Olympiad problems

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Found problems: 2

2023 India EGMO TST, P5

Let $k$ be a positive integer. A sequence of integers $a_1, a_2, \cdots$ is called $k$-pop if the following holds: for every $n \in \mathbb{N}$, $a_n$ is equal to the number of distinct elements in the set $\{a_1, \cdots , a_{n+k} \}$. Determine, as a function of $k$, how many $k$-pop sequences there are. [i]Proposed by Sutanay Bhattacharya[/i]

India EGMO 2023 TST, 5

Tags: combinatorics , sequence , rigid

Let $k$ be a positive integer. A sequence of integers $a_1, a_2, \cdots$ is called $k$-pop if the following holds: for every $n \in \mathbb{N}$, $a_n$ is equal to the number of distinct elements in the set $\{a_1, \cdots , a_{n+k} \}$. Determine, as a function of $k$, how many $k$-pop sequences there are. [i]Proposed by Sutanay Bhattacharya[/i]