HMMT 二月 2009 · 冲刺赛 · 第 25 题

HMMT February 2009 — Guts Round — Problem 25

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

解析

[ 12 ] Four points, A , B , C , and D , are chosen randomly on the circumference of a circle with independent uniform probability. What is the expected number of sides of triangle ABC for which the projection of 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 orthog- onal projection of D onto AB lies interior to the segment. This happens exactly when either ∠ DAB or ∠ DBA is obtuse, which is equivalent to saying that A and B lie on the same side of the diameter through D . This happens with probability 1 / 2. Therefore, desired answer is 3 / 2. ∑ n − 1 2009