IMO Shortlist 2005 problem A4
Dodao/la:
arhiva2. travnja 2012. Find all functions
![f: \mathbb{R}\to\mathbb{R}](/media/m/5/8/e/58ed5d9949596af6c667ae20695d9728.png)
such that
![f\left(x+y\right)+f\left(x\right)f\left(y\right)=f\left(xy\right)+2xy+1](/media/m/c/f/7/cf7e186bdb6b26e471e06de3ce426030.png)
for all real numbers
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y](/media/m/c/c/0/cc082a07a517ebbe9b72fd580832a939.png)
.
%V0
Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f\left(x+y\right)+f\left(x\right)f\left(y\right)=f\left(xy\right)+2xy+1$ for all real numbers $x$ and $y$.
Izvor: Međunarodna matematička olimpijada, shortlist 2005