IMO Shortlist 2010 problem A1
Dodao/la:
arhiva23. lipnja 2013. Find all function
![f:\mathbb{R}\rightarrow\mathbb{R}](/media/m/5/6/2/56257456bca0fc5d753b46f282759409.png)
such that for all
![x,y\in\mathbb{R}](/media/m/5/3/f/53f619364029607afe565494c79b62d8.png)
the following equality holds
![f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor](/media/m/1/a/f/1afb67b62e30be2543c20cf2d3a2ce7f.png)
where
![\left\lfloor a\right\rfloor](/media/m/c/f/e/cfec9ecd126eaff61b8256f41ae271cd.png)
is greatest integer not greater than
![a.](/media/m/d/2/c/d2c84cdedb25528188824b2a35a7c63f.png)
Proposed by Pierre Bornsztein, France
%V0
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds $$f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor$$
where $\left\lfloor a\right\rfloor$ is greatest integer not greater than $a.$
Proposed by Pierre Bornsztein, France
Izvor: Međunarodna matematička olimpijada, shortlist 2010