Problem

Source: ELMO Shortlist 2023 A2

Tags: Elmo, algebra, functional equation



Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\] Proposed by Luke Robitaille