Middle European Mathematical Olympiad 2011 Problem T-1

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that the equality $$y^{2} f(x) + x^{2} f(y) + xy = x y f(x + y) + x^{2} + y^{2}$$ holds for all $x, y \in \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers.