{"status": "success", "data": {"description_md": "A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \\frac{3}{7}$, and\n\n$$a_n=\\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$$for all $n \\geq 3$. Then $a_{2019}$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$\n\n$\\textbf{(A) } 2020 \\qquad\\textbf{(B) } 4039 \\qquad\\textbf{(C) } 6057 \\qquad\\textbf{(D) } 6061 \\qquad\\textbf{(E) } 8078$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>A sequence of numbers is defined recursively by  <span class=\"katex--inline\">a_1 = 1</span> ,  <span class=\"katex--inline\">a_2 = \\frac{3}{7}</span> , and</p>&#10;<p> <span class=\"katex--display\">a_n=\\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}</span> for all  <span class=\"katex--inline\">n \\geq 3</span> . Then  <span class=\"katex--inline\">a_{2019}</span>  can be written as  <span class=\"katex--inline\">\\frac{p}{q}</span> , where  <span class=\"katex--inline\">p</span>  and  <span class=\"katex--inline\">q</span>  are relatively prime positive integers. What is  <span class=\"katex--inline\">p+q ?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 2020 \\qquad\\textbf{(B) } 4039 \\qquad\\textbf{(C) } 6057 \\qquad\\textbf{(D) } 6061 \\qquad\\textbf{(E) } 8078</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 2, "problem_name": "2019 AMC 12A Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12A_p10", "prev": "/problem/19_amc12A_p08"}}