A few weeks ago, I found a fascinating solution to a USAMO combinatorics problem that used group theory. Look at the 2nd solution on this link to view it.
I think there might be a way to use group theory to solve the following IMO problem from 2007:
In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of members of a clique is called its size. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged in two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.
I'm relatively new to group theory, so although I've spent a lot of time trying to solve this problem, I can't find the solution. How can it be solved?
If you know of any similar problems that use group theory to produce an interesting proof for a combinatorics problem, you may post that as well. I'm trying to learn how to solve problems this way, and any other examples will help me understand the technique.