{"status": "success", "data": {"description_md": "Equilateral $\\triangle ABC$ is inscribed in a circle of radius 2. Extend $\\overline{AB}$ through $B$ to point $D$ so that $AD=13$, and extend $\\overline{AC}$ through $C$ to point $E$ so that $AE=11$. Through $D$, draw a line $l_1$ parallel to $\\overline{AE}$, and through $E$, draw a line ${l}_2$ parallel to $\\overline{AD}$. Let $F$ be the intersection of ${l}_1$ and ${l}_2$. Let $G$ be the point on the circle that is collinear with $A$ and $F$ and distinct from $A$. Given that the area of $\\triangle CBG$ can be expressed in the form $\\frac{p\\sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime, find $p+q+r$.\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>Equilateral <span class=\"katex--inline\">\\triangle ABC</span> is inscribed in a circle of radius 2. Extend <span class=\"katex--inline\">\\overline{AB}</span> through <span class=\"katex--inline\">B</span> to point <span class=\"katex--inline\">D</span> so that <span class=\"katex--inline\">AD=13</span>, and extend <span class=\"katex--inline\">\\overline{AC}</span> through <span class=\"katex--inline\">C</span> to point <span class=\"katex--inline\">E</span> so that <span class=\"katex--inline\">AE=11</span>. Through <span class=\"katex--inline\">D</span>, draw a line <span class=\"katex--inline\">l_1</span> parallel to <span class=\"katex--inline\">\\overline{AE}</span>, and through <span class=\"katex--inline\">E</span>, draw a line <span class=\"katex--inline\">{l}_2</span> parallel to <span class=\"katex--inline\">\\overline{AD}</span>. Let <span class=\"katex--inline\">F</span> be the intersection of <span class=\"katex--inline\">{l}_1</span> and <span class=\"katex--inline\">{l}_2</span>. Let <span class=\"katex--inline\">G</span> be the point on the circle that is collinear with <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">F</span> and distinct from <span class=\"katex--inline\">A</span>. Given that the area of <span class=\"katex--inline\">\\triangle CBG</span> can be expressed in the form <span class=\"katex--inline\">\\frac{p\\sqrt{q}}{r}</span>, where <span class=\"katex--inline\">p</span>, <span class=\"katex--inline\">q</span>, and <span class=\"katex--inline\">r</span> are positive integers, <span class=\"katex--inline\">p</span> and <span class=\"katex--inline\">r</span> are relatively prime, and <span class=\"katex--inline\">q</span> is not divisible by the square of any prime, find <span class=\"katex--inline\">p+q+r</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": 5, "problem_name": "2006 AIME II Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/06_aime_II_p13", "prev": "/problem/06_aime_II_p11"}}