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f(f(x)^2*y)=x^3f*(xy)

Source: IMO Shortlist 2010, Algebra 5

July 17, 2011
functionalgebraIMO Shortlistfunctional equation

Problem Statement

Denote by Q+\mathbb{Q}^+ the set of all positive rational numbers. Determine all functions f:Q+↦Q+f : \mathbb{Q}^+ \mapsto \mathbb{Q}^+ which satisfy the following equation for all x,y∈Q+:x, y \in \mathbb{Q}^+: f(f(x)2y)=x3f(xy).f\left( f(x)^2y \right) = x^3 f(xy).
Proposed by Thomas Huber, Switzerland
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