Integer part of polynomial equation's solutions
Problem 438Published on Sunday, 29th September 2013, 01:00 am; Solved by 136; Difficulty rating: 95%
For an n-tuple of integers t = (a1, ..., an), let (x1, ..., xn) be the solutions of the polynomial equation xn + a1xn-1 + a2xn-2 + ... + an-1x + an = 0.
Consider the following two conditions:
- x1, ..., xn are all real.
- If x1, ..., xn are sorted, ⌊xi⌋ = i for 1 ≤ i ≤ n. (⌊·⌋: floor function.)
In the case of n = 4, there are 12 n-tuples 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 n-tuples t which satisfy both conditions.
Find ∑S(t) for n = 7.