## Pythagorean odds

### Problem 285

Published on Saturday, 3rd April 2010, 05:00 am; Solved by 753; Difficulty rating: 55%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?