Question 14
Two circles $\Gamma_{1}$ and $\Gamma_{2}$ of radius $1$ and $2$, respectively, are centered at the origin. A particle is placed at $(2,0)$ and is shot towards $\Gamma_{1}$. When it reaches $\Gamma_{1}$, it bounces off the circumference and heads back towards $\Gamma_{2}$. The particle continues bouncing off the two circles in this fashion.
If the particle is shot at an acute angle $\theta$ above the $x$-axis, it will bounce $11$ times before returning to $(2,0)$ for the first time. If $\cot \theta = a - \sqrt{b}$ for positive integers $a$ and $b$, compute $100a + b$.
If the particle is shot at an acute angle $\theta$ above the $x$-axis, it will bounce $11$ times before returning to $(2,0)$ for the first time. If $\cot \theta = a - \sqrt{b}$ for positive integers $a$ and $b$, compute $100a + b$.