HMMT 二月 2015 · 几何 · 第 9 题

英文解析

9. Let ABCD be a regular tetrahedron with side length 1. Let X be the point in triangle BCD suchthat [ XBC ] = 2[ XBD ] = 4[ XCD ], where [ ̟ ] denotes the area of figure ̟ . Let Y lie on segment AXsuch that 2 AY = Y X . Let M be the midpoint of BD . Let Z be a point on segment AM such that the lines Y Z and BC intersect at some point. Find .AZ
   4ZM
   Answer: We apply three-dimensional barycentric coordinates with reference tetrahedron
   ABCD . The given conditions imply that 7
   X = (0 : 1 : 2 : 4)
   Y = (14 : 1 : 2 : 4)
   M = (0 : 1 : 0 : 1)
   Z = ( t : 1 : 0 : 1)
   ( )
   ( )
   14 1 2 4 t 1 1
   for some real number t . Normalizing, we obtain Y = , , , and Z = , , 0 , . If
   21 21 21 21 t +2 t +2 t +2
   Y Z intersects line BC then there exist parameters α + β = 1 such that αY + βZ has zero A and Dcoordinates, meaning
   14 tα + β = 0
   21 t + 2
   4 1
   α + β = 0
   21 t + 2
   α + β = 1 .
   22 7
   Adding twice the second equation to the first gives α + β = 0, so α = − 22, β = 21, and thus t = .
   21 2
   AZ 2+2 4
   It follows that Z = (7 : 2 : 0 : 2), and = = .
   ZM 7 7