HMMT 二月 2024 · 冲刺赛 · 第 8 题

HMMT February 2024 — Guts Round — Problem 8

[6] Three points, A , B , and C , are selected independently and uniformly at random from the interior of a unit square. Compute the expected value of ∠ ABC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2024, February 17, 2024 — GUTS ROUND Organization Team Team ID# 2

解析

[6] Three points, A , B , and C , are selected independently and uniformly at random from the interior of a unit square. Compute the expected value of ∠ ABC . Proposed by: Akash Das π ◦ Answer: 60 = 3 ◦ Solution: Since ∠ ABC + ∠ BCA + ∠ CAB = 180 for all choices of A , B , and C , the expected value ◦ is 60 . 2