{"status": "success", "data": {"description_md": "Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \\neq C$ is a point of intersection of the circumcircles of $\\triangle ACD$ and $\\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\\tfrac{a+b\\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.\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>Triangle <span class=\"katex--inline\">ABC</span> has side lengths <span class=\"katex--inline\">AB=4</span>, <span class=\"katex--inline\">BC=5</span>, and <span class=\"katex--inline\">CA=6</span>. Points <span class=\"katex--inline\">D</span> and <span class=\"katex--inline\">E</span> are on ray <span class=\"katex--inline\">AB</span> with <span class=\"katex--inline\">AB&lt;AD&lt;AE</span>. The point <span class=\"katex--inline\">F \\neq C</span> is a point of intersection of the circumcircles of <span class=\"katex--inline\">\\triangle ACD</span> and <span class=\"katex--inline\">\\triangle EBC</span> satisfying <span class=\"katex--inline\">DF=2</span> and <span class=\"katex--inline\">EF=7</span>. Then <span class=\"katex--inline\">BE</span> can be expressed as <span class=\"katex--inline\">\\tfrac{a+b\\sqrt{c}}{d}</span>, where <span class=\"katex--inline\">a</span>, <span class=\"katex--inline\">b</span>, <span class=\"katex--inline\">c</span>, and <span class=\"katex--inline\">d</span> are positive integers such that <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">d</span> are relatively prime, and <span class=\"katex--inline\">c</span> is not divisible by the square of any prime. Find <span class=\"katex--inline\">a+b+c+d</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": 6, "problem_name": "2019 AIME I Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/19_aime_I_p14", "prev": "/problem/19_aime_I_p12"}}