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Omni-MATH / Two circles Γ₁ and Γ₂ of radius 1 and 2, respectively, are centered at the origin. A particle is placed at (2,0) and is shot towards Γ₁. When it reaches Γ₁, it bounces off the circ…

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problem

Two circles Γ1\Gamma_{1} and Γ2\Gamma_{2} of radius 1 and 2, respectively, are centered at the origin. A particle is placed at (2,0)(2,0) and is shot towards Γ1\Gamma_{1}. When it reaches Γ1\Gamma_{1}, it bounces off the circumference and heads back towards Γ2\Gamma_{2}. The particle continues bouncing off the two circles in this fashion. If the particle is shot at an acute angle θ\theta above the xx-axis, it will bounce 11 times before returning to (2,0)(2,0) for the first time. If $\cot \theta=a-\sqrt{b}$ for positive integers aa and bb, compute 100a+b100 a+b.
Plain-text mathematical notation (without MathML)
Two circles Γ₁ and Γ₂ of radius 1 and 2, respectively, are centered at the origin. A particle is placed at (2,0) and is shot towards Γ₁. When it reaches Γ₁, it bounces off the circumference and heads back towards Γ₂. The particle continues bouncing off the two circles in this fashion. 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 $\cot \theta=a-\sqrt{b}$ for positive integers a and b, compute 100a+b.
Original LaTeX notation
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 $100 a+b$.

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Source and history

task_id
a0743bd6-12a5-5642-bbb7-0ea11bb41ef6
task_key
test--a0743bd6-12a5-5642-bbb7-0ea11bb41ef6
task_revision_id
1
config / split
/ test

Official source

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