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Found problems: 1

2023 European Mathematical Cup, 4
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be a function such that for all positive integers $x$ and $y$, the number $f(x)+y$ is a perfect square if and only if $x+f(y)$ is a perfect square. Prove that $f$ is injective. [i]Remark.[/i] A function $f\colon\mathbb{N}\rightarrow\mathbb{N}$ is injective if for all pairs $(x,y)$ of distinct positive integers, $f(x)\neq f(y)$ holds. [i]Ivan Novak[/i]
function ivan novak orz functional equation algebra functional equations modular arithmetic number theory