{"status": "success", "data": {"description_md": "There is a unique angle $\\theta$ between $0^{\\circ}$ and $90^{\\circ}$ such that for nonnegative integers $n$, the value of $\\tan{\\left(2^{n}\\theta\\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\\theta$ is $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.\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": "<p>There is a unique angle <span class=\"katex--inline\">\\theta</span> between <span class=\"katex--inline\">0^{\\circ}</span> and <span class=\"katex--inline\">90^{\\circ}</span> such that for nonnegative integers <span class=\"katex--inline\">n</span>, the value of <span class=\"katex--inline\">\\tan{\\left(2^{n}\\theta\\right)}</span> is positive when <span class=\"katex--inline\">n</span> is a multiple of <span class=\"katex--inline\">3</span>, and negative otherwise. The degree measure of <span class=\"katex--inline\">\\theta</span> is <span class=\"katex--inline\">\\tfrac{p}{q}</span>, where <span class=\"katex--inline\">p</span> and <span class=\"katex--inline\">q</span> are relatively prime integers. Find <span class=\"katex--inline\">p+q</span>.</p>&#10;<hr><p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. 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": "2019 AIME II Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/19_aime_II_p11", "prev": "/problem/19_aime_II_p09"}}