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Found problems: 1

2023 Turkey Olympic Revenge, 4
Tags: number theory , functional equation , revenge
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all integers $x$ and $y$, the number $$f(x)^2+2xf(y)+y^2$$ is a perfect square. [i]Proposed by Barış Koyuncu[/i]