Olympiad Combinatorics Problems Solutions (2026 Release)
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.
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.
When stuck, ask: “What’s the smallest/biggest/largest/minimal possible …?” 5. Graph Theory Modeling: Turn the Problem into Vertices & Edges Many combinatorial problems—about friendships, tournaments, networks, or matchings—are secretly graph problems.
When a problem says "prove there exist two such that…", think pigeonhole. 2. Invariants & Monovariants: Finding the Unchanging Invariants are properties that never change under allowed operations. Monovariants are quantities that always increase or decrease (but never go back).
A finite set of points in the plane, not all collinear. Prove there exists a line passing through exactly two of the points.
