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p|a_1^k + ... + a_n^k for infinitely many p

Source: Iranian National Olympiad (3rd Round) 2004

January 9, 2009
number theorygreatest common divisormodular arithmeticprime numbersnumber theory proposed

Problem Statement

a1,a2,…,an a_1, a_2, \ldots, a_n are integers, not all equal. Prove that there exist infinitely many prime numbers p p such that for some k k p\mid a_1^k \plus{} \dots \plus{} a_n^k.
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