Problem

Source: Ukrainian Mathematical Olympiad 2024. Day 1, Problem 10.4

Tags: function, algebra, functional equation



Find all functions $f:\mathbb{R} \to \mathbb{R}$, such that for any $x, y \in \mathbb{R}$ holds the following: $$f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)$$ Proposed by Mykhailo Shtandenko