2023 IMO Shortlist N5

Let $a_{1}<a_{2}<a_{3}<\cdots$ be positive integers such that $a_{k+1}$ divides $2\left(a_{1}+a_{2}+\cdots+a_{k}\right)$ for every $k \geq 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_{k}$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_{k}$. (Netherlands)

令 $a_{1}<a_{2}<a_{3}<\cdots$ 为正整数，使得 $a_{k+1}$ 可以除以 $2\left(a_{1}+a_{2}+\cdots+a_{k}\right)$，每 $k \geq 1$。假设对于无穷多个素数$p$，存在$k$使得$p$整除$a_{k}$。证明对于每个正整数 $n$，都存在 $k$ 使得 $n$ 除以 $a_{k}$。 (荷兰)