{"status": "success", "data": {"description_md": "Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,...$ and a sequence of points on the circles $A_1,A_2,A_3,...$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r=\\frac{11}{60}$, the distance from the center of $C_0$ to $B$ is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

$\\includegraphics[width=142, height=127, totalheight=127]{https://latex.artofproblemsolving.com/4/2/1/42165bad46c0877ff18e5bd3123561e843316d76.png}$\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": "

Circle C_0 has radius 1, and the point A_0 is a point on the circle. Circle C_1 has radius r<1 and is internally tangent to C_0 at point A_0. Point A_1 lies on circle C_1 so that A_1 is located 90^{\\circ} counterclockwise from A_0 on C_1. Circle C_2 has radius r^2 and is internally tangent to C_1 at point A_1. In this way a sequence of circles C_1,C_2,C_3,... and a sequence of points on the circles A_1,A_2,A_3,... are constructed, where circle C_n has radius r^n and is internally tangent to circle C_{n-1} at point A_{n-1}, and point A_n lies on C_n 90^{\\circ} counterclockwise from point A_{n-1}, as shown in the figure below. There is one point B inside all of these circles. When r=\\frac{11}{60}, the distance from the center of C_0 to B 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": 5, "problem_name": "2017 AIME II Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/17_aime_II_p13", "prev": "/problem/17_aime_II_p11"}}