HMMT 十一月 2016 · GEN 赛 · 第 4 题
HMMT November 2016 — GEN Round — Problem 4
题目详情
英文原题
- A rectangular pool table has vertices at (0 , 0) (12 , 0) (0 , 10), and (12 , 10). There are pockets only inthe four corners. A ball is hit from (0 , 0) along the line y = x and bounces off several walls beforeeventually entering a pocket. Find the number of walls that the ball bounces off of before entering apocket.
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解析
英文解析
- A rectangular pool table has vertices at (0 , 0) (12 , 0) (0 , 10), and (12 , 10). There are pockets only inthe four corners. A ball is hit from (0 , 0) along the line y = x and bounces off several walls beforeeventually entering a pocket. Find the number of walls that the ball bounces off of before entering apocket.
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 theball 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 linesx = 12 , 24 , 36 , 48. This gives a total of 9 walls hit before entering a pocket.
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