# 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.

As the statement is true for , by mathematical induction we can conclude

**So, Case 2.1, Case 2.2.1 and Case 2.2.2 are the three independent possible solutions.**

--Dineshram