Problem

Source: Romanian National Olympiad 2024 - Grade 9 - Problem 3

Tags: function, algebra, functional equation, Functional Equations



Find the functions $f: \mathbb{R} \to \mathbb{R}$ that satisfy $$(f(x)-y)f(x+f(y))=f(x^2)-yf(y),$$for all real numbers $x$ and $y.$