Question 99
Let $a_1, a_2, a_3, \dots$ be an infinite sequence of positive integers such that for all positive integers $n, m$, $a_{n+2m}$ divides $a_n + a_{n+m}$. Prove that this sequence is eventually periodic, that is, there exist positive integers $N$ and $d$ such that $a_n = a_{n+d}$ for all $n \gt N$.