2024 IMO Shortlist C7

Let $N$ be a positive integer and let $a_1,a_2,\ldots$ be an infinite sequence of positive integers. Suppose that, for each $n>N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1,a_2,\ldots,a_{n-1}$.

Prove that at least one of the sequences $a_1,a_3,a_5,\ldots$ and $a_2,a_4,a_6,\ldots$ is eventually periodic.

设 $N$ 为正整数，$a_1,a_2,\ldots$ 为一个正整数无限数列。假设对每个 $n>N$，$a_n$ 等于 $a_{n-1}$ 在列表 $a_1,a_2,\ldots,a_{n-1}$ 中出现的次数。

证明数列 $a_1,a_3,a_5,\ldots$ 与 $a_2,a_4,a_6,\ldots$ 中至少有一个最终是周期的。