PUMaC 2020 · 几何（A 组） · 第 2 题

PUMaC 2020 — Geometry (Division A) — Problem 2

Hexagon ABCDEF has an inscribed circle Ω that is tangent to each of its sides. If AB = 12, √ ◦ ◦ ∠ F AB = 120 , and ∠ ABC = 150 , and if the radius of Ω can be written as m + n for positive integers m, n , find m + n .

解析

Hexagon ABCDEF has an inscribed circle Ω that is tangent to each of its sides. If AB = 12, √ ◦ ◦ ∠ F AB = 120 , and ∠ ABC = 150 , and if the radius of Ω can be written as m + n for positive integers m, n , find m + n . Proposed by: Sunay Joshi Answer: 36 Let r denote the radius of Ω, let O denote the center of Ω, and let Ω touch side AB at point 1 ◦ X . Then OX is the altitude from O in 4 AOB . Note that ∠ OAB = ∠ F AB = 60 and 2 √ 1 r 3 ◦ ∠ OBA = ∠ ABC = 75 . Thus by right angle trigonometry, AX = = r and BX = ◦ 2 tan 60 3 √ √ √ √ r 3 = (2 − 3) r . As AB = AX + BX = 12, we have ( + 2 − 3) r = 12 → r = 9 + 27, ◦ tan 75 3 thus our answer is m + n = 36.