HMMT 十一月 2011 · 冲刺赛 · 第 25 题

HMMT November 2011 — Guts Round — Problem 25

[ 12 ] Let XY Z be an equilateral triangle, and let K, L, M be points on sides XY, Y Z, ZX , respectively, such that XK/KY = B , Y L/LZ = 1 /C , and ZM/M X = 1. Determine the ratio of the area of triangle KLM to the area of triangle XY Z .

解析

[ 12 ] Let XY Z be an equilateral triangle, and let K, L, M be points on sides XY, Y Z, ZX , respectively, such that XK/KY = B , Y L/LZ = 1 /C , and ZM/M X = 1. Determine the ratio of the area of triangle KLM to the area of triangle XY Z . 1 Answer: First, we note that 5 [ KLM ] = [ XY Z ] − [ XKM ] − [ Y LK ] − [ ZM L ] . Then, note that XK XM B 1 [ XKM ] = · · [ XY Z ] = · · [ XY Z ] XY XZ B + 1 2 Y L Y K 1 1 [ Y LK ] = · · [ XY Z ] = · · [ XY Z ] Y Z Y X C + 1 B + 1 ZM ZL 1 1 [ ZM L ] = · · [ XY Z ] = · · [ XY Z ] ZX ZY 2 C + 1 Consequently, [ KLM ] A = [ XY Z ] B 1 1 1 C 1 = 1 − · − · − · B + 1 2 C + 1 B + 1 C + 1 2 B + C = ( B + 1)( C + 1)(2) 1 If we solve our system of equations for A, B, C , we get that A = . 5 Y 1 5 1 K 3 L 4 5 2 3 X Z 1 1 M 2 2