HMMT 十一月 2009 · 冲刺赛 · 第 30 题

英文解析

30. [ 17 ] Regular hexagon ABCDEF has side length 2. A laser beam is fired inside the hexagon from point
    A and hits BC at point G . The laser then reflects off BC and hits the midpoint of DE . Find BG .
    Answer: Look at the diagram below, in which points J , K , M , T , and X have been defined. M2
    is the midpoint of DE , BCJK is a rhombus with J lying on the extension of CD , T is the intersection 5
    of lines CD and GM when extended, and X is on JT such that XM ‖ JK .
    °
    It can be shown that m ∠ M DX = m ∠ M XD = 60 , so 4 DM X is equilateral, which yields XM = 1.
    The diagram indicates that JX = 5. One can show by angle-angle similarity that 4 T XM ∼ 4 T JK ,
    which yields T X = 5.
    One can also show by angle-angle similarity that 4 T JK ∼ 4 T CG , which yields the proportion
    T J T C 8 2 = . We know everything except CG , which we can solve for. This yields CG = , so BG = .
    JK CG 5 5
    2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUNDnd