PUMaC 2015 · 数论（B 组） · 第 4 题

PUMaC 2015 — Number Theory (Division B) — Problem 4

[ 4 ] A circle with radius 1 and center (0 , 1) lies on the coordinate plane. Ariel stands at the origin and rolls a ball of paint at an angle of 35 degrees relative to the positive x -axis (counting degrees counterclockwise). The ball repeatedly bounces off the circle and leaves behind a trail of paint where it rolled. After the ball of paint returns to the origin, the paint has traced out a star with n points on the circle. What is n ?

解析

[ 4 ] A circle with radius 1 and center (0 , 1) lies on the coordinate plane. Ariel stands at the origin and rolls a ball of paint at an angle of 35 degrees relative to the positive x -axis (counting degrees counterclockwise). The ball repeatedly bounces off the circle and leaves behind a trail of paint where it rolled. After the ball of paint returns to the origin, the paint has traced out a star with n points on the circle. What is n ? Solution: The ball of paint bounces off the circle such that the angle between two consecutive bounces that is centered at the origin is 2 ˙ 35 = 70 degree as usual. Therefore, the paint keeps ◦ ◦ bouncing until it hits the first multiple of 360 that is also a multiple of 70 . Therefore, the number of unique intersections with the circle is lcm(360 , 70) 360 · 7 = = 36 . 70 10 · 7 Author: Heesu Hwang 1