2011 IMO Shortlist Problems/A6

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