HMMT 二月 2013 · 冲刺赛 · 第 16 题
HMMT February 2013 — Guts Round — Problem 16
题目详情
- [ 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 ballbounces 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#
英文原题
[ 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.
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HMMT 2013, 16 FEBRUARY 2013 — GUTS ROUND
Organization Team Team ID#
解析
英文解析
- [ 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 ballbounces 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