HMMT 十一月 2016 · THM 赛 · 第 6 题

英文解析

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
   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