HMMT 十一月 2016 · THM 赛 · 第 6 题

HMMT November 2016 — THM Round — Problem 6

Let P , P , . . . , P be points in the complex plane, which are also roots of the equation x +6 x − 216 = 0. 1 2 6 Given that P P P P P P is a convex hexagon, determine the area of this hexagon. 1 2 3 4 5 6

解析

Let P , P , . . . , P be points in the complex plane, which are also roots of the equation x +6 x − 216 = 0. 1 2 6 Given that P P P P P P is a convex hexagon, determine the area of this hexagon. 1 2 3 4 5 6 Proposed by: Eshaan Nichani √ Answer: 9 3 6 3 3 3 Factor x + 6 x − 216 = ( x − 12)( x + 18). This gives us 6 points equally spaced in terms of their √ √ 3 3 angles from the origin, alternating in magnitude between 12 and 18. This means our hexagon is √ √ 3 3 composed of 6 triangles, each with sides of length 12 and 18 and with a 60 degree angle in between √ √ 3 3 them. This yields the area of each triangle as , so the total area of the hexagon is 9 3. 2