Australian functional equation, 2017
Problem. Determine all functions such that for all real numbers and .
Origin: Australian Mathematical Olympiad 2017, problem 3.
Solution. The first thing we need to try in problems like this (functional equations) is to set variables to some specific numbers. The most obvious number to try here is . So, when we set , we get for any . Also, when we set , we get for any .
Taking the latter equation, reversing its left and right sides, and replacing with from the former equation (notice that this is for an arbitrary ), we get Now we can apply the original equation from the problem setting (putting instead of there), so we continue the chain of equalities: So, for any .
Now, the idea is to represent an arbitrary as for some and . We have two cases:
- If for any (a special case), then we are done.
- If for some , then for an arbitrary , we have . Therefore, is a constant.
So, for an arbitrary constant . Notice, that this includes the special case mentioned above when .
Finally, verifying that all such functions indeed satisfy the original condition.