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functional equations over positive rationals make me big sad

Source: 2023 HMIC P1

April 25, 2023
algebrafunctional equationHMIC

Problem Statement

Let Q+\mathbb{Q}^{+} denote the set of positive rational numbers. Find, with proof, all functions f:Q+ā†’Q+f:\mathbb{Q}^+ \to \mathbb{Q}^+ such that, for all positive rational numbers xx and y,y, we have f(x)=f(x+y)+f(x+x2f(y)).f(x)=f(x+y)+f(x+x^2f(y)).
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