{"status": "success", "data": {"description_md": "Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\\overline{BC}$, $Y$ on $\\overline{DE}$, and $Z$ on $\\overline{EF}$. Suppose that $AB=40$, and $EF=41(\\sqrt{3}-1)$. What is the side-length of the square?<br><center><img class=\"problem-image\" alt='[asy] size(200); defaultpen(linewidth(1)); pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60); pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A; draw(A--B--C--D--E--F--cycle); draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype(\"2 2\")); dot(\"$A$\",A,W,linewidth(4)); dot(\"$B$\",B,dir(0),linewidth(4)); dot(\"$C$\",C,dir(0),linewidth(4)); dot(\"$D$\",D,dir(20),linewidth(4)); dot(\"$E$\",E,dir(100),linewidth(4)); dot(\"$F$\",F,W,linewidth(4)); dot(\"$X$\",X,dir(0),linewidth(4)); dot(\"$Y$\",Y,N,linewidth(4)); dot(\"$Z$\",Z,W,linewidth(4)); [/asy]' class=\"latexcenter\" height=\"245\" src=\"https://latex.artofproblemsolving.com/2/9/f/29fd640711a70115cf67481ab55a4be235902efc.png\" width=\"335\"/></center>\n\n$\\textbf{(A)}\\ 29\\sqrt{3} \\qquad\\textbf{(B)}\\ \\frac{21}{2}\\sqrt{2}+\\frac{41}{2}\\sqrt{3}\\qquad\\textbf{(C)}\\ 20\\sqrt{3}+16$\n\n$\\textbf{(D)}\\ 20\\sqrt{2}+13\\sqrt{3} \\qquad\\textbf{(E)}\\ 21\\sqrt{6}$\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>Square  <span class=\"katex--inline\">AXYZ</span>  is inscribed in equiangular hexagon  <span class=\"katex--inline\">ABCDEF</span>  with  <span class=\"katex--inline\">X</span>  on  <span class=\"katex--inline\">\\overline{BC}</span> ,  <span class=\"katex--inline\">Y</span>  on  <span class=\"katex--inline\">\\overline{DE}</span> , and  <span class=\"katex--inline\">Z</span>  on  <span class=\"katex--inline\">\\overline{EF}</span> . Suppose that  <span class=\"katex--inline\">AB=40</span> , and  <span class=\"katex--inline\">EF=41(\\sqrt{3}-1)</span> . What is the side-length of the square?<br/><center><img class=\"latexcenter\" alt=\"[asy] size(200); defaultpen(linewidth(1)); pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60); pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A; draw(A--B--C--D--E--F--cycle); draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype(&#34;2 2&#34;)); dot(&#34;$A$&#34;,A,W,linewidth(4)); dot(&#34;$B$&#34;,B,dir(0),linewidth(4)); dot(&#34;$C$&#34;,C,dir(0),linewidth(4)); dot(&#34;$D$&#34;,D,dir(20),linewidth(4)); dot(&#34;$E$&#34;,E,dir(100),linewidth(4)); dot(&#34;$F$&#34;,F,W,linewidth(4)); dot(&#34;$X$&#34;,X,dir(0),linewidth(4)); dot(&#34;$Y$&#34;,Y,N,linewidth(4)); dot(&#34;$Z$&#34;,Z,W,linewidth(4)); [/asy]\" height=\"245\" src=\"https://latex.artofproblemsolving.com/2/9/f/29fd640711a70115cf67481ab55a4be235902efc.png\" width=\"335\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 29\\sqrt{3} \\qquad\\textbf{(B)}\\ \\frac{21}{2}\\sqrt{2}+\\frac{41}{2}\\sqrt{3}\\qquad\\textbf{(C)}\\ 20\\sqrt{3}+16</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(D)}\\ 20\\sqrt{2}+13\\sqrt{3} \\qquad\\textbf{(E)}\\ 21\\sqrt{6}</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": "2012 AMC 12B Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12B_p22", "prev": "/problem/12_amc12B_p20"}}