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2021 EGMO P2: f(xf(x)+y) = f(y) + x^2 for rational x, y

Source: 2021 EGMO P2

April 13, 2021
EGMOfunctional equationEGMO 2021

Problem Statement

Find all functions f:Qā†’Qf:\mathbb{Q}\to\mathbb{Q} such that the equation f(xf(x)+y)=f(y)+x2f(xf(x)+y) = f(y) + x^2holds for all rational numbers xx and yy.
Here, Q\mathbb{Q} denotes the set of rational numbers.
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