Problem

Source: ELMO Shortlist 2023 A6

Tags: Elmo, algebra



Let \(\mathbb R_{>0}\) denote the set of positive real numbers and \(\mathbb R_{\ge0}\) the set of nonnegative real numbers. Find all functions \(f:\mathbb R\times \mathbb R_{>0}\to \mathbb R_{\ge0}\) such that for all real numbers \(a\), \(b\), \(x\), \(y\) with \(x,y>0\), we have \[f(a,x)+f(b,y)=f(a+b,x+y)+f(ay-bx,xy(x+y)).\] Proposed by Luke Robitaille