返回题库

HMMT 二月 2009 · 冲刺赛 · 第 25 题

HMMT February 2009 — Guts Round — Problem 25

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 12 ] Four points, A , B , C , and D , are chosen randomly on the circumference of a circle with independentuniform probability. What is the expected number of sides of triangle ABC for which the projectionof D onto the line containing the side lies between the two vertices?
    ∑
    n − 1
    2009
解析

英文解析

  1. [ 12 ] Four points, A , B , C , and D , are chosen randomly on the circumference of a circle with independentuniform probability. What is the expected number of sides of triangle ABC for which the projectionof D onto the line containing the side lies between the two vertices?
    Answer: 3 / 2
    Solution: By linearity of expectations, the answer is exactly 3 times the probability that the orthogonal projection of D onto AB lies interior to the segment. This happens exactly when either ∠ DABor ∠ DBA is obtuse, which is equivalent to saying that A and B lie on the same side of the diameterthrough D . This happens with probability 1 / 2. Therefore, desired answer is 3 / 2.
    ∑
    n − 1
    2009