{"status": "success", "data": {"description_md": "Circles $\\mathcal{P}$ and $\\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\\mathcal{P}$ and point $C$ is on $\\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\\ell$ through $A$ intersects $\\mathcal{P}$ again at $D$ and intersects $\\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\\ell$, and the areas of $\\triangle DBA$ and $\\triangle ACE$ are equal. This common area is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

$\\includegraphics[width=142, height=102, totalheight=102]{https://latex.artofproblemsolving.com/6/b/7/6b7782afc839b219809c6266cec4abca23e9d026.png}$\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full 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 \\mathcal{P} and \\mathcal{Q} have radii 1 and 4, respectively, and are externally tangent at point A. Point B is on \\mathcal{P} and point C is on \\mathcal{Q} so that line BC is a common external tangent of the two circles. A line \\ell through A intersects \\mathcal{P} again at D and intersects \\mathcal{Q} again at E. Points B and C lie on the same side of \\ell, and the areas of \\triangle DBA and \\triangle ACE are equal. This common area is \\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.

Leading zeroes must be inputted, so if your answer is 34, then input 034. 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": 6, "problem_name": "2015 AIME II Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/15_aime_II_p14"}}