Solutions: Olympiad Combinatorics Problems

At a party, some people shake hands. Prove that the number of people who shake an odd number of hands is even.

A knight starts on a standard chessboard. Is it possible to visit every square exactly once and return to the start (a closed tour)? Olympiad Combinatorics Problems Solutions

Pick one person, say Alex. Among the other 5, either at least 3 are friends with Alex or at least 3 are strangers to Alex. By focusing on that group of 3, you apply the pigeonhole principle again to force a monochromatic triangle in the friendship graph. At a party, some people shake hands

But here’s the secret:

If you’ve ever looked at an International Mathematical Olympiad (IMO) problem and felt your brain do a double backflip, chances are it was a combinatorics question. Unlike algebra or geometry, where formulas and theorems provide a clear roadmap, combinatorics problems often feel like puzzles wrapped in riddles. Is it possible to visit every square exactly

Count the total number of handshakes (sum of all handshake counts divided by 2). The sum of degrees is even. The sum of even degrees is even, so the sum of odd degrees must also be even. Hence, an even number of people have odd degree.

This is equivalent to showing every tournament has a Hamiltonian path. Use induction: Remove a vertex, find a path in the remaining tournament, then insert the vertex somewhere.