Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.
Imagine that the integers are split into sets $A$, $B$, $C$ with the restriction that the negative of an integer belonging to $A$ should belong to $B$, and ineteger that can be represented as the sum of two integers belonging to $B$ should be in $A$. It is fairly simple to show that the converse of both these restrictions should also hold true- the negative of an integer in $B$ is in $A$ and the sum of two integers in $A$ is in $B$.
In what sorts of ways can we arrange all integers in these sets?