CZ-SK-MO 1987 Problem 4

Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$ . If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$ , prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$