{"status": "success", "data": {"description_md": "Circles $A, B$ and $C$ are externally tangent to each other, and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$. What is the radius of circle $B$?\n\n$$<center><img class=\"problem-image\" alt='[asy] unitsize(15mm); pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);  draw(Circle(D,2)); draw(Circle(A,1)); draw(Circle(B,8/9)); draw(Circle(C,8/9));  label(\"\\(A\\)\", A); label(\"\\(B\\)\", B); label(\"\\(C\\)\", C); label(\"\\(D\\)\", (-1.2,1.8)); [/asy]' class=\"latexcenter\" height=\"285\" src=\"https://latex.artofproblemsolving.com/6/9/6/69677da9f9025d93eca8bc2a4d63d4a3b5b11c9c.png\" width=\"285\"/></center>$$\n\n$\\text{(A) } \\frac23 \\qquad \\text{(B) } \\frac {\\sqrt3}{2} \\qquad \\text{(C) } \\frac78 \\qquad \\text{(D) } \\frac89 \\qquad \\text{(E) } \\frac {1 + \\sqrt3}{3}$\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>Circles  <span class=\"katex--inline\">A, B</span>  and  <span class=\"katex--inline\">C</span>  are externally tangent to each other, and internally tangent to circle  <span class=\"katex--inline\">D</span> . Circles  <span class=\"katex--inline\">B</span>  and  <span class=\"katex--inline\">C</span>  are congruent. Circle  <span class=\"katex--inline\">A</span>  has radius  <span class=\"katex--inline\">1</span>  and passes through the center of  <span class=\"katex--inline\">D</span> . What is the radius of circle  <span class=\"katex--inline\">B</span> ?</p>&#10;<p> <span class=\"katex--display\">&lt;center&gt;&lt;img class=&#34;problem-image&#34; alt='[asy] unitsize(15mm); pair A=(-1,0),B=(2/3,8/9),C=(2/3,-8/9),D=(0,0);  draw(Circle(D,2)); draw(Circle(A,1)); draw(Circle(B,8/9)); draw(Circle(C,8/9));  label(&#34;\\(A\\)&#34;, A); label(&#34;\\(B\\)&#34;, B); label(&#34;\\(C\\)&#34;, C); label(&#34;\\(D\\)&#34;, (-1.2,1.8)); [/asy]' class=&#34;latexcenter&#34; height=&#34;285&#34; src=&#34;https://latex.artofproblemsolving.com/6/9/6/69677da9f9025d93eca8bc2a4d63d4a3b5b11c9c.png&#34; width=&#34;285&#34;/&gt;&lt;/center&gt;</span> </p>&#10;<p> <span class=\"katex--inline\">\\text{(A) } \\frac23 \\qquad \\text{(B) } \\frac {\\sqrt3}{2} \\qquad \\text{(C) } \\frac78 \\qquad \\text{(D) } \\frac89 \\qquad \\text{(E) } \\frac {1 + \\sqrt3}{3}</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": "2004 AMC 12A Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/04_amc12A_p20", "prev": "/problem/04_amc12A_p18"}}