HMMT 十一月 2025 · 冲刺赛 · 第 17 题

HMMT November 2025 — Guts Round — Problem 17

[10] Let P be a point inside equilateral triangle ABC such that ∠ BP C = 150 . Given that circumradii of triangle ABP and triangle ACP are 3 and 5, respectively, compute AP .

解析

[10] Let P be a point inside equilateral triangle ABC such that ∠ BP C = 150 . Given that circumradii of triangle ABP and triangle ACP are 3 and 5, respectively, compute AP . Proposed by: Pitchayut Saengrungkongka √ 30 15 34 √ Answer: = 17 34 Solution: A Y X P B C Let X and Y be the centers of ⊙ ( ABP ) and ⊙ ( ACP ). Write ∠ XAY = ∠ XAP + ∠ P AY ◦ ◦ = (90 − ∠ ABP ) + (90 − ∠ ACP ) ◦ ◦ = ( ∠ P BC + 30 ) + ( ∠ P CB + 30 ) ◦ ◦ ◦ = 60 + (180 − ∠ BP C ) = 90 . © 2025 HMMT √ √ 2 2 Thus, XY = 3 + 5 = 34. Note that AP is twice the height from A onto XY by symmetry, so √ 4[ AXY ] 15 34 AP = = . XY 17