2019 AMC 12A Problems/Problem 22

Problem

Circles $\omega$ and $\gamma$ , both centered at $O$ , have radii $20$ and $17$ , respectively. Equilateral triangle $ABC$ , whose interior lies in the interior of $\omega$ but in the exterior of $\gamma$ , has vertex $A$ on $\omega$ , and the line containing side $\overline{BC}$ is tangent to $\gamma$ . Segments $\overline{AO}$ and $\overline{BC}$ intersect at $P$ , and $\dfrac{BP}{CP} = 3$ . Then $AB$ can be written in the form $\dfrac{m}{\sqrt{n}} - \dfrac{p}{\sqrt{q}}$ for positive integers $m$ , $n$ , $p$ , $q$ with $\gcd(m,n) = \gcd(p,q) = 1$ . What is $m+n+p+q$ ? $\phantom{}$

$\textbf{(A) } 42 \qquad \textbf{(B) }86 \qquad \textbf{(C) } 92 \qquad \textbf{(D) } 114 \qquad \textbf{(E) } 130$

Solution

2019 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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All AMC 12 Problems and Solutions

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2019 AMC 12A Problems/Problem 22

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