{"status": "success", "data": {"description_md": "An annulus is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$, with $b>c$. Let $OX$ be a radius of the larger circle, let $XZ$ be tangent to the smaller circle at $Z$, and let $OY$ be the radius of the larger circle that contains $Z$. Let $a=XZ$, $d=YZ$, and $e=XY$. What is the area of the annulus? <br><center><img class=\"problem-image\" alt=\"[asy] import graph; unitsize(1.5cm); defaultpen(0.8); real r1=1.5, r2=2.5; pair O=(0,0); path inner=Circle(O,r1), outer=Circle(O,r2); pair Y=(0,r2), Z=(0,r1), X=intersectionpoint( Z--(Z+(10,0)), outer ); filldraw(outer,lightgray,black); filldraw(inner,white,black); draw(X--O--Y); draw(Y--X--Z); label(&quot;$O$&quot;,O,SW); label(&quot;$X$&quot;,X,E); label(&quot;$Y$&quot;,Y,N); label(&quot;$Z$&quot;,Z,SW); label(&quot;$a$&quot;,X--Z,N); label(&quot;$b$&quot;,0.25*X,SE); label(&quot;$c$&quot;,O--Z,E); label(&quot;$d$&quot;,Y--Z,W); label(&quot;$e$&quot;,Y*0.65 + X*0.35,SW); defaultpen(0.5); dot(O); dot(X); dot(Z); dot(Y); [/asy]\" class=\"latexcenter\" height=\"378\" src=\"https://latex.artofproblemsolving.com/e/8/1/e810cc7b6572d11a1cb77c70ea93bf73a9411484.png\" width=\"358\"/></center>\n\n$\\mathrm{(A) \\ } \\pi a^2 \\qquad \\mathrm{(B) \\ } \\pi b^2 \\qquad \\mathrm{(C) \\ } \\pi c^2 \\qquad \\mathrm{(D) \\ } \\pi d^2 \\qquad \\mathrm{(E) \\ } \\pi e^2$\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>An annulus is the region between two concentric circles. The concentric circles in the figure have radii  <span class=\"katex--inline\">b</span>  and  <span class=\"katex--inline\">c</span> , with  <span class=\"katex--inline\">b&gt;c</span> . Let  <span class=\"katex--inline\">OX</span>  be a radius of the larger circle, let  <span class=\"katex--inline\">XZ</span>  be tangent to the smaller circle at  <span class=\"katex--inline\">Z</span> , and let  <span class=\"katex--inline\">OY</span>  be the radius of the larger circle that contains  <span class=\"katex--inline\">Z</span> . Let  <span class=\"katex--inline\">a=XZ</span> ,  <span class=\"katex--inline\">d=YZ</span> , and  <span class=\"katex--inline\">e=XY</span> . What is the area of the annulus? <br/><center><img class=\"latexcenter\" alt=\"[asy] import graph; unitsize(1.5cm); defaultpen(0.8); real r1=1.5, r2=2.5; pair O=(0,0); path inner=Circle(O,r1), outer=Circle(O,r2); pair Y=(0,r2), Z=(0,r1), X=intersectionpoint( Z--(Z+(10,0)), outer ); filldraw(outer,lightgray,black); filldraw(inner,white,black); draw(X--O--Y); draw(Y--X--Z); label(&#34;$O$&#34;,O,SW); label(&#34;$X$&#34;,X,E); label(&#34;$Y$&#34;,Y,N); label(&#34;$Z$&#34;,Z,SW); label(&#34;$a$&#34;,X--Z,N); label(&#34;$b$&#34;,0.25*X,SE); label(&#34;$c$&#34;,O--Z,E); label(&#34;$d$&#34;,Y--Z,W); label(&#34;$e$&#34;,Y*0.65 + X*0.35,SW); defaultpen(0.5); dot(O); dot(X); dot(Z); dot(Y); [/asy]\" height=\"378\" src=\"https://latex.artofproblemsolving.com/e/8/1/e810cc7b6572d11a1cb77c70ea93bf73a9411484.png\" width=\"358\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A) \\ } \\pi a^2 \\qquad \\mathrm{(B) \\ } \\pi b^2 \\qquad \\mathrm{(C) \\ } \\pi c^2 \\qquad \\mathrm{(D) \\ } \\pi d^2 \\qquad \\mathrm{(E) \\ } \\pi e^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": 2, "problem_name": "2004 AMC 12B Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/04_amc12B_p11", "prev": "/problem/04_amc12B_p09"}}