HMMT 十一月 2016 · GEN 赛 · 第 4 题

HMMT November 2016 — GEN Round — Problem 4

A rectangular pool table has vertices at (0 , 0) (12 , 0) (0 , 10), and (12 , 10). There are pockets only in the four corners. A ball is hit from (0 , 0) along the line y = x and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket. 1 ∞ 2

解析

A rectangular pool table has vertices at (0 , 0) (12 , 0) (0 , 10), and (12 , 10). There are pockets only in the four corners. A ball is hit from (0 , 0) along the line y = x and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket. Proposed by: Allen Liu Answer: 9 Consider the tiling of the plane with the 12 × 10 rectangle to form a grid. Then the reflection of the ball off a wall is equivalent to traveling along the straight line y = x into another 12 × 10 rectangle. Hence we want to find the number of walls of the grid that the line y = x hits before it reaches the first corner it hits, (60 , 60). The line y = x hits each of the horizontal lines y = 10 , 20 , 30 , 40 , 50 and each of the vertical lines x = 12 , 24 , 36 , 48. This gives a total of 9 walls hit before entering a pocket. ∞ 1 2