Olympiad Combinatorics Problems Solutions Access

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

A finite set of points in the plane, not all collinear. Prove there exists a line passing through exactly two of the points. 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. Show that in any group of 6 people,

A knight starts on a standard chessboard. Is it possible to visit every square exactly once and return to the start (a closed tour)? If you’ve ever looked at an International Mathematical

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

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.

Take a classic problem like “Prove that in any set of 10 integers, there exist two whose difference is divisible by 9.” Apply the pigeonhole principle. You’ve just taken the first step into a larger world.