HMMT 二月 2013 · 冲刺赛 · 第 16 题

HMMT February 2013 — Guts Round — Problem 16

[ 8 ] The walls of a room are in the shape of a triangle ABC with ∠ ABC = 90 , ∠ BAC = 60 , and AB = 6. Chong stands at the midpoint of BC and rolls a ball toward AB . Suppose that the ball bounces off AB , then AC , then returns exactly to Chong. Find the length of the path of the ball. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT 2013, 16 FEBRUARY 2013 — GUTS ROUND Organization Team Team ID#

解析

[ 8 ] The walls of a room are in the shape of a triangle ABC with ∠ ABC = 90 , ∠ BAC = 60 , and AB = 6. Chong stands at the midpoint of BC and rolls a ball toward AB . Suppose that the ball bounces off AB , then AC , then returns exactly to Chong. Find the length of the path of the ball. √ ′ ′ Answer: 3 21 Let C be the reflection of C across AB and B be the reflection of B across ′ ′ AC ; note that B , A, C are collinear by angle chasing. The image of the path under these reflections ′ ′ ′ ′ is just the line segment M M , where M is the midpoint of BC and M is the midpoint of B C , so ′ ′ ′ our answer is just the length of M M . Applying the Law of Cosines to triangle M C M , we have √ √ √ ′ 2 1 ′ M M = 27 + 243 − 2 · 3 3 · 9 3 · = 189, so M M = 3 21. 2