HMMT 二月 2021 · 冲刺赛 · 第 15 题

HMMT February 2021 — Guts Round — Problem 15

[11] Two circles Γ and Γ of radius 1 and 2, respectively, are centered at the origin. A particle is placed 1 2 at (2 , 0) and is shot towards Γ . When it reaches Γ , it bounces off the circumference and heads back 1 1 towards Γ . The particle continues bouncing off the two circles in this fashion. 2 If the particle is shot at an acute angle θ above the x -axis, it will bounce 11 times before returning to √ (2 , 0) for the first time. If cot θ = a − b for positive integers a and b , compute 100 a + b . 2

解析

[11] Two circles Γ and Γ of radius 1 and 2, respectively, are centered at the origin. A particle is 1 2 placed at (2 , 0) and is shot towards Γ . When it reaches Γ , it bounces off the circumference and heads 1 1 back towards Γ . The particle continues bouncing off the two circles in this fashion. 2 If the particle is shot at an acute angle θ above the x -axis, it will bounce 11 times before returning to √ (2 , 0) for the first time. If cot θ = a − b for positive integers a and b , compute 100 a + b . Proposed by: Hahn Lheem Answer: 403 ◦ ◦ Solution: By symmetry, the particle must bounce off of Γ at points that make angles of 60 , 120 , 2 ◦ ◦ ◦ 180 , 240 , and 300 with the positive x -axis. Similarly, the particle must bounce off of Γ at points 1 ◦ ◦ ◦ ◦ ◦ ◦ that make angles of 30 , 90 , 150 , 210 , 270 , and 330 with the positive x -axis. ◦ ◦ In particular, the first point that the ball touches on Γ is (cos 30 , sin 30 ). So, 1 ◦ √ 2 − cos 30 cot θ = = 4 − 3 . ◦ sin 30 2