HMMT 二月 2025 · 冲刺赛 · 第 9 题

HMMT February 2025 — Guts Round — Problem 9

[7] Let P and Q be points selected uniformly and independently at random inside a regular hexagon ABCDEF . Compute the probability that segment P Q is entirely contained in at least one of the quadri- laterals ABCD , BCDE , CDEF , DEF A , EF AB , or F ABC .

解析

[7] Let P and Q be points selected uniformly and independently at random inside a regular hexagon ABCDEF . Compute the probability that segment P Q is entirely contained in at least one of the quadrilaterals ABCD , BCDE , CDEF , DEF A , EF AB , or F ABC . Proposed by: Isabella Zhu 5 Answer: 6 Solution: B A P O C F Q D E Let O be the center of the hexagon. Without loss of generality, assume P is in △ ABO . Then, segment P Q is entirely contained in one of the given quadrilaterals if and only if Q is not in △ DEO . The [ DEO ] 1 5 probability that Q is in △ DEO is = , so the answer is . [ ABCDEF ] 6 6