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Functional equation over rationals by Walther Janous

Source: Austrian Math Olympiad 2016, final round, problem 1

May 26, 2016
algebrafunctional equationfunction

Problem Statement

Let α∈Q+\alpha\in\mathbb{Q}^+. Determine all functions f:Q+→Q+f:\mathbb{Q}^+\to\mathbb{Q}^+ that for all x,y∈Q+x,y\in\mathbb{Q}^+ satisfy the equation f(xy+y) = f(x)f(y)+f(y)+αx. f\left(\frac{x}{y}+y\right) ~=~ \frac{f(x)}{f(y)}+f(y)+\alpha x. Here Q+\mathbb{Q}^+ denote the set of positive rational numbers.
(Proposed by Walther Janous)
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