{"status": "success", "data": {"description_md": "The first two terms of a sequence are $a_1 = 1$ and $a_2 = \\frac {1}{\\sqrt3}$. For $n\\ge1$,\n\n$$a_{n + 2} = \\frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.$$<br>What is $|a_{2009}|$?\n\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ 2 - \\sqrt3\\qquad \\textbf{(C)}\\ \\frac {1}{\\sqrt3}\\qquad \\textbf{(D)}\\ 1\\qquad \\textbf{(E)}\\ 2 + \\sqrt3$\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>The first two terms of a sequence are <span class=\"katex--inline\">a_1 = 1</span> and <span class=\"katex--inline\">a_2 = \\frac {1}{\\sqrt3}</span>. For <span class=\"katex--inline\">n\\ge1</span>,</p>&#10;<p><span class=\"katex--display\">a_{n + 2} = \\frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.</span><br/>What is <span class=\"katex--inline\">|a_{2009}|</span>?</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ 2 - \\sqrt3\\qquad \\textbf{(C)}\\ \\frac {1}{\\sqrt3}\\qquad \\textbf{(D)}\\ 1\\qquad \\textbf{(E)}\\ 2 + \\sqrt3</span></p>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2009 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/09_amc12A_p24"}}