HMMT 二月 2011 · ALGCOMB 赛 · 第 8 题

HMMT February 2011 — ALGCOMB Round — Problem 8

Let ABCDEF be a regular hexagon of area 1. Let M be the midpoint of DE . Let X be the intersection of AC and BM , let Y be the intersection of BF and AM , and let Z be the intersection of AC and BF . If [ P ] denotes the area of polygon P for any polygon P in the plane, evaluate [ BXC ] + [ AY F ] + [ ABZ ] − [ M XZY ].

解析

Let ABCDEF be a regular hexagon of area 1. Let M be the midpoint of DE . Let X be the intersection of AC and BM , let Y be the intersection of BF and AM , and let Z be the intersection of AC and BF . If [ P ] denotes the area of polygon P for any polygon P in the plane, evaluate [ BXC ] + [ AY F ] + [ ABZ ] − [ M XZY ]. Answer: 0 A B Z Y X F O C E D M Algebra & Combinatorics Individual Test Let O be the center of the hexagon. The desired area is [ ABCDEF ] − [ ACDM ] − [ BF EM ]. Note ◦ ◦ that [ ADM ] = [ ADE ] / 2 = [ ODE ] = [ ABC ], where the last equation holds because sin 60 = sin 120 . Thus, [ ACDM ] = [ ACD ] + [ ADM ] = [ ACD ] + [ ABC ] = [ ABCD ], but the area of ABCD is half the area of the hexagon. Similarly, the area of [ BF EM ] is half the area of the hexagon, so the answer is zero.