HMMT 二月 2017 · 冲刺赛 · 第 2 题
HMMT February 2017 — Guts Round — Problem 2
题目详情
英文原题
- [ 4 ] Let A, B, C, D, E, F be 6 points on a circle in that order. Let X be the intersection of AD and BE ,
Y is the intersection of AD and CF , and Z is the intersection of CF and BE . X lies on segments BZand AY and Y lies on segment CZ . Given that AX = 3, BX = 2, CY = 4, DY = 10, EZ = 16, and
F Z = 12, find the perimeter of triangle XY Z .
2 2
解析
英文解析
- [ 4 ] Let A, B, C, D, E, F be 6 points on a circle in that order. Let X be the intersection of AD and BE ,
Y is the intersection of AD and CF , and Z is the intersection of CF and BE . X lies on segments BZand AY and Y lies on segment CZ . Given that AX = 3, BX = 2, CY = 4, DY = 10, EZ = 16, and
F Z = 12, find the perimeter of triangle XY Z .
Proposed by: Sam Korsky
Answer:77
Let XY = z, Y Z = x , and ZX = y. By Power of a Point, we have that 6
3( z + 10) = 2( y + 16) , 4( x + 12) = 10( z + 3) , and 12( x + 4) = 16( y + 2) .
11 14 9
Solving this system gives XY = and Y Z = and ZX = . Therefore, our answer if XY + Y Z +
3 3 2
ZX = .77
2 26