Question: Let T be a triangle with angles of 30, 60, and 90, and a hypotenuse of 1. We choose ten points inside T at random. Prove that there will be four points among them that can be covered by a half-circle of radius 0.42.
2026-04-02 12:59:12.1775134752
Challenge (+) Problem 32, Chapter 1 (Pigeonhole Principle), A Walk Through Combinatorics, Miklos Bona
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I'm not convinced by point #4. Suppose the ten points placed originally happened to be in triangle T, and so did the circle. Then how do I know if cutting the circle in half won't cut out some of the points?