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Found problems: 1

2019 USAMO, 1
Tags: yes , function
Let $\mathbb{N}$ be the set of positive integers. A function $f:\mathbb{N}\to\mathbb{N}$ satisfies the equation \[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\] for all positive integers $n$. Given this information, determine all possible values of $f(1000)$. [i]Proposed by Evan Chen[/i]