Problem

Source: 2020 Kazakhstan MO grades XI P2

Tags: algebra, functional equation



Find all functions $ f: \mathbb {R} ^ + \to \mathbb {R} ^ + $ such that for any $ x, y \in \mathbb {R} ^ + $ the following equality holds: \[f (x) f (y) = f \left (\frac {xy} {x f (x) + y} \right). \]$ \mathbb {R} ^ + $ denotes the set of positive real numbers.