HMMT 十一月 2009 · 团队赛 · 第 3 题

HMMT November 2009 — Team Round — Problem 3

[ 3 ] Find the total area contained in all the circles. Bouncy Balls In the following problems, you will consider the trajectories of balls moving and bouncing off of the boundaries of various containers. The balls are small enough that you can treat them as points. Let us suppose that a ball starts at a point X , strikes a boundary (indicated by the line segment AB ) at Y , and then continues, moving along the ray Y Z . Balls always bounce in such a way that ∠ XY A = ∠ BY Z . This is indicated in the above diagram. ball bounces off of AB at the point Y Z B X Y A Balls bounce off of boundaries in the same way light reflects off of mirrors - if the ball hits the boundary at point P , the trajectory after P is the reflection of the trajectory before P through the perpendicular to the 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 no other sides between its launch and its impact with the right side).

解析

[ 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 is 13 2 equal to πr . Since the radii form a geometric sequence, the areas do as well. Specifically, the areas i 100 100 16 180 π 9 form a sequence with initial term π · and common ratio , so their sum is then π · = . 65 9 81 13 81 Bouncy Balls In the following problems, you will consider the trajectories of balls moving and bouncing off of the boundaries of various containers. The balls are small enough that you can treat them as points. Let us suppose that a ball starts at a point X , strikes a boundary (indicated by the line segment AB ) at Y , and then continues, moving along the ray Y Z . Balls always bounce in such a way that ∠ XY A = ∠ BY Z . This is indicated in the above diagram. ball bounces off of AB at the point Y Z B X Y A Balls bounce off of boundaries in the same way light reflects off of mirrors - if the ball hits the boundary at point P , the trajectory after P is the reflection of the trajectory before P through the perpendicular to the 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 no other sides between its launch and its impact with the right side).