HMMT 二月 2025 · COMB 赛 · 第 9 题
HMMT February 2025 — COMB Round — Problem 9
题目详情
- Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon.
解析
- Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon. Proposed by: Albert Wang 4 Answer: 9 Solution: P P Q Q First, we compute the probability that the line through two random points in a triangle ABC passes through segments AB and AC . We can take an affine transform of the two random points and the triangle such that ABC becomes equilateral. Since the distribution of the two points is still uniform 1 and independent, the probability of the line intersecting any two given sides is by symmetry. 3 Next, we compute the probability that the line through two random points in a rectangle ABCD passes through opposite edges AB and CD . B C Q P A If the line passes through AB and BC , the points must both lie in triangle ABC , whose area is half 1 that of ABCD . Given this, the probability the line passes through those two sides is , as computed 3 2 1 1 1 before. Thus the probability the line passes through AB and BC is · = . The same goes for 2 3 12 the other pairs of adjacent edges. By symmetry, the line is equally likely to pass through either pair 1 1 1 of opposite edges, each with probability 1 − 4 · = . 2 12 3 P Q We now return to the original problem. If the line passes through a pair of opposite edges, then 2 both points must be in the rectangle formed by these edges, which has area that of the hexagon. 3 1 Given this, the probability the line passes through those two edges is as computed before. Thus, the 3 2 2 1 4 probability that the line passes through the given pair of opposite edges is · = . Hence, the 3 3 27 4 4 probability the line passes through any of the three pairs of opposite edges is 3 · = . 27 9