International Mathematical Olympiad 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 $x \leq 0$.