HMMT 十一月 2022 · THM 赛 · 第 7 题

HMMT November 2022 — THM Round — Problem 7

Alice and Bob are playing in the forest. They have six sticks of length 1 , 2 , 3 , 4 , 5 , 6 inches. Somehow, they have managed to arrange these sticks, such that they form the sides of an equiangular hexagon. Compute the sum of all possible values of the area of this hexagon.

解析

Alice and Bob are playing in the forest. They have six sticks of length 1 , 2 , 3 , 4 , 5 , 6 inches. Somehow, they have managed to arrange these sticks, such that they form the sides of an equiangular hexagon. Compute the sum of all possible values of the area of this hexagon. Proposed by: Vidur Jasuja √ Answer: 33 3 Solution: Let the side lengths, in counterclockwise order, be a , b , c , d , e , f . Place the hexagon on the coordinate plane with edge a parallel to the x -axis and the intersection between edge a and edge f at the origin (oriented so that edge b lies in the first quadrant). If you travel along all six sides of the hexagon starting from the origin, we get that the final x coordinate must be a + b/ 2 − c/ 2 − d − e/ 2 + f / 2 = 0 by vector addition. Identical arguments tell us that we must also have b + c/ 2 − d/ 2 − e − f / 2 + a/ 2 = 0 and c + d/ 2 − e/ 2 − f − a/ 2 + b/ 2 = 0. Combining these linear equations tells us that a − d = e − b = c − f . This is a necessary and sufficient condition for the side lengths to form an equiangular hexagon. WLOG say that a = 1 and b < f (otherwise, you can rotate/reflect it to get it to this case). Thus, we must either have ( a, b, c, d, e, f ) = (1 , 5 , 3 , 4 , 2 , 6) or (1 , 4 , 5 , 2 , 3 , 6). Calculating the areas of √ √ √ these two cases gets either 67 3 / 4 or 65 3 / 4, for a sum of 33 3.