HMMT 十一月 2024 · 冲刺赛 · 第 25 题

HMMT November 2024 — Guts Round — Problem 25

[13] Let ABC be an equilateral triangle. A regular hexagon P XQY RZ of side length 2 is placed so that P , Q , and R lie on segments BC , CA , and AB , respectively. If points A , X , and Y are collinear, compute BC .

解析

[13] Let ABC be an equilateral triangle. A regular hexagon P XQY RZ of side length 2 is placed so that P , Q , and R lie on segments BC , CA , and AB , respectively. If points A , X , and Y are collinear, compute BC . Proposed by: Rishabh Das √ √ √ √ Answer: 6 + 3 2 = 6 + 18 Solution: A Y R Q O Z X B C P ◦ ◦ Notice that ∠ QAR = 60 , and △ Y AR is isosceles with base angle 120 . This implies that Y is the √ ◦ circumcenter of △ AQR . Thus, Y A = Y R = Y Q = 2. We have ∠ AY R = 90 , so AR = 2 2. More- √ √ ◦ ◦ ◦ over, ∠ AY Q = 150 , so ∠ Y AQ = 15 , which implies that AQ = 4 sin 75 = 6 + 2. By symmetry, √ √ √ √ we also get that BR = AQ = 6 + 2. Hence, the answer is AR + BR = 6 + 3 2 .