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.
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. Olympiad Combinatorics Problems Solutions
But here’s the secret:
Let’s break down the most common types of Olympiad combinatorics problems and the strategies to solve them. The principle is deceptively simple: If you put (n) items into (m) boxes and (n > m), at least one box contains two items. Take a classic problem like “Prove that in
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). Use induction: Remove a vertex, find a path
At a party, some people shake hands. Prove that the number of people who shake an odd number of hands is even.