HMMT 二月 2021 · 冲刺赛 · 第 9 题

HMMT February 2021 — Guts Round — Problem 9

[10] Let AD , BE , and CF be segments sharing a common midpoint, with AB < AE and BC < BF . ◦ Suppose that each pair of segments forms a 60 angle, and that AD = 7, BE = 10, and CF = 18. Let K denote the sum of the areas of the six triangles 4 ABC , 4 BCD , 4 CDE , 4 DEF , 4 EF A , and 4 F AB. √ Compute K 3.

解析

[10] Let AD , BE , and CF be segments sharing a common midpoint, with AB < AE and BC < BF . ◦ Suppose that each pair of segments forms a 60 angle, and that AD = 7, BE = 10, and CF = 18. Let K denote the sum of the areas of the six triangles 4 ABC , 4 BCD , 4 CDE , 4 DEF , 4 EF A , and √ 4 F AB. Compute K 3. Proposed by: Milan Haiman Answer: 141 Solution: Let M be the common midpoint, and let x = 7 , y = 10 , z = 18. One can verify that hexagon ABCDEF is convex. We have √ √ √ √ 1 3 x y 1 3 y z 1 3 x z 3( xy + yz − zx ) [ ABC ] = [ ABM ]+[ BCM ] − [ ACM ] = · · · + · · · − · · · = . 2 2 2 2 2 2 2 2 2 2 2 2 16 Summing similar expressions for all 6 triangles, we have √ 3(2 xy + 2 yz + 2 zx ) K = . 16 √ Substituting x, y, z gives K = 47 3, for an answer of 141. Remark: As long as hexagon ABCDEF is convex, K is the area of this hexagon.