HMMT 十一月 2010 · GEN1 赛 · 第 10 题

HMMT November 2010 — GEN1 Round — Problem 10

[ 8 ] You are given two diameters AB and CD of circle Ω with radius 1. A circle is drawn in one of the smaller sectors formed such that it is tangent to AB at E , tangent to CD at F , and tangent to Ω at 2 P . Lines P E and P F intersect Ω again at X and Y . What is the length of XY , given that AC = ? 3

解析

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