# 2015 IMO Problems/Problem 5

Let be the set of real numbers. Determine all functions : satisfying the equation

for all real numbers and .

Proposed by Dorlir Ahmeti, Albania

:

for all real numbers and .

(1) Put in the equation, We get or Let , then

(2) Put in the equation, We get But so, or Hence

Case :

Put in the equation, We get or, Say , we get

So, is a solution

Case : Again put in the equation, We get or,

We observe that must be a polynomial of power as any other power (for that matter, any other function) will make the and of different powers and will not have any non-trivial solutions.

Also, if we put in the above equation we get

satisfies both the above.

Hence, the solutions are and .