{"status": "success", "data": {"description_md": "Lines $l_1$ and $l_2$ both pass through the origin and make first-quadrant angles of $\\frac{\\pi}{70}$ and $\\frac{\\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l$, the transformation $R(l)$ produces another line as follows: $l$ is reflected in $l_1$, and the resulting line is reflected in $l_2$. Let $R^{(1)}(l)=R(l)$ and $R^{(n)}(l)=R\\left(R^{(n-1)}(l)\\right)$. Given that $l$ is the line $y=\\frac{19}{92}x$, find the smallest positive integer $m$ for which $R^{(m)}(l)=l$.\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": "

Lines l_1 and l_2 both pass through the origin and make first-quadrant angles of \\frac{\\pi}{70} and \\frac{\\pi}{54} radians, respectively, with the positive x-axis. For any line l, the transformation R(l) produces another line as follows: l is reflected in l_1, and the resulting line is reflected in l_2. Let R^{(1)}(l)=R(l) and R^{(n)}(l)=R\\left(R^{(n-1)}(l)\\right). Given that l is the line y=\\frac{19}{92}x, find the smallest positive integer m for which R^{(m)}(l)=l.

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": "1992 AIME Problem 11", "can_next": true, "can_prev": true, "nxt": "/problem/92_aime_p12", "prev": "/problem/92_aime_p10"}}