{"status": "success", "data": {"description_md": "Let $z_1,z_2,z_3,\\ldots,z_{12}$ be the 12 zeroes of the polynomial $z^{12}-2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $i z_j$. Then the maximum possible value of the real part of $\\displaystyle\\sum_{j=1}^{12} w_j$ can be written as $m+\\sqrt{n}$ where $m$ and $n$ are positive integers. Find $m+n$.\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>Let <span class=\"katex--inline\">z_1,z_2,z_3,\\ldots,z_{12}</span> be the 12 zeroes of the polynomial <span class=\"katex--inline\">z^{12}-2^{36}</span>. For each <span class=\"katex--inline\">j</span>, let <span class=\"katex--inline\">w_j</span> be one of <span class=\"katex--inline\">z_j</span> or <span class=\"katex--inline\">i z_j</span>. Then the maximum possible value of the real part of <span class=\"katex--inline\">\\displaystyle\\sum_{j=1}^{12} w_j</span> can be written as <span class=\"katex--inline\">m+\\sqrt{n}</span> where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are positive integers. Find <span class=\"katex--inline\">m+n</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": 4, "problem_name": "2011 AIME II Problem 8", "can_next": true, "can_prev": true, "nxt": "/problem/11_aime_II_p09", "prev": "/problem/11_aime_II_p07"}}