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An FE. Who woulda thunk it?

Source: 2021 USAJMO Problem 1

April 15, 2021
functionUSAJMOfunctional equation

Problem Statement

Let N\mathbb{N} denote the set of positive integers. Find all functions f:N→Nf : \mathbb{N} \rightarrow \mathbb{N} such that for positive integers aa and b,b, f(a2+b2)=f(a)f(b) and f(a2)=f(a)2.f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.
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