2007 BMO Problems/Problem 2
(Bulgaria) Find all functions such that
, for any .
We first note that is a solution to the equation. Henceforth we shall consider other solutions to the equations, i.e., functions such that for some , .
Setting gives us .
We note that for any , , i.e., as and assume all real values, assume all real values.
Now, setting and , we obtain
Since takes on all real values, it follows that for all , . It is easy to see that any value of will satisfy the desired condition. Thus the only solutions to the functional equation are and , an arbitrary constant.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.