IMO 2011 Problem 3

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies $$f(x + y) \leq yf(x) + f(f(x))$$ for all real numbers $x$ and $y$ . Prove that $f(x) = 0$ for all $x \leq 0$ .

Proposed by Igor Voronovich, Belarus