{"status": "success", "data": {"description_md": "Semicircle $\\Gamma$ has diameter $\\overline{AB}$ of length $14$. Circle $\\Omega$ lies tangent to $\\overline{AB}$ at a point $P$ and intersects $\\Gamma$ at points $Q$ and $R$. If $QR=3\\sqrt3$ and $\\angle QPR=60^\\circ$, then the area of $\\triangle PQR$ equals $\\tfrac{a\\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$?\n\n$\\textbf{(A) }110 \\qquad \\textbf{(B) }114 \\qquad \\textbf{(C) }118 \\qquad \\textbf{(D) }122\\qquad \\textbf{(E) }126$\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>Semicircle  <span class=\"katex--inline\">\\Gamma</span>  has diameter  <span class=\"katex--inline\">\\overline{AB}</span>  of length  <span class=\"katex--inline\">14</span> . Circle  <span class=\"katex--inline\">\\Omega</span>  lies tangent to  <span class=\"katex--inline\">\\overline{AB}</span>  at a point  <span class=\"katex--inline\">P</span>  and intersects  <span class=\"katex--inline\">\\Gamma</span>  at points  <span class=\"katex--inline\">Q</span>  and  <span class=\"katex--inline\">R</span> . If  <span class=\"katex--inline\">QR=3\\sqrt3</span>  and  <span class=\"katex--inline\">\\angle QPR=60^\\circ</span> , then the area of  <span class=\"katex--inline\">\\triangle PQR</span>  equals  <span class=\"katex--inline\">\\tfrac{a\\sqrt{b}}{c}</span> , where  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">c</span>  are relatively prime positive integers, and  <span class=\"katex--inline\">b</span>  is a positive integer not divisible by the square of any prime. What is  <span class=\"katex--inline\">a+b+c</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }110 \\qquad \\textbf{(B) }114 \\qquad \\textbf{(C) }118 \\qquad \\textbf{(D) }122\\qquad \\textbf{(E) }126</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": 5, "problem_name": "2021 AMC 12A Problem 24", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc12A_p25", "prev": "/problem/21_amc12A_p23"}}