Problem

Source: Iranian National Olympiad (3rd Round) 2004

Tags: number theory, greatest common divisor, modular arithmetic, prime numbers, number theory proposed



$ a_1, a_2, \ldots, a_n$ are integers, not all equal. Prove that there exist infinitely many prime numbers $ p$ such that for some $ k$ \[ p\mid a_1^k + \dots + a_n^k.\]