{"status": "success", "data": {"description_md": "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 $m+n+p+q$?\n\n$\\phantom{ }$\n\n$\\textbf{(A) } 42 \\qquad \\textbf{(B) }86 \\qquad \\textbf{(C) } 92 \\qquad \\textbf{(D) } 114 \\qquad \\textbf{(E) } 130$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

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

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2019 AMC 12A Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12A_p23", "prev": "/problem/19_amc12A_p21"}}