HMMT 二月 2016 · 几何 · 第 8 题

英文解析

8. For i = 0 , 1 , . . . , 5 let l be the ray on the Cartesian plane starting at the origin, an angle θ = iicounterclockwise from the positive x -axis. For each i , point P is chosen uniformly at random from 3
   the intersection of l with the unit disk. Consider the convex hull of the points P , which will (withii iprobability 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 byithe origin and the two adjacent vertices. Let the probability that a given vertex is contained inthe 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 abthat | P | < ab − ab = ; alternatively this is easy to derive using coordinate methods.
   1 2
   ( a + b ) a + b
   The corresponding double integral evaluates to p = (1 − ln(2)), thus telling us our answer.2 3