ILL 1977 Problem 60

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