HMMT 十一月 2009 · 团队赛 · 第 7 题

英文解析

7. [ 4 ] How many times does the ball bounce before it returns to a vertex? (The final contact with a vertexdoes not count as a bounce.)
   Answer: 7 The key idea is that, instead of reflecting the line AY off of BC , we will reflect ABCabout BC and extend AY beyond 4 ABC . We keep doing this until the extension of AY hits a vertexof one of our reflected triangles. This is illustrated in the diagram below:
   √
   √
   3 3
   3 3
   We can calculate that the line AY has slope = , so that (as indicated in the diagram), AY2
   ( )277
   √
   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.