## Integer part of polynomial equation's solutions

### Problem 438

Published on Sunday, 29th September 2013, 01:00 am; Solved by 150; Difficulty rating: 95%
For an `n`-tuple of integers `t` = (`a`_{1}, ..., `a`_{n}), let (`x`_{1}, ..., `x`_{n}) be the solutions of the polynomial equation `x`^{n} + `a`_{1}`x`^{n-1} + `a`_{2}`x`^{n-2} + ... + `a`_{n-1}`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 `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.