Problem

Source: Macedonian Mathematical Olympiad 2023 P1

Tags: algebra, functional equation, Macedonia, national olympiad



Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ we have: $$xf(x+y)+yf(y-x) = f(x^2+y^2)\,.$$Proposed by Nikola Velov