IMO Shortlist 2011 problem A6
Dodao/la:
arhiva23. lipnja 2013. Let
![f : \mathbb R \to \mathbb R](/media/m/a/c/7/ac7b7cb5a2b26dbe3eda7b7e0e5d73e3.png)
be a real-valued function defined on the set of real numbers that satisfies
![f(x + y) \leq yf(x) + f(f(x))](/media/m/0/f/2/0f211e9e41d7290d84d573fb92f45c0b.png)
for all real numbers
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
. Prove that
![f(x) = 0](/media/m/1/a/0/1a0332a8813a1110e758c328354c4121.png)
for all
![x \leq 0](/media/m/2/5/d/25d02aeed203c6a3fded3077cafba4d2.png)
.
Proposed by Igor Voronovich, Belarus
%V0
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
Izvor: Međunarodna matematička olimpijada, shortlist 2011