HMMT 二月 2012 · 冲刺赛 · 第 16 题

HMMT February 2012 — Guts Round — Problem 16

[ 7 ] Let A , B , C , and D be points randomly selected independently and uniformly within the unit square. What is the probability that the six lines AB , AC , AD , BC , BD , and CD all have positive slope? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TH 15 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 11 FEBRUARY 2012 — GUTS ROUND

解析

[ 7 ] Let A , B , C , and D be points randomly selected independently and uniformly within the unit square. What is the probability that the six lines AB , AC , AD , BC , BD , and CD all have positive slope? 1 Answer: Consider the sets of x -coordinates and y -coordinates of the points. In order to 24 make 6 lines of positive slope, we must have smallest x-coordinate must be paired with the smallest y-coordinate, the second smallest together, and so forth. If we fix the order of the x -coordinates, the probability that the corresponding y -coordinates are in the same order is 1 / 24.