IMO Shortlist 2009 problem A7
Dodao/la:
arhiva2. travnja 2012. Find all functions
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
from the set of real numbers into the set of real numbers which satisfy for all
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
,
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
the identity
![f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2](/media/m/5/3/7/53760469d5e7b6f657530857e07e1a11.png)
Proposed by Japan
%V0
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity $$f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2$$
Proposed by Japan
Izvor: Međunarodna matematička olimpijada, shortlist 2009