstrates:
However, one can make them rigid by adding diagonal edges to the cells. For example, for the 2x3 grid graph, there are 19 ways to make the graph rigid:
Note that for the purposes of this problem, we do not consider changing the orientation of a diagonal edge or adding both diagonal edges to a cell as a different way of making a grid graph rigid.
Let R(m,n) be the number of ways to make the m × n grid graph rigid.
E.g. R(2,3) = 19 and R(5,5) = 23679901
Define S(N) as ∑ R(i,j) for 1 ≤ i, j ≤ N.
E.g. S(5) = 25021721.
Find S(100), give your answer modulo 1000000033.
The Fibonacci numbers $\{f_n, n \ge 0\}$ are defined recursively as $f_n = f_{n1} + f_{n2}$ with base cases $f_0 = 0$ and $f_1 = 1$.
Define the polynomials $\{F_n, n \ge 0\}$ as $F_n(x) = \displaystyle{\sum_{i=0}^n f_i x^i}$.
For example, $F_7(x) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + 13x^7$, and $F_7(11) = 268\,357\,683$.
Let $n = 10^{15}$. Find the sum $\displaystyle{\sum_{x=0}^{100} F_n(x)}$ and give your answer modulo $1\,307\,674\,368\,000 \ (= 15!)$.
Julie proposes the following wager to her sister Louise.
She suggests they play a game of chance to determine who will wash the dishes.
For this game, they shall use a generator of independent random numbers uniformly distributed between 0 and 1.
The game starts with S = 0.
The first player, Louise, adds to S different random numbers from the generator until S > 1 and records her last random number 'x'.
The second player, Julie, continues adding to S different random numbers from the generator until S > 2 and records her last random number 'y'.
The player with the highest number wins and the loser washes the dishes, i.e. if y > x the second player wins.
For example, if the first player draws 0.62 and 0.44, the first player turn ends since 0.62+0.44 > 1 and x = 0.44.
If the second players draws 0.1, 0.27 and 0.91, the second player turn ends since 0.62+0.44+0.1+0.27+0.91 > 2 and y = 0.91.
Since y > x, the second player wins.
Louise thinks about it for a second, and objects: "That's not fair".
What is the probability that the second player wins?
Give your answer rounded to 10 places behind the decimal point in the form 0.abcdefghij
When we calculate 8^{n} modulo 11 for n=0 to 9 we get: 1, 8, 9, 6, 4, 10, 3, 2, 5, 7.
As we see all possible values from 1 to 10 occur. So 8 is a primitive root of 11.
But there is more:
If we take a closer look we see:
1+8=9
8+9=17≡6 mod 11
9+6=15≡4 mod 11
6+4=10
4+10=14≡3 mod 11
10+3=13≡2 mod 11
3+2=5
2+5=7
5+7=12≡1 mod 11.
8 is called a Fibonacci primitive root of 11.
Not every prime has a Fibonacci primitive root.
There are 323 primes less than 10000 with one or more Fibonacci primitive roots and the sum of these primes is 1480491.
Find the sum of the primes less than 100,000,000 with at least one Fibonacci primitive root.
For an ntuple of integers t = (a_{1}, ..., a_{n}), let (x_{1}, ..., x_{n}) be the solutions of the polynomial equation x^{n} + a_{1}x^{n1} + a_{2}x^{n2} + ... + a_{n1}x + a_{n} = 0.
Consider the following two conditions:
 x_{1}, ..., x_{n} are all real.
 If x_{1}, ..., x_{n} are sorted, ⌊x_{i}⌋ = i for 1 ≤ i ≤ n. (⌊·⌋: floor function.)
In the case of n = 4, there are 12 ntuples of integers which satisfy both conditions.
We define S(t) as the sum of the absolute values of the integers in t.
For n = 4 we can verify that ∑ S(t) = 2087 for all ntuples t which satisfy both conditions.
Find ∑ S(t) for n = 7.
Let d(k) be the sum of all divisors of k.
We define the function S(N) = $\sum_{i=1}^N \sum_{j=1}^Nd(i \cdot j)$.
For example, S(3) = d(1) + d(2) + d(3) + d(2) + d(4) + d(6) + d(3) + d(6) + d(9) = 59.
You are given that S(10^{3}) = 563576517282 and S(10^{5}) mod 10^{9} = 215766508.
Find S(10^{11}) mod 10^{9}.
We want to tile a board of length n and height 1 completely, with either 1 × 2 blocks or 1 × 1 blocks with a single decimal digit on top:
For example, here are some of the ways to tile a board of length n = 8:
Let T(n) be the number of ways to tile a board of length n as described above.
For example, T(1) = 10 and T(2) = 101.
Let S(L) be the triple sum ∑_{a,b,c} gcd(T(c^{a}), T(c^{b})) for 1 ≤ a, b, c ≤ L.
For example:
S(2) = 10444
S(3) = 1292115238446807016106539989
S(4) mod 987 898 789 = 670616280.
Find S(2000) mod 987 898 789.
For an integer M, we define R(M) as the sum of 1/(p·q) for all the integer pairs p and q which satisfy all of these conditions:
 1 ≤ p < q ≤ M
 p + q ≥ M
 p and q are coprime.
We also define S(N) as the sum of R(i) for 2 ≤ i ≤ N.
We can verify that S(2) = R(2) = 1/2, S(10) ≈ 6.9147 and S(100) ≈ 58.2962.
Find S(10^{7}). Give your answer rounded to four decimal places.
An integer is called elevenfree if its decimal expansion does not contain any substring representing a power of 11 except 1.
For example, 2404 and 13431 are elevenfree, while 911 and 4121331 are not.
Let E(n) be the nth positive elevenfree integer. For example, E(3) = 3, E(200) = 213 and E(500 000) = 531563.
Find E(10^{18}).
Let g(n) be a sequence defined as follows:
g(4) = 13,
g(n) = g(n1) + gcd(n, g(n1)) for n > 4.
The first few values are:
n  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  ... 
g(n)  13  14  16  17  18  27  28  29  30  31  32  33  34  51  54  55  60  ... 
You are given that g(1 000) = 2524 and g(1 000 000) = 2624152.
Find g(10^{15}).
A group of p people decide to sit down at a round table and play a lotteryticket trading game. Each person starts off with a randomlyassigned, unscratched lottery ticket. Each ticket, when scratched, reveals a wholepound prize ranging anywhere from £1 to £p, with no two tickets alike. The goal of the game is for each person to maximize his ticket winnings upon leaving the game.
An arbitrary person is chosen to be the first player. Going around the table, each player has only one of two options:
1. The player can scratch his ticket and reveal its worth to everyone at the table.
2. The player can trade his unscratched ticket for a previous player's scratched ticket, and then leave the game with that ticket. The previous player then scratches his newlyacquired ticket and reveals its worth to everyone at the table.
The game ends once all tickets have been scratched. All players still remaining at the table must leave with their currentlyheld tickets.
Assume that each player uses the optimal strategy for maximizing the expected value of his ticket winnings.
Let E(p) represent the expected number of players left at the table when the game ends in a game consisting of p players (e.g. E(111) = 5.2912 when rounded to 5 significant digits).
Let $S_1(N) = \sum \limits_{p = 1}^{N} {E(p)}$.
Let $S_k(N) = \sum \limits_{p = 1}^{N} {S_{k1}(p)}$ for $k \gt 1$.
Find S_{20}(10^{14}) and write the answer in scientific notation rounded to 10 significant digits. Use a lowercase e to separate mantissa and exponent (e.g. S_{3}(100) = 5.983679014e5).
For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$.
We will call $f_{n,a,b}$ a retraction if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n \,\,\,$ for every $0 \le x < n$.
Let $R(n)$ be the number of retractions for $n$.
You are given that
$\displaystyle \sum_{k=1}^{99\,999} R(\binom {100\,000} k) \equiv 628701600 \mod 1\,000\,000\,007$.
Find $\displaystyle \sum_{k=1}^{9\,999\,999} R(\binom {10\,000\,000} k)$.
Give your answer modulo $1\,000\,000\,007$.
For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$.
We will call $f_{n,a,b}$ a retraction if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n \,\,\,$ for every $0 \le x < n$.
Let $R(n)$ be the number of retractions for $n$.
$\displaystyle F(N)=\sum_{n=1}^NR(n^4+4)$.
$F(1024)=77532377300600$.
Find $F(10^7)$.
Give your answer modulo $1\,000\,000\,007$.
For every integer $n>1$, the family of functions $f_{n,a,b}$ is defined
by
$f_{n,a,b}(x)\equiv a x + b \mod n\,\,\, $ for $a,b,x$ integer and $0< a <n, 0 \le b < n,0 \le x < n$.
We will call $f_{n,a,b}$ a retraction if $\,\,\, f_{n,a,b}(f_{n,a,b}(x)) \equiv f_{n,a,b}(x) \mod n \,\,\,$ for every $0 \le x < n$.
Let $R(n)$ be the number of retractions for $n$.
$\displaystyle F(N)=\sum_{n=2}^N R(n)$.
$F(10^7)\equiv 638042271 \mod 1\,000\,000\,007$.
Find $F(10^{14})$.
Give your answer modulo $1\,000\,000\,007$.
The function lcm(a,b) denotes the least common multiple of a and b.
Let A(n) be the average of the values of lcm(n,i) for 1≤i≤n.
E.g: A(2)=(2+2)/2=2 and A(10)=(10+10+30+20+10+30+70+40+90+10)/10=32.
S(100)=122726.
Find S(99999999019) mod 999999017.
Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation: b^{2}x^{2} + b^{2}y^{2} + a^{2}z^{2} = a^{2}b^{2}.
Phil wants to know how much chocolate is needed to cover one candy centre with a uniform coat of chocolate one millimeter thick.
If a=1 mm and b=1 mm, the amount of chocolate required is 

π mm^{3} 
Find the amount of chocolate in mm^{3} required if a=3 mm and b=1 mm. Give your answer as the number rounded to 8 decimal places behind the decimal point.
A hypocycloid is the curve drawn by a point on a small circle rolling inside a larger circle. The parametric equations of a hypocycloid centered at the origin, and starting at the right most point is given by:
$x(t) = (R  r) \cos(t) + r \cos(\frac {R  r} r t)$
$y(t) = (R  r) \sin(t)  r \sin(\frac {R  r} r t)$
Where R is the radius of the large circle and r the radius of the small circle.
Let $C(R, r)$ be the set of distinct points with integer coordinates on the hypocycloid with radius R and r and for which there is a corresponding value of t such that $\sin(t)$ and $\cos(t)$ are rational numbers.
Let $S(R, r) = \sum_{(x,y) \in C(R, r)} x + y$ be the sum of the absolute values of the x and y coordinates of the points in $C(R, r)$.
Let $T(N) = \sum_{R = 3}^N \sum_{r=1}^{\lfloor \frac {R  1} 2 \rfloor} S(R, r)$ be the sum of $S(R, r)$ for R and r positive integers, $R\leq N$ and $2r < R$.
C(3, 1) = 
{(3, 0), (1, 2), (1,0), (1,2)} 
C(2500, 1000) = 
{(2500, 0), (772, 2376), (772, 2376), (516, 1792),
(516, 1792), (500, 0), (68, 504), (68, 504), (1356, 1088), (1356, 1088), (1500, 1000), (1500, 1000)} 
Note: (625, 0) is not an element of C(2500, 1000) because $\sin(t)$ is not a rational number for the corresponding values of t.
S(3, 1) = (3 + 0) + (1 + 2) + (1 + 0) + (1 + 2) = 10
T(3) = 10; T(10) = 524 ;T(100) = 580442; T(10^{3}) = 583108600.
Find T(10^{6}).
Consider the number 15.
There are eight positive numbers less than 15 which are coprime to 15: 1, 2, 4, 7, 8, 11, 13, 14.
The modular inverses of these numbers modulo 15 are: 1, 8, 4, 13, 2, 11, 7, 14
because
1 · 1 mod 15=1
2 · 8=16 mod 15=1
4 · 4=16 mod 15=1
7 · 13=91 mod 15=1
11 · 11=121 mod 15=1
14 · 14=196 mod 15=1
Let I(n) be the largest positive number m smaller than n1 such that the modular inverse of m modulo n equals m itself.
So I(15)=11.
Also I(100)=51 and I(7)=1.
Find ∑ I(n) for 3≤n≤2×10^{7}
Define F(m,n) as the number of ntuples of positive integers for which the product of the elements doesn't exceed m.
F(10, 10) = 571.
F(10^{6}, 10^{6}) mod 1 234 567 891 = 252903833.
Find F(10^{9}, 10^{9}) mod 1 234 567 891.
A simple quadrilateral is a polygon that has four distinct vertices, has no straight angles and does not selfintersect.
Let Q(m, n) be the number of simple quadrilaterals whose vertices are lattice points with coordinates (x,y) satisfying 0 ≤ x ≤ m and 0 ≤ y ≤ n.
For example, Q(2, 2) = 94 as can be seen below:
It can also be verified that Q(3, 7) = 39590, Q(12, 3) = 309000 and Q(123, 45) = 70542215894646.
Find Q(12345, 6789) mod 135707531.
In the following equation x, y, and n are positive integers.
$$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$$For a limit L we define F(L) as the number of solutions which satisfy x < y ≤ L.
We can verify that F(15) = 4 and F(1000) = 1069.
Find F(10^{12}).
Let f(n) be the largest positive integer x less than 10^{9} such that the last 9 digits of n^{x} form the number x (including leading zeros), or zero if no such integer exists.
For example:
 f(4) = 411728896 (4^{411728896} = ...490411728896)
 f(10) = 0
 f(157) = 743757 (157^{743757} = ...567000743757)
 ∑ f(n), 2 ≤ n ≤ 10^{3} = 442530011399
Find ∑ f(n), 2 ≤ n ≤ 10^{6}.
Define:
x_{n} = (1248^{n} mod 32323)  16161
y_{n} = (8421^{n} mod 30103)  15051
P_{n} = {(x_{1}, y_{1}), (x_{2}, y_{2}), ..., (x_{n}, y_{n})}
For example, P_{8} = {(14913, 6630), (10161, 5625), (5226, 11896), (8340, 10778), (15852, 5203), (15165, 11295), (1427, 14495), (12407, 1060)}.
Let C(n) be the number of triangles whose vertices are in P_{n} which contain the origin in the interior.
Examples:
C(8) = 20
C(600) = 8950634
C(40 000) = 2666610948988
Find C(2 000 000).
Let f(n) = n^{2}  3n  1.
Let p be a prime.
Let R(p) be the smallest positive integer n such that f(n) mod p^{2} = 0 if such an integer n exists, otherwise R(p) = 0.
Let SR(L) be ∑ R(p) for all primes not exceeding L.
Find SR(10^{7}).
Consider the alphabet A made out of the letters of the word "project": A={c,e,j,o,p,r,t}.
Let T(n) be the number of strings of length n consisting of letters from A that do not have a substring that is one of the 5040 permutations of "project".
Find T(10^{12}). Give the last 9 digits of your answer.
The flipping game is a two player game played on a N by N square board.
Each square contains a disk with one side white and one side black.
The game starts with all disks showing their white side.
A turn consists of flipping all disks in a rectangle with the following properties:
 the upper right corner of the rectangle contains a white disk
 the rectangle width is a perfect square (1, 4, 9, 16, ...)
 the rectangle height is a triangular number (1, 3, 6, 10, ...)
Players alternate turns. A player wins by turning the grid all black.
Let W(N) be the number of winning moves for the first player on a N by N board with all disks white, assuming perfect play.
W(1) = 1, W(2) = 0, W(5) = 8 and W(10^{2}) = 31395.
For N=5, the first player's eight winning first moves are:
Find W(10^{6}).
On the Euclidean plane, an ant travels from point A(0, 1) to point B(d, 1) for an integer d.
In each step, the ant at point (x_{0}, y_{0}) chooses one of the lattice points (x_{1}, y_{1}) which satisfy x_{1} ≥ 0 and y_{1} ≥ 1 and goes straight to (x_{1}, y_{1}) at a constant velocity v. The value of v depends on y_{0} and y_{1} as follows:
 If y_{0} = y_{1}, the value of v equals y_{0}.
 If y_{0} ≠ y_{1}, the value of v equals (y_{1}  y_{0}) / (ln(y_{1})  ln(y_{0})).
The left image is one of the possible paths for d = 4. First the ant goes from A(0, 1) to P_{1}(1, 3) at velocity (3  1) / (ln(3)  ln(1)) ≈ 1.8205. Then the required time is sqrt(5) / 1.8205 ≈ 1.2283.
From P_{1}(1, 3) to P_{2}(3, 3) the ant travels at velocity 3 so the required time is 2 / 3 ≈ 0.6667. From P_{2}(3, 3) to B(4, 1) the ant travels at velocity (1  3) / (ln(1)  ln(3)) ≈ 1.8205 so the required time is sqrt(5) / 1.8205 ≈ 1.2283.
Thus the total required time is 1.2283 + 0.6667 + 1.2283 = 3.1233.
The right image is another path. The total required time is calculated as 0.98026 + 1 + 0.98026 = 2.96052. It can be shown that this is the quickest path for d = 4.
Let F(d) be the total required time if the ant chooses the quickest path. For example, F(4) ≈ 2.960516287.
We can verify that F(10) ≈ 4.668187834 and F(100) ≈ 9.217221972.
Find F(10000). Give your answer rounded to nine decimal places.
Let f_{n}(k) = e^{k/n}  1, for all nonnegative integers k.
Remarkably, f_{200}(6) + f_{200}(75) + f_{200}(89) + f_{200}(226) = 3.141592644529… ≈ π.
In fact, it is the best approximation of π of the form f_{n}(a) + f_{n}(b) + f_{n}(c) + f_{n}(d) for n = 200.
Let g(n) = a^{2} + b^{2} + c^{2} + d^{ 2} for a, b, c, d that minimize the error:  f_{n}(a) + f_{n}(b) + f_{n}(c) + f_{n}(d)  π
(where x denotes the absolute value of x).
You are given g(200) = 6^{2} + 75^{2} + 89^{2} + 226^{2} = 64658.
Find g(10000).^{}
A 3smooth number is an integer which has no prime factor larger than 3. For an integer N, we define S(N) as the set of 3smooth numbers less than or equal to N . For example, S(20) = { 1, 2, 3, 4, 6, 8, 9, 12, 16, 18 }.
We define F(N) as the number of permutations of S(N) in which each element comes after all of its proper divisors.
This is one of the possible permutations for N = 20.
 1, 2, 4, 3, 9, 8, 16, 6, 18, 12.
This is not a valid permutation because 12 comes before its divisor 6.
 1, 2, 4, 3, 9, 8, 12, 16, 6, 18.
We can verify that F(6) = 5, F(8) = 9, F(20) = 450 and F(1000) ≈ 8.8521816557e21.
Find F(10^{18}). Give as your answer its scientific notation rounded to ten digits after the decimal point.
When giving your answer, use a lowercase e to separate mantissa and exponent. E.g. if the answer is 112,233,445,566,778,899 then the answer format would be 1.1223344557e17.
The function $f$ is defined for all positive integers as follows:
 $f(1)=1$
 $f(3)=3$
 $f(2n)=f(n)$
 $f(4n + 1)=2f(2n + 1)  f(n)$
 $f(4n + 3)=3f(2n + 1)  2f(n)$
The function $S(n)$ is defined as $\sum_{i=1}^{n}f(i)$.
$S(8)=22$ and $S(100)=3604$.
Find $S(3^{37})$. Give the last 9 digits of your answer.
The Möbius function, denoted μ(n), is defined as:
 μ(n) = (1)^{ω(n)} if n is squarefree (where ω(n) is the number of distinct prime factors of n)
 μ(n) = 0 if n is not squarefree.
Let P(a,b) be the number of integers n in the interval [a,b] such that μ(n) = 1.
Let N(a,b) be the number of integers n in the interval [a,b] such that μ(n) = 1.
For example, P(2,10) = 2 and N(2,10) = 4.
Let C(n) be the number of integer pairs (a,b) such that:
 1 ≤ a ≤ b ≤ n,
 99·N(a,b) ≤ 100·P(a,b), and
 99·P(a,b) ≤ 100·N(a,b).
For example, C(10) = 13, C(500) = 16676 and C(10 000) = 20155319.
Find C(20 000 000).
The kernel of a polygon is defined by the set of points from which the entire polygon's boundary is visible. We define a polar polygon as a polygon for which the origin is strictly contained inside its kernel.
For this problem, a polygon can have collinear consecutive vertices. However, a polygon still cannot have selfintersection and cannot have zero area.
For example, only the first of the following is a polar polygon (the kernels of the second, third, and fourth do not strictly contain the origin, and the fifth does not have a kernel at all):
Notice that the first polygon has three consecutive collinear vertices.
Let P(n) be the number of polar polygons such that the vertices (x, y) have integer coordinates whose absolute values are not greater than n.
Note that polygons should be counted as different if they have different set of edges, even if they enclose the same area. For example, the polygon with vertices [(0,0),(0,3),(1,1),(3,0)] is distinct from the polygon with vertices [(0,0),(0,3),(1,1),(3,0),(1,0)].
For example, P(1) = 131, P(2) = 1648531, P(3) = 1099461296175 and P(343) mod 1 000 000 007 = 937293740.
Find P(7^{13}) mod 1 000 000 007.
Let P(m,n) be the number of distinct terms in an m×n multiplication table.
For example, a 3×4 multiplication table looks like this:
×  1  2  3  4 

1  1  2  3  4 
2  2  4  6  8 
3  3  6  9  12 
There are 8 distinct terms {1,2,3,4,6,8,9,12}, therefore P(3,4) = 8.
You are given that:
P(64,64) = 1263,
P(12,345) = 1998, and
P(32,10^{15}) = 13826382602124302.
Find P(64,10^{16}).
An integer s is called a superinteger of another integer n if the digits of n form a subsequence of the digits of s.
For example, 2718281828 is a superinteger of 18828, while 314159 is not a superinteger of 151.
Let p(n) be the nth prime number, and let c(n) be the nth composite number. For example, p(1) = 2, p(10) = 29, c(1) = 4 and c(10) = 18.
{p(i) : i ≥ 1} = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...}
{c(i) : i ≥ 1} = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ...}
Let P^{D} the sequence of the digital roots of {p(i)} (C^{D} is defined similarly for {c(i)}):
P^{D} = {2, 3, 5, 7, 2, 4, 8, 1, 5, 2, ...}
C^{D} = {4, 6, 8, 9, 1, 3, 5, 6, 7, 9, ...}
Let P_{n} be the integer formed by concatenating the first n elements of P^{D} (C_{n} is defined similarly for C^{D}).
P_{10} = 2357248152
C_{10} = 4689135679
Let f(n) be the smallest positive integer that is a common superinteger of P_{n} and C_{n}.
For example, f(10) = 2357246891352679, and f(100) mod 1 000 000 007 = 771661825.
Find f(10 000) mod 1 000 000 007.
An integer is called Bsmooth if none of its prime factors is greater than $B$.
Let $S_B(n)$ be the largest $B$smooth divisor of $n$.
Examples:
$S_1(10)=1$
$S_4(2100) = 12$
$S_{17}(2496144) = 5712$
Define $\displaystyle F(n)=\sum_{B=1}^n \sum_{r=0}^n S_B(\binom n r)$. Here, $\displaystyle \binom n r$ denotes the binomial coefficient.
Examples:
$F(11) = 3132$
$F(1111) \mod 1\,000\,000\,993 = 706036312$
$F(111\,111) \mod 1\,000\,000\,993 = 22156169$
Find $F(11\,111\,111) \mod 1\,000\,000\,993$.
In a room N chairs are placed around a round table.
Knights enter the room one by one and choose at random an available empty chair.
To have enough elbow room the knights always leave at least one empty chair between each other.
When there aren't any suitable chairs left, the fraction C of empty chairs is determined.
We also define E(N) as the expected value of C.
We can verify that E(4) = 1/2 and E(6) = 5/9.
Find E(10^{18}). Give your answer rounded to fourteen decimal places in the form 0.abcdefghijklmn.
Consider a single game of Ramvok:
Let t represent the maximum number of turns the game lasts. If t = 0, then the game ends immediately. Otherwise, on each turn i, the player rolls a die. After rolling, if i < t the player can either stop the game and receive a prize equal to the value of the current roll, or discard the roll and try again next turn. If i = t, then the roll cannot be discarded and the prize must be accepted. Before the game begins, t is chosen by the player, who must then pay an upfront cost ct for some constant c. For c = 0, t can be chosen to be infinite (with an upfront cost of 0). Let R(d, c) be the expected profit (i.e. net gain) that the player receives from a single game of optimallyplayed Ramvok, given a fair dsided die and cost constant c. For example, R(4, 0.2) = 2.65. Assume that the player has sufficient funds for paying any/all upfront costs.
Now consider a game of Super Ramvok:
In Super Ramvok, the game of Ramvok is played repeatedly, but with a slight modification. After each game, the die is altered. The alteration process is as follows: The die is rolled once, and if the resulting face has its pips visible, then that face is altered to be blank instead. If the face is already blank, then it is changed back to its original value. After the alteration is made, another game of Ramvok can begin (and during such a game, at each turn, the die is rolled until a face with a value on it appears). The player knows which faces are blank and which are not at all times. The game of Super Ramvok ends once all faces of the die are blank.
Let S(d, c) be the expected profit that the player receives from an optimallyplayed game of Super Ramvok, given a fair dsided die to start (with all sides visible), and cost constant c. For example, S(6, 1) = 208.3.
Let F(n) = ∑ _{4≤d≤n} ∑ _{0≤c≤n} S(d, c).
Calculate F(20), rounded to the nearest integer.
The triangle ΔABC is inscribed in an ellipse with equation $\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$, 0 < 2b < a, a and b integers.
Let r(a,b) be the radius of the incircle of ΔABC when the incircle has center (2b, 0) and A has coordinates $\left( \frac a 2, \frac {\sqrt 3} 2 b\right)$.
For example, r(3,1) = ½, r(6,2) = 1, r(12,3) = 2.
Let $G(n) = \sum_{a=3}^n \sum_{b=1}^{\lfloor \frac {a  1} 2 \rfloor} r(a, b)$
You are given G(10) = 20.59722222, G(100) = 19223.60980 (rounded to 10 significant digits).
Find G(10^{11}).
Give your answer in scientific notation rounded to 10 significant digits. Use a lowercase e to separate mantissa and exponent.
For G(10) the answer would have been 2.059722222e1.
There are N seats in a row. N people come one after another to fill the seats according to the following rules:
 No person sits beside another.
 The first person chooses any seat.
 Each subsequent person chooses the seat furthest from anyone else already seated, as long as it does not violate rule 1. If there is more than one choice satisfying this condition, then the person chooses the leftmost choice.
Note that due to rule 1, some seats will surely be left unoccupied, and the maximum number of people that can be seated is less than N (for N > 1).
Here are the possible seating arrangements for N = 15:
We see that if the first person chooses correctly, the 15 seats can seat up to 7 people.
We can also see that the first person has 9 choices to maximize the number of people that may be seated.
Let f(N) be the number of choices the first person has to maximize the number of occupants for N seats in a row. Thus, f(1) = 1, f(15) = 9, f(20) = 6, and f(500) = 16.
Also, ∑ f(N) = 83 for 1 ≤ N ≤ 20 and ∑ f(N) = 13343 for 1 ≤ N ≤ 500.
Find ∑ f(N) for 1 ≤ N ≤ 10^{12}. Give the last 8 digits of your answer.
Let $\varphi$ be the golden ratio: $\varphi=\frac{1+\sqrt{5}}{2}.$
Remarkably it is possible to write every positive integer as a sum of powers of $\varphi$ even if we require that every power of $\varphi$ is used at most once in this sum.
Even then this representation is not unique.
We can make it unique by requiring that no powers with consecutive exponents are used and that the representation is finite.
E.g:
$2=\varphi+\varphi^{2}$ and $3=\varphi^{2}+\varphi^{2}$
To represent this sum of powers of $\varphi$ we use a string of 0's and 1's with a point to indicate where the negative exponents start.
We call this the representation in the phigital numberbase.
So $1=1_{\varphi}$, $2=10.01_{\varphi}$, $3=100.01_{\varphi}$ and $14=100100.001001_{\varphi}$.
The strings representing 1, 2 and 14 in the phigital number base are palindromic, while the string representing 3 is not.
(the phigital point is not the middle character).
The sum of the positive integers not exceeding 1000 whose phigital representation is palindromic is 4345.
Find the sum of the positive integers not exceeding $10^{10}$ whose phigital representation is palindromic.
For a positive integer n and digits d, we define F(n, d) as the number of the divisors of n whose last digits equal d.
For example, F(84, 4) = 3. Among the divisors of 84 (1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84), three of them (4, 14, 84) have the last digit 4.
We can also verify that F(12!, 12) = 11 and F(50!, 123) = 17888.
Find F(10^{6}!, 65432) modulo (10^{16} + 61).
12n musicians participate at a music festival. On the first day, they form 3n quartets and practice all day.
It is a disaster. At the end of the day, all musicians decide they will never again agree to play with any member of their quartet.
On the second day, they form 4n trios, each musician avoiding his previous quartet partners.
Let f(12n) be the number of ways to organize the trios amongst the 12n musicians.
You are given f(12) = 576 and f(24) mod 1 000 000 007 = 509089824.
Find f(600) mod 1 000 000 007.
Let R(a, b, c) be the maximum area covered by three nonoverlapping circles inside a triangle with edge lengths a, b and c.
Let S(n) be the average value of R(a, b, c) over all integer triplets (a, b, c) such that 1 ≤ a ≤ b ≤ c < a + b ≤ n
You are given S(2) = R(1, 1, 1) ≈ 0.31998, S(5) ≈ 1.25899.
Find S(1803) rounded to 5 decimal places behind the decimal point.
The number sequence game starts with a sequence S of N numbers written on a line.
Two players alternate turns. At his turn, a player must select and remove either the first or the last number remaining in the sequence.
The player score is the sum of all the numbers he has taken. Each player attempts to maximize his own sum.
If N = 4 and S = {1, 2, 10, 3}, then each player maximizes his score as follows: Player 1: removes the first number (1)
 Player 2: removes the last number from the remaining sequence (3)
 Player 1: removes the last number from the remaining sequence (10)
 Player 2: removes the remaining number (2)
Player 1 score is 1 + 10 = 11.
Let F(N) be the score of player 1 if both players follow the optimal strategy for the sequence S = {s_{1}, s_{2}, ..., s_{N}} defined as:
 s_{1} = 0
 s_{i+1} = (s_{i}^{2} + 45) modulo 1 000 000 007
The sequence begins with S = {0, 45, 2070, 4284945, 753524550, 478107844, 894218625, ...}.
You are given F(2) = 45, F(4) = 4284990, F(100) = 26365463243, F(10^{4}) = 2495838522951.
Find F(10^{8}).
Let us consider mixtures of three substances: A, B and C. A mixture can be described by a ratio of the amounts of A, B, and C in it, i.e., (a : b : c). For example, a mixture described by the ratio (2 : 3 : 5) contains 20% A, 30% B and 50% C.
For the purposes of this problem, we cannot separate the individual components from a mixture. However, we can combine different amounts of different mixtures to form mixtures with new ratios.
For example, say we have three mixtures with ratios (3 : 0 : 2), (3 : 6 : 11) and (3 : 3 : 4). By mixing 10 units of the first, 20 units of the second and 30 units of the third, we get a new mixture with ratio (6 : 5 : 9), since:
(10·^{3}/_{5} + 20·^{3}/_{20} + 30·^{3}/_{10} : 10·^{0}/_{5} + 20·^{6}/_{20} + 30·^{3}/_{10} : 10·^{2}/_{5} + 20·^{11}/_{20} + 30·^{4}/_{10})
= (18 : 15 : 27) = (6 : 5 : 9)
However, with the same three mixtures, it is impossible to form the ratio (3 : 2 : 1), since the amount of B is always less than the amount of C.
Let n be a positive integer. Suppose that for every triple of integers (a, b, c) with 0 ≤ a, b, c ≤ n and gcd(a, b, c) = 1, we have a mixture with ratio (a : b : c). Let M(n) be the set of all such mixtures.
For example, M(2) contains the 19 mixtures with the following ratios:
{(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (0 : 1 : 2), (0 : 2 : 1),
(1 : 0 : 0), (1 : 0 : 1), (1 : 0 : 2), (1 : 1 : 0), (1 : 1 : 1),
(1 : 1 : 2), (1 : 2 : 0), (1 : 2 : 1), (1 : 2 : 2), (2 : 0 : 1),
(2 : 1 : 0), (2 : 1 : 1), (2 : 1 : 2), (2 : 2 : 1)}.
Let E(n) be the number of subsets of M(n) which can produce the mixture with ratio (1 : 1 : 1), i.e., the mixture with equal parts A, B and C.
We can verify that E(1) = 103, E(2) = 520447, E(10) mod 11^{8} = 82608406 and E(500) mod 11^{8} = 13801403.
Find E(10 000 000) mod 11^{8}.
Let $a_k$, $b_k$, and $c_k$ represent the three solutions (real or complex numbers) to the equation $\frac 1 x = (\frac k x)^2(k+x^2)k x$.
For instance, for $k=5$, we see that $\{a_5, b_5, c_5 \}$ is approximately $\{5.727244, 0.363622+2.057397i, 0.3636222.057397i\}$.
Let $\displaystyle S(n) = \sum_{p=1}^n\sum_{k=1}^n(a_k+b_k)^p(b_k+c_k)^p(c_k+a_k)^p$.
Interestingly, $S(n)$ is always an integer. For example, $S(4) = 51160$.
Find $S(10^6)$ modulo $1\,000\,000\,007$.
Consider all the words which can be formed by selecting letters, in any order, from the phrase:
Suppose those with 15 letters or less are listed in alphabetical order and numbered sequentially starting at 1.
The list would include:
 1 : a
 2 : aa
 3 : aaa
 4 : aaaa
 5 : aaaaa
 6 : aaaaaa
 7 : aaaaaac
 8 : aaaaaacd
 9 : aaaaaacde
 10 : aaaaaacdee
 11 : aaaaaacdeee
 12 : aaaaaacdeeee
 13 : aaaaaacdeeeee
 14 : aaaaaacdeeeeee
 15 : aaaaaacdeeeeeef
 16 : aaaaaacdeeeeeeg
 17 : aaaaaacdeeeeeeh
 ...
 28 : aaaaaacdeeeeeey
 29 : aaaaaacdeeeeef
 30 : aaaaaacdeeeeefe
 ...
 115246685191495242: euleoywuttttsss
 115246685191495243: euler
 115246685191495244: eulera
 ...
 525069350231428029: ywuuttttssssrrr
Define P(w) as the position of the word w.
Define W(p) as the word in position p.
We can see that P(w) and W(p) are inverses: P(W(p)) = p and W(P(w)) = w.
Examples:
 W(10) = aaaaaacdee
 P(aaaaaacdee) = 10
 W(115246685191495243) = euler
 P(euler) = 115246685191495243
Find W(P(legionary) + P(calorimeters)  P(annihilate) + P(orchestrated)  P(fluttering)).
Give your answer using lowercase characters (no punctuation or space).
A group of chefs (numbered #1, #2, etc) participate in a turnbased strategic cooking competition. On each chef's turn, he/she cooks up a dish to the best of his/her ability and gives it to a separate panel of judges for tastetesting. Let S(k) represent chef #k's skill level (which is publicly known). More specifically, S(k) is the probability that chef #k's dish will be assessed favorably by the judges (on any/all turns). If the dish receives a favorable rating, then the chef must choose one other chef to be eliminated from the competition. The last chef remaining in the competition is the winner.
The game always begins with chef #1, with the turn order iterating sequentially over the rest of the chefs still in play. Then the cycle repeats from the lowestnumbered chef. All chefs aim to optimize their chances of winning within the rules as stated, assuming that the other chefs behave in the same manner. In the event that a chef has more than one equallyoptimal elimination choice, assume that the chosen chef is always the one with the nextclosest turn.
Define W_{n}(k) as the probability that chef #k wins in a competition with n chefs. If we have S(1) = 0.25, S(2) = 0.5, and S(3) = 1, then W_{3}(1) = 0.29375.
Going forward, we assign S(k) = F_{k}/F_{n+1} over all 1 ≤ k ≤ n, where F_{k} is a Fibonacci number: F_{k} = F_{k1} + F_{k2} with base cases F_{1} = F_{2} = 1. Then, for example, when considering a competition with n = 7 chefs, we have W_{7}(1) = 0.08965042, W_{7}(2) = 0.20775702, W_{7}(3) = 0.15291406, W_{7}(4) = 0.14554098, W_{7}(5) = 0.15905291, W_{7}(6) = 0.10261412, and W_{7}(7) = 0.14247050, rounded to 8 decimal places each.
Let E(n) represent the expected number of dishes cooked in a competition with n chefs. For instance, E(7) = 42.28176050.
Find E(14) rounded to 8 decimal places.
ABC is an integer sided triangle with incenter I and perimeter p.
The segments IA, IB and IC have integral length as well.
Let L = p + IA + IB + IC.
Let S(P) = ∑ L for all such triangles where p ≤ P. For example, S(10^{3}) = 3619.
Find S(10^{7}).
We define a permutation as an operation that rearranges the order of the elements {1, 2, 3, ..., n}.
There are n! such permutations, one of which leaves the elements in their initial order.
For n = 3 we have 3! = 6 permutations:
 P_{1} = keep the initial order
 P_{2} = exchange the 1^{st} and 2^{nd} elements
 P_{3} = exchange the 1^{st} and 3^{rd} elements
 P_{4} = exchange the 2^{nd} and 3^{rd} elements
 P_{5} = rotate the elements to the right
 P_{6} = rotate the elements to the left
If we select one of these permutations, and we reapply the same permutation repeatedly, we eventually restore the initial order.
For a permutation P_{i}, let f(P_{i}) be the number of steps required to restore the initial order by applying the permutation P_{i} repeatedly.
For n = 3, we obtain:
 f(P_{1}) = 1 : (1,2,3) → (1,2,3)
 f(P_{2}) = 2 : (1,2,3) → (2,1,3) → (1,2,3)
 f(P_{3}) = 2 : (1,2,3) → (3,2,1) → (1,2,3)
 f(P_{4}) = 2 : (1,2,3) → (1,3,2) → (1,2,3)
 f(P_{5}) = 3 : (1,2,3) → (3,1,2) → (2,3,1) → (1,2,3)
 f(P_{6}) = 3 : (1,2,3) → (2,3,1) → (3,1,2) → (1,2,3)
Let g(n) be the average value of f^{2}(P_{i}) over all permutations P_{i} of length n.
g(3) = (1^{2} + 2^{2} + 2^{2} + 2^{2} + 3^{2} + 3^{2})/3! = 31/6 ≈ 5.166666667e0
g(5) = 2081/120 ≈ 1.734166667e1
g(20) = 12422728886023769167301/2432902008176640000 ≈ 5.106136147e3
Find g(350) and write the answer in scientific notation rounded to 10 significant digits, using a lowercase e to separate mantissa and exponent, as in the examples above.
The arithmetic derivative is defined by
 p' = 1 for any prime p
 (ab)' = a'b + ab' for all integers a, b (Leibniz rule)
For example, 20' = 24.
Find ∑ gcd(k,k') for 1 < k ≤ 5×10^{15}.
Note: gcd(x,y) denotes the greatest common divisor of x and y.
Let d(n) be the number of divisors of n.
Let M(n,k) be the maximum value of d(j) for n ≤ j ≤ n+k1.
Let S(u,k) be the sum of M(n,k) for 1 ≤ n ≤ uk+1.
You are given that S(1000,10)=17176.
Find S(100 000 000,100 000).
Let F_{5}(n) be the number of strings s such that:
 s consists only of '0's and '1's,
 s has length at most n, and
 s contains a palindromic substring of length at least 5.
For example, F_{5}(4) = 0, F_{5}(5) = 8, F_{5}(6) = 42 and F_{5}(11) = 3844.
Let D(L) be the number of integers n such that 5 ≤ n ≤ L and F_{5}(n) is divisible by 87654321.
For example, D(10^{7}) = 0 and D(5·10^{9}) = 51.
Find D(10^{18}).
Let f_{k}(n) be the sum of the k^{th} powers of the first n positive integers.
For example, f_{2}(10) = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} + 7^{2} + 8^{2} + 9^{2} + 10^{2} = 385.
Let S_{k}(n) be the sum of f_{k}(i) for 1 ≤ i ≤ n. For example, S_{4}(100) = 35375333830.
What is ∑ (S_{10000}(10^{12}) mod p) over all primes p between 2 ⋅ 10^{9} and 2 ⋅ 10^{9} + 2000?
Alice and Bob have enjoyed playing Nim every day. However, they finally got bored of playing ordinary threeheap Nim.
So, they added an extra rule:
 Must not make two heaps of the same size.
The triple (a,b,c) indicates the size of three heaps.
Under this extra rule, (2,4,5) is one of the losing positions for the next player.
To illustrate:
 Alice moves to (2,4,3)
 Bob moves to (0,4,3)
 Alice moves to (0,2,3)
 Bob moves to (0,2,1)
Unlike ordinary threeheap Nim, (0,1,2) and its permutations are the end states of this game.
For an integer N, we define F(N) as the sum of a+b+c for all the losing positions for the next player, with 0 < a < b < c < N.
For example, F(8) = 42, because there are 4 losing positions for the next player, (1,3,5), (1,4,6), (2,3,6) and (2,4,5).
We can also verify that F(128) = 496062.
Find the last 9 digits of F(10^{18}).
Let G(a, b) be the smallest nonnegative integer n for which gcd(n^{3} + b, (n + a)^{3} + b) is maximized.
For example, G(1, 1) = 5 because gcd(n^{3} + 1, (n + 1)^{3} + 1) reaches its maximum value of 7 for n = 5, and is smaller for 0 ≤ n < 5.
Let H(m, n) = ∑ G(a, b) for 1 ≤ a ≤ m, 1 ≤ b ≤ n.
You are given H(5, 5) = 128878 and H(10, 10) = 32936544.
Find H(18, 1900).
There are n stones in a pond, numbered 1 to n. Consecutive stones are spaced one unit apart.
A frog sits on stone 1. He wishes to visit each stone exactly once, stopping on stone n. However, he can only jump from one stone to another if they are at most 3 units apart. In other words, from stone i, he can reach a stone j if 1 ≤ j ≤ n and j is in the set {i3, i2, i1, i+1, i+2, i+3}.
Let f(n) be the number of ways he can do this. For example, f(6) = 14, as shown below:
1 → 2 → 3 → 4 → 5 → 6
1 → 2 → 3 → 5 → 4 → 6
1 → 2 → 4 → 3 → 5 → 6
1 → 2 → 4 → 5 → 3 → 6
1 → 2 → 5 → 3 → 4 → 6
1 → 2 → 5 → 4 → 3 → 6
1 → 3 → 2 → 4 → 5 → 6
1 → 3 → 2 → 5 → 4 → 6
1 → 3 → 4 → 2 → 5 → 6
1 → 3 → 5 → 2 → 4 → 6
1 → 4 → 2 → 3 → 5 → 6
1 → 4 → 2 → 5 → 3 → 6
1 → 4 → 3 → 2 → 5 → 6
1 → 4 → 5 → 2 → 3 → 6
Other examples are f(10) = 254 and f(40) = 1439682432976.
Let S(L) = ∑ f(n)^{3} for 1 ≤ n ≤ L.
Examples:
S(10) = 18230635
S(20) = 104207881192114219
S(1 000) mod 10^{9} = 225031475
S(1 000 000) mod 10^{9} = 363486179
Find S(10^{14}) mod 10^{9}.
We call a positive integer double pandigital if it uses all the digits 0 to 9 exactly twice (with no leading zero). For example, 40561817703823564929 is one such number.
How many double pandigital numbers are divisible by 11?
Define the sequence a_{1}, a_{2}, a_{3}, ... as:
 a_{1} = 1
 a_{n+1} = 6a_{n}^{2} + 10a_{n} + 3 for n ≥ 1.
Examples:
a_{3} = 2359
a_{6} = 269221280981320216750489044576319
a_{6} mod 1 000 000 007 = 203064689
a_{100} mod 1 000 000 007 = 456482974
Define B(x,y,n) as ∑ (a_{n} mod p) for every prime p such that x ≤ p ≤ x+y.
Examples:
B(10^{9}, 10^{3}, 10^{3}) = 23674718882
B(10^{9}, 10^{3}, 10^{15}) = 20731563854
Find B(10^{9}, 10^{7}, 10^{15}).
70 coloured balls are placed in an urn, 10 for each of the seven rainbow colours.
What is the expected number of distinct colours in 20 randomly picked balls?
Give your answer with nine digits after the decimal point (a.bcdefghij).
The Collatz sequence is defined as: $a_{i+1} = \left\{ \large{\frac {a_i} 2 \atop 3 a_i+1} {\text{if }a_i\text{ is even} \atop \text{if }a_i\text{ is odd}} \right.$.
The Collatz conjecture states that starting from any positive integer, the sequence eventually reaches the cycle 1,4,2,1....
We shall define the sequence prefix p(n) for the Collatz sequence starting with a_{1} = n as the subsequence of all numbers not a power of 2 (2^{0}=1 is considered a power of 2 for this problem). For example:
p(13) = {13, 40, 20, 10, 5}
p(8) = {}
Any number invalidating the conjecture would have an infinite length sequence prefix.
Let S_{m} be the set of all sequence prefixes of length m. Two sequences {a_{1}, a_{2}, ..., a_{m}} and {b_{1}, b_{2}, ..., b_{m}} in S_{m} are said to belong to the same prefix family if a_{i} < a_{j} if and only if b_{i} < b_{j} for all 1 ≤ i,j ≤ m.
For example, in S_{4}, {6, 3, 10, 5} is in the same family as {454, 227, 682, 341}, but not {113, 340, 170, 85}.
Let f(m) be the number of distinct prefix families in S_{m}.
You are given f(5) = 5, f(10) = 55, f(20) = 6771.
Find f(90).
Let W(n,k) be the number of ways in which n can be written as the product of k distinct positive integers.
For example, W(144,4) = 7. There are 7 ways in which 144 can be written as a product of 4 distinct positive integers:
 144 = 1×2×4×18
 144 = 1×2×8×9
 144 = 1×2×3×24
 144 = 1×2×6×12
 144 = 1×3×4×12
 144 = 1×3×6×8
 144 = 2×3×4×6
Note that permutations of the integers themselves are not considered distinct.
Furthermore, W(100!,10) modulo 1 000 000 007 = 287549200.
Find W(10000!,30) modulo 1 000 000 007.
Given an integer sided triangle ABC:
Let I be the incenter of ABC.
Let D be the intersection between the line AI and the circumcircle of ABC (A ≠ D).
We define F(L) as the sum of BC for the triangles ABC that satisfy AC = DI and BC ≤ L.
For example, F(15) = 45 because the triangles ABC with (BC,AC,AB) = (6,4,5), (12,8,10), (12,9,7), (15,9,16) satisfy the conditions.
Find F(10^{9}).
Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hanoi," which consists of three upright rods and disks of different sizes that can slide onto any of the rods. The game begins with a stack of n disks placed on the leftmost rod in descending order by size. The objective of the game is to move all of the disks from the leftmost rod to the rightmost rod, given the following restrictions:
 Only one disk can be moved at a time.
 A valid move consists of taking the top disk from one stack and placing it onto another stack (or an empty rod).
 No disk can be placed on top of a smaller disk.
Moving on to a variant of this game, consider a long room k units (square tiles) wide, labeled from 1 to k in ascending order. Three rods are placed at squares a, b, and c, and a stack of n disks is placed on the rod at square a.
Bob begins the game standing at square b. His objective is to play the Tower of Hanoi game by moving all of the disks to the rod at square c. However, Bob can only pick up or set down a disk if he is on the same square as the rod / stack in question.
Unfortunately, Bob is also drunk. On a given move, Bob will either stumble one square to the left or one square to the right with equal probability, unless Bob is at either end of the room, in which case he can only move in one direction. Despite Bob's inebriated state, he is still capable of following the rules of the game itself, as well as choosing when to pick up or put down a disk.
The following animation depicts a sideview of a sample game for n = 3, k = 7, a = 2, b = 4, and c = 6:
Let E(n,k,a,b,c) be the expected number of squares that Bob travels during a single optimallyplayed game. A game is played optimally if the number of diskpickups is minimized.
Interestingly enough, the result is always an integer. For example, E(2,5,1,3,5) = 60 and E(3,20,4,9,17) = 2358.
Find the last nine digits of ∑_{1≤n≤10000} E(n,10^{n},3^{n},6^{n},9^{n}).
For positive integers n and m, we define two polynomials F_{n}(x) = x^{n} and G_{m}(x) = (x1)^{m}.
We also define a polynomial R_{n,m}(x) as the remainder of the division of F_{n}(x) by G_{m}(x).
For example, R_{6,3}(x) = 15x^{2}  24x + 10.
Let C(n, m, d) be the absolute value of the coefficient of the dth degree term of R_{n,m}(x).
We can verify that C(6, 3, 1) = 24 and C(100, 10, 4) = 227197811615775.
Find C(10^{13}, 10^{12}, 10^{4}) mod 999999937.
A gambler decides to participate in a special lottery. In this lottery the gambler plays a series of one or more games.
Each game costs m pounds to play and starts with an initial pot of 1 pound. The gambler flips an unbiased coin. Every time a head appears, the pot is doubled and the gambler continues. When a tail appears, the game ends and the gambler collects the current value of the pot. The gambler is certain to win at least 1 pound, the starting value of the pot, at the cost of m pounds, the initial fee.
The gambler cannot continue to play if his fortune falls below m pounds.
Let p_{m}(s) denote the probability that the gambler will never run out of money in this lottery given his initial fortune s and the cost per game m.
For example p_{2}(2) ≈ 0.2522, p_{2}(5) ≈ 0.6873 and p_{6}(10 000) ≈ 0.9952 (note: p_{m}(s) = 0 for s < m).
Find p_{15}(10^{9}) and give your answer rounded to 7 decimal places behind the decimal point in the form 0.abcdefg.
The number of divisors of 120 is 16.
In fact 120 is the smallest number having 16 divisors.
Find the smallest number with 2^{500500} divisors.
Give your answer modulo 500500507.
The eight divisors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
The ten numbers not exceeding 100 having exactly eight divisors are 24, 30, 40, 42, 54, 56, 66, 70, 78 and 88.
Let f(n) be the count of numbers not exceeding n with exactly eight divisors.
You are given f(100) = 10, f(1000) = 180 and f(10^{6}) = 224427.
Find f(10^{12}).
We define a block to be a rectangle with a height of 1 and an integervalued length. Let a castle be a configuration of stacked blocks.
Given a game grid that is w units wide and h units tall, a castle is generated according to the following rules:
 Blocks can be placed on top of other blocks as long as nothing sticks out past the edges or hangs out over open space.
 All blocks are aligned/snapped to the grid.
 Any two neighboring blocks on the same row have at least one unit of space between them.
 The bottom row is occupied by a block of length w.
 The maximum achieved height of the entire castle is exactly h.
 The castle is made from an even number of blocks.
The following is a sample castle for w=8 and h=5:
Let F(w,h) represent the number of valid castles, given grid parameters w and h.
For example, F(4,2) = 10, F(13,10) = 3729050610636, F(10,13) = 37959702514, and F(100,100) mod 1 000 000 007 = 841913936.
Find (F(10^{12},100) + F(10000,10000) + F(100,10^{12})) mod 1 000 000 007.
Alice is playing a game with n cards numbered 1 to n.
A game consists of iterations of the following steps.
(1) Alice picks one of the cards at random.
(2) Alice cannot see the number on it. Instead, Bob, one of her friends, sees the number and tells Alice how many previouslyseen numbers are bigger than the number which he is seeing.
(3) Alice can end or continue the game. If she decides to end, the number becomes her score. If she decides to continue, the card is removed from the game and she returns to (1). If there is no card left, she is forced to end the game.
Let F(n) be the Alice's expected score if she takes the optimized strategy to minimize her score.
For example, F(3) = 5/3. At the first iteration, she should continue the game. At the second iteration, she should end the game if Bob says that one previouslyseen number is bigger than the number which he is seeing, otherwise she should continue the game.
We can also verify that F(4) = 15/8 and F(10) ≈ 2.5579365079.
Find F(10^{6}). Give your answer rounded to 10 decimal places behind the decimal point.
Let ABCD be a quadrilateral whose vertices are lattice points lying on the coordinate axes as follows:
A(a, 0), B(0, b), C(−c, 0), D(0, −d), where 1 ≤ a, b, c, d ≤ m and a, b, c, d, m are integers.
It can be shown that for m = 4 there are exactly 256 valid ways to construct ABCD. Of these 256 quadrilaterals, 42 of them strictly contain a square number of lattice points.
How many quadrilaterals ABCD strictly contain a square number of lattice points for m = 100?
Let:
$\begin{array}{ll} x(0)&=0 \\ x(1)&=1 \\ x(2k)&=(3x(k)+2x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \text {, where } \lfloor \text { } \rfloor \text { is the floor function} \\ x(2k+1)&=(2x(k)+3x(\lfloor \frac k 2 \rfloor)) \text{ mod } 2^{60} \text{ for } k \ge 1 \\ y_n(k)&=\left\{{\begin{array}{lc} x(k) && \text{if } k \ge n \\ 2^{60}  1  max(y_n(2k),y_n(2k+1)) && \text{if } k < n \end{array}} \right. \\ A(n)&=y_n(1) \end{array}$
You are given:
$\begin{array}{ll} x(2)&=3 \\ x(3)&=2 \\ x(4)&=11 \\ y_4(4)&=11 \\ y_4(3)&=2^{60}9\\ y_4(2)&=2^{60}12 \\ y_4(1)&=A(4)=8 \\ A(10)&=2^{60}34\\ A(10^3)&=101881 \end{array}$
Find $A(10^{12})$.Consider the infinite repeating sequence of digits:
1234321234321234321...
Amazingly, you can break this sequence of digits into a sequence of integers such that the sum of the digits in the n'th value is n.
The sequence goes as follows:
1, 2, 3, 4, 32, 123, 43, 2123, 432, 1234, 32123, ...
Let v_{n} be the n'th value in this sequence. For example, v_{2} = 2, v_{5} = 32 and v_{11} = 32123.
Let S(n) be v_{1} + v_{2} + ... + v_{n}. For example, S(11) = 36120, and S(1000) mod 123454321 = 18232686.
Find S(10^{14}) mod 123454321.
Let $t_n$ be the tribonacci numbers defined as:
$t_0 = t_1 = 0$;
$t_2 = 1$;
$t_n = t_{n1} + t_{n2} + t_{n3}$ for $n \ge 3$
and let $r_n = t_n \text{ mod } 10^7$.
For each pair of Vectors $V_n=(v_1,v_2,v_3)$ and $W_n=(w_1,w_2,w_3)$ with $v_1=r_{12n11}r_{12n10}, v_2=r_{12n9}+r_{12n8}, v_3=r_{12n7} \cdot r_{12n6}$ and
$w_1=r_{12n5}r_{12n4}, w_2=r_{12n3}+r_{12n2}, w_3=r_{12n1} \cdot r_{12n}$
we define $S(n)$ as the minimal value of the manhattan length of the vector $D=k \cdot V_n+l \cdot W_n$ measured as $k \cdot v_1+l \cdot w_1+k \cdot v_2+l \cdot w_2+k \cdot v_3+l \cdot w_3$
for any integers $k$ and $l$ with $(k,l)\neq (0,0)$.
The first vector pair is (1, 3, 28), (11, 125, 40826).
You are given that $S(1)=32$ and $\sum_{n=1}^{10} S(n)=130762273722$.
Find $\sum_{n=1}^{20000000} S(n)$.
Consider the Gaussian integer i1. A base i1 representation of a Gaussian integer a+bi is a finite sequence of digits d_{n1}d_{n2}...d_{1}d_{0} such that:
 a+bi = d_{n1}(i1)^{n1} + d_{n2}(i1)^{n2} + ... + d_{1}(i1) + d_{0}
 Each d_{k} is in {0,1}
 There are no leading zeroes, i.e. d_{n1} ≠ 0, unless a+bi is itself 0
Here are base i1 representations of a few Gaussian integers:
11+24i → 111010110001101
2411i → 110010110011
8+0i → 111000000
5+0i → 11001101
0+0i → 0
Define f(a+bi) as the number of 1s in the unique base i1 representation of a+bi. For example, f(11+24i) = 9 and f(2411i) = 7.
Define B(L) as the sum of f(a+bi) for all integers a, b such that a ≤ L and b ≤ L. For example, B(500) = 10795060.
Find B(10^{15}) mod 1 000 000 007.
Anton and Bertrand love to play three pile Nim.
However, after a lot of games of Nim they got bored and changed the rules somewhat.
They may only take a number of stones from a pile that is a proper divisor of the number of stones present in the pile.
E.g. if a pile at a certain moment contains 24 stones they may take only 1,2,3,4,6,8 or 12 stones from that pile.
So if a pile contains one stone they can't take the last stone from it as 1 isn't a proper divisor of 1.
The first player that can't make a valid move loses the game.
Of course both Anton and Bertrand play optimally.
The triple (a,b,c) indicates the number of stones in the three piles.
Let S(n) be the number of winning positions for the next player for 1 ≤ a, b, c ≤ n.
S(10) = 692 and S(100) = 735494.
Find S(123456787654321) modulo 1234567890.
Circles A and B are tangent to each other and to line L at three distinct points.
Circle C is inside the space between A, B and L, and tangent to all three.
Let r_{A}, r_{B} and r_{C} be the radii of A, B and C respectively.
Let S(n) = ∑ r_{A} + r_{B} + r_{C}, for 0 < r_{A} ≤ r_{B} ≤ n where r_{A}, r_{B} and r_{C} are integers. The only solution for 0 < r_{A} ≤ r_{B} ≤ 5 is r_{A} = 4, r_{B} = 4 and r_{C} = 1, so S(5) = 4 + 4 + 1 = 9. You are also given S(100) = 3072.
Find S(10^{9}).
Let Seq(n,k) be the number of positiveinteger sequences {a_{i}}_{1≤i≤n} of length n such that:
 n is divisible by a_{i} for 1 ≤ i ≤ n, and
 n + a_{1} + a_{2} + ... + a_{n} is divisible by k.
Examples:
Seq(3,4) = 4, and the 4 sequences are:
{1, 1, 3}
{1, 3, 1}
{3, 1, 1}
{3, 3, 3}
Seq(4,11) = 8, and the 8 sequences are:
{1, 1, 1, 4}
{1, 1, 4, 1}
{1, 4, 1, 1}
{4, 1, 1, 1}
{2, 2, 2, 1}
{2, 2, 1, 2}
{2, 1, 2, 2}
{1, 2, 2, 2}
The last nine digits of Seq(1111,24) are 840643584.
Find the last nine digits of Seq(1234567898765,4321).
Let $\varphi(n)$ be Euler's totient function.
Let $f(n)=(\sum_{i=1}^{n}\varphi(n^i)) \text{ mod } (n+1)$.
Let $g(n)=\sum_{i=1}^{n} f(i)$.
$g(100)=2007$.
Find $g(5 \times 10^8)$.
ABC is an integral sided triangle with sides a≤b≤c.
m_{c} is the median connecting C and the midpoint of AB.
F(n) is the number of such triangles with c≤n for which m_{c} has integral length as well.
F(10)=3 and F(50)=165.
Find F(100000).
A geoboard (of order N) is a square board with equallyspaced pins protruding from the surface, representing an integer point lattice for coordinates 0 ≤ x,y ≤ N.
John begins with a pinless geoboard. Each position on the board is a hole that can be filled with a pin. John decides to generate a random integer between 1 and N+1 (inclusive) for each hole in the geoboard. If the random integer is equal to 1 for a given hole, then a pin is placed in that hole.
After John is finished generating numbers for all (N+1)^{2} holes and placing any/all corresponding pins, he wraps a tight rubberband around the entire group of pins protruding from the board. Let S represent the shape that is formed. S can also be defined as the smallest convex shape that contains all the pins.
The above image depicts a sample layout for N = 4. The green markers indicate positions where pins have been placed, and the blue lines collectively represent the rubberband. For this particular arrangement, S has an area of 6. If there are fewer than three pins on the board (or if all pins are collinear), S can be assumed to have zero area.
Let E(N) be the expected area of S given a geoboard of order N. For example, E(1) = 0.18750, E(2) = 0.94335, and E(10) = 55.03013 when rounded to five decimal places each.
Calculate E(100) rounded to five decimal places.
Let d(p,n,0) be the multiplicative inverse of n modulo prime p, defined as n × d(p,n,0) = 1 mod p.
Let d(p,n,k) = $\sum_{i=1}^n$d(p,i,k−1) for k ≥ 1.
Let D(a,b,k) = $\sum$(d(p,p1,k) mod p) for all primes a ≤ p < a + b.
You are given:
 D(101,1,10) = 45
 D(10^{3},10^{2},10^{2}) = 8334
 D(10^{6},10^{3},10^{3}) = 38162302
Find D(10^{9},10^{5},10^{5}).
5smooth numbers are numbers whose largest prime factor doesn't exceed 5.
5smooth numbers are also called Hamming numbers.
Let S(L) be the sum of the numbers n not exceeding L such that Euler's totient function φ(n) is a Hamming number.
S(100)=3728.
Find S(10^{12}). Give your answer modulo 2^{32}.
For every real number $a \gt 1$ is given the sequence $g_a$ by:
$g_{a}(x)=1$ for $x \lt a$
$g_{a}(x)=g_{a}(x1)+g_a(xa)$ for $x \ge a$
$G(n)=g_{\sqrt {n}}(n)$
$G(90)=7564511$.
Find $\sum G(p)$ for $p$ prime and $10000000 \lt p \lt 10010000$
Give your answer modulo 1000000007.
Let S(n) = Σ a+b+c over all triples (a,b,c) such that:
 a, b, and c are prime numbers.
 a < b < c < n.
 a+1, b+1, and c+1 form a geometric sequence.
For example, S(100) = 1035 with the following triples:
(2, 5, 11), (2, 11, 47), (5, 11, 23), (5, 17, 53), (7, 11, 17), (7, 23, 71), (11, 23, 47), (17, 23, 31), (17, 41, 97), (31, 47, 71), (71, 83, 97)
Find S(10^{8}).
An arrangement of coins in one or more rows with the bottom row being a block without gaps and every coin in a higher row touching exactly two coins in the row below is called a fountain of coins. Let f(n) be the number of possible fountains with n coins. For 4 coins there are three possible arrangements:
Therefore f(4) = 3 while f(10) = 78.
Let T(n) be the number of all possible colourings with three colours for all f(n) different fountains with n coins, given the condition that no two touching coins have the same colour. Below you see the possible colourings for one of the three valid fountains for 4 coins:
You are given that T(4) = 48 and T(10) = 17760.
Find the last 9 digits of T(20000).
We define a simber to be a positive integer in which any odd digit, if present, occurs an odd number of times, and any even digit, if present, occurs an even number of times.
For example, 141221242 is a 9digit simber because it has three 1's, four 2's and two 4's.
Let Q(n) be the count of all simbers with at most n digits.
You are given Q(7) = 287975 and Q(100) mod 1 000 000 123 = 123864868.
Find (∑_{1≤u≤39} Q(2^{u})) mod 1 000 000 123.
Let smpf(n) be the smallest prime factor of n.
smpf(91)=7 because 91=7×13 and smpf(45)=3 because 45=3×3×5.
Let S(n) be the sum of smpf(i) for 2 ≤ i ≤ n.
E.g. S(100)=1257.
Find S(10^{12}) mod 10^{9}.
Despite the popularity of Hilbert's infinite hotel, Hilbert decided to try managing extremely large finite hotels, instead.
To cut costs, Hilbert wished to power the new hotel with his own special generator. Each floor would send power to the floor above it, with the top floor sending power back down to the bottom floor. That way, Hilbert could have the generator placed on any given floor (as he likes having the option) and have electricity flow freely throughout the entire hotel.
Unfortunately, the contractors misinterpreted the schematics when they built the hotel. They informed Hilbert that each floor sends power to another floor at random, instead. This may compromise Hilbert's freedom to have the generator placed anywhere, since blackouts could occur on certain floors.
For example, consider a sample flow diagram for a threestory hotel:
If the generator were placed on the first floor, then every floor would receive power. But if it were placed on the second or third floors instead, then there would be a blackout on the first floor. Note that while a given floor can receive power from many other floors at once, it can only send power to one other floor.
To resolve the blackout concerns, Hilbert decided to have a minimal number of floors rewired. To rewire a floor is to change the floor it sends power to. In the sample diagram above, all possible blackouts can be avoided by rewiring the second floor to send power to the first floor instead of the third floor.
Let F(n) be the sum of the minimum number of floor rewirings needed over all possible powerflow arrangements in a hotel of n floors. For example, F(3) = 6, F(8) = 16276736, and F(100) mod 135707531 = 84326147.
Find F(12344321) mod 135707531.
Consider the following algorithm for sorting a list:
 1. Starting from the beginning of the list, check each pair of adjacent elements in turn.
 2. If the elements are out of order:
 a. Move the smallest element of the pair at the beginning of the list.
 b. Restart the process from step 1.
 3. If all pairs are in order, stop.
For example, the list { 4 1 3 2 } is sorted as follows:
 4 1 3 2 (4 and 1 are out of order so move 1 to the front of the list)
 1 4 3 2 (4 and 3 are out of order so move 3 to the front of the list)
 3 1 4 2 (3 and 1 are out of order so move 1 to the front of the list)
 1 3 4 2 (4 and 2 are out of order so move 2 to the front of the list)
 2 1 3 4 (2 and 1 are out of order so move 1 to the front of the list)
 1 2 3 4 (The list is now sorted)
Let F(L) be the number of times step 2a is executed to sort list L. For example, F({ 4 1 3 2 }) = 5.
Let E(n) be the expected value of F(P) over all permutations P of the integers {1, 2, ..., n}.
You are given E(4) = 3.25 and E(10) = 115.725.
Find E(30). Give your answer rounded to two digits after the decimal point.
Consider the following algorithm for sorting a list:
 1. Starting from the beginning of the list, check each pair of adjacent elements in turn.
 2. If the elements are out of order:
 a. Move the smallest element of the pair at the beginning of the list.
 b. Restart the process from step 1.
 3. If all pairs are in order, stop.
For example, the list { 4 1 3 2 } is sorted as follows:
 4 1 3 2 (4 and 1 are out of order so move 1 to the front of the list)
 1 4 3 2 (4 and 3 are out of order so move 3 to the front of the list)
 3 1 4 2 (3 and 1 are out of order so move 1 to the front of the list)
 1 3 4 2 (4 and 2 are out of order so move 2 to the front of the list)
 2 1 3 4 (2 and 1 are out of order so move 1 to the front of the list)
 1 2 3 4 (The list is now sorted)
Let F(L) be the number of times step 2a is executed to sort list L. For example, F({ 4 1 3 2 }) = 5.
We can list all permutations P of the integers {1, 2, ..., n} in lexicographical order, and assign to each permutation an index I_{n}(P) from 1 to n! corresponding to its position in the list.
Let Q(n, k) = min(I_{n}(P)) for F(P) = k, the index of the first permutation requiring exactly k steps to sort with First Sort. If there is no permutation for which F(P) = k, then Q(n, k) is undefined.
For n = 4 we have:
P  I_{4}(P)  F(P)  

{1, 2, 3, 4}  1  0  Q(4, 0) = 1 
{1, 2, 4, 3}  2  4  Q(4, 4) = 2 
{1, 3, 2, 4}  3  2  Q(4, 2) = 3 
{1, 3, 4, 2}  4  2  
{1, 4, 2, 3}  5  6  Q(4, 6) = 5 
{1, 4, 3, 2}  6  4  
{2, 1, 3, 4}  7  1  Q(4, 1) = 7 
{2, 1, 4, 3}  8  5  Q(4, 5) = 8 
{2, 3, 1, 4}  9  1  
{2, 3, 4, 1}  10  1  
{2, 4, 1, 3}  11  5  
{2, 4, 3, 1}  12  3  Q(4, 3) = 12 
{3, 1, 2, 4}  13  3  
{3, 1, 4, 2}  14  3  
{3, 2, 1, 4}  15  2  
{3, 2, 4, 1}  16  2  
{3, 4, 1, 2}  17  3  
{3, 4, 2, 1}  18  2  
{4, 1, 2, 3}  19  7  Q(4, 7) = 19 
{4, 1, 3, 2}  20  5  
{4, 2, 1, 3}  21  6  
{4, 2, 3, 1}  22  4  
{4, 3, 1, 2}  23  4  
{4, 3, 2, 1}  24  3 
Let R(k) = min(Q(n, k)) over all n for which Q(n, k) is defined.
Find R(12^{12}).
An ellipse E(a, b) is given at its initial position by equation:
$\frac {x^2} {a^2} + \frac {(y  b)^2} {b^2} = 1$
The ellipse rolls without slipping along the x axis for one complete turn. Interestingly, the length of the curve generated by a focus is independent from the size of the minor axis:
$F(a,b) = 2 \pi \text{ } max(a,b)$
This is not true for the curve generated by the ellipse center. Let C(a,b) be the length of the curve generated by the center of the ellipse as it rolls without slipping for one turn.
You are given C(2, 4) ~ 21.38816906.
Find C(1, 4) + C(3, 4). Give your answer rounded to 8 digits behind the decimal point in the form ab.cdefghij.
Let f(n) be the largest prime factor of n.
Let g(n) = f(n) + f(n+1) + f(n+2) + f(n+3) + f(n+4) + f(n+5) + f(n+6) + f(n+7) + f(n+8), the sum of the largest prime factor of each of nine consecutive numbers starting with n.
Let h(n) be the maximum value of g(k) for 2 ≤ k ≤ n.
You are given:
 f(100) = 5
 f(101) = 101
 g(100) = 409
 h(100) = 417
 h(10^{9}) = 4896292593
Find h(10^{16}).
A secret integer t is selected at random within the range 1 ≤ t ≤ n.
The goal is to guess the value of t by making repeated guesses, via integer g. After a guess is made, there are three possible outcomes, in which it will be revealed that either g < t, g = t, or g > t. Then the process can repeat as necessary.
Normally, the number of guesses required on average can be minimized with a binary search: Given a lower bound L and upper bound H (initialized to L = 1 and H = n), let g = ⌊(L+H)/2⌋ where ⌊⋅⌋ is the integer floor function. If g = t, the process ends. Otherwise, if g < t, set L = g+1, but if g > t instead, set H = g−1. After setting the new bounds, the search process repeats, and ultimately ends once t is found. Even if t can be deduced without searching, assume that a search will be required anyway to confirm the value.
Your friend Bob believes that the standard binary search is not that much better than his randomized variant: Instead of setting g = ⌊(L+H)/2⌋, simply let g be a random integer between L and H, inclusive. The rest of the algorithm is the same as the standard binary search. This new search routine will be referred to as a random binary search.
Given that 1 ≤ t ≤ n for random t, let B(n) be the expected number of guesses needed to find t using the standard binary search, and let R(n) be the expected number of guesses needed to find t using the random binary search. For example, B(6) = 2.33333333 and R(6) = 2.71666667 when rounded to 8 decimal places.
Find R(10^{10}) − B(10^{10}) rounded to 8 decimal places.
Let S(n,k,b) represent the number of valid solutions to x_{1} + x_{2} + ... + x_{k} ≤ n, where 0 ≤ x_{m} ≤ b^{m} for all 1 ≤ m ≤ k.
For example, S(14,3,2) = 135, S(200,5,3) = 12949440, and S(1000,10,5) mod 1 000 000 007 = 624839075.
Find (∑_{10 ≤ k ≤ 15} S(10^{k},k,k)) mod 1 000 000 007.
A 10substring of a number is a substring of its digits that sum to 10. For example, the 10substrings of the number 3523014 are:
 3523014
 3523014
 3523014
 3523014
A number is called 10substringfriendly if every one of its digits belongs to a 10substring. For example, 3523014 is 10substringfriendly, but 28546 is not.
Let T(n) be the number of 10substringfriendly numbers from 1 to 10^{n} (inclusive).
For example T(2) = 9 and T(5) = 3492.
Find T(10^{18}) mod 1 000 000 007.
Every divisor d of a number n has a complementary divisor n/d.
Let f(n) be the sum of the greatest common divisor of d and n/d over all positive divisors d of n, that is $f(n)=\displaystyle\sum\limits_{dn}\, \text{gcd}(d,\frac n d)$.
Let F be the summatory function of f, that is $F(k)=\displaystyle\sum\limits_{n=1}^k \, f(n)$.
You are given that F(10)=32 and F(1000)=12776.
Find F(10^{15}).
Let g(a,n,b,m) be the smallest nonnegative solution x to the system:
x = a mod n
x = b mod m
if such a solution exists, otherwise 0.
E.g. g(2,4,4,6)=10, but g(3,4,4,6)=0.
Let φ(n) be Euler's totient function.
Let f(n,m)=g(φ(n),n,φ(m),m)
Find ∑ f(n,m) for 1000000 ≤ n < m < 1005000
Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots as a present.
His nanobots can be programmed to select and locate exactly one other bot precisely and, after activation, move towards this bot along the shortest possible path and draw a coloured line onto the surface while moving. Placed on a plane, the bots will start to move towards their selected bots in a straight line. In contrast, being placed on a ball, they will start to move along a geodesic as the shortest possible path. However, in both cases, whenever their target moves they will adjust their direction instantaneously to the new shortest possible path. All bots will move at the same speed after their simultaneous activation until each bot reaches its goal.
Now Bob places n bots on the ball (with radius 1) equidistantly on a small circle with radius 0.999 and programs each of them to move toward the next nanobot sitting counterclockwise on that small circle. After activation, the bots move in a sort of spiral until they finally meet at one point on the ball.
Using three bots, Bob finds that every bot will draw a line of length 2.84, resulting in a total length of 8.52 for all three bots, each time rounded to two decimal places. The coloured ball looks like this:
In order to show off a little with his presents, Bob decides to use just enough bots to make sure that the line each bot draws is longer than 1000. What is the total length of all lines drawn with this number of bots, rounded to two decimal places?
The Carmichael function λ(n) is defined as the smallest positive integer m such that a^{m} = 1 modulo n for all integers a coprime with n.
For example λ(8) = 2 and λ(240) = 4.
Define L(n) as the smallest positive integer m such that λ(k) ≥ n for all k ≥ m.
For example, L(6) = 241 and L(100) = 20 174 525 281.
Find L(20 000 000). Give the last 9 digits of your answer.
The classical eight queens puzzle is the well known problem of placing eight chess queens on a 8×8 chessboard so that no two queens threaten each other. Allowing configurations to reappear in rotated or mirrored form, a total of 92 distinct configurations can be found for eight queens. The general case asks for the number of distinct ways of placing n queens on a n×n board, e.g. you can find 2 distinct configurations for n=4.
Let's define a weak queen on a n×n board to be a piece which can move any number of squares if moved horizontally, but a maximum of n−1−w squares if moved vertically or diagonally, 0≤w<n being the "weakness factor". For example, a weak queen on a n×n board with a weakness factor of w=1 located in the bottom row will not be able to threaten any square in the top row as the weak queen would need to move n−1 squares vertically or diagonally to get there, but may only move n−2 squares in these directions. In contrast, the weak queen is not handicapped horizontally, thus threatening every square in its own row, independently from its current position in that row.
Let Q(n,w) be the number of ways n weak queens with weakness factor w can be placed on a n×n board so that no two queens threaten each other. It can be shown, for example, that Q(4,0)=2, Q(4,2)=16 and Q(4,3)=256.
Let $S(n)=\displaystyle\sum_{w=0}^{n1} Q(n,w)$.
You are given that S(4)=276 and S(5)=3347.
Find S(14).
Consider the infinite integer sequence S starting with:
S = 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 8, 4, 9, 1, 10, 11, 5, ...
Circle the first occurrence of each integer.
S = ①, 1, ②, 1, ③, 2, ④, 1, ⑤, 3, ⑥, 2, ⑦, ⑧, 4, ⑨, 1, ⑩, ⑪, 5, ...
The sequence is characterized by the following properties:
 The circled numbers are consecutive integers starting with 1.
 Immediately preceding each noncircled numbers a_{i}, there are exactly ⌊√a_{i}⌋ adjacent circled numbers, where ⌊⌋ is the floor function.
 If we remove all circled numbers, the remaining numbers form a sequence identical to S, so S is a fractal sequence.
Let T(n) be the sum of the first n elements of the sequence.
You are given T(1) = 1, T(20) = 86, T(10^{3}) = 364089 and T(10^{9}) = 498676527978348241.
Find T(10^{18}). Give the last 9 digits of your answer.
Let S(n) be the sum of all positive integers m not exceeding n having the following property:
a ^{m+4} ≡ a (mod m) for all integers a.
The values of m ≤ 100 that satisfy this property are 1, 2, 3, 5 and 21, thus S(100) = 1+2+3+5+21 = 32.
You are given S(10^{6}) = 22868117.
Find S(10^{12}).
Let π(x) be the prime counting function, i.e. the number of prime numbers less than or equal to x.
For example, π(1)=0, π(2)=1, π(100)=25.
Let T(n,k) be the number of ktuples (x_{1},…,x_{k}) which satisfy:
1. every x_{i} is a positive integer;
2. $\displaystyle \sum_{i=1}^k \pi(x_i)=n$
For example T(3,3)=19.
The 19 tuples are (1,1,5), (1,5,1), (5,1,1), (1,1,6), (1,6,1), (6,1,1), (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1), (1,2,4), (1,4,2), (2,1,4), (2,4,1), (4,1,2), (4,2,1), (2,2,2).
You are given T(10,10) = 869 985 and T(10^{3},10^{3}) ≡ 578 270 566 (mod 1 004 535 809).
Find T(20 000, 20 000) mod 1 004 535 809.
Consider a positive integer sequence S = (s_{1}, s_{2}, ..., s_{n}).
Let f(S) be the perimeter of the maximumarea quadrilateral whose side lengths are 4 elements (s_{i}, s_{j}, s_{k}, s_{l}) of S (all i, j, k, l distinct). If there are many quadrilaterals with the same maximum area, then choose the one with the largest perimeter.
For example, if S = (8, 9, 14, 9, 27), then we can take the elements (9, 14, 9, 27) and form an isosceles trapezium with parallel side lengths 14 and 27 and both leg lengths 9. The area of this quadrilateral is 127.611470879... It can be shown that this is the largest area for any quadrilateral that can be formed using side lengths from S. Therefore, f(S) = 9 + 14 + 9 + 27 = 59.
Let u_{n} = 2^{B(3n)} + 3^{B(2n)} + B(n+1), where B(k) is the number of 1 bits of k in base 2.
For example, B(6) = 2, B(10) = 2 and B(15) = 4, and u_{5} = 2^{4} + 3^{2} + 2 = 27.
Also, let U_{n} be the sequence (u_{1}, u_{2}, ..., u_{n}).
For example, U_{10} = (8, 9, 14, 9, 27, 16, 36, 9, 27, 28).
It can be shown that f(U_{5}) = 59, f(U_{10}) = 118, f(U_{150}) = 3223.
It can also be shown that ∑ f(U_{n}) = 234761 for 4 ≤ n ≤ 150.
Find ∑ f(U_{n}) for 4 ≤ n ≤ 3 000 000.
Start from an ordered list of all integers from 1 to n. Going from left to right, remove the first number and every other number afterward until the end of the list. Repeat the procedure from right to left, removing the right most number and every other number from the numbers left. Continue removing every other numbers, alternating left to right and right to left, until a single number remains.
Starting with n = 9, we have:
1 2 3 4 5 6 7 8 9
2 4 6 8
2 6
6
Let P(n) be the last number left starting with a list of length n.
Let $\displaystyle S(n) = \sum_{k=1}^n P(k)$.
You are given P(1)=1, P(9) = 6, P(1000)=510, S(1000)=268271.
Find S(10^{18}) mod 987654321.
A Pythagorean triple consists of three positive integers $a, b$ and $c$ satisfying $a^2+b^2=c^2$.
The triple is called primitive if $a, b$ and $c$ are relatively prime.
Let P($n$) be the number of primitive Pythagorean triples with $a < b < c \le n$.
For example P(20) = 3, since there are three triples: (3,4,5), (5,12,13) and (8,15,17).
You are given that P(10^{6}) = 159139.
Find P(3141592653589793).
The n^{th} harmonic number H_{n} is defined as the sum of the multiplicative inverses of the first n positive integers, and can be written as a reduced fraction a_{n}/b_{n}.
$H_n = \displaystyle \sum_{k=1}^n \frac 1 k = \frac {a_n} {b_n}$, with $\text {gcd}(a_n, b_n)=1$.
Let M(p) be the largest value of n such that b_{n} is not divisible by p.
For example, M(3) = 68 because $H_{68} = \frac {a_{68}} {b_{68}} = \frac {14094018321907827923954201611} {2933773379069966367528193600}$, b_{68}=2933773379069966367528193600 is not divisible by 3, but all larger harmonic numbers have denominators divisible by 3.
You are given M(7) = 719102.
Find M(137).
Let S(k) be the sum of three or more distinct positive integers having the following properties:
 No value exceeds k.
 The values form a geometric progression.
 The sum is maximal.
S(4) = 4 + 2 + 1 = 7
S(10) = 9 + 6 + 4 = 19
S(12) = 12 + 6 + 3 = 21
S(1000) = 1000 + 900 + 810 + 729 = 3439
Let $T(n) = \sum_{k=4}^n (1)^k S(k)$.
T(1000) = 2268
Find T(10^{17}).
Define function P(n,k) = 1 if n can be written as the sum of k prime numbers (with repetitions allowed), and P(n,k) = 0 otherwise.
For example, P(10,2) = 1 because 10 can be written as either 3 + 7 or 5 + 5, but P(11,2) = 0 because no two primes can sum to 11.
Let S(n) be the sum of all P(i,k) over 1 ≤ i,k ≤ n.
For example, S(10) = 20, S(100) = 2402, and S(1000) = 248838.
Let F(k) be the kth Fibonacci number (with F(0) = 0 and F(1) = 1).
Find the sum of all S(F(k)) over 3 ≤ k ≤ 44
Let F(r,c,n) be the number of ways to colour a rectangular grid with r rows and c columns using at most n colours such that no two adjacent cells share the same colour. Cells that are diagonal to each other are not considered adjacent.
For example, F(2,2,3) = 18, F(2,2,20) = 130340, and F(3,4,6) = 102923670.
Let S(r,c,n) = $\sum_{k=1}^{n}$ F(r,c,k).
For example, S(4,4,15) mod 10^{9}+7 = 325951319.
Find S(9,10,1112131415) mod 10^{9}+7.
The sum of the k^{th} powers of the first n positive integers can be expressed as a polynomial of degree k+1 with rational coefficients, the Faulhaber's Formulas:
$1^k + 2^k + ... + n^k = \sum_{i=1}^n i^k = \sum_{i=1}^{k+1} a_{i} n^i = a_{1} n + a_{2} n^2 + ... + a_{k} n^k + a_{k+1} n^{k + 1}$,
where a_{i}'s are rational coefficients that can be written as reduced fractions p_{i}/q_{i} (if a_{i} = 0, we shall consider q_{i} = 1).
For example, $1^4 + 2^4 + ... + n^4 = \frac 1 {30} n + \frac 1 3 n^3 + \frac 1 2 n^4 + \frac 1 5 n^5.$
Define D(k) as the value of q_{1} for the sum of k^{th} powers (i.e. the denominator of the reduced fraction a_{1}).
Define F(m) as the m^{th} value of k ≥ 1 for which D(k) = 20010.
You are given D(4) = 30 (since a_{1} = 1/30), D(308) = 20010, F(1) = 308, F(10) = 96404.
Find F(10^{5}).
Define f_{k}(n) = $\sum_{i=0}^{n}$ f_{k}($\lfloor\frac{i}{k}\rfloor$) where f_{k}(0) = 1 and $\lfloor x \rfloor$ denotes the floor function.
For example, f_{5}(10) = 18, f_{7}(100) = 1003, and f_{2}(10^{3}) = 264830889564.
Find $(\sum_{k=2}^{10}$ f_{k}(10^{14})$)$ mod (10^{9}+7).
Assuming that two points are chosen randomly (with uniform distribution) within a rectangle, it is possible to determine the expected value of the distance between these two points.
For example, the expected distance between two random points in a unit square is about 0.521405, while the expected distance between two random points in a rectangle with side lengths 2 and 3 is about 1.317067.
Now we define a hollow square lamina of size n to be an integer sized square with side length n ≥ 3 consisting of n^{2} unit squares from which a rectangle consisting of x × y unit squares (1 ≤ x,y ≤ n  2) within the original square has been removed.
For n = 3 there exists only one hollow square lamina:
For n = 4 you can find 9 distinct hollow square laminae, allowing shapes to reappear in rotated or mirrored form:
Let S(n) be the sum of the expected distance between two points chosen randomly within each of the possible hollow square laminae of size n. The two points have to lie within the area left after removing the inner rectangle, i.e. the graycolored areas in the illustrations above.
For example, S(3) = 1.6514 and S(4) = 19.6564, rounded to four digits after the decimal point.
Find S(40) rounded to four digits after the decimal point.
A gozinta chain for n is a sequence {1,a,b,...,n} where each element properly divides the next.
There are eight gozinta chains for 12:
{1,12} ,{1,2,12}, {1,2,4,12}, {1,2,6,12}, {1,3,12}, {1,3,6,12}, {1,4,12} and {1,6,12}.
Let g(n) be the number of gozinta chains for n, so g(12)=8.
g(48)=48 and g(120)=132.
Find the sum of the numbers n not exceeding 10^{16} for which g(n)=n.
The smallest number m such that 10 divides m! is m=5.
The smallest number m such that 25 divides m! is m=10.
Let s(n) be the smallest number m such that n divides m!.
So s(10)=5 and s(25)=10.
Let S(n) be ∑s(i) for 2 ≤ i ≤ n.
S(100)=2012.
Find S(10^{8}).
Two players are playing a game. There are k piles of stones. When it is his turn a player has to choose a pile and replace it by two piles of stones under the following two conditions:
 Both new piles must have a number of stones more than one and less than the number of stones of the original pile.
 The number of stones of each of the new piles must be a divisor of the number of stones of the original pile.
The first player unable to make a valid move loses.
Let f(n,k) be the number of winning positions for the first player, assuming perfect play, when the game is played with k piles each having between 2 and n stones (inclusively).
f(10,5)=40085.
Find f(10^{7},10^{12}).
Give your answer modulo 987654321.
Let a_{0}, a_{1}, a_{2}, ... be an integer sequence defined by:
 a_{0} = 1;
 for n ≥ 1, a_{n} is the sum of the digits of all preceding terms.
The sequence starts with 1, 1, 2, 4, 8, 16, 23, 28, 38, 49, ...
You are given a_{106} = 31054319.
Find a_{1015}.
Let A_{n} be the smallest positive integer satisfying A_{n} mod p_{i} = i for all 1 ≤ i ≤ n, where p_{i} is the
ith prime.
For example A_{2} = 5, since this is the smallest positive solution of the system of equations
 A_{2} mod 2 = 1
 A_{2} mod 3 = 2
The system of equations for A_{3} adds another constraint. That is, A_{3} is the smallest positive solution of
 A_{3} mod 2 = 1
 A_{3} mod 3 = 2
 A_{3} mod 5 = 3
and hence A_{3} = 23. Similarly, one gets A_{4} = 53 and A_{5} = 1523.
Let S(n) be the sum of all primes up to n that divide at least one element in the sequence A.
For example, S(50) = 69 = 5 + 23 + 41, since 5 divides A_{2}, 23 divides A_{3} and 41 divides A_{10} = 5765999453. No other prime number up to 50 divides an element in A.
Find S(300000).
Let P(n) be the set of the first n positive integers {1, 2, ..., n}.
Let Q(n) be the set of all the nonempty subsets of P(n).
Let R(n) be the set of all the nonempty subsets of Q(n).
An element X ∈ R(n) is a nonempty subset of Q(n), so it is itself a set.
From X we can construct a graph as follows:
 Each element Y ∈ X corresponds to a vertex and labeled with Y;
 Two vertices Y_{1} and Y_{2} are connected if Y_{1} ∩ Y_{2} ≠ ∅.
For example, X = {{1}, {1,2,3}, {3}, {5,6}, {6,7}} results in the following graph:
This graph has two connected components.
Let C(n,k) be the number of elements of R(n) that have exactly k connected components in their graph.
You are given C(2,1) = 6, C(3,1) = 111, C(4,2) = 486, C(100,10) mod 1 000 000 007 = 728209718.
Find C(10^{4},10) mod 1 000 000 007.
On a chess board, a centaur moves like a king or a knight. The diagram below shows the valid moves of a centaur (represented by an inverted king) on an 8x8 board.
It can be shown that at most n^{2} nonattacking centaurs can be placed on a board of size 2n×2n.
Let C(n) be the number of ways to place n^{2} centaurs on a 2n×2n board so that no centaur attacks another directly.
For example C(1) = 4, C(2) = 25, C(10) = 1477721.
Let F_{i} be the i^{th} Fibonacci number defined as F_{1} = F_{2} = 1 and F_{i} = F_{i1} + F_{i2} for i > 2.
Find $\displaystyle \left( \sum_{i=2}^{90} C(F_i) \right) \text{mod } (10^8+7)$.
The McCarthy 91 function is defined as follows: $$ M_{91}(n) = \begin{cases} n  10 & \text{if } n > 100 \\ M_{91}(M_{91}(n+11)) & \text{if } 0 \leq n \leq 100 \end{cases} $$
We can generalize this definition by abstracting away the constants into new variables: $$ M_{m,k,s}(n) = \begin{cases} n  s & \text{if } n > m \\ M_{m,k,s}(M_{m,k,s}(n+k)) & \text{if } 0 \leq n \leq m \end{cases} $$
This way, we have $M_{91} = M_{100,11,10}$.
Let $F_{m,k,s}$ be the set of fixed points of $M_{m,k,s}$. That is, $$F_{m,k,s}= \left\{ n \in \mathbb{N} \,  \, M_{m,k,s}(n) = n \right\}$$
For example, the only fixed point of $M_{91}$ is $n = 91$. In other words, $F_{100,11,10}= \{91\}$.
Now, define $SF(m,k,s)$ as the sum of the elements in $F_{m,k,s}$ and let $S(p,m) = \displaystyle \sum_{1 \leq s < k \leq p}{SF(m,k,s)}$.
For example, $S(10, 10) = 225$ and $S(1000, 1000)=208724467$.
Find $S(10^6, 10^6)$.
A Gaussian integer is a number z = a + bi where a, b are integers and i^{2} = 1.
Gaussian integers are a subset of the complex numbers, and the integers are the subset of Gaussian integers for which b = 0.
A Gaussian integer unit is one for which a^{2} + b^{2} = 1, i.e. one of 1, i, 1, i.
Let's define a proper Gaussian integer as one for which a > 0 and b ≥ 0.
A Gaussian integer z_{1} = a_{1} + b_{1}i is said to be divisible by z_{2} = a_{2} + b_{2}i if z_{3} = a_{3} + b_{3}i = z_{1}/z_{2} is a Gaussian integer.
$\frac {z_1} {z_2} = \frac {a_1 + b_1 i} {a_2 + b_2 i} = \frac {(a_1 + b_1 i)(a_2  b_2 i)} {(a_2 + b_2 i)(a_2  b_2 i)} = \frac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^2} + \frac {a_2 b_1  a_1 b_2} {a_2^2 + b_2^2}i = a_3 + b_3 i$
So, z_{1} is divisible by z_{2} if $\frac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^2}$ and $\frac {a_2 b_1  a_1 b_2} {a_2^2 + b_2^2}$ are integers.
For example, 2 is divisible by 1 + i because 2/(1 + i) = 1  i is a Gaussian integer.
A Gaussian prime is a Gaussian integer that is divisible only by a unit, itself or itself times a unit.
For example, 1 + 2i is a Gaussian prime, because it is only divisible by 1, i, 1, i, 1 + 2i, i(1 + 2i) = i  2, (1 + 2i) = 1  2i and i(1 + 2i) = 2  i.
2 is not a Gaussian prime as it is divisible by 1 + i.
A Gaussian integer can be uniquely factored as the product of a unit and proper Gaussian primes.
For example 2 = i(1 + i)^{2} and 1 + 3i = (1 + i)(2 + i).
A Gaussian integer is said to be squarefree if its prime factorization does not contain repeated proper Gaussian primes.
So 2 is not squarefree over the Gaussian integers, but 1 + 3i is.
Units and Gaussian primes are squarefree by definition.
Let f(n) be the count of proper squarefree Gaussian integers with a^{2} + b^{2} ≤ n.
For example f(10) = 7 because 1, 1 + i, 1 + 2i, 1 + 3i = (1 + i)(2 + i), 2 + i, 3 and 3 + i = i(1 + i)(1 + 2i) are squarefree, while 2 = i(1 + i)^{2} and 2 + 2i = i(1 + i)^{3} are not.
You are given f(10^{2}) = 54, f(10^{4}) = 5218 and f(10^{8}) = 52126906.
Find f(10^{14}).
A triangle is cut into four pieces by two straight lines, each starting at one vertex and ending on the opposite edge. This results in forming three smaller triangular pieces, and one quadrilateral. If the original triangle has an integral area, it is often possible to choose cuts such that all of the four pieces also have integral area. For example, the diagram below shows a triangle of area 55 that has been cut in this way.
Representing the areas as a, b, c and d, in the example above, the individual areas are a = 22, b = 8, c = 11 and d = 14. It is also possible to cut a triangle of area 55 such that a = 20, b = 2, c = 24, d = 9.
Define a triangle cutting quadruple (a, b, c, d) as a valid integral division of a triangle, where a is the area of the triangle between the two cut vertices, d is the area of the quadrilateral and b and c are the areas of the two other triangles, with the restriction that b ≤ c. The two solutions described above are (22,8,11,14) and (20,2,24,9). These are the only two possible quadruples that have a total area of 55.
Define S(n) as the sum of the area of the uncut triangles represented by all valid quadruples with a+b+c+d ≤ n.
For example, S(20) = 259.
Find S(10000).
Let r be the real root of the equation x^{3} = x^{2} + 1.
Every positive integer can be written as the sum of distinct increasing powers of r.
If we require the number of terms to be finite and the difference between any two exponents to be three or more, then the representation is unique.
For example, 3 = r^{ 10} + r^{ 5} + r^{ 1} + r^{ 2} and 10 = r^{ 10} + r^{ 7} + r^{ 6}.
Interestingly, the relation holds for the complex roots of the equation.
Let w(n) be the number of terms in this unique representation of n. Thus w(3) = 4 and w(10) = 3.
More formally, for all positive integers n, we have:
n = $\displaystyle \sum_{k=\infty}^{\infty}$ b_{k} r^{k}
under the conditions that:
b_{k} is 0 or 1 for all k;
b_{k} + b_{k+1} + b_{k+2} ≤ 1 for all k;
w(n) = $\displaystyle \sum_{k=\infty}^{\infty}$ b_{k} is finite.
Let S(m) = $\displaystyle \sum_{j=1}^{m}$ w(j^{2}).
You are given S(10) = 61 and S(1000) = 19403.
Find S(5 000 000).
An ascent of a column j in a matrix occurs if the value of column j is smaller than the value of column j+1 in all rows.
Let P(k, r, n) be the number of r x n matrices with the following properties:
 The rows are permutations of {1, 2, 3, ... , n}.
 Numbering the first column as 1, a column ascent occurs at column j<n if and only if j is not a multiple of k.
For example, P(1, 2, 3) = 19, P(2, 4, 6) = 65508751 and P(7, 5, 30) mod 1000000123 = 161858102.
Let Q(n) =$\, \displaystyle \sum_{k=1}^n\,$ P(k, n, n).For example, Q(5) = 21879393751 and Q(50) mod 1000000123 = 819573537.
Find Q(50000) mod 1000000123.
Coprime Nim is just like ordinary normal play Nim, but the players may only remove a number of stones from a pile that is coprime with the current size of the pile. Two players remove stones in turn. The player who removes the last stone wins.
Let L(n, k) be the number of losing starting positions for the first player, assuming perfect play, when the game is played with k piles, each having between 1 and n  1 stones inclusively.
For example, L(5, 2) = 6 since the losing initial positions are (1, 1), (2, 2), (2, 4), (3, 3), (4, 2) and (4, 4).
You are also given L(10, 5) = 9964, L(10, 10) = 472400303, L(10^{3}, 10^{3}) mod 1 000 000 007 = 954021836.
Find L(10^{7}, 10^{7}) mod 1 000 000 007
Let $S(n)$ be the number of pairs $(a,b)$ of distinct divisors of $n$ such that $a$ divides $b$.
For $n=6$ we get the following pairs: $(1,2), (1,3), (1,6),( 2,6)$ and $(3,6)$. So $S(6)=5$.
Let $p_m\#$ be the product of the first $m$ prime numbers, so $p_2\# = 2*3 = 6$.
Let $E(m, n)$ be the highest integer $k$ such that $2^k$ divides $S((p_m\#)^n)$.
$E(2,1) = 0$ since $2^0$ is the highest power of 2 that divides S(6)=5.
Let $Q(n)=\sum_{i=1}^{n} E(904961, i)$
$Q(8)=2714886$.
Evaluate $Q(10^{12})$.
Construct triangle ABC such that:
 Vertices A, B and C are lattice points inside or on the circle of radius r centered at the origin;
 the triangle contains no other lattice point inside or on its edges;
 the perimeter is maximum.
Let R be the circumradius of triangle ABC and T(r) = R/r.
For r = 5, one possible triangle has vertices (4,3), (4,2) and (1,0) with perimeter $\sqrt{13}+\sqrt{34}+\sqrt{89}$ and circumradius R = $\sqrt {\frac {19669} 2 }$, so T(5) =$\sqrt {\frac {19669} {50} }$.
You are given T(10) ~ 97.26729 and T(100) ~ 9157.64707.
Find T(10^{7}). Give your answer rounded to the nearest integer.
A company specialises in producing large rectangular metal sheets, starting from unit square metal plates. The welding is performed by a range of robots of increasing size. Unfortunately, the programming options of these robots are rather limited. Each one can only process up to 25 identical rectangles of metal, which they can weld along either edge to produce a larger rectangle. The only programmable variables are the number of rectangles to be processed (up to and including 25), and whether to weld the long or short edge.
For example, the first robot could be programmed to weld together 11 raw unit square plates to make a 11×1 strip. The next could take 10 of these 11×1 strips, and weld them either to make a longer 110×1 strip, or a 11×10 rectangle. Many, but not all, possible dimensions of metal sheets can be constructed in this way.
One regular customer has a particularly unusual order. He always demands that the finished product should have an exact area, and that the long side must not be more than 10% larger than the short side. If these requirements can be met in more than one way, in terms of the exact dimensions of the two sides, then he demands that all variants are produced. For example, if he were to ask for metal sheet of area 889200, then there are three final dimensions that can be produced: 900×988, 912×975 and 936×950. The target area of 889200 is the smallest area which can be manufactured in three different variants, within the limitations of the robot welders.
Let M(n) be the minimal area that can be manufactured in exactly n variants with the longer edge not greater than 10% bigger than the shorter edge. Hence M(3) = 889200.
Find $ \sum_{n=2}^{100} M(n)$.
A line segment of length $2n3$ is randomly split into $n$ segments of integer length ($n \ge 3$). In the sequence given by this split, the segments are then used as consecutive sides of a convex $n$polygon, formed in such a way that its area is maximal. All of the $\binom{2n4} {n1}$ possibilities for splitting up the initial line segment occur with the same probability.
Let $E(n)$ be the expected value of the area that is obtained by this procedure.
For example, for $n=3$ the only possible split of the line segment of length $3$ results in three line segments with length $1$, that form an equilateral triangle with an area of $\frac 1 4 \sqrt{3}$. Therefore $E(3)=0.433013$, rounded to $6$ decimal places.
For $n=4$ you can find $4$ different possible splits, each of which is composed of three line segments with length $1$ and one line segment with length $2$. All of these splits lead to the same maximal quadrilateral with an area of $\frac 3 4 \sqrt{3}$, thus $E(4)=1.299038$, rounded to $6$ decimal places.
Let $S(k)=\displaystyle \sum_{n=3}^k E(n)$.
For example, $S(3)=0.433013$, $S(4)=1.732051$, $S(5)=4.604767$ and $S(10)=66.955511$, rounded to $6$ decimal places each.
Find $S(50)$, rounded to $6$ decimal places.
Let $\sigma(n)$ be the sum of the divisors of $n$.
E.g. the divisors of 4 are 1, 2 and 4, so $\sigma(4)=7$.
The numbers $n$ not exceeding 20 such that 7 divides $\sigma(n)$ are: 4,12,13 and 20, the sum of these numbers being 49.
Let $S(n , d)$ be the sum of the numbers $i$ not exceeding $n$ such that $d$ divides $\sigma(i)$.
So $S(20 , 7)=49$.
You are given: $S(10^6,2017)=150850429$ and $S(10^9 , 2017)=249652238344557$.
Find $S(10^{11} , 2017)$
Adam plays the following game with his birthday cake.
He cuts a piece forming a circular sector of 60 degrees and flips the piece upside down, with the icing on the bottom.
He then rotates the cake by 60 degrees counterclockwise, cuts an adjacent 60 degree piece and flips it upside down.
He keeps repeating this, until after a total of twelve steps, all the icing is back on top.
Amazingly, this works for any piece size, even if the cutting angle is an irrational number: all the icing will be back on top after a finite number of steps.
Now, Adam tries something different: he alternates cutting pieces of size $x=\frac{360}{9}$ degrees, $y=\frac{360}{10}$ degrees and $z=\frac{360 }{\sqrt{11}}$ degrees. The first piece he cuts has size x and he flips it. The second has size y and he flips it. The third has size z and he flips it. He repeats this with pieces of size x, y and z in that order until all the icing is back on top, and discovers he needs 60 flips altogether.
Let F(a, b, c) be the minimum number of piece flips needed to get all the icing back on top for pieces of size $x=\frac{360}{a}$ degrees, $y=\frac{360}{b}$ degrees and $z=\frac{360}{\sqrt{c}}$ degrees.
Let $G(n) = \sum_{9 \le a < b < c \le n} F(a,b,c)$, for integers a, b and c.
You are given that F(9, 10, 11) = 60, F(10, 14, 16) = 506, F(15, 16, 17) = 785232.
You are also given G(11) = 60, G(14) = 58020 and G(17) = 1269260.
Find G(53).
Tom has built a random generator that is connected to a row of $n$ light bulbs. Whenever the random generator is activated each of the $n$ lights is turned on with the probability of $\frac 1 2$, independently of its former state or the state of the other light bulbs.
While discussing with his friend Jerry how to use his generator, they invent two different games, they call the reciprocal games:
Both games consist of $n$ turns. Each turn is started by choosing a number $k$ randomly between (and including) $1$ and $n$, with equal probability of $\frac 1 n$ for each number, while the possible win for that turn is the reciprocal of $k$, that is $\frac 1 k$.
In game A, Tom activates his random generator once in each turn. If the number of lights turned on is the same as the previously chosen number $k$, Jerry wins and gets $\frac 1 k$, otherwise he will receive nothing for that turn. Jerry's expected win after playing the total game A consisting of $n$ turns is called $J_A(n)$. For example $J_A(6)=0.39505208$, rounded to 8 decimal places.
For each turn in game B, after $k$ has been randomly selected, Tom keeps reactivating his random generator until exactly $k$ lights are turned on. After that Jerry takes over and reactivates the random generator until he, too, has generated a pattern with exactly $k$ lights turned on. If this pattern is identical to Tom's last pattern, Jerry wins and gets $\frac 1 k$, otherwise he will receive nothing. Jerry's expected win after the total game B consisting of $n$ turns is called $J_B(n)$. For example $J_B(6)=0.43333333$, rounded to 8 decimal places.
Let $\displaystyle S(m)=\sum_{n=1}^m (J_A(n)+J_B(n))$. For example $S(6)=7.58932292$, rounded to 8 decimal places.
Find S(123456789), rounded to 8 decimal places.
Tom has built a random generator that is connected to a row of $n$ light bulbs. Whenever the random generator is activated each of the $n$ lights is turned on with the probability of $\frac 1 2$, independently of its former state or the state of the other light bulbs.
While discussing with his friend Jerry how to use his generator, they invent two different games, they call the reciprocal games:
Both games consist of $n$ turns. Each turn is started by choosing a number $k$ randomly between (and including) $1$ and $n$, with equal probability of $\frac 1 n$ for each number, while the possible win for that turn is the reciprocal of $k$, that is $\frac 1 k$.
In game A, Tom activates his random generator once in each turn. If the number of lights turned on is the same as the previously chosen number $k$, Jerry wins and gets $\frac 1 k$, otherwise he will receive nothing for that turn. Jerry's expected win after playing the total game A consisting of $n$ turns is called $J_A(n)$. For example $J_A(6)=0.39505208$, rounded to 8 decimal places.
For each turn in game B, after $k$ has been randomly selected, Tom keeps reactivating his random generator until exactly $k$ lights are turned on. After that Jerry takes over and reactivates the random generator until he, too, has generated a pattern with exactly $k$ lights turned on. If this pattern is identical to Tom's last pattern, Jerry wins and gets $\frac 1 k$, otherwise he will receive nothing. Jerry's expected win after the total game B consisting of $n$ turns is called $J_B(n)$. For example $J_B(6)=0.43333333$, rounded to 8 decimal places.
Let $D(n)=J_B(n)−J_A(n)$. For example, $D(6) = 0.03828125$.
Find the 7 most significant digits of $D(123456789)$ after removing all leading zeros.
(If, for example, we had asked for the 7 most significant digits of $D(6)$, the answer would have been 3828125.)
A mountain range consists of a line of mountains with slopes of exactly 45°, and heights governed by the prime numbers, p_{n}. The upslope of the k^{th} mountain is of height p_{2k−1}, and the downslope is p_{2k}. The first few foothills of this range are illustrated below.
Tenzing sets out to climb each one in turn, starting from the lowest. At the top of each peak, he looks back and counts how many of the previous peaks he can see. In the example above, the eyeline from the third mountain is drawn in red, showing that he can only see the peak of the second mountain from this viewpoint. Similarly, from the 9^{th} mountain, he can see three peaks, those of the 5^{th}, 7^{th} and 8^{th} mountain.
Let P(k) be the number of peaks that are visible looking back from the k^{th} mountain. Hence P(3)=1 and P(9)=3.
Also $\displaystyle \sum_{k=1}^{100} P(k) = 227$.
Find $\displaystyle \sum_{k=1}^{2500000} P(k)$.
A snowflake of order n is formed by overlaying an equilateral triangle (rotated by 180 degrees) onto each equilateral triangle of the same size in a snowflake of order n1. A snowflake of order 1 is a single equilateral triangle.
Some areas of the snowflake are overlaid repeatedly. In the above picture, blue represents the areas that are one layer thick, red two layers thick, yellow three layers thick, and so on.
For an order n snowflake, let A(n) be the number of triangles that are one layer thick, and let B(n) be the number of triangles that are three layers thick. Define G(n) = gcd(A(n), B(n)).
E.g. A(3) = 30, B(3) = 6, G(3)=6
A(11) = 3027630, B(11) = 19862070, G(11) = 30
Further, G(500) = 186 and $\sum_{n=3}^{500}G(n)=5124$
Find $\displaystyle \sum_{n=3}^{10^7}G(n)$.
A positive number is pandigital in base b if it contains all digits from 0 to b  1 at least once when written in base b.
A nsuperpandigital number is a number that is simultaneously pandigital in all bases from 2 to n inclusively.
For example 978 = 1111010010_{2} = 1100020_{3} = 33102_{4} = 12403_{5} is the smallest 5superpandigital number.
Similarly, 1093265784 is the smallest 10superpandigital number.
The sum of the 10 smallest 10superpandigital numbers is 20319792309.
What is the sum of the 10 smallest 12superpandigital numbers?
A matrix $M$ is called idempotent if $M^2 = M$.
Let $M$ be a three by three matrix :
$M=\begin{pmatrix}
a & b & c\\
d & e & f\\
g &h &i\\
\end{pmatrix}$.
Let C(n) be the number of idempotent three by three matrices $M$ with integer elements such that
$ n \le a,b,c,d,e,f,g,h,i \le n$.
C(1)=164 and C(2)=848.
Find C(200).
$n$ runners in very different training states want to compete in a race. Each one of them is given a different starting number $k$ $(1\leq k \leq n)$ according to his (constant) individual racing speed being $v_k=\frac{k}{n}$.
In order to give the slower runners a chance to win the race, $n$ different starting positions are chosen randomly (with uniform distribution) and independently from each other within the racing track of length $1$. After this, the starting position nearest to the goal is assigned to runner $1$, the next nearest starting position to runner $2$ and so on, until finally the starting position furthest away from the goal is assigned to runner $n$. The winner of the race is the runner who reaches the goal first.
Interestingly, the expected running time for the winner is $\frac{1}{2}$, independently of the number of runners. Moreover, while it can be shown that all runners will have the same expected running time of $\frac{n}{n+1}$, the race is still unfair, since the winning chances may differ significantly for different starting numbers:
Let $P_{n,k}$ be the probability for runner $k$ to win a race with $n$ runners and $E_n = \sum_{k=1}^n k P_{n,k}$ be the expected starting number of the winner in that race. It can be shown that, for example,
$P_{3,1}=\frac{4}{9}$, $P_{3,2}=\frac{2}{9}$, $P_{3,3}=\frac{1}{3}$ and $E_3=\frac{17}{9}$ for a race with $3$ runners.
You are given that $E_4=2.21875$, $E_5=2.5104$ and $E_{10}=3.66021568$.
Find $E_{1000000}$ rounded to 4 digits after the decimal point.
Let $q$ be a prime and $A \ge B >0$ be two integers with the following properties:
 $A$ and $B$ have no prime factor in common, that is $\text{gcd}(A,B)=1$.
 The product $AB$ is divisible by every prime less than q.
It can be shown that, given these conditions, any sum $A+B<q^2$ and any difference $1<AB<q^2$ has to be a prime number. Thus you can verify that a number $p$ is prime by showing that either $p=A+B<q^2$ or $p=AB<q^2$ for some $A,B,q$ fulfilling the conditions listed above.
Let $V(p)$ be the smallest possible value of $A$ in any sum $p=A+B$ and any difference $p=AB$, that verifies $p$ being prime. Examples:
$V(2)=1$, since $2=1+1< 2^2$.
$V(37)=22$, since $37=22+15=2 \cdot 11+3 \cdot 5< 7^2$ is the associated sum with the smallest possible $A$.
$V(151)=165$ since $151=16514=3 \cdot 5 \cdot 11  2 \cdot 7<13^2$ is the associated difference with the smallest possible $A$.
Let $S(n)$ be the sum of $V(p)$ for all primes $p<n$. For example, $S(10)=10$ and $S(200)=7177$.
Find $S(3800)$.
It was quite an ordinary day when a mysterious alien vessel appeared as if from nowhere. After waiting several hours and receiving no response it is decided to send a team to investigate, of which you are included. Upon entering the vessel you are met by a friendly holographic figure, Katharina, who explains the purpose of the vessel, Eulertopia.
She claims that Eulertopia is almost older than time itself. Its mission was to take advantage of a combination of incredible computational power and vast periods of time to discover the answer to life, the universe, and everything. Hence the resident cleaning robot, Leonhard, along with his housekeeping responsibilities, was built with a powerful computational matrix to ponder the meaning of life as he wanders through a massive 1000 by 1000 square grid of rooms. She goes on to explain that the rooms are numbered sequentially from left to right, row by row. So, for example, if Leonhard was wandering around a 5 by 5 grid then the rooms would be numbered in the following way.
Many millenia ago Leonhard reported to Katharina to have found the answer and he is willing to share it with any life form who proves to be worthy of such knowledge.
Katharina further explains that the designers of Leonhard were given instructions to program him with equal probability of remaining in the same room or travelling to an adjacent room. However, it was not clear to them if this meant (i) an equal probability being split equally between remaining in the room and the number of available routes, or, (ii) an equal probability (50%) of remaining in the same room and then the other 50% was to be split equally between the number of available routes.
The records indicate that they decided to flip a coin. Heads would mean that the probability of remaining was dynamically related to the number of exits whereas tails would mean that they program Leonhard with a fixed 50% probability of remaining in a particular room. Unfortunately there is no record of the outcome of the coin, so without further information we would need to assume that there is equal probability of either of the choices being implemented.
Katharina suggests it should not be too challenging to determine that the probability of finding him in a square numbered room in a 5 by 5 grid after unfathomable periods of time would be approximately 0.177976190476 [12 d.p.].
In order to prove yourself worthy of visiting the great oracle you must calculate the probability of finding him in a square numbered room in the 1000 by 1000 lair in which he has been wandering.
(Give your answer rounded to 12 decimal places)
A bouncing point moves counterclockwise along a circle with circumference $1$ with jumps of constant length $l<1$, until it hits a gap of length $g<1$, that is placed in a distance $d$ counterclockwise from the starting point. The gap does not include the starting point, that is $g+d<1$.
Let $S(l,g,d)$ be the sum of the length of all jumps, until the point falls into the gap. It can be shown that $S(l,g,d)$ is finite for any irrational jump size $l$, regardless of the values of $g$ and $d$.
Examples:
$S(\sqrt{\frac 1 2}, 0.06, 0.7)=0.7071 \dots$, $S(\sqrt{\frac 1 2}, 0.06, 0.3543)=1.4142 \dots$ and
$S(\sqrt{\frac 1 2}, 0.06, 0.2427)=16.2634 \dots$.
Let $M(n, g)$ be the maximum of $ \sum S(\sqrt{\frac 1 p}, g, d)$ for all primes $p \le n$ and any valid value of $d$.
Examples:
$M(3, 0.06) =29.5425 \dots$, since $S(\sqrt{\frac 1 2}, 0.06, 0.2427)+S(\sqrt{\frac 1 3}, 0.06, 0.2427)=29.5425 \dots$ is the maximal reachable sum for $g=0.06$.
$M(10, 0.01)=266.9010 \dots$
Find $M(100, 0.00002)$, rounded to 4 decimal places.
An equilateral triangle with integer side length $n \ge 3$ is divided into $n^2$ equilateral triangles with side length 1 as shown in the diagram below.
The vertices of these triangles constitute a triangular lattice with $\frac{(n+1)(n+2)} 2$ lattice points.
Let $H(n)$ be the number of all regular hexagons that can be found by connecting 6 of these points.
For example, $H(3)=1$, $H(6)=12$ and $H(20)=966$.
Find $\displaystyle \sum_{n=3}^{12345} H(n)$.
Any positive integer can be written as a product of prime powers: p_{1}^{a1} × p_{2}^{a2} × ... × p_{k}^{ak},
where p_{i} are distinct prime integers, a_{i} > 0 and p_{i} < p_{j} if i < j.
A decreasing prime power positive integer is one for which a_{i} ≥ a_{j} if i < j.
For example, 1, 2, 15=3×5, 360=2^{3}×3^{2}×5 and 1000=2^{3}×5^{3} are decreasing prime power integers.
Let C(n) be the count of decreasing prime power positive integers not exceeding n.
C(100) = 94 since all positive integers not exceeding 100 have decreasing prime powers except 18, 50, 54, 75, 90 and 98.
You are given C(10^{6}) = 922052.
Find C(10^{13}).
A lattice cube is a cube in which all vertices have integer coordinates. Let C(n) be the number of different lattice cubes in which the coordinates of all vertices range between (and including) 0 and n. Two cubes are hereby considered different if any of their vertices have different coordinates.
For example, C(1)=1, C(2)=9, C(4)=100, C(5)=229, C(10)=4469 and C(50)=8154671.
Different cubes may contain different numbers of lattice points.
For example, the cube with the vertices
(0, 0, 0), (3, 0, 0), (0, 3, 0), (0, 0, 3), (0, 3, 3), (3, 0, 3), (3, 3, 0), (3, 3, 3) contains 64 lattice points (56 lattice points on the surface including the 8 vertices and 8 points within the cube).
In contrast, the cube with the vertices
(0, 2, 2), (1, 4, 4), (2, 0, 3), (2, 3, 0), (3, 2, 5), (3, 5, 2), (4, 1, 1), (5, 3, 3) contains only 40 lattice points (20 points on the surface and 20 points within the cube), although both cubes have the same side length 3.
Let S(n) be the sum of the lattice points contained in the different lattice cubes in which the coordinates of all vertices range between (and including) 0 and n.
For example, S(1)=8, S(2)=91, S(4)=1878, S(5)=5832, S(10)=387003 and S(50)=29948928129.
Find S(5000) mod 10^{9}.
A Hilbert number is any positive integer of the form $4k+1$ for integer $k\geq 0$. We shall define a squarefree Hilbert number as a Hilbert number which is not divisible by the square of any Hilbert number other than one. For example, $117$ is a squarefree Hilbert number, equaling $9\times13$. However $6237$ is a Hilbert number that is not squarefree in this sense, as it is divisible by $9^2$. The number $3969$ is also not squarefree, as it is divisible by both $9^2$ and $21^2$.
There are $2327192$ squarefree Hilbert numbers below $10^7$.
How many squarefree Hilbert numbers are there below $10^{16}$?
A number is psmooth if it has no prime factors larger than p.
Let T be the sequence of triangular numbers, ie T(n)=n(n+1)/2.
Find the sum of all indices n such that T(n) is 47smooth.
Let a, b and c be the sides of an integer sided triangle with one angle of 120 degrees, a≤b≤c and ba≤100.
Let T(n) be the number of such triangles with c≤n.
T(1000)=235 and T(10^{8})=1245.
Find T(10^{100}).
A standard envelope shape is a convex figure consisting of an isosceles triangle (the flap) placed on top of a rectangle. An example of an envelope with integral sides is shown below. Note that to form a sensible envelope, the perpendicular height of the flap (BCD) must be smaller than the height of the rectangle (ABDE).
In the envelope illustrated, not only are all the sides integral, but also all the diagonals (AC, AD, BD, BE and CE) are integral too. Let us call an envelope with these properties a Heron envelope.
Let S(p) be the sum of the perimeters of all the Heron envelopes with a perimeter less than or equal to p.
You are given that S(10^{4}) = 884680. Find S(10^{7}).
A long long time ago in a galaxy far far away, the Wimwians, inhabitants of planet WimWi, discovered an unmanned drone that had landed on their planet. On examining the drone, they uncovered a device that sought the answer for the so called "Birthday Problem". The description of the problem was as follows:
If people on your planet were to enter a very large room one by one, what will be the expected number of people in the room when you first find 3 people with Birthdays within 1 day from each other.
The description further instructed them to enter the answer into the device and send the drone into space again. Startled by this turn of events, the Wimwians consulted their best mathematicians. Each year on Wimwi has 10 days and the mathematicians assumed equally likely birthdays and ignored leap years (leap years in Wimwi have 11 days), and found 5.78688636 to be the required answer. As such, the Wimwians entered this answer and sent the drone back into space.
After traveling light years away, the drone then landed on planet Joka. The same events ensued except this time, the numbers in the device had changed due to some unknown technical issues. The description read:
If people on your planet were to enter a very large room one by one, what will be the expected number of people in the room when you first find 3 people with Birthdays within 7 days from each other.
With a 100day year on the planet, the Jokars (inhabitants of Joka) found the answer to be 8.48967364 (rounded to 8 decimal places because the device allowed only 8 places after the decimal point) assuming equally likely birthdays. They too entered the answer into the device and launched the drone into space again.
This time the drone landed on planet Earth. As before the numbers in the problem description had changed. It read:
If people on your planet were to enter a very large room one by one, what will be the expected number of people in the room when you first find 4 people with Birthdays within 7 days from each other.
What would be the answer (rounded to eight places after the decimal point) the people of Earth have to enter into the device for a year with 365 days? Ignore leap years. Also assume that all birthdays are equally likely and independent of each other.
Consider the term $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$ that is representing a nested square root. $x$, $y$ and $z$ are positive integers and $y$ and $z$ are not allowed to be perfect squares, so the number below the outer square root is irrational. Still it can be shown that for some combinations of $x$, $y$ and $z$ the given term can be simplified into a sum and/or difference of simple square roots of integers, actually denesting the square roots in the initial expression.
Here are some examples of this denesting:
$\small \sqrt{3+\sqrt{2}+\sqrt{2}}=\sqrt{2}+\sqrt{1}=\sqrt{2}+1$
$\small \sqrt{8+\sqrt{15}+\sqrt{15}}=\sqrt{5}+\sqrt{3}$
$\small \sqrt{20+\sqrt{96}+\sqrt{12}}=\sqrt{9}+\sqrt{6}+\sqrt{3}\sqrt{2}=3+\sqrt{6}+\sqrt{3}\sqrt{2}$
$\small \sqrt{28+\sqrt{160}+\sqrt{108}}=\sqrt{15}+\sqrt{6}+\sqrt{5}\sqrt{2}$
As you can see the integers used in the denested expression may also be perfect squares resulting in further simplification.
Let F($n$) be the number of different terms $\small \sqrt{x+\sqrt{y}+\sqrt{z}}$, that can be denested into the sum and/or difference of a finite number of square roots, given the additional condition that $0<x \le n$. That is,
$\small \displaystyle \sqrt{x+\sqrt{y}+\sqrt{z}}=\sum_{i=1}^k s_i\sqrt{a_i}$
with $k$, $x$, $y$, $z$ and all $a_i$ being positive integers, all $s_i =\pm 1$ and $x\le n$.
Furthermore $y$ and $z$ are not allowed to be perfect squares.
Nested roots with the same value are not considered different, for example $\small \sqrt{7+\sqrt{3}+\sqrt{27}}$, $\small \sqrt{7+\sqrt{12}+\sqrt{12}}$ and $\small \sqrt{7+\sqrt{27}+\sqrt{3}}$, that can all three be denested into $\small 2+\sqrt{3}$, would only be counted once.
You are given that F(10)=17, F(15)=46, F(20)=86, F(30)=213 and F(100)=2918 and F(5000)=11134074.
Find F(5000000).
The number 209 can be expressed as $a^2 + 3ab + b^2$ in two distinct ways:
$ \qquad 209 = 8^2 + 3\cdot 8\cdot 5 + 5^2$
$ \qquad 209 = 13^2 + 3\cdot13\cdot 1 + 1^2$
Let $f(n,r)$ be the number of integers $k$ not exceeding $n$ that can be expressed as $k=a^2 + 3ab + b^2$, with $a\gt b>0$ integers, in exactly $r$ different ways.
You are given that $f(10^5, 4) = 237$ and $f(10^8, 6) = 59517$.
Find $f(10^{15}, 40)$.
A square is drawn around a circle as shown in the diagram below on the left.
We shall call the blue shaded region the Lsection.
A line is drawn from the bottom left of the square to the top right as shown in the diagram on the right.
We shall call the orange shaded region a concave triangle.
It should be clear that the concave triangle occupies exactly half of the Lsection.
Two circles are placed next to each other horizontally, a rectangle is drawn around both circles, and a line is drawn from the bottom left to the top right as shown in the diagram below.
This time the concave triangle occupies approximately 36.46% of the Lsection.
If n circles are placed next to each other horizontally, a rectangle is drawn around the n circles, and a line is drawn from the bottom left to the top right, then it can be shown that the least value of n for which the concave triangle occupies less than 10% of the Lsection is n = 15.
What is the least value of n for which the concave triangle occupies less than 0.1% of the Lsection?
The coefficients in the expansion of $(x+1)^k$ are called binomial coefficients.
Analoguously the coefficients in the expansion of $(x^4+x^3+x^2+x+1)^k$ are called quintinomial coefficients.
(quintus= Latin for fifth).
Consider the expansion of $(x^4+x^3+x^2+x+1)^3$:
$x^{12}+3x^{11}+6x^{10}+10x^9+15x^8+18x^7+19x^6+18x^5+15x^4+10x^3+6x^2+3x+1$
As we can see 7 out of the 13 quintinomial coefficients for $k=3$ are odd.
Let $Q(k)$ be the number of odd coefficients in the expansion of $(x^4+x^3+x^2+x+1)^k$.
So $Q(3)=7$.
You are given $Q(10)=17$ and $Q(100)=35$.
Find $\sum_{k=1}^{18}Q(10^k) $.
Christopher Robin and Pooh Bear love the game of Poohsticks so much that they invented a new version which allows them to play for longer before one of them wins and they have to go home for tea. The game starts as normal with both dropping a stick simultaneously on the upstream side of a bridge. But rather than the game ending when one of the sticks emerges on the downstream side, instead they fish their sticks out of the water, and drop them back in again on the upstream side. The game only ends when one of the sticks emerges from under the bridge ahead of the other one having also 'lapped' the other stick  that is, having made one additional journey under the bridge compared to the other stick.
On a particular day when playing this game, the time taken for a stick to travel under the bridge varies between a minimum of 30 seconds, and a maximum of 60 seconds. The time taken to fish a stick out of the water and drop it back in again on the other side is 5 seconds. The current under the bridge has the unusual property that the sticks' journey time is always an integral number of seconds, and it is equally likely to emerge at any of the possible times between 30 and 60 seconds (inclusive). It turns out that under these circumstances, the expected time for playing a single game is 1036.15 seconds (rounded to 2 decimal places). This time is measured from the point of dropping the sticks for the first time, to the point where the winning stick emerges from under the bridge having lapped the other.
The stream flows at different rates each day, but maintains the property that the journey time in seconds is equally distributed amongst the integers from a minimum, $n$, to a maximum, $m$, inclusive. Let the expected time of play in seconds be $E(m,n)$. Hence $E(60,30)=1036.15...$
Let $S(k)=\sum_{m=2}^k\sum_{n=1}^{m1}E(m,n)$.
For example $S(5)=7722.82$ rounded to 2 decimal places.
Find $S(100)$ and give your answer rounded to 2 decimal places.
Let H(n) denote the number of sets of positive integers such that the least common multiple of the integers in the set equals n.
E.g.:
The integers in the following ten sets all have a least common multiple of 6:
{2,3}, {1,2,3}, {6}, {1,6}, {2,6} ,{1,2,6}, {3,6}, {1,3,6}, {2,3,6} and {1,2,3,6}.
Thus H(6)=10.
Let L(n) denote the least common multiple of the numbers 1 through n.
E.g. L(6) is the least common multiple of the numbers 1,2,3,4,5,6 and L(6) equals 60.
Let HL(n) denote H(L(n)).
You are given HL(4)=H(12)=44.
Find HL(50000). Give your answer modulo 10^{9}.
Given a nonsquare integer $d$, any real $x$ can be approximated arbitrarily close by quadratic integers $a+b\sqrt{d}$, where $a,b$ are integers. For example, the following inequalities approximate $\pi$ with precision $10^{13}$:
$$4375636191520\sqrt{2}6188084046055 < \pi < 721133315582\sqrt{2}1019836515172 $$
We call $BQA_d(x,n)$ the quadratic integer closest to $x$ with the absolute values of $a,b$ not exceeding $n$.
We also define the integral part of a quadratic integer as $I_d(a+b\sqrt{d}) = a$.
You are given that:
 $BQA_2(\pi,10) = 6  2\sqrt{2}$
 $BQA_5(\pi,100)=26\sqrt{5}55$
 $BQA_7(\pi,10^6)=560323  211781\sqrt{7}$
 $I_2(BQA_2(\pi,10^{13}))=6188084046055$
Find the sum of $I_d(BQA_d(\pi,10^{13}))$ for all nonsquare positive integers less than 100.
For any N, let f(N) be the last twelve hexadecimal digits before the trailing zeroes in N!.
For example, the hexadecimal representation of 20! is 21C3677C82B40000,
so f(20) is the digit sequence 21C3677C82B4.
Find f(20!). Give your answer as twelve hexadecimal digits, using uppercase for the digits A to F.
We define two sequences $S = \{S(1), S(2), ..., S(n)\}$ and $S_2 = \{S_2(1), S_2(2), ..., S_2(n)\}$:
$S(k) = (p_k)^k$ mod $10007$ where $p_k$ is the $k$th prime number.
$S_2(k) = S(k) + S(\lfloor\frac{k}{10000}\rfloor + 1)$ where $\lfloor \cdot \rfloor$ denotes the floor function.
Then let $M(i, j)$ be the median of elements $S_2(i)$ through $S_2(j)$, inclusive. For example, $M(1, 10) = 2021.5$ and $M(10^2, 10^3) = 4715.0$.
Let $F(n, k) = \sum_{i=1}^{nk+1} M(i, i + k  1)$. For example, $F(100, 10) = 463628.5$ and $F(10^5, 10^4) = 675348207.5$.
Find $F(10^7, 10^5)$. If the sum is not an integer, use $.5$ to denote a half. Otherwise, use $.0$ instead.
For a polygon $P$, let $t(P)$ be the number of ways in which $P$ can be tiled using rhombi and squares with edge length 1. Distinct rotations and reflections are counted as separate tilings.
For example, if $O$ is a regular octagon with edge length 1, then $t(O) = 8$. As it happens, all these 8 tilings are rotations of one another:
Let $O_{a,b}$ be the equalangled convex octagon whose edges alternate in length between $a$ and $b$.
For example, here is $O_{2,1}$, with one of its tilings:
You are given that $t(O_{1,1})=8$, $t(O_{2,1})=76$ and $t(O_{3,2})=456572$.
Find $t(O_{4,2})$.
A deck of cards numbered from 1 to n is shuffled randomly such that each permutation is equally likely.
The cards are to be sorted into ascending order using the following technique:
 Look at the initial sequence of cards. If it is already sorted, then there is no need for further action. Otherwise, if any subsequences of cards happen to be in the correct place relative to one another (ascending with no gaps), then those subsequences are fixed by attaching the cards together. For example, with 7 cards initially in the order 4123756, the cards labelled 1, 2 and 3 would be attached together, as would 5 and 6.
 The cards are 'shuffled' by being thrown into the air, but note that any correctly sequenced cards remain attached, so their orders are maintained. The cards (or bundles of attached cards) are then picked up randomly. You should assume that this randomisation is unbiased, despite the fact that some cards are single, and others are grouped together.
 Repeat steps 1 and 2 until the cards are sorted.
Let S(n) be the expected number of shuffles needed to sort the cards. Since the order is checked before the first shuffle, S(1) = 0. You are given that S(2) = 1, and S(5) = 4213/871.
Find S(52), and give your answer rounded to 8 decimal places.
Let T(r) be the number of integer quadruplets x, y, z, t such that x^{2} + y^{2} + z^{2} + t^{2} ≤ r^{2}. In other words, T(r) is the number of lattice points in the fourdimensional hyperball of radius r.
You are given that T(2) = 89, T(5) = 3121, T(100) = 493490641 and T(10^{4}) = 49348022079085897.
Find T(10^{8}) mod 1000000007.
 A division consists of $n$ boats (typically 13), placed in order based on past performance.
 All boats within a division start at 40 metre intervals along the river, in order with the highestplaced boat starting furthest upstream.
 The boats all start rowing simultaneously, upstream, trying to catch the boat in front while avoiding being caught by boats behind.
 Each boat continues rowing until either it reaches the finish line or it catches up with ("bumps") a boat in front.
 The finish line is a distance $L$ metres (the course length, in reality about 1800 metres) upstream from the starting position of the lowestplaced boat. (Because of the staggered starting positions, higherplaced boats row a slightly shorter course than lowerplaced boats.)
 When a "bump" occurs, the "bumping" boat takes no further part in the race. The "bumped" boat must continue, however, and may even be "bumped" again by boats that started two or more places behind it.

After the race, boats are assigned new places within the division, based on the bumps that occurred. Specifically, for any boat $A$ that started in a lower place than $B$, then $A$ will be placed higher than $B$ in the new order if and only if one of the following occurred:
 $A$ bumped $B$ directly
 $A$ bumped another boat that went on to bump $B$
 $A$ bumped another boat, that bumped yet another boat, that bumped $B$
 etc
Suppose that, in a particular race, each boat $B_j$ rows at a steady speed $v_j = $log$X_j$ metres per second, where the $X_j$ are chosen randomly (with uniform distribution) between 0 and 1, independently from one another. These speeds are relative to the riverbank: you may disregard the flow of the river.
Let $p(n,L)$ be the probability that the new order is an even permutation of the starting order, when there are $n$ boats in the division and $L$ is the course length.
For example, with $n=3$ and $L=160$, labelling the boats as $A$,$B$,$C$ in starting order with $C$ highest, the different possible outcomes of the race are as follows:
Bumps occurring  New order  Permutation  Probability 

none  $A$, $B$, $C$  even  $4/15$ 
$B$ bumps $C$  $A$, $C$, $B$  odd  $8/45$ 
$A$ bumps $B$  $B$, $A$, $C$  odd  $1/3$ 
$B$ bumps $C$, then $A$ bumps $C$  $C$, $A$, $B$  even  $4/27$ 
$A$ bumps $B$, then $B$ bumps $C$  $C$, $B$, $A$  odd  $2/27$ 
Therefore, $p(3,160) = 4/15 + 4/27 = 56/135$.
You are also given that $p(4,400)=0.5107843137$, rounded to 10 digits after the decimal point.
Find $p(13,1800)$ rounded to 10 digits after the decimal point.
Consider the number 48.
There are five pairs of integers $a$ and $b$ ($a \leq b$) such that $a \times b=48$: (1,48), (2,24), (3,16), (4,12) and (6,8).
It can be seen that both 6 and 8 have 4 divisors.
So of those five pairs one consists of two integers with the same number of divisors.
In general:
Let $C(n)$ be the number of pairs of positive integers $a \times b=n$, ($a \leq b$) such that $a$ and $b$ have the same number of divisors;
so $C(48)=1$.
You are given $C(10!)=3$: (1680, 2160), (1800, 2016) and (1890,1920).
Find $C(100!)$
The wellknown Rubik's Cube puzzle has many fascinating mathematical properties. The 2×2×2 variant has 8 cubelets with a total of 24 visible faces, each with a coloured sticker. Successively turning faces will rearrange the cubelets, although not all arrangements of cubelets are reachable without dismantling the puzzle.
Suppose that we wish to apply new stickers to a 2×2×2 Rubik's cube in a nonstandard colouring. Specifically, we have $n$ different colours available (with an unlimited supply of stickers of each colour), and we place one sticker on each of the 24 faces in any arrangement that we please. We are not required to use all the colours, and if desired the same colour may appear in more than one face of a single cubelet.
We say that two such colourings $c_1,c_2$ are essentially distinct if a cube coloured according to $c_1$ cannot be made to match a cube coloured according to $c_2$ by performing mechanically possible Rubik's Cube moves.
For example, with two colours available, there are 183 essentially distinct colourings.
How many essentially distinct colourings are there with 10 different colours available?
Let H(n) be the number of distinct integer sided equiangular convex hexagons with perimeter not exceeding n.
Hexagons are distinct if and only if they are not congruent.
You are given H(6) = 1, H(12) = 10, H(100) = 31248.
Find H(55106).
Equiangular hexagons with perimeter not exceeding 12
For every positive number $n$ we define the function $streak(n)=k$ as the smallest positive integer $k$ such that $n+k$ is not divisible by $k+1$.
E.g:
13 is divisible by 1
14 is divisible by 2
15 is divisible by 3
16 is divisible by 4
17 is NOT divisible by 5
So $streak(13) = 4$.
Similarly:
120 is divisible by 1
121 is NOT divisible by 2
So $streak(120) = 1$.
Define $P(s, N)$ to be the number of integers $n$, $1 < n < N$, for which $streak(n) = s$.
So $P(3, 14) = 1$ and $P(6, 10^6) = 14286$.
Find the sum, as $i$ ranges from 1 to 31, of $P(i, 4^i)$.
Alice enlists the help of some friends to generate a random number, using a single unfair coin. She and her friends sit around a table and, starting with Alice, they take it in turns to toss the coin. Everyone keeps a count of how many heads they obtain individually. The process ends as soon as Alice obtains a Head. At this point, Alice multiplies all her friends' Head counts together to obtain her random number.
As an illustration, suppose Alice is assisted by Bob, Charlie, and Dawn, who are seated round the table in that order, and that they obtain the sequence of Head/Tail outcomes THHH—TTTT—THHT—H beginning and ending with Alice. Then Bob and Charlie each obtain 2 heads, and Dawn obtains 1 head. Alice's random number is therefore $2\times 2\times 1 = 4$.
Define $e(n, p)$ to be the expected value of Alice's random number, where $n$ is the number of friends helping (excluding Alice herself), and $p$ is the probability of the coin coming up Tails.
It turns out that, for any fixed $n$, $e(n, p)$ is always a polynomial in $p$. For example, $e(3, p) = p^3 + 4p^2 + p$.
Define $c(n, k)$ to be the coefficient of $p^k$ in the polynomial $e(n, p)$. So $c(3, 1) = 1$, $c(3, 2) = 4$, and $c(3, 3) = 1$.
You are given that $c(100, 40) \equiv 986699437 \text{ } (\text{mod } 10^9+7)$.
Find $c(10000000, 4000000) \mod 10^9+7$.
Let $S(n)$ be the sum of all contiguous integersubstrings that can be formed from the integer $n$. The substrings need not be distinct.
For example, $S(2024) = 2 + 0 + 2 + 4 + 20 + 02 + 24 + 202 + 024 + 2024 = 2304$.
Let $P(n)$ be the integer formed by concatenating the first $n$ primes together. For example, $P(7) = 2357111317$.
Let $C(n, k)$ be the integer formed by concatenating $k$ copies of $P(n)$ together. For example, $C(7, 3) = 235711131723571113172357111317$.
Evaluate $S(C(10^6, 10^{12}))$ mod $(10^9 + 7)$.
Let $F(N)$ be the maximum number of lattice points in an axisaligned $N\times N$ square that the graph of a single strictly convex increasing function can pass through.
You are given that $F(1) = 2$, $F(3) = 3$, $F(9) = 6$, $F(11) = 7$, $F(100) = 30$ and $F(50000) = 1898$.
Below is the graph of a function reaching the maximum 3 for $N=3$:
Find $F(10^{18})$.
Consider an $n$player game played in consecutive pairs: Round $1$ takes place between players $1$ and $2$, round $2$ takes place between players $2$ and $3$, and so on and so forth, all the way up to round $n$, which takes place between players $n$ and $1$. Then round $n+1$ takes place between players $1$ and $2$ as the entire cycle starts again.
In other words, during round $r$, player $((r1) \bmod n) + 1$ faces off against player $(r \bmod n) + 1$.
During each round, a fair coin is tossed to decide which of the two players wins that round. If any given player wins both rounds $r$ and $r+1$, then that player wins the entire game.
Let $P_n(k)$ be the probability that player $k$ wins in an $n$player game, in the form of a reduced fraction. For example, $P_3(1) = 12/49$ and $P_6(2) = 368/1323$.
Let $M_n(k)$ be the product of the reduced numerator and denominator of $P_n(k)$. For example, $M_3(1) = 588$ and $M_6(2) = 486864$.
Find the last $8$ digits of $M_{10^8+7}(10^4+7)$.
A gozinta chain for n is a sequence {1,a,b,...,n} where each element properly divides the next.
For example, there are eight distinct gozinta chains for 12:
{1,12}, {1,2,12}, {1,2,4,12}, {1,2,6,12}, {1,3,12}, {1,3,6,12}, {1,4,12} and {1,6,12}.
Let S(n) be the sum of all numbers, k, not exceeding n, which have 252 distinct gozinta chains.
You are given S(10^{6})=8462952 and S(10^{12})=623291998881978.
Find S(10^{36}), giving the last nine digits of your answer.
Frodo and Sam need to travel 100 leagues due East from point A to point B. On normal terrain, they can cover 10 leagues per day, and so the journey would take 10 days. However, their path is crossed by a long marsh which runs exactly SouthWest to NorthEast, and walking through the marsh will slow them down. The marsh is 50 leagues wide at all points, and the midpoint of AB is located in the middle of the marsh. A map of the region is shown in the diagram below:
The marsh consists of 5 distinct regions, each 10 leagues across, as shown by the shading in the map. The strip closest to point A is relatively light marsh, and can be crossed at a speed of 9 leagues per day. However, each strip becomes progressively harder to navigate, the speeds going down to 8, 7, 6 and finally 5 leagues per day for the final region of marsh, before it ends and the terrain becomes easier again, with the speed going back to 10 leagues per day.
If Frodo and Sam were to head directly East for point B, they would travel exactly 100 leagues, and the journey would take approximately 13.4738 days. However, this time can be shortened if they deviate from the direct path.
Find the shortest possible time required to travel from point A to B, and give your answer in days, rounded to 10 decimal places.
Let $D(m,n)=\displaystyle\sum_{dm}\sum_{k=1}^n\sigma_{\small 0}(kd)$ where $d$ runs through all divisors of $m$ and $\sigma_{\small 0}(n)$ is the number of divisors of $n$.
You are given $D(3!,10^2)=3398$ and $D(4!,10^6)=268882292$.
Find $D(200!,10^{12}) \text{ mod } (10^9 + 7)$.
For every $n \ge 1$ the primecounting function $\pi(n)$ is equal to the number of primes
not exceeding $n$.
E.g. $\pi(6)=3$ and $\pi(100)=25$.
We say that a sequence of integers $u = (u_0,\cdots,u_m)$ is a $\pi$ sequence if
 $u_n \ge 1$ for every $n$
 $u_{n+1}= \pi(u_n)$
 $u$ has two or more elements
For $u_0=10$ there are three distinct $\pi$ sequences: (10,4), (10,4,2) and (10,4,2,1).
Let $c(u)$ be the number of elements of $u$ that are not prime.
Let $p(n,k)$ be the number of $\pi$ sequences $u$ for which $u_0\le n$ and $c(u)=k$.
Let $P(n)$ be the product of all $p(n,k)$ that are larger than 0.
You are given: P(10)=3×8×9×3=648 and P(100)=31038676032.
Find $P(10^8)$. Give your answer modulo 1000000007.
A random generator produces a sequence of symbols drawn from the set {I, V, X, L, C, D, M, #}. Each item in the sequence is determined by selecting one of these symbols at random, independently of the other items in the sequence. At each step, the seven letters are equally likely to be selected, with probability 14% each, but the # symbol only has a 2% chance of selection.
We write down the sequence of letters from left to right as they are generated, and we stop at the first occurrence of the # symbol (without writing it). However, we stipulate that what we have written down must always (when nonempty) be a valid Roman numeral representation in minimal form. If appending the next letter would contravene this then we simply skip it and try again with the next symbol generated.
Please take careful note of About... Roman Numerals for the definitive rules for this problem on what constitutes a "valid Roman numeral representation" and "minimal form". For example, the (only) sequence that represents 49 is XLIX. The subtractive combination IL is invalid because of rule (ii), while XXXXIX is valid but not minimal. The rules do not place any restriction on the number of occurrences of M, so all integers have a valid representation. These are the same rules as were used in Problem 89, and members are invited to solve that problem first.
Find the expected value of the number represented by what we have written down when we stop. (If nothing is written down then count that as zero.) Give your answer rounded to 8 places after the decimal point.
Peter moves in a hallway with N+1 doors consecutively numbered from 0 through N. All doors are initially closed. Peter starts in front of door 0, and repeatedly performs the following steps:
 First, he walks a positive square number of doors away from his position.
 Then he walks another, larger square number of doors away from his new position.
 He toggles the door he faces (opens it if closed, closes it if open).
 And finally returns to door 0.
We call an action any sequence of those steps. Peter never performs the exact same action twice, and makes sure to perform all possible actions that don't bring him past the last door.
Let F(N) be the number of doors that are open after Peter has performed all possible actions. You are given that F(5) = 1, F(100) = 27, F(1000) = 233 and F(10^{6}) = 112168.
Find F(10^{12}).
Let's call two numbers friend numbers if their representation in base 10 has at least one common digit.
E.g. 1123 and 3981 are friend numbers.
Let $f(n)$ be the number of pairs $(p,q)$ with $1\le p \lt q \lt n$ such that $p$ and $q$ are friend numbers.
$f(100)=1539$.
Find $f(10^{18})$ mod $1000267129$.
Dave is doing his homework on the balcony and, preparing a presentation about Pythagorean triangles, has just cut out a triangle with side lengths 30cm, 40cm and 50cm from some cardboard, when a gust of wind blows the triangle down into the garden.
Another gust blows a small ant straight onto this triangle. The poor ant is completely disoriented and starts to crawl straight ahead in random direction in order to get back into the grass.
Assuming that all possible positions of the ant within the triangle and all possible directions of moving on are equiprobable, what is the probability that the ant leaves the triangle along its longest side?
Give your answer rounded to 10 digits after the decimal point.
An integer partition of a number $n$ is a way of writing $n$ as a sum of positive integers. Partitions that differ only by the order of their summands are considered the same.
We call an integer partition special if 1) all its summands are distinct, and 2) all its even summands are also divisible by 4.
For example, the special partitions of $10$ are: \[10 = 1+4+5=3+7=1+9\]
The number $10$ admits many more integer partitions (a total of 42), but only those three are special.
Let be $P(n)$ the number of special integer partitions of $n$. You are given that $P(1) = 1$, $P(2) = 0$, $P(3) = 1$, $P(6) = 1$, $P(10)=3$, $P(100) = 37076$ and $P(1000)=3699177285485660336$.
Find $\displaystyle \sum_{i=1}^{10^7}{P(i)}$. Give the result modulo $10^9+7$.
Consider the natural numbers having at least 5 prime factors, which don't have to be distinct.
Sorting these numbers by size gives a list which starts with:
 32=2⋅2⋅2⋅2⋅2
 48=2⋅2⋅2⋅2⋅3
 64=2⋅2⋅2⋅2⋅2⋅2
 72=2⋅2⋅2⋅3⋅3
 80=2⋅2⋅2⋅2⋅5
 96=2⋅2⋅2⋅2⋅2⋅3
 ...
So, for example, the fifth number with at least 5 prime factors is 80.
Find the millionth number with at least one million prime factors.
Give your answer modulo 123454321.
Alice plays the following game, she starts with a list of integers $L$ and on each step she can either:
 remove two elements $a$ and $b$ from $L$ and add $a^b$ to $L$
 or conversely remove an element $c$ from $L$ that can be written as $a^b$, with $a$ and $b$ being two integers such that $a, b > 1$, and add both $a$ and $b$ to $L$
For example starting from the list $L=\{8\}$, Alice can remove $8$ and add $2$ and $3$ resulting in $L=\{2,3\}$ in a first step. Then she can obtain $L=\{9\}$ in a second step.
Note that the same integer is allowed to appear multiple times in the list.
An integer $n>1$ is said to be creative if for any integer $m>1$ Alice can obtain a list that contains $m$ starting from $L=\{n\}$.
Find the sum of all creative integers less than or equal to $10^{12}$.
For two integers $n,e > 1$, we define a $(n,e)$MPS (Mirror Power Sequence) to be an infinite sequence of integers $(a_i)_{i\ge 0}$ such that for all $i\ge 0$, $a_{i+1} = min(a_i^e,na_i^e)$ and $a_i > 1$.
Examples of such sequences are the two $(18,2)$MPS sequences made of alternating $2$ and $4$.
Note that even though such a sequence is uniquely determined by $n,e$ and $a_0$, for most values such a sequence does not exist. For example, no $(n,e)$MPS exists for $n < 6$.
Define $C(n)$ to be the number of $(n,e)$MPS for some $e$, and $\displaystyle D(N) = \sum_{n=2}^N {C(n)}$.
You are given that $D(10) = 2$, $D(100) = 21$, $D(1000) = 69$, $D(10^6) = 1303$ and $D(10^{12}) = 1014800$.
Find $D(10^{18})$.
Consider the numbers 15, 16 and 18:
$15=3\times 5$ and $3+5=8$.
$16 = 2\times 2\times 2\times 2$ and $2+2+2+2=8$.
$18 = 2\times 3\times 3$ and $2+3+3=8$.
15, 16 and 18 are the only numbers that have 8 as sum of the prime factors (counted with multiplicity).
We define $S(k)$ to be the sum of all numbers $n$ where the sum of the prime factors (with multiplicity) of $n$ is $k$.
Hence $S(8) = 15+16+18 = 49$.
Other examples: $S(1) = 0$, $S(2) = 2$, $S(3) = 3$, $S(5) = 5 + 6 = 11$.
The Fibonacci sequence is $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, F_5 = 5$, ....
Find the last nine digits of $\displaystyle\sum_{k=2}^{24}{S(F_k)}$.
For a set of positive integers $\{a, a+1, a+2, \dots , b\}$, let $C(a,b)$ be the number of nonempty subsets in which the product of all elements is a perfect square.
For example $C(5,10)=3$, since the products of all elements of $\{5, 8, 10\}$, $\{5, 8, 9, 10\}$ and $\{9\}$ are perfect squares, and no other subsets of $\{5, 6, 7, 8, 9, 10\}$ have this property.
You are given that $C(40,55) =15$, and $C(1000,1234) \text{ mod } 1000000007=975523611$.
Find $C(1000000,1234567) \text{ mod } 1000000007$.
A circle $C$ of circumference $c$ centimetres has a smaller circle $S$ of circumference $s$ centimetres lying offcentre within it. Four other distinct circles, which we call "planets", with circumferences $p$, $p$, $q$, $q$ centimetres respectively ($p<q$), are inscribed within $C$ but outside $S$, with each planet touching both $C$ and $S$ tangentially. The planets are permitted to overlap one another, but the boundaries of $S$ and $C$ must be at least 1cm apart at their closest point.
Now suppose that these circles are actually gears with perfectly meshing teeth at a pitch of 1cm. $C$ is an internal gear with teeth on the inside. We require that $c$, $s$, $p$, $q$ are all integers (as they are the numbers of teeth), and we further stipulate that any gear must have at least 5 teeth.
Note that "perfectly meshing" means that as the gears rotate, the ratio between their angular velocities remains constant, and the teeth of one gear perfectly align with the groves of the other gear and vice versa. Only for certain gear sizes and positions will it be possible for $S$ and $C$ each to mesh perfectly with all the planets. Arrangements where not all gears mesh perfectly are not valid.
Define $g(c,s,p,q)$ to be the number of such gear arrangements for given values of $c$, $s$, $p$, $q$: it turns out that this is finite as only certain discrete arrangements are possible satisfying the above conditions. For example, $g(16,5,5,6)=9$.
Here is one such arrangement:
Let $G(n) = \sum_{s+p+q\le n} g(s+p+q,s,p,q)$ where the sum only includes cases with $p<q$, $p\ge 5$, and $s\ge 5$, all integers. You are given that $G(16)=9$ and $G(20)=205$.
Find $G(500)$.
Gauss famously proved that every positive integer can be expressed as the sum of three triangular numbers (including 0 as the lowest triangular number). In fact most numbers can be expressed as a sum of three triangular numbers in several ways.
Let $G(n)$ be the number of ways of expressing $n$ as the sum of three triangular numbers, regarding different arrangements of the terms of the sum as distinct.
For example, $G(9) = 7$, as 9 can be expressed as: 3+3+3, 0+3+6, 0+6+3, 3+0+6, 3+6+0, 6+0+3, 6+3+0.
You are given $G(1000) = 78$ and $G(10^6) = 2106$.
Find $G(17 526 \times 10^9)$.
A riffle shuffle is executed as follows: a deck of cards is split into two equal halves, with the top half taken in the left hand and the bottom half taken in the right hand. Next, the cards are interleaved exactly, with the top card in the right half inserted just after the top card in the left half, the 2nd card in the right half just after the 2nd card in the left half, etc. (Note that this process preserves the location of the top and bottom card of the deck)
Let $s(n)$ be the minimum number of consecutive riffle shuffles needed to restore a deck of size $n$ to its original configuration, where $n$ is a positive even number.
Amazingly, a standard deck of $52$ cards will first return to its original configuration after only $8$ perfect shuffles, so $s(52) = 8$. It can be verified that a deck of $86$ cards will also return to its original configuration after exactly $8$ shuffles, and the sum of all values of $n$ that satisfy $s(n) = 8$ is $412$.
Find the sum of all values of n that satisfy $s(n) = 60$.
The lambdacalculus is a universal model of computation at the core of functional programming languages. It is based on lambdaterms, a minimal programming language featuring only function definitions, function calls and variables. Lambdaterms are built according to the following rules:
 Any variable $x$ (single letter, from some infinite alphabet) is a lambdaterm.
 If $M$ and $N$ are lambdaterms, then $(M N)$ is a lambdaterm, called the application of $M$ to $N$.
 If $x$ is a variable and $M$ is a term, then $(\lambda x. M)$ is a lambdaterm, called an abstraction. An abstraction defines an anonymous function, taking $x$ as parameter and sending back $M$.
A lambdaterm $T$ is said to be closed if for all variables $x$, all occurrences of $x$ within $T$ are contained within some abstraction $(\lambda x. M)$ in $T$. The smallest such abstraction is said to bind the occurrence of the variable $x$. In other words, a lambdaterm is closed if all its variables are bound to parameters of enclosing functions definitions. For example, the term $(\lambda x. x)$ is closed, while the term $(\lambda x. (x y))$ is not because $y$ is not bound.
Also, we can rename variables as long as no binding abstraction changes. This means that $(\lambda x. x)$ and $(\lambda y. y)$ should be considered equivalent since we merely renamed a parameter. Two terms equivalent modulo such renaming are called $\alpha$equivalent. Note that $(\lambda x. (\lambda y. (x y)))$ and $(\lambda x. (\lambda x. (x x)))$ are not $\alpha$equivalent, since the abstraction binding the first variable was the outer one and becomes the inner one. However, $(\lambda x. (\lambda y. (x y)))$ and $(\lambda y. (\lambda x. (y x)))$ are $\alpha$equivalent.
The following table regroups the lambdaterms that can be written with at most $15$ symbols, symbols being parenthesis, $\lambda$, dot and variables.
\[\begin{array}{cccc} \hline (\lambda x.x) & (\lambda x.(x x)) & (\lambda x.(\lambda y.x)) & (\lambda x.(\lambda y.y)) \\ \hline (\lambda x.(x (x x))) & (\lambda x.((x x) x)) & (\lambda x.(\lambda y.(x x))) & (\lambda x.(\lambda y.(x y))) \\ \hline (\lambda x.(\lambda y.(y x))) & (\lambda x.(\lambda y.(y y))) & (\lambda x.(x (\lambda y.x))) & (\lambda x.(x (\lambda y.y))) \\ \hline (\lambda x.((\lambda y.x) x)) & (\lambda x.((\lambda y.y) x)) & ((\lambda x.x) (\lambda x.x)) & (\lambda x.(x (x (x x)))) \\ \hline (\lambda x.(x ((x x) x))) & (\lambda x.((x x) (x x))) & (\lambda x.((x (x x)) x)) & (\lambda x.(((x x) x) x)) \\ \hline \end{array}\]Let be $\Lambda(n)$ the number of distinct closed lambdaterms that can be written using at most $n$ symbols, where terms that are $\alpha$equivalent to one another should be counted only once. You are given that $\Lambda(6) = 1$, $\Lambda(9) = 2$, $\Lambda(15) = 20$ and $\Lambda(35) = 3166438$.
Find $\Lambda(2000)$. Give the answer modulo $1\,000\,000\,007$.
An unbiased coin is tossed repeatedly until two consecutive heads are obtained. Suppose these occur on the $(M1)$th and $M$th toss.
Let $P(n)$ be the probability that $M$ is divisible by $n$. For example, the outcomes HH, HTHH, and THTTHH all count towards $P(2)$, but THH and HTTHH do not.
You are given that $P(2) =\frac 3 5$ and $P(3)=\frac 9 {31}$. Indeed, it can be shown that $P(n)$ is always a rational number.
For a prime $p$ and a fully reduced fraction $\frac a b$, define $Q(\frac a b,p)$ to be the smallest positive $q$ for which $a \equiv b q \pmod{p}$.
For example $Q(P(2), 109) = Q(\frac 3 5, 109) = 66$, because $5 \cdot 66 = 330 \equiv 3 \pmod{109}$ and 66 is the smallest positive such number.
Similarly $Q(P(3),109) = 46$.
Find $Q(P(10^{18}),1\,000\,000\,009)$.
$G(N)=\sum_{j=1}^N\sum_{i=1}^j \text{gcd}(i,j)$.
You are given: $G(10)=122$.
Find $G(10^{11})$. Give your answer modulo 998244353
A binary matrix is a matrix consisting entirely of 0s and 1s. Consider the following transformations that can be performed on a binary matrix:
 Swap any two rows
 Swap any two columns
 Flip all elements in a single row (1s become 0s, 0s become 1s)
 Flip all elements in a single column
Two binary matrices $A$ and $B$ will be considered equivalent if there is a sequence of such transformations that when applied to $A$ yields $B$. For example, the following two matrices are equivalent:
$A=\begin{pmatrix} 1 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{pmatrix} \quad B=\begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$via the sequence of two transformations "Flip all elements in column 3" followed by "Swap rows 1 and 2".
Define $c(n)$ to be the maximum number of $n\times n$ binary matrices that can be found such that no two are equivalent. For example, $c(3)=3$. You are also given that $c(5)=39$ and $c(8)=656108$.
Find $c(20)$, and give your answer modulo $1\,001\,001\,011$.
Consider the set $S$ of all possible products of $n$ positive integers not exceeding $m$, that is
$S=\{ x_1x_2\dots x_n \,  \, 1 \le x_1, x_2, ..., x_n \le m \}$.
Let $F(m,n)$ be the number of the distinct elements of the set $S$.
For example, $F(9, 2) = 36$ and $F(30,2)=308$.
Find $F(30, 10001)\text{ mod }1\,000\,000\,007$.
A position in chess is an (orientated) arrangement of chess pieces placed on a chessboard of given size. In the following, we consider all positions in which $n$ pawns are placed on a $n \times n$ board in such a way, that there is a single pawn in every row and every column.
We call such a position an open position, if a rook, starting at the (empty) lower left corner and using only moves towards the right or upwards, can reach the upper right corner without moving onto any field occupied by a pawn.
Let $f(n)$ be the number of open positions for a $n \times n$ chessboard.
For example, $f(3)=2$, illustrated by the two open positions for a $3 \times 3$ chessboard below.
You are also given $f(5)=70$.
Find $f(10^8)$ modulo $1\,008\,691\,207$.
Alice and Bob are playing a modified game of Nim called Scatterstone Nim, with Alice going first, alternating turns with Bob. The game begins with an arbitrary set of stone piles with a total number of stones equal to $n$.
During a player's turn, he/she must pick a pile having at least $2$ stones and perform a split operation, dividing the pile into an arbitrary set of $p$ nonempty, arbitrarilysized piles where $2 \leq p \leq k$ for some fixed constant $k$. For example, a pile of size $4$ can be split into $\{1, 3\}$ or $\{2, 2\}$, or $\{1, 1, 2\}$ if $k = 3$ and in addition $\{1, 1, 1, 1\}$ if $k = 4$.
If no valid move is possible on a given turn, then the other player wins the game.
A winning position is defined as a set of stone piles where a player can ultimately ensure victory no matter what the other player does.
Let $f(n,k)$ be the number of winning positions for Alice on her first turn, given parameters $n$ and $k$. For example, $f(5, 2) = 3$ with winning positions $\{1, 1, 1, 2\}, \{1, 4\}, \{2, 3\}$. In contrast, $f(5, 3) = 5$ with winning positions $\{1, 1, 1, 2\}, \{1, 1, 3\}, \{1, 4\}, \{2, 3\}, \{5\}$.
Let $g(n)$ be the sum of $f(n,k)$ over all $2 \leq k \leq n$. For example, $g(7)=66$ and $g(10)=291$.
Find $g(200)$ mod $(10^9 + 7)$.
Given a set, $L$, of unique lines, let $M(L)$ be the number of lines in the set and let $S(L)$ be the sum over every line of the number of times that line is crossed by another line in the set. For example, two sets of three lines are shown below:
In both cases M(L) is 3 and S(L) is 6: each of the three lines is crossed by two other lines. Note that even if the lines cross at a single point, all of the separate crossings of lines are counted.
Consider points ($T_{2k−1}$, $T_{2k}$), for integer $k >= 1$, generated in the following way:
$S_0 = 290797$
$S_{n+1} = {S_n}^2 \:\: \rm{mod} \:\: 50515093$
$T_n = ( S_n \:\: \rm{mod} \:\: 2000 ) − 1000$
For example, the first three points are: (527, 144), (−488, 732), (−454, −947). Given the first $n$ points generated in this manner, let $L_n$ be the set of unique lines that can be formed by joining each point with every other point, the lines being extended indefinitely in both directions. We can then define $M(L_n)$ and $S(L_n)$ as described above.
For example, $M(L_3) = 3$ and $S(L_3) = 6$. Also $M(L_{100}) = 4948$ and $S(L_{100}) = 24477690$.
Find $S(L_{2500})$.
Let $(p_1 p_2 \ldots p_k)$ denote the permutation of the set ${1, ..., k}$ that maps $p_i\mapsto i$. Define the length of the permutation to be $k$; note that the empty permutation $()$ has length zero.
Define an occurrence of a permutation $p=(p_1 p_2 \ldots p_k)$ in a permutation $P=(P_1 P_2 \ldots P_n)$ to be a sequence $1\leq t_1 < t_2 < \cdots < t_k \leq n$ such that $p_i < p_j$ if and only if $P_{t_i} < P_{t_j}$ for all $i,j \in \{1, ..., k\}$.
For example, $(1243)$ occurs twice in the permutation $(314625)$: once as the 1st, 3rd, 4th and 6th elements $(3\,\,46\,\,5)$, and once as the 2nd, 3rd, 4th and 6th elements $(\,\,146\,\,5)$.
Let $f(n, m)$ be the number of permutations $P$ of length at most $n$ such that there is no occurrence of the permutation $1243$ in $P$ and there are at most $m$ occurrences of the permutation $21$ in $P$.
For example, $f(2,0) = 3$, with the permutations $()$, $(1)$, $(1,2)$ but not $(2,1)$.
You are also given that $f(4, 5) = 32$ and $f(10, 25) = 294\,400$.
Find $f(10^{18}, 40)$ modulo $1\,000\,000\,007$.
For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of $1500=2^2 \times 3 \times 5^3$ are $2$ and $5$.
Let $C_k(N)$ be the number of integers between $1$ and $N$ inclusive with exactly $k$ square prime factors. You are given some values of $C_k(N)$ in the table below.
\[\begin{array}{ccccccc} \hline & k = 0 & k = 1 & k = 2 & k = 3 & k = 4 & k = 5 \\ \hline N=10 & 7 & 3 & 0 & 0 & 0 & 0 \\ \hline N=10^2 & 61 & 36 & 3 & 0 & 0 & 0 \\ \hline N=10^3 & 608 & 343 & 48 & 1 & 0 & 0 \\ \hline N=10^4 & 6083 & 3363 & 533 & 21 & 0 & 0 \\ \hline N=10^5 & 60794 & 33562 & 5345 & 297 & 2 & 0 \\ \hline N=10^6 & 607926 & 335438 & 53358 & 3218 & 60 & 0 \\ \hline N=10^7 & 6079291 & 3353956 & 533140 & 32777 & 834 & 2 \\ \hline N=10^8 & 60792694 & 33539196 & 5329747 & 329028 & 9257 & 78 \\ \hline \end{array}\]Find the product of all nonzero $C_k(10^{16})$. Give the result reduced modulo $1\,000\,000\,007$.
For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of $1500=2^2 \times 3 \times 5^3$ are $2$ and $5$.
Let $C_k(N)$ be the number of integers between $1$ and $N$ inclusive with exactly $k$ square prime factors. It can be shown that with growing $N$ the ratio $\frac{C_k(N)}{N}$ gets arbitrarily close to a constant $c_{k}^{\infty}$, as suggested by the table below.
\[\begin{array}{cccccc} \hline & k = 0 & k = 1 & k = 2 & k = 3 & k = 4 \\ \hline C_k(10) & 7 & 3 & 0 & 0 & 0 \\ \hline C_k(10^2) & 61 & 36 & 3 & 0 & 0 \\ \hline C_k(10^3) & 608 & 343 & 48 & 1 & 0 \\ \hline C_k(10^4) & 6083 & 3363 & 533 & 21 & 0 \\ \hline C_k(10^5) & 60794 & 33562 & 5345 & 297 & 2 \\ \hline C_k(10^6) & 607926 & 335438 & 53358 & 3218 & 60 \\ \hline C_k(10^7) & 6079291 & 3353956 & 533140 & 32777 & 834 \\ \hline C_k(10^8) & 60792694 & 33539196 & 5329747 & 329028 & 9257 \\ \hline C_k(10^9) & 607927124 & 335389706 & 53294365 & 3291791 & 95821 \\ \hline c_k^{\infty} & \frac{6}{\pi^2} & 3.3539\times 10^{1} & 5.3293\times 10^{2} & 3.2921\times 10^{3} & 9.7046\times 10^{5}\\ \hline \end{array}\] Find $c_{7}^{\infty}$. Give the result in scientific notation rounded to 5 signficant digits, using a $e$ to separate mantissa and exponent. E.g. if the answer were $0.000123456789$, then the answer format would be $1.2346e\text{}4$.Define $F(n)$ to be the number of integers $x≤n$ that can be written in the form $x=a^2b^3$, where $a$ and $b$ are integers not necessarily different and both greater than 1.
For example, $32=2^2\times 2^3$ and $72=3^2\times 2^3$ are the only two integers less than 100 that can be written in this form. Hence, $F(100)=2$.
Further you are given $F(2\times 10^4)=130$ and $F(3\times 10^6)=2014$.
Find $F(9\times 10^{18})$.
Let $A_q(n)$ be the number of subsets, $B$, of the set $\{1, 2, ..., q \cdot n\}$ that satisfy two conditions:
1) $B$ has exactly $n$ elements;
2) the sum of the elements of $B$ is divisible by $n$.
E.g. $A_2(5)=52$ and $A_3(5)=603$.
Let $S_q(L)$ be $\sum A_q(p)$ where the sum is taken over all primes $p \le L$.E.g. $S_2(10)=554$, $S_2(100)$ mod $1\,000\,000\,009=100433628$ and
$S_3(100)$ mod $1\,000\,000\,009=855618282$.
Find $S_2(10^8)+S_3(10^8)$. Give your answer modulo $1\,000\,000\,009$.
Consider writing a natural number as product of powers of natural numbers with given exponents, additionally requiring different base numbers for each power.
For example, $256$ can be written as a product of a square and a fourth power in three ways such that the base numbers are different.
That is, $256=1^2\times 4^4=4^2\times 2^4=16^2\times 1^4$
Though $4^2$ and $2^4$ are both equal, we are concerned only about the base numbers in this problem. Note that permutations are not considered distinct, for example $16^2\times 1^4$ and $1^4 \times 16^2$ are considered to be the same.
Similarly, $10!$ can be written as a product of one natural number, two squares and three cubes in two ways ($10!=42\times5^2\times4^2\times3^3\times2^3\times1^3=21\times5^2\times2^2\times4^3\times3^3\times1^3$) whereas $20!$ can be given the same representation in $41680$ ways.
Let $F(n)$ denote the number of ways in which $n$ can be written as a product of one natural number, two squares, three cubes and four fourth powers.
You are given that $F(25!)=4933$, $F(100!) \bmod 1\,000\,000\,007=693\,952\,493$,
and $F(1\,000!) \bmod 1\,000\,000\,007=6\,364\,496$.
Find $F(1\,000\,000!) \bmod 1\,000\,000\,007$.
Given any positive integer $n$, we can construct a new integer by inserting plus signs between some of the digits of the base $B$ representation of $n$, and then carrying out the additions.
For example, from $n=123_{10}$ ($n$ in base 10) we can construct the four base 10 integers $123_{10}$, $1+23=24_{10}$, $12+3=15_{10}$ and $1+2+3=6_{10}$
Let $f(n,B)$ be the smallest number of steps needed to arrive at a singledigit number in base $B$. For example, $f(7,10)=0$ and $f(123,10)=1$.
Let $g(n,B_1,B_2)$ be the sum of the positive integers $i$ not exceeding $n$ such that $f(i,B_1)=f(i,B_2)$.
You are given $g(100,10,3)=3302$.
Find $g(10^7,10,3)$
 The path begins at $(0,0)$ and ends at $(a,b)$.
 The path consists only of unit moves upwards or to the right; that is the coordinates are increasing with every move.
Define $G(P_{a,b},k)=k^{A(P_{a,b})}$. Let $C(a,b,k)$ equal the sum of $G(P_{a,b},k)$ over all valid paths in a $a\times b$ lattice grid.
You are given that
 $C(2,2,1)=6$
 $C(2,2,2)=35$
 $C(10,10,1)=184\,756$
 $C(15,10,3) \equiv 880\,419\,838 \mod 1\,000\,000\,007$
 $C(10\,000,10\,000,4) \equiv 395\,913\,804 \mod 1\,000\,000\,007$
A multiplicative function $f(x)$ is a function over positive integers satisfying $f(1)=1$ and $f(a b)=f(a) f(b)$ for any two coprime positive integers $a$ and $b$.
For integer $k$ let $f_k(n)$ be a multiplicative function additionally satisfying $f_k(p^e)=p^k$ for any prime $p$ and any integer $e>0$.
For example, $f_1(2)=2$, $f_1(4)=2$, $f_1(18)=6$ and $f_2(18)=36$.
Let $\displaystyle S_k(n)=\sum_{i=1}^{n} f_k(i)$. For example, $S_1(10)=41$, $S_1(100)=3512$, $S_2(100)=208090$, $S_1(10000)=35252550$ and $\displaystyle \sum_{k=1}^{3} S_k(10^{8}) \equiv 338787512 \pmod{ 1\,000\,000\,007}$.
Find $\displaystyle \sum_{k=1}^{50} S_k(10^{12}) \bmod 1\,000\,000\,007$.
Bob plays a singleplayer game of chance using two standard 6sided dice and twelve cards numbered 1 to 12. When the game starts, all cards are placed face up on a table.
Each turn, Bob rolls both dice, getting numbers $x$ and $y$ respectively, each in the range 1,...,6. He must choose amongst three options: turn over card $x$, card $y$, or card $x+y$. (If the chosen card is already face down, it is turned to face up, and vice versa.)
If Bob manages to have all twelve cards face down at the same time, he wins.
Alice plays a similar game, except that instead of dice she uses two fair coins, counting heads as 2 and tails as 1, and that she uses four cards instead of twelve. Alice finds that, with the optimal strategy for her game, the expected number of turns taken until she wins is approximately 5.673651.
Assuming that Bob plays with an optimal strategy, what is the expected number of turns taken until he wins? Give your answer rounded to 6 places after the decimal point.
Consider a row of $n$ dice all showing 1.
First turn every second die,$ (2,4,6,\ldots)$, so that the number showing is increased by 1. Then turn every third die. The sixth die will now show a 3. Then turn every fourth die and so on until every $n$th die (only the last die) is turned. If the die to be turned is showing a 6 then it is changed to show a 1.
Let $f(n)$ be the number of dice that are showing a 1 when the process finishes. You are given $f(100)=2$ and $f(10^8) = 69$.
Find $f(10^{36})$.
Let $f(n)$ be the largest prime factor of $n$ and $\displaystyle F(n) = \sum_{i=2}^{n}f(i)$.
For example $F(10)=32$, $F(100)=1915$ and $F(10000)=10118280$.
Find $F(201820182018)$. Give your answer modulus $10^9$.
Two positive integers $a$ and $b$ are 2friendly when $\gcd(a,b) = 2^t, t>0$. For example, 24 and 40 are 2friendly because $\gcd(24,40) = 8 = 2^3$ while 24 and 36 are not because $\gcd(24,36) = 12 = 2^2\cdot 3$ not a power of 2.
Let $f(n)$ be the number of pairs, $(p,q)$, of positive integers with $1\le p\lt q\le n$ such that $p$ and $q$ are 2friendly. You are given $f(10^2) = 1031$ and $f(10^6) = 321418433$ modulo $1\,000\,000\,007$.
Find $f(10^{11})$ modulo $1\,000\,000\,007$.
Sam and Tom are trying a game of (partially) covering a given line segment of length $L$ by taking turns in placing unit squares onto the line segment.
As illustrated below, the squares may be positioned in two different ways, either "straight" by placing the midpoints of two opposite sides on the line segment, or "diagonal" by placing two opposite corners on the line segment. Newly placed squares may touch other squares, but are not allowed to overlap any other square laid down before.
The player who is able to place the last unit square onto the line segment wins.
With Sam starting each game by placing the first square, they quickly realise that Sam can easily win every time by placing the first square in the middle of the line segment, making the game boring.
Therefore they decide to randomise Sam's first move, by first tossing a fair coin to determine whether the square will be placed straight or diagonal onto the line segment and then choosing the actual position on the line segment randomly with all possible positions being equally likely. Sam's gain of the game is defined to be 0 if he loses the game and $L$ if he wins. Assuming optimal play of both players after Sam's initial move, you can see that Sam's expected gain, called $e(L)$, is only dependent on the length of the line segment.
For example, if $L=2$, Sam will win with a probability of 1, so $e(2)= 2$.
Choosing $L=4$, the winning probability will be 0.33333333 for the straight case and 0.22654092 for the diagonal case, leading to $e(4)=1.11974851$ (rounded to 8 digits after the decimal point each).
Being interested in the optimal value of $L$ for Sam, let's define $f(a,b)$ to be the maximum of $e(L)$ for some $L \in [a,b]$.
You are given $f(2,10)=2.61969775$, being reached for $L= 7.82842712$, and $f(10,20)=
5.99374121$ (rounded to 8 digits each).
Find $f(200,500)$, rounded to 8 digits after the decimal point.
On planet J, a year lasts for $D$ days. Holidays are defined by the two following rules.
 At the beginning of the reign of the current Emperor, his birthday is declared a holiday from that year onwards.
 If both the day before and after a day $d$ are holidays, then $d$ also becomes a holiday.
Initially there are no holidays. Let $E(D)$ be the expected number of Emperors to reign before all the days of the year are holidays, assuming that their birthdays are independent and uniformly distributed throughout the $D$ days of the year.
You are given $E(2)=1$, $E(5)=31/6$, $E(365)\approx 1174.3501$.
Find $E(10000)$. Give your answer rounded to 4 digits after the decimal point.
Let $n$ be a natural number and $p_1^{\alpha_1}\cdot p_2^{\alpha_2}...p_k^{\alpha_k}$ its prime factorisation.
Define the Liouville function $\lambda(n)$ as $\lambda(n) = (1)^{\sum\limits_{i=1}^{k}\alpha_i}$.
(i.e. $1$ if the sum of the exponents $\alpha_i$ is odd and $1$ if the sum of the exponents is even. )
Let $S(n,L,H)$ be the sum $\lambda(d) \cdot d$ over all divisors $d$ of $n$ for which $L \leq d \leq H$.
You are given:
$\bullet\, S(10! , 100, 1000) = 1457$
$\bullet\, S(15!, 10^3, 10^5) = 107974$
$\bullet\, S(30!,10^8, 10^{12}) = 9766732243224$.
Find $S(70!,10^{20}, 10^{60})$ and give your answer modulo $1\,000\,000\,007$.
It is possible to find positive integers $A$ and $B$ such that given any triangular number, $T_n$, then $AT_n +B$ is always a triangular number. We define $F_3(N)$ to be the sum of $(A+B)$ over all such possible pairs $(A,B)$ with $\max(A,B)\le N$. For example $F_3(100) = 184$.
Polygonal numbers are generalisations of triangular numbers. Polygonal numbers with parameter $k$ we call $k$gonal numbers. The formula for the $n$th $k$gonal number is $\frac 12n\big(n(k2)+4k\big)$ where $n \ge 1$. For example when $k = 3$ we get $\frac 12n(n+1)$ the formula for triangular numbers.
The statement above is true for pentagonal, heptagonal and in fact any $k$gonal number with $k$ odd. For example when $k=5$ we get the pentagonal numbers and we can find positive integers $A$ and $B$ such that given any pentagonal number, $P_n$, then $AP_n+B$ is always a pentagonal number. We define $F_5(N)$ to be the sum of $(A+B)$ over all such possible pairs $(A,B)$ with $\max(A,B)\le N$.
Similarly we define $F_k(N)$ for odd $k$. You are given $\sum_{k} F_k(10^3) = 14993$ where the sum is over all odd $k = 3,5,7,\ldots$.
Find $\sum_{k} F_k(10^{12})$ where the sum is over all odd $k = 3,5,7,\ldots$
For some fixed $\rho \in [0, 1]$, we begin a sum $s$ at $0$ and repeatedly apply a process: With probability $\rho$, we add $1$ to $s$, otherwise we add $2$ to $s$.
The process ends when either $s$ is a perfect square or $s$ exceeds $10^{18}$, whichever occurs first. For example, if $s$ goes through $0, 2, 3, 5, 7, 9$, the process ends at $s=9$, and two squares $1$ and $4$ were skipped over.
Let $f(\rho)$ be the expected number of perfect squares skipped over when the process finishes.
It can be shown that the power series for $f(\rho)$ is $\sum_{k=0}^\infty a_k \rho^k$ for a suitable (unique) choice of coefficients $a_k$. Some of the first few coefficients are $a_0=1$, $a_1=0$, $a_5=18$, $a_{10}=45176$.
Let $F(n) = \sum_{k=0}^n a_k$. You are given that $F(10) = 53964$ and $F(50) \equiv 842418857 \pmod{10^9}$.
Find $F(1000)$, and give your answer modulo $10^9$.
The game may start with any arrangement of $c$ coins in squares on the board. It is possible at any time for more than one coin to occupy the same square on the board at the same time. The coins are distinguishable, so swapping two coins gives a different arrangement if (and only if) they are on different squares.
On a given turn, the player must choose a coin and move it either left or up $2$, $3$, $5$, or $7$ spaces in a single direction. The only restriction is that the coin cannot move off the edge of the board.
The game ends when a player is unable to make a valid move, thereby granting the other player the victory.
Assuming that Alice goes first and that both players are playing optimally, let $M(n, c)$ be the number of possible starting arrangements for which Alice can ensure her victory, given a board of size $n$ by $n$ with $c$ distinct coins.
For example, $M(3, 1) = 4$, $M(3, 2) = 40$, and $M(9, 3) = 450304$.
What are the last $9$ digits of $M(10\,000\,019, 100)$?
Let $B(n) = \displaystyle \prod_{k=0}^n {n \choose k}$, a product of binomial coefficients.
For example, $B(5) = {5 \choose 0} \times {5 \choose 1} \times {5 \choose 2} \times {5 \choose 3} \times {5 \choose 4} \times {5 \choose 5} = 1 \times 5 \times 10 \times 10 \times 5 \times 1 = 2500$.
Let $D(n) = \displaystyle \sum_{dB(n)} d$, the sum of the divisors of $B(n)$.
For example, the divisors of B(5) are 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500, 625, 1250 and 2500,
so D(5) = 1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 + 100 + 125 + 250 + 500 + 625 + 1250 + 2500 = 5467.
Let $S(n) = \displaystyle \sum_{k=1}^n D(k)$.
You are given $S(5) = 5736$, $S(10) = 141740594713218418$ and $S(100)$ mod $1\,000\,000\,007 = 332792866$.
Find $S(20\,000)$ mod $1\,000\,000\,007$.
An infinitely long cylinder has its curved surface fully covered with different coloured but otherwise identical rectangular stickers, without overlapping. The stickers are aligned with the cylinder, so two of their edges are parallel with the cylinder's axis, with four stickers meeting at each corner.
Let $a>0$ and suppose that the colouring is periodic along the cylinder, with the pattern repeating every $a$ stickers. (The period is allowed to be any divisor of $a$.) Let $b$ be the number of stickers that fit round the circumference of the cylinder.
Let $f(m, a, b)$ be the number of different such periodic patterns that use exactly $m$ distinct colours of stickers. Translations along the axis, reflections in any plane, rotations in any axis, (or combinations of such operations) applied to a pattern are to be counted as the same as the original pattern.
You are given that $f(2, 2, 3) = 11$, $f(3, 2, 3) = 56$, and $f(2, 3, 4) = 156$. Furthermore, $f(8, 13, 21) \equiv 49718354 \pmod{1\,000\,000\,007}$, and $f(13, 144, 233) \equiv 907081451 \pmod{1\,000\,000\,007}$.
Find $\sum_{i=4}^{40} f(i, F_{i1}, F_i) \bmod 1\,000\,000\,007$, where $F_i$ are the Fibonacci numbers starting at $F_0=0$, $F_1=1$.
Consider the values of $\log_2(8)$, $\text{log}_4(64)$ and $\text{log}_3(27)$. All three are equal to $3$.
Generally, the function $f(m,n)=\text{log}_m(n)$ over integers $m,n \ge 2$ has the property that
$f(m_1,n_1)=f(m_2,n_2)$ if
 $\, m_1=a^e, n_1=a^f, m_2=b^e,n_2=b^f \,$ for some integers $a,b,e,f \, \,$ or
 $ \, m_1=a^e, n_1=b^e, m_2=a^f,n_2=b^f \,$ for some integers $a,b,e,f \,$
We call a function $g(m,n)$ over integers $m,n \ge 2$ protologarithmic if
 $\quad \, \, \, \, g(m_1,n_1)=g(m_2,n_2)$ if any integers $a,b,e,f$ fulfilling 1. or 2. can be found
 and $\, g(m_1,n_1) \ne g(m_2,n_2)$ if no integers $a,b,e,f$ fulfilling 1. or 2. can be found
Let $D(N)$ be the number of distinct values that any protologarithmic function $g(m,n)$ attains over $2\le m, n\le N$.
For example, $D(5)=13$, $D(10)=69$, $D(100)=9607$ and $D(10000)=99959605$.
Find $D(10^{18})$, and give the last 9 digits as answer.
Note: According to the four exponentials conjecture the function $\text{log}_m(n)$ is protologarithmic.
While this conjecture is yet unproven in general, $\text{log}_m(n)$ can be used to calculate $D(N)$ for small values of $N$.
Consider a horizontal frictionless tube with length $L$ millimetres, and a diameter of 20 millimetres. The east end of the tube is open, while the west end is sealed. The tube contains $N$ marbles of diameter 20 millimetres at designated starting locations, each one initially moving either westward or eastward with common speed $v$.
Since there are marbles moving in opposite directions, there are bound to be some collisions. We assume that the collisions are perfectly elastic, so both marbles involved instantly change direction and continue with speed $v$ away from the collision site. Similarly, if the westmost marble collides with the sealed end of the tube, it instantly changes direction and continues eastward at speed $v$. On the other hand, once a marble reaches the unsealed east end, it exits the tube and has no further interaction with the remaining marbles.
To obtain the starting positions and initial directions, we use the pseudorandom sequence $r_j$ defined by:
$r_1 = 6\,563\,116$
$r_{j+1} = r_j^2 \bmod 32\,745\,673$
The westmost marble is initially positioned with a gap of $(r_1 \bmod 1000) + 1$ millimetres between it and the sealed end of the tube, measured from the westmost point of the surface of the marble. Then, for $2\le j\le N$, counting from the west, the gap between the $(j1)$th and $j$th marbles, as measured from their closest points, is given by $(r_j \bmod 1000) + 1$ millimetres.
Furthermore, the $j$th marble is initially moving eastward if $r_j \le 10\,000\,000$, and westward if $r_j > 10\,000\,000$.
For example, with $N=3$, the sequence specifies gaps of 117, 432, and 173 millimetres. The marbles' centres are therefore 127, 579, and 772 millimetres from the sealed west end of the tube. The westmost marble initially moves eastward, while the other two initially move westward.
Under this setup, and with a five metre tube ($L=5000$), it turns out that the middle (second) marble travels 5519 millimetres before its centre reaches the eastmost end of the tube.
Let $d(L, N, j)$ be the distance in millimetres that the $j$th marble travels before its centre reaches the eastern end of the tube. So $d(5000, 3, 2) = 5519$. You are also given that $d(10\,000, 11, 6) = 11\,780$ and $d(100\,000, 101, 51) = 114\,101$.
Find $d(1\,000\,000\,000, 1\,000\,001, 500\,001)$.
Let $T(n, m)$ be the number of $m$tuples of positive integers such that the sum of any two neighbouring elements of the tuple is $\le n$.
For example, $T(3, 4)=8$, via the following eight $4$tuples:
$(1, 1, 1, 1)$
$(1, 1, 1, 2)$
$(1, 1, 2, 1)$
$(1, 2, 1, 1)$
$(1, 2, 1, 2)$
$(2, 1, 1, 1)$
$(2, 1, 1, 2)$
$(2, 1, 2, 1)$
You are also given that $T(5, 5)=246$, $T(10, 10^{2}) \equiv 862820094 \pmod{1\,000\,000\,007}$ and $T(10^2, 10) \equiv 782136797 \pmod{1\,000\,000\,007}$.
Find $T(5000, 10^{12}) \bmod 1\,000\,000\,007$.
The numbers $545$, $5\,995$ and $15\,151$ are the three smallest palindromes divisible by $109$. There are nine palindromes less than $100\,000$ which are divisible by $109$.
How many palindromes less than $10^{32}$ are divisible by $10\,000\,019\,$ ?
Given an irrational number $\alpha$, let $S_\alpha(n)$ be the sequence $S_\alpha(n)=\lfloor {\alpha \cdot n} \rfloor  \lfloor {\alpha \cdot (n1)} \rfloor$ for $n \ge 1$.
($\lfloor ... \rfloor$ is the floorfunction.)
It can be proven that for any irrational $\alpha$ there exist infinitely many values of $n$ such that the subsequence $ \{S_\alpha(1),S_\alpha(2)...S_\alpha(n) \} $ is palindromic.
The first 20 values of $n$ that give a palindromic subsequence for $\alpha = \sqrt{31}$ are: 1, 3, 5, 7, 44, 81, 118, 273, 3158, 9201, 15244, 21287, 133765, 246243, 358721, 829920, 9600319, 27971037, 46341755, 64712473.
Let $H_g(\alpha)$ be the sum of the first $g$ values of $n$ for which the corresponding subsequence is palindromic.
So $H_{20}(\sqrt{31})=150243655$.
Let $T=\{2,3,5,6,7,8,10,...,1000\}$ be the set of positive integers, not exceeding 1000, excluding perfect squares.
Calculate the sum of $H_{100}(\sqrt \beta)$ for $\beta \in T$. Give the last 15 digits of your answer.
In the context of formal languages, any finite sequence of letters of a given alphabet $\Sigma$ is called a word over $\Sigma$. We call a word incomplete if it does not contain every letter of $\Sigma$.
For example, using the alphabet $\Sigma=\{ a, b, c\}$, '$ab$', '$abab$' and '$\,$' (the empty word) are incomplete words over $\Sigma$, while '$abac$' is a complete word over $\Sigma$.
Given an alphabet $\Sigma$ of $\alpha$ letters, we define $I(\alpha,n)$ to be the number of incomplete words over $\Sigma$ with a length not exceeding $n$.
For example, $I(3,0)=1$, $I(3,2)=13$ and $I(3,4)=79$.
Find $I(10^7,10^{12})$. Give your answer modulo $1\,000\,000\,007$.
In the context of formal languages, any finite sequence of letters of a given alphabet $\Sigma$ is called a word over $\Sigma$. We call a word incomplete if it does not contain every letter of $\Sigma$.
For example, using the alphabet $\Sigma=\{ a, b, c\}$, '$ab$', '$abab$' and '$\,$' (the empty word) are incomplete words over $\Sigma$, while '$abac$' is a complete word over $\Sigma$.
Given an alphabet $\Sigma$ of $\alpha$ letters, we define $I(\alpha,n)$ to be the number of incomplete words over $\Sigma$ with a length not exceeding $n$.
For example, $I(3,0)=1$, $I(3,2)=13$ and $I(3,4)=79$.
Let $\displaystyle S(k,n)=\sum_{\alpha=1}^k I(\alpha,n)$.
For example, $S(4,4)=406$, $S(8,8)=27902680$ and $S (10,100) \equiv 983602076 \text { mod } 1\,000\,000\,007$.
Find $S(10^7,10^{12})$. Give your answer modulo $1\,000\,000\,007$.
Consider the sequence $n^2+3$ with $n \ge 1$.
If we write down the first terms of this sequence we get:
$4, 7, 12, 19, 28, 39, 52, 67, 84, 103, 124, 147, 172, 199, 228, 259, 292, 327, 364,$... .
We see that the terms for $n=6$ and $n=7$ ($39$ and $52$) are both divisible by $13$.
In fact $13$ is the largest prime dividing any two successive terms of this sequence.
Let $P(k)$ be the largest prime that divides any two successive terms of the sequence $n^2+k^2$.
Find the last 18 digits of $\displaystyle \sum_{k=1}^{10\,000\,000} P(k)$.
We call an integer sided triangle $n$pandigital if it contains one angle of 120 degrees and, when the sides of the triangle are written in base $n$, together they use all $n$ digits of that base exactly once.
For example, the triangle (217, 248, 403) is 9pandigital because it contains one angle of 120 degrees and the sides written in base 9 are $261_9, 305_9, 487_9$ using each of the 9 digits of that base once.
Find the sum of the largest sides of all npandigital triangles with $9 \le n \le 18$.
Two friends $A$ and $B$ are great fans of Chess. They both enjoy playing the game, but after each game the player who lost the game would like to continue (to get back at his opponent) and the player who won would prefer to stop (to finish on a high).
So they come up with a plan. After every game, they would toss a (biased) coin with probability $p$ of Heads (and hence probability $1p$ of Tails). If they get Tails, they will continue with the next game. Otherwise they end the match. Also, after every game the players make a note of who is leading in the match.
Let $p_A$ denote the probability of $A$ winning a game and $p_B$ the probability of $B$ winning a game. Accordingly $1p_Ap_B$ is the probability that a game ends in a draw. Let $\mathbb{E}_A(p_A,p_B,p)$ denote the expected number of times $A$ was leading in the match.
For example, $\mathbb{E}_A(0.25,0.25,0.5)\approx 0.585786$ and $\mathbb{E}_A(0.47,0.48,0.001)\approx 377.471736$, both rounded to six places after the decimal point.
Let $\displaystyle H(n)=\sum_{k=3}^n \mathbb{E}_A\left(\frac 1 {\sqrt{k+3}},\frac 1 {\sqrt{k+3}}+\frac 1 {k^2},\frac 1 {k^3}\right)$
For example $H(3) \approx 6.8345$, rounded to 4 digits after the decimal point.
Find $H(50)$, rounded to 4 digits after the decimal point.
Alice walks on a lattice grid. She can step from one lattice point $A (a,b)$ to another $B (a+x,b+y)$ providing distance $AB = \sqrt{x^2+y^2}$ is a Fibonacci number $\{1,2,3,5,8,13,\ldots\}$ and $x\ge 0,$ $y\ge 0$.
In the lattice grid below Alice can step from the blue point to any of the red points.
Let $F(W,H)$ be the number of paths Alice can take from $(0,0)$ to $(W,H)$.
You are given $F(3,4) = 278$ and $F(10,10) = 215846462$.
Find $F(10\,000,10\,000) \bmod 1\,000\,000\,007$.
Let $t_k$ be the tribonacci numbers defined as:
$\quad t_0 = t_1 = 0$;
$\quad t_2 = 1$;
$\quad t_k = t_{k1} + t_{k2} + t_{k3} \quad \text{ for } k \ge 3$.
For a given integer $n$, let $A_n$ be an array of length $n$ (indexed from 0 to $n1$), that is initially filled with zeros.
The array is changed iteratively by replacing $A_n[(t_{2 i2} \text{ mod } n)]$ with $A_n[(t_{2 i2} \text{ mod } n)]+2 (t_{2 i1} \text{ mod } n)n+1$ in each step $i$.
After each step $i$, define $M_n(i)$ to be $\displaystyle \max\{\sum_{j=p}^q A_n[j]: 0\le p\le q<n\}$, the maximal sum of any contiguous subarray of $A_n$.
The first 6 steps for $n=5$ are illustrated below:
Initial state: $\, A_5=\{0,0,0,0,0\}$
Step 1: $\quad \Rightarrow A_5=\{4,0,0,0,0\}$ , $M_5(1)=0$
Step 2: $\quad \Rightarrow A_5=\{4, 2, 0, 0, 0\}$ , $M_5(2)=0$
Step 3: $\quad \Rightarrow A_5=\{4, 2, 4, 0, 0\}$ , $M_5(3)=4$
Step 4: $\quad \Rightarrow A_5=\{4, 2, 6, 0, 0\}$ , $M_5(4)=6$
Step 5: $\quad \Rightarrow A_5=\{4, 2, 6, 0, 4\}$ , $M_5(5)=10$
Step 6: $\quad \Rightarrow A_5=\{4, 2, 6, 0, 4\}$ , $M_5(6)=12$
Let $\displaystyle S(n,l)=\sum_{i=1}^l M_n(i)$. Thus $S(5,6)=32$.
You are given $S(5,100)=2416$, $S(14,100)=3881$ and $S(107,1000)=1618572$.
Find $S(10\,000\,003,10\,200\,000)S(10\,000\,003,10\,000\,000)$.
Peter is playing a solitaire game on an infinite checkerboard, each square of which can hold an unlimited number of tokens.
Each move of the game consists of the following steps:
 Choose one token $T$ to move. This may be any token on the board, as long as not all of its four adjacent squares are empty.
 Select and discard one token $D$ from a square adjacent to that of $T$.
 Move $T$ to any one of its four adjacent squares (even if that square is already occupied).
The board is marked with a line called the dividing line. Initially, every square to the left of the dividing line contains a token, and every square to the right of the dividing line is empty:
Peter's goal is to get a token as far as possible to the right in a finite number of moves. However, he quickly finds out that, even with his infinite supply of tokens, he cannot move a token more than four squares beyond the dividing line.
Peter then considers starting configurations with larger supplies of tokens: each square in the $d$th column to the left of the dividing line starts with $d^n$ tokens instead of 1. This is illustrated below for $n=1$:
Let $F(n)$ be the maximum number of squares Peter can move a token beyond the dividing line. For example, $F(0)=4$. You are also given that $F(1)=6$, $F(2)=9$, $F(3)=13$, $F(11)=58$ and $F(123)=1173$.
Find $F(1234567)$.
Two players play a game with two piles of stones.
On his or her turn, a player chooses a positive integer $n$ and does one of the following:
 removes $n$ stones from one pile;
 removes $n$ stones from both piles; or
 removes $n$ stones from one pile and $2n$ stones from the other pile.
The player who removes the last stone wins.
We denote by $(n,m)$ the position in which the piles have $n$ and $m$ stones remaining. Note that $(n,m)$ is considered to be the same position as $(m,n)$.
Then, for example, if the position is $(2,6)$, the next player may reach the following positions:
$(0,2)$, $(0,4)$, $(0,5)$, $(0,6)$, $(1,2)$, $(1,4)$, $(1,5)$, $(1,6)$, $(2,2)$, $(2,3)$, $(2,4)$, $(2,5)$
A position is a losing position if the player to move next cannot force a win. For example, $(1,3)$, $(2,6)$, $(4,5)$ are the first few losing positions.
Let $f(M)$ be the sum of $n+m$ for all losing positions $(n,m)$ with $n\le m$ and $n+m \le M$. For example, $f(10) = 21$, by considering the losing positions $(1,3)$, $(2,6)$, $(4,5)$.
You are given that $f(100) = 1164$ and $f(1000) = 117002$.
Find $f(10^7)$.
 Every minute, each individual will simultaneously undergo some kind of transformation.
 Each individual $A$ of type $\alpha$ will, independently, do one of the following (at random with equal probability):
 clone itself, resulting in a new bacterium of type $\alpha$ (alongside $A$ who remains)
 split into 3 new bacteria of type $\beta$ (replacing $A$)
 Each individual $B$ of type $\beta$ will, independently, do one of the following (at random with equal probability):
 spawn a new bacterium of type $\alpha$ (alongside $B$ who remains)
 die
If a population starts with a single bacterium of type $\alpha$, then it can be shown that there is a 0.07243802 probability that the population will eventually die out, and a 0.92756198 probability that the population will last forever. These probabilities are given rounded to 8 decimal places.
Now consider another species of bacteria, $S_{k,m}$ (where $k$ and $m$ are positive integers), which occurs in $k$ different types $\alpha_i$ for $0\le i< k$. The rules governing this species' lifecycle involve the sequence $r_n$ defined by:
 $r_0 = 306$
 $r_{n+1} = r_n^2 \bmod 10\,007$
Every minute, for each $i$, each bacterium $A$ of type $\alpha_i$ will independently choose an integer $j$ uniformly at random in the range $0\le j<m$. What it then does depends on $q = r_{im+j} \bmod 5$:
 If $q=0$, $A$ dies.
 If $q=1$, $A$ clones itself, resulting in a new bacterium of type $\alpha_i$ (alongside $A$ who remains).
 If $q=2$, $A$ mutates, changing into type $\alpha_{(2i) \bmod k}$.
 If $q=3$, $A$ splits into 3 new bacteria of type $\alpha_{(i^2+1) \bmod k}$ (replacing $A$).
 If $q=4$, $A$ spawns a new bacterium of type $\alpha_{(i+1) \bmod k}$ (alongside $A$ who remains).
In fact, our original species was none other than $S_{2,2}$, with $\alpha=\alpha_0$ and $\beta=\alpha_1$.
Let $P_{k,m}$ be the probability that a population of species $S_{k,m}$, starting with a single bacterium of type $\alpha_0$, will eventually die out. So $P_{2,2} = 0.07243802$. You are also given that $P_{4,3} = 0.18554021$ and $P_{10,5} = 0.53466253$, all rounded to 8 decimal places.
Find $P_{500,10}$, and give your answer rounded to 8 decimal places.
After buying a Gerver Sofa from the Moving Sofa Company, Jack wants to buy a matching cocktail table from the same company. Most important for him is that the table can be pushed through his Lshaped corridor into the living room without having to be lifted from its table legs.
Unfortunately, the simple square model offered to him is too small for him, so he asks for a bigger model.
He is offered the new pentagonal model illustrated below:
Note, while the shape and size can be ordered individually, due to the production process, all edges of the pentagonal table have to have the same length.
Given optimal form and size, what is the biggest pentagonal cocktail table (in terms of area) that Jack can buy that still fits through his unit wide Lshaped corridor?
Give your answer rounded to 10 digits after the decimal point (if Jack had choosen the square model instead the answer would have been 1.0000000000).
A positive integer is called square root smooth if all of its prime factors are strictly less than its square root.
Including the number $1$, there are $29$ square root smooth numbers not exceeding $100$.
How many square root smooth numbers are there not exceeding $10\,000\,000\,000$?
The Knights of the Order of Fibonacci are preparing a grand feast for their king. There are $n$ knights, and each knight is assigned a distinct number from 1 to $n$.
When the knights sit down at the roundtable for their feast, they follow a peculiar seating rule: two knights can only sit next to each other if their respective numbers sum to a Fibonacci number.
When the $n$ knights all try to sit down around a circular table with $n$ chairs, they are unable to find a suitable seating arrangement for any $n>2$ despite their best efforts. Just when they are about to give up, they remember that the king will sit on his throne at the table as well.
Suppose there are $n=7$ knights and 7 chairs at the roundtable, in addition to the king’s throne. After some trial and error, they come up with the following seating arrangement (K represents the king):
Notice that the sums $4+1$, $1+7$, $7+6$, $6+2$, $2+3$, and $3+5$ are all Fibonacci numbers, as required. It should also be mentioned that the king always prefers an arrangement where the knight to the his left has a smaller number than the knight to his right. With this additional rule, the above arrangement is unique for $n=7$, and the knight sitting in the 3rd chair from the king’s left is knight number 7.
Later, several new knights are appointed to the Order, giving 34 knights and chairs in addition to the king's throne. The knights eventually determine that there is a unique seating arrangement for $n=34$ satisfying the above rules, and this time knight number 30 is sitting in the 3rd chair from the king's left.
Now suppose there are $n=99\,194\,853\,094\,755\,497$ knights and the same number of chairs at the roundtable (not including the king’s throne). After great trials and tribulations, they are finally able to find the unique seating arrangement for this value of $n$ that satisfies the above rules.
Find the number of the knight sitting in the $10\,000\,000\,000\,000\,000$th chair from the king’s left.
A certain type of tile comes in three different sizes  1×1, 1×2, and 1×3  and in four different colours: blue, green, red and yellow. There is an unlimited number of tiles available in each combination of size and colour.
These are used to tile a $2\times n$ rectangle, where $n$ is a positive integer, subject to the following conditions:
 The rectangle must be fully covered by nonoverlapping tiles.
 It is not permitted for four tiles to have their corners meeting at a single point.
 Adjacent tiles must be of different colours.
For example, the following is an acceptable tiling of a $2\times 12$ rectangle:
but the following is not an acceptable tiling, because it violates the "no four corners meeting at a point" rule:
Let $F(n)$ be the number of ways the $2\times n$ rectangle can be tiled subject to these rules. Where reflecting horizontally or vertically would give a different tiling, these tilings are to be counted separately.
For example, $F(2) = 120$, $F(5) = 45876$, and $F(100)\equiv 53275818 \pmod{1\,000\,004\,321}$.
Find $F(10^{16}) \bmod 1\,000\,004\,321$.
A certain type of flexible tile comes in three different sizes  1×1, 1×2, and 1×3  and in $k$ different colours. There is an unlimited number of tiles available in each combination of size and colour.
These are used to tile a closed loop of width $2$ and length (circumference) $n$, where $n$ is a positive integer, subject to the following conditions:
 The loop must be fully covered by nonoverlapping tiles.
 It is not permitted for four tiles to have their corners meeting at a single point.
 Adjacent tiles must be of different colours.
For example, the following is an acceptable tiling of a $2\times 23$ loop with $k=4$ (blue, green, red and yellow):
but the following is not an acceptable tiling, because it violates the "no four corners meeting at a point" rule:
Let $F_k(n)$ be the number of ways the $2\times n$ loop can be tiled subject to these rules when $k$ colours are available. (Not all $k$ colours have to be used.) Where reflecting horizontally or vertically would give a different tiling, these tilings are to be counted separately.
For example, $F_4(3) = 104$, $F_5(7) = 3327300$, and $F_6(101)\equiv 75309980 \pmod{1\,000\,004\,321}$.
Find $F_{10}(10\,004\,003\,002\,001) \bmod 1\,000\,004\,321$.
Consider the following process that can be applied recursively to any positive integer $n$:
 if $n = 1$ do nothing and the process stops,
 if $n$ is divisible by 7 divide it by 7,
 otherwise add 1.
Define $g(n)$ to be the number of 1's that must be added before the process ends. For example:
Eight 1's are added so $g(125) = 8$. Similarly $g(1000) = 9$ and $g(10000) = 21$.
Define $S(N) = \sum_{n=1}^{N} g(n)$ and $H(K) = S\left(\frac{7^K1}{11}\right)$. You are given $H(10) = 690409338$.
Find $H(10^9)$ modulo $1\,117\,117\,717$.
At Euler University, each of the $n$ students (numbered from 1 to $n$) occupies a bed in the dormitory and uses a desk in the classroom.
Some of the beds are in private rooms which a student occupies alone, while the others are in double rooms occupied by two students as roommates. Similarly, each desk is either a single desk for the sole use of one student, or a twin desk at which two students sit together as desk partners.
We represent the bed and desk sharing arrangements each by a list of pairs of student numbers. For example, with $n=4$, if $(2,3)$ represents the bed pairing and $(1,3)(2,4)$ the desk pairing, then students 2 and 3 are roommates while 1 and 4 have single rooms, and students 1 and 3 are desk partners, as are students 2 and 4.
The new chancellor of the university decides to change the organisation of beds and desks: he will choose a permutation $\sigma$ of the numbers $1,2,\ldots,n$ and each student $k$ will be given both the bed and the desk formerly occupied by student number $\sigma(k)$.
The students agree to this change, under the conditions that:
 Any two students currently sharing a room will still be roommates.
 Any two students currently sharing a desk will still be desk partners.
In the example above, there are only two ways to satisfy these conditions: either take no action ($\sigma$ is the identity permutation), or reverse the order of the students.
With $n=6$, for the bed pairing $(1,2)(3,4)(5,6)$ and the desk pairing $(3,6)(4,5)$, there are 8 permutations which satisfy the conditions. One example is the mapping $(1, 2, 3, 4, 5, 6) \mapsto (1, 2, 5, 6, 3, 4)$.
With $n=36$, if we have bed pairing:
$(2,13)(4,30)(5,27)(6,16)(10,18)(12,35)(14,19)(15,20)(17,26)(21,32)(22,33)(24,34)(25,28)$
and desk pairing
$(1,35)(2,22)(3,36)(4,28)(5,25)(7,18)(9,23)(13,19)(14,33)(15,34)(20,24)(26,29)(27,30)$
then among the $36!$ possible permutations (including the identity permutation), 663552 of them satisfy the conditions stipulated by the students.
The downloadable text files beds.txt and desks.txt contain pairings for $n=500$. Each pairing is written on its own line, with the student numbers of the two roommates (or desk partners) separated with a comma. For example, the desk pairing in the $n=4$ example above would be represented in this file format as:
1,3 2,4
With these pairings, find the number of permutations that satisfy the students' conditions. Give your answer modulo $999\,999\,937$.
We define the $\mathcal{I}$ operator as the function \[\mathcal{I}(x,y) = (1+x+y)^2+yx\] and $\mathcal{I}$expressions as arithmetic expressions built only from variables names and applications of $\mathcal{I}$. A variable name may consist of one or more letters. For example, the three expressions $x$, $\mathcal{I}(x,y)$, and $\mathcal{I}(\mathcal{I}(x,ab),x)$ are all $\mathcal{I}$expressions.
For two $\mathcal{I}$expressions $e_1$ and $e_2$ such that the equation $e_1=e_2$ has a solution in nonnegative integers, we define the least simultaneous value of $e_1$ and $e_2$ to be the minimum value taken by $e_1$ and $e_2$ on such a solution. If the equation $e_1=e_2$ has no solution in nonnegative integers, we define the least simultaneous value of $e_1$ and $e_2$ to be $0$. For example, consider the following three $\mathcal{I}$expressions: \[\begin{array}{l}A = \mathcal{I}(x,\mathcal{I}(z,t))\\ B = \mathcal{I}(\mathcal{I}(y,z),y)\\ C = \mathcal{I}(\mathcal{I}(x,z),y)\end{array}\] The least simultaneous value of $A$ and $B$ is $23$, attained for $x=3,y=1,z=t=0$. On the other hand, $A=C$ has no solutions in nonnegative integers, so the least simultaneous value of $A$ and $C$ is $0$. The total sum of least simultaneous pairs made of $\mathcal{I}$expressions from $\{A,B,C\}$ is $26$.
Find the sum of least simultaneous values of all $\mathcal{I}$expressions pairs made of distinct expressions from file Iexpressions.txt (pairs $(e_1,e_2)$ and $(e_2,e_1)$ are considered to be identical). Give the last nine digits of the result as the answer.
Let $\omega(n)$ denote the number of distinct prime divisors of a positive integer $n$.
So $\omega(1) = 0$ and $\omega(360) = \omega(2^{3} \times 3^{2} \times 5) = 3$.
Let $S(n)$ be $ \Sigma_{d  n} 2^{\omega(d)} $.
E.g. $S(6) = 2^{\omega(1)}+2^{\omega(2)}+2^{\omega(3)}+2^{\omega(6)} = 2^0+2^1+2^1+2^2 = 9$.
Let $F(n)=\Sigma_{i=2}^n S(i!)$. $F(10)=4821.$
Find $F(10\,000\,000)$. Give your answer modulo $1\,000\,000\,087$.
Let $d(i,b)$ be the digit sum of the number $i$ in base $b$. For example $d(9,2)=2$, since $9=1001_2$. When using different bases, the respective digit sums most of the time deviate from each other, for example $d(9,4)=3 \ne d(9,2)$.
However, for some numbers $i$ there will be a match, like $d(17,4)=d(17,2)=2$. Let $ M(n,b_1,b_2)$ be the sum of all natural numbers $i \le n$ for which $d(i,b_1)=d(i,b_2)$. For example, $M(10,8,2)=18$, $M(100,8,2)=292$ and $M(10^6,8,2)=19173952$.
Find $\displaystyle \sum_{k=3}^6 \sum_{l=1}^{k2}M(10^{16},2^k,2^l)$, giving the last 16 digits as the answer.
Let $g(n)$ be the number of undirected graphs with $n$ nodes satisfying the following properties:
 The graph is connected and has no cycles or multiple edges.
 Each node is either red, blue, or yellow.
 A red node may have no more than 4 edges connected to it.
 A blue or yellow node may have no more than 3 edges connected to it.
 An edge may not directly connect a yellow node to a yellow node.
For example, $g(2)=5$, $g(3)=15$, and $g(4) = 57$.
You are also given that $g(10) = 710249$ and $g(100) \equiv 919747298 \pmod{1\,000\,000\,007}$.
Find $g(10\,000) \bmod 1\,000\,000\,007$.
If a triple of positive integers $(a, b, c)$ satisfies $a^2+b^2=c^2$, it is called a Pythagorean triple. No triple $(a, b, c)$ satisfies $a^e+b^e=c^e$ when $e \ge 3$ (Fermat's Last Theorem). However, if the exponents of the lefthand side and righthand side differ, this is not true. For example, $3^3+6^3=3^5$.
Let $a, b, c, e, f$ be all positive integers, $0 \lt a \lt b$, $e \ge 2$, $f \ge 3$ and $c^f \le N$. Let $F(N)$ be the number of $(a, b, c, e, f)$ such that $a^e+b^e=c^f$. You are given $F(10^3) = 7$, $F(10^5) = 53$ and $F(10^7) = 287$.
Find $F(10^{18})$.
Let $S$ be the set consisting of the four letters $\{\texttt{`A'},\texttt{`E'},\texttt{`F'},\texttt{`R'}\}$.
For $n\ge 0$, let $S^*(n)$ denote the set of words of length $n$ consisting of letters belonging to $S$.
We designate the words $\texttt{FREE}, \texttt{FARE}, \texttt{AREA}, \texttt{REEF}$ as keywords.
Let $f(n)$ be the number of words in $S^*(n)$ that contains all four keywords exactly once.
This first happens for $n=9$, and indeed there is a unique 9 lettered word that contain each of the keywords once: $\texttt{FREEFAREA}$
So, $f(9)=1$.
You are also given that $f(15)=72863$.
Find $f(30)$.
Let $N$ and $K$ be two positive integers.
$F_n$ is the $n$th Fibonacci number: $F_1 = F_2 = 1$, $F_n = F_{n  1} + F_{n  2}$ for all $n \geq 3$.
Let $s_n = F_{2n  1} \mod N$ and let $t_n = F_{2n} \mod N$.
Start with an array of integers $A = (A[0], \cdots, A[N  1])$ where initially every $A\text{[}i]$ is equal to $i$. Now perform $K$ successive operations on $A$, where the $j$th operation consists of reversing the order of those elements in $A$ with indices between $s_j$ and $t_j$ (both ends inclusive).
Define $R(N,K)$ to be $\sum_{i = 0}^{N  1}i \times A\text {[}i]$ after $K$ operations.
For example, $R(5, 4) = 27$, as can be seen from the following procedure:
Initial position: $(0, 1, 2, 3, 4)$
Step 1  Reverse $A[1]$ to $A[1]$: $(0, 1, 2, 3, 4)$
Step 2  Reverse $A[2]$ to $A[3]$: $(0, 1, 3, 2, 4)$
Step 3  Reverse $A[0]$ to $A[3]$: $(2, 3, 1, 0, 4)$
Step 4  Reverse $A[3]$ to $A[1]$: $(2, 0, 1, 3, 4)$
$R(5, 4) = 0 \times 2 + 1 \times 0 + 2 \times 1 + 3 \times 3 + 4 \times 4 = 27$
Also, $R(10^2, 10^2) = 246597$ and $R(10^4, 10^4) = 249275481640$.
Find $R(10^{18}, 10^6)$ giving your answer modulo $10^9$.
Given positive integers $a \le b \le c \le d$, it may be possible to form quadrilaterals with edge lengths $a,b,c,d$ (in any order). When this is the case, let $M(a,b,c,d)$ denote the maximal area of such a quadrilateral.
For example, $M(2,2,3,3)=6$, attained e.g. by a $2\times 3$ rectangle.
Let $SP(n)$ be the sum of $a+b+c+d$ over all choices $a \le b \le c \le d$ for which $M(a,b,c,d)$ is a positive integer not exceeding $n$.
$SP(10)=186$ and $SP(100)=23238$.
Find $SP(1\,000\,000)$.
5smooth numbers are numbers whose largest prime factor doesn't exceed 5.
5smooth numbers are also called Hamming numbers.
Let $\Omega(a)$ be the count of prime factors of $a$ (counted with multiplicity).
Let $s(a)$ be the sum of the prime factors of $a$ (with multiplicity).
For example, $\Omega(300) = 5$ and $s(300) = 2+2+3+5+5 = 17$.
Let $f(n)$ be the number of pairs, $(p,q)$, of Hamming numbers such that $\Omega(p)=\Omega(q)$ and $s(p)+s(q)=n$.
You are given $f(10)=4$ (the pairs are $(4,9),(5,5),(6,6),(9,4)$) and $f(10^2)=3629$.
Find $f(10^7) \bmod 1\,000\,000\,007$.
Consider the following variant of "The Chase" game. This game is played between $n$ players sitting around a circular table using two dice. It consists of $n1$ rounds, and at the end of each round one player is eliminated and has to pay a certain amount of money into a pot. The last player remaining is the winner and receives the entire contents of the pot.
At the beginning of a round, each die is given to a randomly selected player. A round then consists of a number of turns.
During each turn, each of the two players with a die rolls it. If a player rolls a 1 or a 2, she passes the die to her neighbour on the left; if she rolls a 5 or a 6, she passes the die to her neighbour on the right; otherwise, she keeps the die for the next turn.
The round ends when one player has both dice at the beginning of a turn. The turn is aborted, that player is eliminated, and she has to pay $s^2$ where $s$ is the number of completed turns in this round. Note that if both dice happen to be handed to the same player at the beginning of a round, then no turns are completed, so the player is eliminated without having to pay any money into the pot.
Let $G(n)$ be the expected amount that the winner will receive. For example $G(5)$ is approximately 96.544, and $G(50)$ is 2.82491788e6 in scientific notation rounded to 9 significant digits.
Find $G(500)$, giving your answer in scientific notation rounded to 9 significant digits.
Define $s(n)$ to be the smallest number that has a digit sum of $n$. For example $s(10) = 19$.
Let $\displaystyle S(k) = \sum_{n=1}^k s(n)$. You are given $S(20) = 1074$.
Further let $f_i$ be the Fibonacci sequence defined by $f_0=0, f_1=1$ and $f_i=f_{i2}+f_{i1}$ for all $i \ge 2$.
Find $\displaystyle \sum_{i=2}^{90} S(f_i)$. Give your answer modulo $1\,000\,000\,007$.
Writing down the numbers which have a digit sum of 10 in ascending order, we get: $19, 28, 37, 46,55,64,73,82,91,109, 118,\dots$
Let $f(n,m)$ be the $m^{\text{th}}$ occurrence of the digit sum $n$. For example, $f(10,1)=19$, $f(10,10)=109$ and $f(10,100)=1423$.
Let $\displaystyle S(k)=\sum_{n=1}^k f(n^3,n^4)$. For example $S(3)=7128$ and $S(10)\equiv 32287064 \mod 1\,000\,000\,007$.
Find $S(10\,000)$ modulo $1\,000\,000\,007$.
$2^7=128$ is the first power of two whose leading digits are "12".
The next power of two whose leading digits are "12" is $2^{80}$.
Define $p(L, n)$ to be the $n$thsmallest value of $j$ such that the base 10 representation of $2^j$ begins with the digits of $L$.
So $p(12, 1) = 7$ and $p(12, 2) = 80$.
You are also given that $p(123, 45) = 12710$.
Find $p(123, 678910)$.
A standard deck of 52 playing cards, which consists of thirteen ranks (Ace, Two, ..., Ten, King, Queen and Jack) each in four suits (Clubs, Diamonds, Hearts and Spades), is randomly shuffled. Let us call a rank perfect if no two cards of that same rank appear next to each other after the shuffle.
It can be seen that the expected number of ranks that are perfect after a random shuffle equals $\frac {4324} {425} \approx 10.1741176471$.
Find the probability that the number of perfect ranks is prime. Give your answer rounded to 10 decimal places.
We stack $n$ plates into $k$ nonempty piles where each pile is a different size. Define $f(n,k)$ to be the maximum number of plates possible in the smallest pile. For example when $n = 10$ and $k = 3$ the piles $2,3,5$ is the best that can be done and so $f(10,3) = 2$. It is impossible to divide 10 into 5 nonempty differentlysized piles and hence $f(10,5) = 0$.
Define $F(n)$ to be the sum of $f(n,k)$ for all possible pile sizes $k\ge 1$. For example $F(100) = 275$.
Further define $S(N) = \displaystyle\sum_{n=1}^N F(n)$. You are given $S(100) = 12656$.
Find $S(10^{16})$ giving your answer modulo $1\,000\,000\,007$.
For $0 \le x \lt 1$, define $d_i(x)$ to be the $i$th digit after the binary point of the binary representation of $x$.
For example $d_2(0.25) = 1$, $d_i(0.25) = 0$ for $i \ne 2$.
Let $f(x) = \displaystyle{\sum_{i=1}^{\infty}\frac{d_i(x)}{i^2}}$.
Let $p(a)$ be probability that $f(x) \gt a$, given that $x$ is uniformly distributed between 0 and 1.
Find $p(0.5)$. Give your answer rounded to 8 digits after the decimal point.
Tom (the cat) and Jerry (the mouse) are playing on a simple graph $G$.
Every vertex of $G$ is a mousehole, and every edge of $G$ is a tunnel connecting two mouseholes.
Originally, Jerry is hiding in one of the mouseholes.
Every morning, Tom can check one (and only one) of the mouseholes. If Jerry happens to be hiding there then Tom catches Jerry and the game is over.
Every evening, if the game continues, Jerry moves to a mousehole which is adjacent (i.e. connected by a tunnel, if there is one available) to his current hiding place. The next morning Tom checks again and the game continues like this.
Let us call a graph $G$ a Tom graph, if our supersmart Tom, who knows the configuration of the graph but does not know the location of Jerry, can guarantee to catch Jerry in finitely many days. For example consider all graphs on 3 nodes:
For graphs 1 and 2, Tom will catch Jerry in at most three days. For graph 3 Tom can check the middle connection on two consecutive days and hence guarantee to catch Jerry in at most two days. These three graphs are therefore Tom Graphs. However, graph 4 is not a Tom Graph because the game could potentially continue forever.
Let $T(n)$ be the number of different Tom graphs with $n$ vertices. Two graphs are considered the same if there is a bijection $f$ between their vertices, such that $(v,w)$ is an edge if and only if $(f(v),f(w))$ is an edge.
We have $T(3) = 3$, $T(7) = 37$, $T(10) = 328$ and $T(20) = 1416269$.
Find $T(2019)$ giving your answer modulo $1\,000\,000\,007$.
Given a character string $s$, we define $L(k,s)$ to be the length of the longest substring of $s$ which appears at least $k$ times in $s$, or $0$ if such a substring does not exist. For example, $L(3,\text{“bbabcabcabcacba”})=4$ because of the three occurrences of the substring $\text{“abca”}$, and $L(2,\text{“bbabcabcabcacba”})=7$ because of the repeated substring $\text{“abcabca”}$. Note that the occurrences can overlap.
Let $a_n$, $b_n$ and $c_n$ be the $0/1$ sequences defined by:
 $a_0 = 0$
 $a_{2n} = a_{n}$
 $a_{2n+1} = 1a_{n}$
 $b_n = \lfloor\frac{n+1}{\varphi}\rfloor  \lfloor\frac{n}{\varphi}\rfloor$ (where $\varphi$ is the golden ratio)
 $c_n = a_n + b_n  2a_nb_n$
and $S_n$ the character string $c_0\ldots c_{n1}$. You are given that $L(2,S_{10})=5$, $L(3,S_{10})=2$, $L(2,S_{100})=14$, $L(4,S_{100})=6$, $L(2,S_{1000})=86$, $L(3,S_{1000}) = 45$, $L(5,S_{1000}) = 31$, and that the sum of nonzero $L(k,S_{1000})$ for $k\ge 1$ is $2460$.
Find the sum of nonzero $L(k,S_{5000000})$ for $k\ge 1$.
Siegbert and Jo take turns playing a game with a heap of $N$ pebbles:
1. Siegbert is the first to take some pebbles. He can take as many pebbles as he wants. (Between 1 and $N$ inclusive.)
2. In each of the following turns the current player must take at least one pebble and at most twice the amount of pebbles taken by the previous player.
3. The player who takes the last pebble wins.
Although Siegbert can always win by taking all the pebbles on his first turn, to make the game more interesting he chooses to take the smallest number of pebbles that guarantees he will still win (assuming both Siegbert and Jo play optimally for the rest of the game).
Let $H(N)$ be that minimal amount for a heap of $N$ pebbles.
$H(1)=1$, $H(4)=1$, $H(17)=1$, $H(8)=8$ and $H(18)=5$ .
Let $G(n)$ be $\displaystyle{\sum_{k=1}^n H(k)}$.
$G(13)=43$.
Find $G(23416728348467685)$.
Two positive integers $x$ and $y$ ($x > y$) can generate a sequence in the following manner:
 $a_x = y$ is the first term,
 $a_{z+1} = a_z^2 \bmod z$ for $z = x, x+1,x+2,\ldots$ and
 the generation stops when a term becomes 0 or 1.
The number of terms in this sequence is denoted $l(x,y)$.
For example, with $x = 5$ and $y = 3$, we get $a_5 = 3$, $a_6 = 3^2 \bmod 5 = 4$, $a_7 = 4^2\bmod 6 = 4$, etc. Giving the sequence of 29 terms:
$ 3,4,4,2,4,7,9,4,4,3,9,6,4,16,4,16,16,4,16,3,9,6,10,19,25,16,16,8,0 $
Hence $l(5,3) = 29$.
$g(x)$ is defined to be the maximum value of $l(x,y)$ for $y < x$. For example, $g(5) = 29$.
Further, define $f(n)$ to be the maximum value of $g(x)$ for $x \le n$. For example, $f(100) = 145$ and $f(10\,000) = 8824$.
Find $f(3\,000\,000)$.
A positive integer $n$ is considered cubefull, if for every prime $p$ that divides $n$, so does $p^3$. Note that $1$ is considered cubefull.
Let $s(n)$ be the function that counts the number of cubefull divisors of $n$. For example, $1$, $8$ and $16$ are the three cubefull divisors of $16$. Therefore, $s(16)=3$.
Let $S(n)$ represent the summatory function of $s(n)$, that is $S(n)=\displaystyle\sum_{i=1}^n s(i)$.
You are given $S(16) = 19$, $S(100) = 126$ and $S(10000) = 13344$.
Find $S(10^{18})$.
Three points, $P_1$, $P_2$ and $P_3$, are randomly selected within a unit square. Consider the three rectangles with sides parallel to the sides of the unit square and a diagonal that is one of the three line segments $\overline{P_1P_2}$, $\overline{P_1P_3}$ or $\overline{P_2P_3}$ (see picture below).
We are interested in the rectangle with the second biggest area. In the example above that happens to be the green rectangle defined with the diagonal $\overline{P_2P_3}$.
Find the expected value of the area of the second biggest of the three rectangles. Give your answer rounded to 10 digits after the decimal point.
The game of Mahjong is played with tiles belonging to $s$ suits. Each tile also has a number in the range $1\ldots n$, and for each suit/number combination there are exactly four indistinguishable tiles with that suit and number. (The real Mahjong game also contains other bonus tiles, but those will not feature in this problem.)
A winning hand is a collection of $3t+2$ Tiles (where $t$ is a fixed integer) that can be arranged as $t$ Triples and one Pair, where:
 A Triple is either a Chow or a Pung
 A Chow is three tiles of the same suit and consecutive numbers
 A Pung is three identical tiles (same suit and same number)
 A Pair is two identical tiles (same suit and same number)
For example, here is a winning hand with $n=9$, $s=3$, $t=4$, consisting in this case of two Chows, two Pungs, and one Pair:
Note that sometimes the same collection of tiles can be represented as $t$ Triples and one Pair in more than one way. This only counts as one winning hand. For example, this is considered to be the same winning hand as above, because it consists of the same tiles:
Let $w(n, s, t)$ be the number of distinct winning hands formed of $t$ Triples and one Pair, where there are $s$ suits available and tiles are numbered up to $n$.
For example, with a single suit and tiles numbered up to 4, we have $w(4, 1, 1) = 20$: there are 12 winning hands consisting of a Pung and a Pair, and another 8 containing a Chow and a Pair. You are also given that $w(9, 1, 4) = 13259$, $w(9, 3, 4) = 5237550$, and $w(1000, 1000, 5) \equiv 107662178 \pmod{1\,000\,000\,007}$.
Find $w(10^8, 10^8, 30)$. Give your answer modulo $1\,000\,000\,007$.
Given a fixed real number $c$, define a random sequence $(X_n)_{n\ge 0}$ by the following random process:
 $X_0 = c$ (with probability 1).
 For $n>0$, $X_n = U_n X_{n1}$ where $U_n$ is a real number chosen at random between zero and one, uniformly, and independently of all previous choices $(U_m)_{m<n}$.
If we desire there to be precisely a 25% probability that $X_{100}<1$, then this can be arranged by fixing $c$ such that $\log_{10} c \approx 46.27$.
Suppose now that $c$ is set to a different value, so that there is precisely a 25% probability that $X_{10\,000\,000}<1$.
Find $\log_{10} c$ and give your answer rounded to two places after the decimal point.
We define 123numbers as follows:
 1 is the smallest 123number.
 When written in base 10 the only digits that can be present are "1", "2" and "3" and if present the number of times they each occur is also a 123number.
So 2 is a 123number, since it consists of one digit "2" and 1 is a 123number. Therefore, 33 is a 123number as well since it consists of two digits "3" and 2 is a 123number.
On the other hand, 1111 is not a 123number, since it contains 4 digits "1" and 4 is not a 123number.
In ascending order, the first 123numbers are:
$1, 2, 3, 11, 12, 13, 21, 22, 23, 31, 32, 33, 111, 112, 113, 121, 122, 123, 131, \ldots$
Let $F(n)$ be the $n$th 123number. For example $F(4)=11$, $F(10)=31$, $F(40)=1112$, $F(1000)=1223321$ and $F(6000)= 2333333333323$.
Find $F(111\,111\,111\,111\,222\,333)$. Give your answer modulo $123\,123\,123$.