{"status": "success", "data": {"description_md": "In $\\triangle{ABC}$ medians $\\overline{AD}$ and $\\overline{BE}$ intersect at $G$ and $\\triangle{AGE}$ is equilateral. Then $\\cos(C)$ can be written as $\\frac{m\\sqrt p}n$, where $m$ and $n$ are relatively prime positive integers and $p$ is a positive integer not divisible by the square of any prime. What is $m+n+p?$\n\n$\\textbf{(A) }44 \\qquad \\textbf{(B) }48 \\qquad \\textbf{(C) }52 \\qquad \\textbf{(D) }56 \\qquad \\textbf{(E) }60$\n___\nFull 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>In  <span class=\"katex--inline\">\\triangle{ABC}</span>  medians  <span class=\"katex--inline\">\\overline{AD}</span>  and  <span class=\"katex--inline\">\\overline{BE}</span>  intersect at  <span class=\"katex--inline\">G</span>  and  <span class=\"katex--inline\">\\triangle{AGE}</span>  is equilateral. Then  <span class=\"katex--inline\">\\cos(C)</span>  can be written as  <span class=\"katex--inline\">\\frac{m\\sqrt p}n</span> , where  <span class=\"katex--inline\">m</span>  and  <span class=\"katex--inline\">n</span>  are relatively prime positive integers and  <span class=\"katex--inline\">p</span>  is a positive integer not divisible by the square of any prime. What is  <span class=\"katex--inline\">m+n+p?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }44 \\qquad \\textbf{(B) }48 \\qquad \\textbf{(C) }52 \\qquad \\textbf{(D) }56 \\qquad \\textbf{(E) }60</span> </p>&#10;<hr><p>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": 3, "problem_name": "2022 AMC 12B Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc12B_p20", "prev": "/problem/22_amc12B_p18"}}