{"status": "success", "data": {"description_md": "Circles $\\mathcal{C}_1$, $\\mathcal{C}_2$, and $\\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\\mathcal{C}_1$ and $\\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\\mathcal{C}_2$ and $\\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y)$, and that $x=p-q\\sqrt{r}$, where $p$, $q$, and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.\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{C}_1, \\mathcal{C}_2, and \\mathcal{C}_3 have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line t_1 is a common internal tangent to \\mathcal{C}_1 and \\mathcal{C}_2 and has a positive slope, and line t_2 is a common internal tangent to \\mathcal{C}_2 and \\mathcal{C}_3 and has a negative slope. Given that lines t_1 and t_2 intersect at (x,y), and that x=p-q\\sqrt{r}, where p, q, and r are positive integers and r is not divisible by the square of any prime, find p+q+r.

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": 4, "problem_name": "2006 AIME II Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/06_aime_II_p10", "prev": "/problem/06_aime_II_p08"}}