Happy counting! 🧩 Do you have a favorite Olympiad combinatorics problem or a clever solution that blew your mind? Share it in the comments below!
Consider all lines through at least two points. Pick the line with the smallest positive distance to a point not on it. Show that line must contain exactly two points, otherwise you’d get a smaller distance. Olympiad Combinatorics Problems Solutions
At a party, some people shake hands. Prove that the number of people who shake an odd number of hands is even. Happy counting
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. Consider all lines through at least two points
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.
Color the board black and white in the usual pattern. A knight always moves from a black square to a white square and vice versa. For a closed tour, the knight must make an equal number of black and white moves, but there are 64 squares. Since 64 is even, a closed knight’s tour is possible in theory—but parity alone doesn’t guarantee it; it’s a starting point for deeper invariants.