Let be a real-valued function defined on the set of real numbers that satisfies for all real numbers and . Prove that for all .
Proposed by Igor Voronovich, Belarus
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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
Source: Međunarodna matematička olimpijada, shortlist 2011
Školjka 
be a real-valued function defined on the set of real numbers that satisfies
and 
. Prove that 
for all 
.