{"status": "success", "data": {"description_md": "There is a unique sequence of integers $a_1, a_2, \\cdots a_{2023}$ such that\n\n$$\\tan2023x = \\frac{a_1 \\tan x + a_3 \\tan^3 x + a_5 \\tan^5 x + \\cdots + a_{2023} \\tan^{2023} x}{1 + a_2 \\tan^2 x + a_4 \\tan^4 x \\cdots + a_{2022} \\tan^{2022} x}$$whenever $\\tan 2023x$ is defined. What is $a_{2023}?$\n\n$\\textbf{(A) } -2023 \\qquad\\textbf{(B) } -2022 \\qquad\\textbf{(C) } -1 \\qquad\\textbf{(D) } 1 \\qquad\\textbf{(E) } 2023$\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>There is a unique sequence of integers  <span class=\"katex--inline\">a_1, a_2, \\cdots a_{2023}</span>  such that</p>&#10;<p> <span class=\"katex--display\">\\tan2023x = \\frac{a_1 \\tan x + a_3 \\tan^3 x + a_5 \\tan^5 x + \\cdots + a_{2023} \\tan^{2023} x}{1 + a_2 \\tan^2 x + a_4 \\tan^4 x \\cdots + a_{2022} \\tan^{2022} x}</span> whenever  <span class=\"katex--inline\">\\tan 2023x</span>  is defined. What is  <span class=\"katex--inline\">a_{2023}?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } -2023 \\qquad\\textbf{(B) } -2022 \\qquad\\textbf{(C) } -1 \\qquad\\textbf{(D) } 1 \\qquad\\textbf{(E) } 2023</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": 5, "problem_name": "2023 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/23_amc12A_p24"}}