{"status": "success", "data": {"description_md": "The figure below shows $13$ circles of radius $1$ within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius $1$?<br><center><img class=\"problem-image\" alt=\"[asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy]\" class=\"latexcenter\" height=\"302\" src=\"https://latex.artofproblemsolving.com/f/c/4/fc485e1337895925d85375cce1a9d3908ca8b7e3.png\" width=\"302\"/></center>\n\n$\\textbf{(A) } 4 \\pi \\sqrt{3} \\qquad\\textbf{(B) } 7 \\pi \\qquad\\textbf{(C) } \\pi\\left(3\\sqrt{3} +2\\right) \\qquad\\textbf{(D) } 10 \\pi \\left(\\sqrt{3} - 1\\right) \\qquad\\textbf{(E) } \\pi\\left(\\sqrt{3} + 6\\right)$\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>The figure below shows  <span class=\"katex--inline\">13</span>  circles of radius  <span class=\"katex--inline\">1</span>  within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius  <span class=\"katex--inline\">1</span> ?<br/><center><img class=\"latexcenter\" alt=\"[asy]unitsize(20);filldraw(circle((0,0),2*sqrt(3)+1),rgb(0.5,0.5,0.5));filldraw(circle((-2,0),1),white);filldraw(circle((0,0),1),white);filldraw(circle((2,0),1),white);filldraw(circle((1,sqrt(3)),1),white);filldraw(circle((3,sqrt(3)),1),white);filldraw(circle((-1,sqrt(3)),1),white);filldraw(circle((-3,sqrt(3)),1),white);filldraw(circle((1,-1*sqrt(3)),1),white);filldraw(circle((3,-1*sqrt(3)),1),white);filldraw(circle((-1,-1*sqrt(3)),1),white);filldraw(circle((-3,-1*sqrt(3)),1),white);filldraw(circle((0,2*sqrt(3)),1),white);filldraw(circle((0,-2*sqrt(3)),1),white);[/asy]\" height=\"302\" src=\"https://latex.artofproblemsolving.com/f/c/4/fc485e1337895925d85375cce1a9d3908ca8b7e3.png\" width=\"302\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 4 \\pi \\sqrt{3} \\qquad\\textbf{(B) } 7 \\pi \\qquad\\textbf{(C) } \\pi\\left(3\\sqrt{3} +2\\right) \\qquad\\textbf{(D) } 10 \\pi \\left(\\sqrt{3} - 1\\right) \\qquad\\textbf{(E) } \\pi\\left(\\sqrt{3} + 6\\right)</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": 2, "problem_name": "2019 AMC 12A Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12A_p11", "prev": "/problem/19_amc12A_p09"}}