HMMT 二月 2017 · 冲刺赛 · 第 2 题

HMMT February 2017 — Guts Round — Problem 2

[ 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 BZ and 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

[ 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 BZ and 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 77 Answer: 6 Let XY = z, Y Z = x , and ZX = y. By Power of a Point, we have that 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 77 ZX = . 6 2 2