HMMT 十一月 2016 · 团队赛 · 第 4 题

HMMT November 2016 — Team Round — Problem 4

[ 5 ] Meghal is playing a game with 2016 rounds 1 , 2 , · · · , 2016. In round n , two rectangular double-sided 2 π mirrors 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 two mirrors at which the laser beam touches a mirror. What is the maximum possible score Meghal could have after she finishes the game?

解析

[ 5 ] Meghal is playing a game with 2016 rounds 1 , 2 , · · · , 2016. In round n , two rectangular double-sided 2 π mirrors 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 two mirrors at which the laser beam touches a mirror. What is the maximum possible score Meghal could have 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 , the 2 next point of contact, if it exists, is the point on OA that is a reflection of the intersection of l with 1 OA . If we continue this logic, we find that the maximum score for round n is equal to the maximum 3 number of intersection points between l and OA for some i . We do casework on whether n is even or i n +2 odd. If n is even, there are at most spokes such that l can hit OA , and if n is odd, there are at most i 2 n +3 such spokes. Then we must sum 2 + 2 + 3 + 3 + · · · + 1009 + 1009 = 1009 · 1010 − 1 − 1 = 1019088 . 2