HMMT 十一月 2009 · 团队赛 · 第 8 题

英文解析

8. [ 6 ] In how many ways can the ball be launched so that it will return again to a vertex for the first timeafter 2009 bounces?
   Answer: 502 We will use the same idea as in the previous problem. We first note that every vertex
   ( )
   √
   5 5 3
   of a triangle can be written uniquely in the form a (5 , 0) + b , , where a and b are non-negative
   2 2
   ( )
   √
   5 5 3
   integers. Furthermore, if a ball ends at a (5 , 0) + b , , then it bounches off of a wall 2( a + b ) − 3
   2 2
   times. Therefore, the possible directions that you can launch the ball in correspond to solutions to
   2( a + b ) − 3 = 2009, or a + b = 1006. However, if a and b have a common factor, say, k , then the ball willa bpass through the vertex corresponding to and before it passes through the vertex correspondingk kto a and b . Therefore, we must discount all such pairs a, b . This corresponds to when a is even ora = 503, so after removing these we are left with 502 remaining possible values of a , hence 502 possibledirections in which to launch the ball.
   Super Mario 64!
   Mario is once again on a quest to save Princess Peach. Mario enters Peach’s castle and finds himself in aroom with 4 doors. This room is the first in a sequence of 2 indistinugishable rooms. In each room, 1 doorleads to the next room in the sequence (or, for the second room, into Bowser’s level), while the other 3 doorslead to the first room.