2012 IMO Problems/Problem 4
Find all functions such that, for all integers and that satisfy , the following equality holds: (Here denotes the set of integers.)
Solution
Consider Then Now we look at
We can write
If , then
Case 1:
Case 2: , we will have or
Case 2.1: if is odd, if is even.
Case 2.2:
or
Case 2.2.1:
and
or and or
Case 2.2.2:
or
and or
If then
We will prove by induction
If then is true for some .
and if the statement is true for
or
and or
the statement is true for as well.
--Dineshram