Olympiad Combinatorics | Problems Solutions

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.

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

When a problem involves moves or transformations, look for what doesn’t change modulo 2, modulo 3, or some clever coloring. 3. Double Counting: Two Ways to Tell the Same Story One of the most elegant weapons in the Olympiad arsenal. Count the same set of objects in two different ways to derive an identity. If you’ve ever looked at an International Mathematical

A knight starts on a standard chessboard. Is it possible to visit every square exactly once and return to the start (a closed tour)? Prove that the number of people who shake

Show that in any group of 6 people, there are either 3 mutual friends or 3 mutual strangers.

Happy counting! 🧩 Do you have a favorite Olympiad combinatorics problem or a clever solution that blew your mind? Share it in the comments below!

In a tournament (every pair of players plays one game, no ties), prove there is a ranking such that each player beats the next player in the ranking.