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Product inequality with a prime

Source: Czech and Slovak Olympiad 1987, National Round, Problem 4

April 10, 2020
algebranumber theoryinequalitiesInequalityprime numbersfactorialnational olympiad

Problem Statement

Given an integer n≥3n\ge3 consider positive integers x1,…,xnx_1,\ldots,x_n such that x1<x2<⋯<xn<2x1x_1<x_2<\cdots<x_n<2x_1. If pp is a prime and rr is a positive integer such that prp^r divides the product x1⋯xnx_1\cdots x_n, prove that x1⋯xnpr>n!.\frac{x_1\cdots x_n}{p^r}>n!.
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