International Mathematical Olympiad 2010 Problem 1

Determine all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that the equality $$f \big (\lfloor x \rfloor y \big) = f (x) \big \lfloor f (y) \rfloor$$ holds for all $x, y \in \mathbb{R}$. (Here $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.)