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Inequality on integers [Poland 2017, P3]

Source: Polish Mathematical Olympiad Finals, Problem 3

April 4, 2017
number theoryPoland

Problem Statement

Integers a1,a2,…,ana_1, a_2, \ldots, a_n satisfy 1<a1<a2<…<an<2a1.1<a_1<a_2<\ldots < a_n < 2a_1. If mm is the number of distinct prime factors of a1a2⋯ana_1a_2\cdots a_n, then prove that (a1a2⋯an)m−1≥(n!)m.(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.
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