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Show that at least one of the numbers is greater than n!/2^n

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January 11, 2011
algebra unsolvedalgebra

Problem Statement

Suppose x0,x1,…,xnx_0, x_1, \ldots , x_n are integers and x0>x1>⋯>xn.x_0 > x_1 > \cdots > x_n. Prove that at least one of the numbers ∣F(x0)∣,∣F(x1)∣,∣F(x2)∣,…,∣F(xn)∣,|F(x_0)|, |F(x_1)|, |F(x_2)|, \ldots, |F(x_n)|, where F(x) = x^n + a_1x^{n-1} + \cdots+ a_n,   a_i \in \mathbb R,   i = 1, \ldots , n, is greater than n!2n.\frac{n!}{2^n}.
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