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f(x)^2+2xf(y)+y^2 is a square

Source: Turkey Olympic Revenge 2023 P4

March 6, 2023
functional equationnumber theoryRevenge

Problem Statement

Find all functions f:Z→Zf: \mathbb{Z}\to \mathbb{Z} such that for all integers xx and yy, the number f(x)2+2xf(y)+y2f(x)^2+2xf(y)+y^2 is a perfect square.
Proposed by Barış Koyuncu
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