HMMT 十一月 2009 · 团队赛 · 第 3 题

英文解析

3. [ 3 ] Find the total area contained in all the circles.
   180 π
   Answer: Using the notation from the previous solution, the area contained in the i th circle isequal to πr . Since the radii form a geometric sequence, the areas do as well. Specifically, the areas 213
   100 i
   100 16 180 πform a sequence with initial term π · and common ratio , so their sum is then π · = .9
   9 81 1365
   Bouncy Balls 81
   In the following problems, you will consider the trajectories of balls moving and bouncing off of theboundaries of various containers. The balls are small enough that you can treat them as points. Let ussuppose that a ball starts at a point X , strikes a boundary (indicated by the line segment AB ) at Y , andthen continues, moving along the ray Y Z . Balls always bounce in such a way that ∠ XY A = ∠ BY Z . Thisis indicated in the above diagram.
   ball bounces off of AB at the point Y
   YXBZ
   Balls bounce off of boundaries in the same way light reflects off of mirrors - if the ball hits the boundary aat point P , the trajectory after P is the reflection of the trajectory before P through the perpendicular tothe boundary at P .
   A ball inside a rectangular container of width 7 and height 12 is launched from the lower-left vertex of
   √
   the container. It first strikes the right side of the container after traveling a distance of 53 (and strikes noother sides between its launch and its impact with the right side).