{"status": "success", "data": {"description_md": "As shown in the figure, line segment $\\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\\overarc{AEB},$ $\\overarc{BFC},$ and $\\overarc{CGD},$ have their diameters on $\\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F.$ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form\n\n$$\\frac{a}{b}\\cdot\\pi-\\sqrt{c}+d,$$where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$?<br><center><img class=\"problem-image\" alt='[asy] size(6cm); filldraw(circle((0,0),2), grey); filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); dot((-3,-1)); label(\"$A$\",(-3,-1),S); dot((-2,0)); label(\"$E$\",(-2,0),NW); dot((-1,-1)); label(\"$B$\",(-1,-1),S); dot((0,0)); label(\"$F$\",(0,0),N); dot((1,-1)); label(\"$C$\",(1,-1), S); dot((2,0)); label(\"$G$\", (2,0),NE); dot((3,-1)); label(\"$D$\", (3,-1), S); [/asy]' class=\"latexcenter\" height=\"178\" src=\"https://latex.artofproblemsolving.com/a/c/9/ac9c68e87cdfb2dde115a063764bd2d48bf835e2.png\" width=\"285\"/></center>\n\n$\\textbf{(A) } 13 \\qquad\\textbf{(B) } 14 \\qquad\\textbf{(C) } 15 \\qquad\\textbf{(D) } 16\\qquad\\textbf{(E) } 17$\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>As shown in the figure, line segment  <span class=\"katex--inline\">\\overline{AD}</span>  is trisected by points  <span class=\"katex--inline\">B</span>  and  <span class=\"katex--inline\">C</span>  so that  <span class=\"katex--inline\">AB=BC=CD=2.</span>  Three semicircles of radius  <span class=\"katex--inline\">1,</span>   <span class=\"katex--inline\">\\overarc{AEB},</span>   <span class=\"katex--inline\">\\overarc{BFC},</span>  and  <span class=\"katex--inline\">\\overarc{CGD},</span>  have their diameters on  <span class=\"katex--inline\">\\overline{AD},</span>  and are tangent to line  <span class=\"katex--inline\">EG</span>  at  <span class=\"katex--inline\">E,F,</span>  and  <span class=\"katex--inline\">G,</span>  respectively. A circle of radius  <span class=\"katex--inline\">2</span>  has its center on  <span class=\"katex--inline\">F.</span>  The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form</p>&#10;<p> <span class=\"katex--display\">\\frac{a}{b}\\cdot\\pi-\\sqrt{c}+d,</span> where  <span class=\"katex--inline\">a,b,c,</span>  and  <span class=\"katex--inline\">d</span>  are positive integers and  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  are relatively prime. What is  <span class=\"katex--inline\">a+b+c+d</span> ?<br/><center><img class=\"latexcenter\" alt=\"[asy] size(6cm); filldraw(circle((0,0),2), grey); filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0)); filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0)); dot((-3,-1)); label(&#34;$A$&#34;,(-3,-1),S); dot((-2,0)); label(&#34;$E$&#34;,(-2,0),NW); dot((-1,-1)); label(&#34;$B$&#34;,(-1,-1),S); dot((0,0)); label(&#34;$F$&#34;,(0,0),N); dot((1,-1)); label(&#34;$C$&#34;,(1,-1), S); dot((2,0)); label(&#34;$G$&#34;, (2,0),NE); dot((3,-1)); label(&#34;$D$&#34;, (3,-1), S); [/asy]\" height=\"178\" src=\"https://latex.artofproblemsolving.com/a/c/9/ac9c68e87cdfb2dde115a063764bd2d48bf835e2.png\" width=\"285\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } 13 \\qquad\\textbf{(B) } 14 \\qquad\\textbf{(C) } 15 \\qquad\\textbf{(D) } 16\\qquad\\textbf{(E) } 17</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": "2019 AMC 12B Problem 15", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12B_p16", "prev": "/problem/19_amc12B_p14"}}