HMMT 十一月 2016 · 团队赛 · 第 4 题

英文解析

4. [ 5 ] Meghal is playing a game with 2016 rounds 1 , 2 , · · · , 2016. In round n , two rectangular double-sided
   2 πmirrors are arranged such that they share a common edge and the angle between the faces is .
   n +2
   Meghal shoots a laser at these mirrors and her score for the round is the number of points on the twomirrors at which the laser beam touches a mirror. What is the maximum possible score Meghal couldhave after she finishes the game?
   Proposed by: Rachel Zhang
   Answer: 1019088
   2 π
   Let points O, A , A lie in a plane such that ∠ A OA = . We represent the mirrors as line segments
   1 2 1 2
   n +2
   extending between O and A , and O and A . Also let points A , A , · · · , A lie in the plane such
   1 2 3 4 n +2
   that A is the reflection of A over OA .
   i +1 i − 1 i
   If Meghal shoots a laser along line l such that the first point of contact with a mirror is along OA , thenext point of contact, if it exists, is the point on OA that is a reflection of the intersection of l with 2
   OA . If we continue this logic, we find that the maximum score for round n is equal to the maximum 1
   number of intersection points between l and OA for some i . We do casework on whether n is even or 3
   n +2 iodd. If n is even, there are at most spokes such that l can hit OA , and if n is odd, there are at most
   2 in +3
   such spokes. Then we must sum 2 + 2 + 3 + 3 + · · · + 1009 + 1009 = 1009 · 1010 − 1 − 1 = 1019088 . 2