{"status": "success", "data": {"description_md": "Let $w_{1}$ and $w_{2}$ denote the circles $x^{2}+y^{2}+10x-24y-87=0$ and $x^{2}+y^{2}-10x-24y+153=0$, respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_{2}$ and internally tangent to $w_{1}$. Given that $m^{2}=p/q$, where $p$ and $q$ are relatively prime integers, find $p+q$.\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": "

Let w_{1} and w_{2} denote the circles x^{2}+y^{2}+10x-24y-87=0 and x^{2}+y^{2}-10x-24y+153=0, respectively. Let m be the smallest positive value of a for which the line y=ax contains the center of a circle that is externally tangent to w_{2} and internally tangent to w_{1}. Given that m^{2}=p/q, where p and q are relatively prime integers, find p+q.

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": "2005 AIME II Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/05_aime_II_p14"}}