Problem

Source: IMO Shortlist 2010, Algebra 5

Tags: function, algebra, IMO Shortlist, functional equation



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