## Integer part of polynomial equation's solutions

### Problem 438

Published on Sunday, 29th September 2013, 01:00 am; Solved by 84

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.