E5. Points in Convex Figures

Abstract

During the summer of 1987, I taught at the International Summer Institute in Orange, California. Students came from the U.S., Japan, Israel, Hungary, Switzerland, and France. For their test I decided to create a problem requiring the use of the Pigeonhole Principle in geometry. I came up with Problem 5.4(A).When you solve a problem like that, you ask yourself, can I prove a stronger result, i.e., a result with a smaller n? This train of thought led me to problem 5.4(B), and consequently to problem 5.5(A). The problem became too good to be used for a test. I saved it for the Fifth Colorado Mathematical Olympiad. Problem 5.5(B) shows that the result of problem 5.5(A) is best possible: you cannot reduce n to below 5. Does it mean that we have reached the end of the road? Not at all! Instead of looking at triangles alone, we can include all convex figures.