2018 USAMO 第 3 题

For a given integer $n\ge 2,$ let $\{a_1,a_2,…,a_m\}$ be the set of positive integers less than $n$ that are relatively prime to $n.$ Prove that if every prime that divides $m$ also divides $n,$ then $a_1^k+a_2^k + \dots + a_m^k$ is divisible by $m$ for every positive integer $k.$

对于给定的整数 $n\ge 2,$ 令 $\{a_1,a_2,…,a_m\}$ 为小于 $n$ 且与 $n 互质的正整数集合。$ 证明，如果除 $m$ 的每个素数也整除 $n,$，则对于每个正整数 $k，$a_1^k+a_2^k + \dots + a_m^k$可被$m$整除。$