{"status": "success", "data": {"description_md": "For $1\\leq i\\leq 215$ let $a_i=\\frac{1}{2^i}$ and $a_{216}=\\frac{1}{2^{215}}$. Let $x_1,x_2,\\ldots,x_{216}$ be positive real numbers such that $$ \\sum\\limits_{i=1}^{216} x_i=1 \\text{ and } \\sum\\limits_{1\\leq i<j \\leq 216} x_ix_j = \\frac{107}{215}+ \\sum\\limits_{i=1}^{216} \\frac{a_ix_i^2}{2(1-a_i)}. $$The maximum possible value of $x_2=\\frac{m}{n}$, where $m$ and $n$ are relatively prime 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>For <span class=\"katex--inline\">1\\leq i\\leq 215</span> let <span class=\"katex--inline\">a_i=\\frac{1}{2^i}</span> and <span class=\"katex--inline\">a_{216}=\\frac{1}{2^{215}}</span>. Let <span class=\"katex--inline\">x_1,x_2,\\ldots,x_{216}</span> be positive real numbers such that <span class=\"katex--display\"> \\sum\\limits_{i=1}^{216} x_i=1 \\text{ and } \\sum\\limits_{1\\leq i&lt;j \\leq 216} x_ix_j = \\frac{107}{215}+ \\sum\\limits_{i=1}^{216} \\frac{a_ix_i^2}{2(1-a_i)}. </span>The maximum possible value of <span class=\"katex--inline\">x_2=\\frac{m}{n}</span>, where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are relatively prime positive integers. Find <span class=\"katex--inline\">m+n</span>.</p>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "2016 AIME II Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/16_aime_II_p14"}}