2024 IMO Problems/Problem 3

Let $a_1, a_2, a_3, \dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n > N$ , $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1, a_2, \dots, a_{n-1}$ .

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

(An infinite sequence $b_1, b_2, b_3, \dots$ is eventually periodic if there exist positive integers $p$ and $M$ such that $b_{m+p} = b_m$ for all $m \ge M$ .)

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2024 IMO Problems/Problem 3