HMMT 十一月 2009 · 冲刺赛 · 第 30 题

HMMT November 2009 — Guts Round — Problem 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nd 2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUND

解析

[ 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 . 2 Answer: Look at the diagram below, in which points J , K , M , T , and X have been defined. M 5 is the midpoint of DE , BCJK is a rhombus with J lying on the extension of CD , T is the intersection 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nd 2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUND