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Fun combinatorial geo

Source: IMC 2022 Day 2 Problem 8

August 5, 2022
probability and statsexpected valuecombinatoricscombinatorial geometryIMC 2022

Problem Statement

Let n,k≥3n, k \geq 3 be integers, and let SS be a circle. Let nn blue points and kk red points be chosen uniformly and independently at random on the circle SS. Denote by FF the intersection of the convex hull of the red points and the convex hull of the blue points. Let mm be the number of vertices of the convex polygon FF (in particular, m=0m=0 when FF is empty). Find the expected value of mm.
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