Olympiad Combinatorics Problems Solutions Apr 2026

A finite set of points in the plane, not all collinear. Prove there exists a line passing through exactly two of the points.

Whenever you see sums of numbers counting relationships, try counting the total number of pairs or triples in two ways. 4. Extremal Principle: Look at the Extreme Pick an object that maximizes or minimizes some quantity. Then show that if the desired condition isn’t met, you can find a contradiction by modifying that extreme object. 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. A finite set of points in the plane, not all collinear

Count the total number of handshakes (sum of all handshake counts divided by 2). The sum of degrees is even. The sum of even degrees is even, so the sum of odd degrees must also be even. Hence, an even number of people have odd degree. At a party, some people shake hands

Pick one person, say Alex. Among the other 5, either at least 3 are friends with Alex or at least 3 are strangers to Alex. By focusing on that group of 3, you apply the pigeonhole principle again to force a monochromatic triangle in the friendship graph.