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# Problem 438

Integer part of polynomial equation’s solutions

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.

• x1, …, xn均为实数。
• 如果 x1, …, xn是从小到大排序的，则[xi] = i对1 ≤ i ≤ n恒成立。（[·]指向下取整函数。）