2021 IMO Shortlist N7

Let $a_{1}, a_{2}, a_{3}, \ldots$ be an infinite sequence of positive integers such that $a_{n+2 m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m$. Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_{n}=a_{n+d}$ for all $n>N$.

令$a_{1}、a_{2}、a_{3}、\ldots$为正整数的无限序列，使得$a_{n+2 m}$除以所有正整数$n$和$m$。证明这个序列最终是周期性的，即存在正整数$N$和$d$，使得对于所有$n>N$，$a_{n}=a_{n+d}$。