HMMT 二月 2016 · 几何 · 第 8 题

HMMT February 2016 — Geometry — Problem 8

For i = 0 , 1 , . . . , 5 let l be the ray on the Cartesian plane starting at the origin, an angle θ = i i 3 counterclockwise from the positive x -axis. For each i , point P is chosen uniformly at random from i the intersection of l with the unit disk. Consider the convex hull of the points P , which will (with i i probability 1) be a convex polygon with n vertices for some n . What is the expected value of n ?

解析

For i = 0 , 1 , . . . , 5 let l be the ray on the Cartesian plane starting at the origin, an angle θ = i i 3 counterclockwise from the positive x -axis. For each i , point P is chosen uniformly at random from i the intersection of l with the unit disk. Consider the convex hull of the points P , which will (with i i probability 1) be a convex polygon with n vertices for some n . What is the expected value of n ? Proposed by: Answer: 2 + 4 ln(2) A vertex P is part of the convex hull if and only if it is not contained in the triangle formed by i the origin and the two adjacent vertices. Let the probability that a given vertex is contained in the aforementioned triangle be p . By linearity of expectation, our answer is simply 6(1 − p ). Say | P | = a, | P | = b . Stewart’s Theorem and the Law of Cosines give that p is equal to the probability 0 2 √ 2 2 a + b + ab ab that | P | < ab − ab = ; alternatively this is easy to derive using coordinate methods. 1 2 ( a + b ) a + b 2 The corresponding double integral evaluates to p = (1 − ln(2)), thus telling us our answer. 3