# 2019 IMO Problems/Problem 1

**Problem:**

*Let be the set of integers. Determine all functions such that, for all*
*integers and , *

**Solution 1:**

Let us substitute in for to get

Now, since the domain and range of are the same, we can let and equal some constant to get
Therefore, we have found that **all** solutions must be of the form

Plugging back into the original equation, we have: which is true. Therefore, we know that satisfies the above for any **integral** constant c, and that this family of equations is unique.

**Solution 2:**
We plug in and to get
respectively.

Setting them equal to each other, we have the equation and moving "like terms" to one side of the equation yields Seeing that this is a difference of outputs of we can relate this to slope by dividing by on both sides. This gives us which means that is linear.

Let Plugging our expression into our original equation yields and letting be constant, this can only be true if If then which implies However, the output is then not all integers, so this doesn't work. If we have Plugging this in works, so the answer is for some integer