Problem 1: Determine all functions satisfying

   

for all positive integers and .

idea and solution: We could attempt some functions. We find that constant function works, does not work. does not work. This prompt us, maybe the only possible solution is ? This is because the left and right hand side has different order, so only a function with zero order works.

We could do some operation. For , , we get . This expression could be simplified a lot if . So let’s just assume on is greater, let’s say , then

   

We know it is impossible since we cannot add infinitely many natural numbers in . Q.E.D.