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HMMT 二月 2011 · ALGCOMB 赛 · 第 8 题

HMMT February 2011 — ALGCOMB Round — Problem 8

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Let ABCDEF be a regular hexagon of area 1. Let M be the midpoint of DE . Let X be theintersection of AC and BM , let Y be the intersection of BF and AM , and let Z be the intersectionof 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

英文解析

  1. Let ABCDEF be a regular hexagon of area 1. Let M be the midpoint of DE . Let X be theintersection of AC and BM , let Y be the intersection of BF and AM , and let Z be the intersectionof 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
    Y XZ
    F O C
    E D
    Algebra & Combinatorics Individual Test m
    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 iszero.