2018 USAMO Problems/Problem 2
Find all functions such that
for all with
The only such function is .
Letting gives , hence . Now observe that even if we fix , is not fixed. Specifically, This is continuous on the interval and has an asymptote at . Since it takes the value 2 when , it can take on all values greater than or equal to 2. So for any , we can find such that . Therefore, for all .
Now, for any , if we let , , and , then . Since , , hence . Therefore, for all .