International Mathematics Competition
for University Students
2019

IMC2019: Day 2, Problem 10

Problem 10. \(\displaystyle 2019\) points are chosen at random, independently, and distributed uniformly in the unit disc \(\displaystyle \{(x,y)\in\RR^2\colon x^2+y^2\leq 1\}\). Let \(\displaystyle C\) be the convex hull of the chosen points. Which probability is larger: that \(\displaystyle C\) is a polygon with three vertices, or a polygon with four vertices?

Proposed by Fedor Petrov, St. Petersburg State University

Hint: Consider the case when 4 ponts, \(\displaystyle X_1,X_2,X_3,X_4\) form a convex quadrilateral and the remaining points lie in trh triangle \(\displaystyle X_1X_2X_3\).

  Solution