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.
(This solution does not work though because we don't know that is surjective)
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