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Problem 438


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.


多项式方程式解的整数部分

对于整数n元组t = (a1, …, an),记(x1, …, xn)是多项式方程xn + a1xn-1 + a2xn-2 + … + an-1x + an = 0的解集。

考虑下面两个条件:

  • 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。)

当n = 4时,有12个整数n元组同时满足这两个条件。
记S(t)是t中整数的绝对值的和。
当n = 4时,对于所有同时满足这两个条件的整数n元组t,可以验证∑S(t) = 2087。

当n = 7时,求∑S(t)。