HMMT 二月 2022 · 冲刺赛 · 第 25 题

HMMT February 2022 — Guts Round — Problem 25

[14] Let ABC be an acute scalene triangle with circumcenter O and centroid G . Given that AGO is a right triangle, AO = 9, and BC = 15, let S be the sum of all possible values for the area of triangle AGO . 2 Compute S .

解析

[14] Let ABC be an acute scalene triangle with circumcenter O and centroid G . Given that AGO is a right triangle, AO = 9, and BC = 15, let S be the sum of all possible values for the area of triangle 2 AGO . Compute S . Proposed by: Akash Das Answer: 288 Solution: Note that we know that O, H, and G are collinear and that HG = 2 OG . Thus, let OG = x √ BC 5 11 and HG = 2 x . We also have sin A = = , so cos A = . Then, if AG ⊥ OG, then we have 2 R 6 6 2 2 2 2 2 2 2 2 2 2 2 x + AG = OG + AG = AO = 81 and HG + AG = 4 x + AG = AH = (2 R cos A ) = 99. √ √ √ √ √ 1 5 3 Solving gives us x = 6 and AG = 5 3. Thus, the area of AGO in this case is · 6 · 5 3 = . 2 2 √ 2 2 2 2 If we have AO ⊥ OG, then we have 99 = AH = AO + OH = 81 + 9 x . This gives us x = 2. In √ √ √ 1 9 2 this case, we have the area of AGO is · 2 · 9 = . Adding up the two areas gives us S = 12 2. 2 2 2 Squaring gives S = 288.