Middle European Mathematical Olympiad 2009 Problem I-1

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that

$$f (x f (y)) + f (f (x) + f (y)) = y f (x) + f (x + f (y))$$

for all $x,y\in \mathbb{R}$, where $\mathbb{R}$ denotes the set of real numbers.