HMMT 十一月 2011 · 冲刺赛 · 第 25 题
HMMT November 2011 — Guts Round — Problem 25
题目详情
英文原题
[12] LetXY 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/MX = 1. Determine the ratio of the area of triangle KLM to the area of triangle XY Z .
解析
英文解析
- [ 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 oftriangle KLM to the area of triangle XY Z .
Answer: First, we note that 1
[ KLM ] = [ XY Z ] − [ XKM ] − [ Y LK ] − [ ZM L ] .5
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)=
If we solve our system of equations for A, B, C , we get that A = .1
Y5
K151
L3
3254
X Z
1 1
2 2M