HMMT 二月 2026 · 冲刺赛 · 第 23 题

HMMT February 2026 — Guts Round — Problem 23

[12] Let Γ be a sphere of radius 5 . Let A , B , C , and D be points on Γ such that AB = BC = CD = DA = 8 and ∠ ABC = ∠ BCD = ∠ CDA = ∠ DAB . Compute AC .

解析

[12] Let Γ be a sphere of radius 5 . Let A , B , C , and D be points on Γ such that AB = BC = CD = DA = 8 and ∠ ABC = ∠ BCD = ∠ CDA = ∠ DAB . Compute AC . Proposed by: Jason Mao √ √ Answer: 6 2 = 72 Solution: Let O be the center of Γ , and let M and N be the midpoints of AC and BD , respectively. Then M N is perpendicular to both AC and BD , and O is the midpoint of M . Let a = OM and b = M A = N B . By Pythagoras, 2 2 2 2 2 a + b = OM + M A = OA = 25 and 2 2 2 2 2 2 b + (2 a ) + b = AM + M N + N B = 64 √ √ 2 2 . Solving for a and b , we get b = 18 , so AC = 2 b = 6 2 .