{"status": "success", "data": {"description_md": "Three semicircles of radius 1 are constructed on diameter AB of a semicircle of<br>radius 2. The centers of the small semicircles divide AB into four line segments<br>of equal length, as shown. What is the area of the shaded region that lies within<br>the large semicircle but outside the smaller semicircles?<br><center><img class=\"problem-image\" alt='[asy] import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dashed=linetype(\"4 4\"); dotfactor=3; pair A=(-2,0), B=(2,0); fill(Arc((0,0),2,0,180)--cycle,mediumgray); fill(Arc((-1,0),1,0,180)--cycle,white); fill(Arc((0,0),1,0,180)--cycle,white); fill(Arc((1,0),1,0,180)--cycle,white); draw(Arc((-1,0),1,60,180)); draw(Arc((0,0),1,0,60),dashed); draw(Arc((0,0),1,60,120)); draw(Arc((0,0),1,120,180),dashed); draw(Arc((1,0),1,0,120)); draw(Arc((0,0),2,0,180)--cycle); dot((0,0)); dot((-1,0)); dot((1,0)); draw((-2,-0.1)--(-2,-0.3),gray); draw((-1,-0.1)--(-1,-0.3),gray); draw((1,-0.1)--(1,-0.3),gray); draw((2,-0.1)--(2,-0.3),gray); label(\"$A$\",A,W); label(\"$B$\",B,E); label(\"1\",(-1.5,-0.1),S); label(\"2\",(0,-0.1),S); label(\"1\",(1.5,-0.1),S);[/asy]' class=\"latexcenter\" height=\"155\" src=\"https://latex.artofproblemsolving.com/b/c/6/bc697df5c9078d8ccf0708f353185c707d3275d4.png\" width=\"298\"/></center>\n\n$\\textbf{(A) } \\pi - \\sqrt{3} \\qquad\\textbf{(B) } \\pi - \\sqrt{2} \\qquad\\textbf{(C) } \\frac{\\pi + \\sqrt{2}}{2} \\qquad\\textbf{(D) } \\frac{\\pi +\\sqrt{3}}{2} \\qquad\\textbf{(E) } \\frac{7}{6}\\pi - \\frac{\\sqrt{3}}{2}$\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>Three semicircles of radius 1 are constructed on diameter AB of a semicircle of<br/>radius 2. The centers of the small semicircles divide AB into four line segments<br/>of equal length, as shown. What is the area of the shaded region that lies within<br/>the large semicircle but outside the smaller semicircles?<br/><center><img class=\"latexcenter\" alt=\"[asy] import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dashed=linetype(&#34;4 4&#34;); dotfactor=3; pair A=(-2,0), B=(2,0); fill(Arc((0,0),2,0,180)--cycle,mediumgray); fill(Arc((-1,0),1,0,180)--cycle,white); fill(Arc((0,0),1,0,180)--cycle,white); fill(Arc((1,0),1,0,180)--cycle,white); draw(Arc((-1,0),1,60,180)); draw(Arc((0,0),1,0,60),dashed); draw(Arc((0,0),1,60,120)); draw(Arc((0,0),1,120,180),dashed); draw(Arc((1,0),1,0,120)); draw(Arc((0,0),2,0,180)--cycle); dot((0,0)); dot((-1,0)); dot((1,0)); draw((-2,-0.1)--(-2,-0.3),gray); draw((-1,-0.1)--(-1,-0.3),gray); draw((1,-0.1)--(1,-0.3),gray); draw((2,-0.1)--(2,-0.3),gray); label(&#34;$A$&#34;,A,W); label(&#34;$B$&#34;,B,E); label(&#34;1&#34;,(-1.5,-0.1),S); label(&#34;2&#34;,(0,-0.1),S); label(&#34;1&#34;,(1.5,-0.1),S);[/asy]\" height=\"155\" src=\"https://latex.artofproblemsolving.com/b/c/6/bc697df5c9078d8ccf0708f353185c707d3275d4.png\" width=\"298\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } \\pi - \\sqrt{3} \\qquad\\textbf{(B) } \\pi - \\sqrt{2} \\qquad\\textbf{(C) } \\frac{\\pi + \\sqrt{2}}{2} \\qquad\\textbf{(D) } \\frac{\\pi +\\sqrt{3}}{2} \\qquad\\textbf{(E) } \\frac{7}{6}\\pi - \\frac{\\sqrt{3}}{2}</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": "2003 AMC 12B Problem 16", "can_next": true, "can_prev": true, "nxt": "/problem/03_amc12B_p17", "prev": "/problem/03_amc12B_p15"}}