HMMT 十一月 2009 · 团队赛 · 第 7 题

HMMT November 2009 — Team Round — Problem 7

[ 4 ] How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.) Page 1 of 2 In this final problem, a ball is again launched from the vertex of an equilateral triangle with side length 5.

解析

[ 4 ] How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.) Answer: 7 The key idea is that, instead of reflecting the line AY off of BC , we will reflect ABC about BC and extend AY beyond 4 ABC . We keep doing this until the extension of AY hits a vertex of one of our reflected triangles. This is illustrated in the diagram below: √ √ 3 3 3 3 2 We can calculate that the line AY has slope = , so that (as indicated in the diagram), AY 7 7 2 ( ) √ 35 15 3 first intersects a vertex at the point , . To get there, it has to travel through 2 horizontal 2 2 lines, 1 upward sloping line, and 4 downward sloping lines, so it bounces 2 + 1 + 4 = 7 times total. In this final problem, a ball is again launched from the vertex of an equilateral triangle with side length 5.