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Functions f(x)f(y) = y^a*f(x/2) + x^b*f(y/2)

Source: IMO Shortlist 1994, A4

August 10, 2008
functionalgebrafunctional equationIMO Shortlist

Problem Statement

Let R \mathbb{R} denote the set of all real numbers and \mathbb{R}^\plus{} the subset of all positive ones. Let α \alpha and β \beta be given elements in R, \mathbb{R}, not necessarily distinct. Find all functions f: \mathbb{R}^\plus{} \mapsto \mathbb{R} such that f(x)f(y) \equal{} y^{\alpha} f \left( \frac{x}{2} \right) \plus{} x^{\beta} f \left( \frac{y}{2} \right) \forall x,y \in \mathbb{R}^\plus{}.
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