ELMO 2021 Problem 2

Let $n > 1$ be an integer and let $a_1, a_2, \ldots, a_n$ be integers such that $n \mid a_i-i$ for all integers $1 \leq i \leq n$ . Prove there exists an infinite sequence $b_1,b_2, \ldots$ such that

• $b_k\in\{a_1,a_2,\ldots, a_n\}$ for all positive integers $k$ , and
• $\sum\limits_{k=1}^{\infty}\frac{b_k}{n^k}$ is an integer.