2021 USAJMO Problems/Problem 1

Problem

Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for positive integers $a$ and $b,$ $f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.$

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2021 USAJMO Problems/Problem 1

Problem

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