{"status": "success", "data": {"description_md": "Circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\\omega_1$, $\\omega_2$, and $\\omega_3$ respectively such that $P_1P_2=P_2P_3=P_3P_1$ and line $P_iP_{i+1}$ is tangent to $\\omega_i$ for each $i=1,2,3$, where $P_4 = P_1$. See the figure below. The area of $\\triangle P_1P_2P_3$ can be written in the form $\\sqrt{a}+\\sqrt{b}$ for positive integers $a$ and $b$. What is $a+b$?
\"[asy]
\n\n$\\textbf{(A) }546\\qquad\\textbf{(B) }548\\qquad\\textbf{(C) }550\\qquad\\textbf{(D) }552\\qquad\\textbf{(E) }554$\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_1 , \\omega_2 , and \\omega_3 each have radius 4 and are placed in the plane so that each circle is externally tangent to the other two. Points P_1 , P_2 , and P_3 lie on \\omega_1 , \\omega_2 , and \\omega_3 respectively such that P_1P_2=P_2P_3=P_3P_1 and line P_iP_{i+1} is tangent to \\omega_i for each i=1,2,3 , where P_4 = P_1 . See the figure below. The area of \\triangle P_1P_2P_3 can be written in the form \\sqrt{a}+\\sqrt{b} for positive integers a and b . What is a+b ?

\"[asy]

\\textbf{(A) }546\\qquad\\textbf{(B) }548\\qquad\\textbf{(C) }550\\qquad\\textbf{(D) }552\\qquad\\textbf{(E) }554

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": "2018 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/18_amc12B_p24"}}