HMMT 十一月 2010 · GEN1 赛 · 第 10 题

英文解析

10. [ 8 ] You are given two diameters AB and CD of circle Ω with radius 1. A circle is drawn in one of thesmaller sectors formed such that it is tangent to AB at E , tangent to CD at F , and tangent to Ω at
    P . Lines P E and P F intersect Ω again at X and Y . What is the length of XY , given that AC = ?2
    √3
    4 2
    Answer: Let O denote the center of circle Ω. We first prove that OX ⊥ AB and OY ⊥ CD .
    Consider the homothety about P which maps the smaller circle to Ω. This homothety takes E to X3
    and also takes AB to the line tangent to circle Ω parallel to AB . Therefore, X is the midpoint of thearc AB , and so OX ⊥ AB . Similarly, OY ⊥ CD .
    √
    θ θ 1 θ 1
    Let θ = ∠ AOC . By the Law of Sines, we have AC = 2 sin . Thus, sin = , and cos = 1 − ( ) =2
    2 2 3 2 3
    √
    2 2
    . Therefore,
    ∠ XOY3
    XY = 2 sin
    ( )2
    °θ = 2 sin 90 −
    θ2 = 2 cos
    √2
    4 2 = .
    General Test 3