The International Math Olympiad 2019 had the following question:
Find all functions such that .
The reason that I decided to record this is because I thought I’d made an interesting observation that allowed me to solve the problem in only a couple of steps. However, I later realized that at least one other person has solved the problem the same way.
The right hand side is symmetric in . Clearly, . Hence, symmetrizing the left side as well, we get . This implies that . Assuming , we get .
Now use and to show that is linear. This shows us that or are the only solutions to this question.