{"status": "success", "data": {"description_md": "Equilateral triangle $T$ is inscribed in circle $A$, which has radius $10$. Circle $B$ with radius $3$ is internally tangent to circle $A$ at one vertex of $T$. Circles $C$ and $D$, both with radius $2$, are internally tangent to circle $A$ at the other two vertices of $T$. Circles $B$, $C$, and $D$ are all externally tangent to circle $E$, which has radius $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.<br>  \n![[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=4;  pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90); pair Ep=(0,4-27/5); pair[] dotted={A,B,C,D,Ep};  draw(Circle(A,10)); draw(Circle(B,3)); draw(Circle(C,2)); draw(Circle(D,2)); draw(Circle(Ep,27/5));  dot(dotted); label(\"$E$\",Ep,E); label(\"$A$\",A,W); label(\"$B$\",B,W); label(\"$C$\",C,W); label(\"$D$\",D,E); [/asy]](https://latex.artofproblemsolving.com/2/c/9/2c9d04c4ba8f11721bf58a4058fa3c4a849369c4.png)\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full 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>Equilateral triangle <span class=\"katex--inline\">T</span> is inscribed in circle <span class=\"katex--inline\">A</span>, which has radius <span class=\"katex--inline\">10</span>. Circle <span class=\"katex--inline\">B</span> with radius <span class=\"katex--inline\">3</span> is internally tangent to circle <span class=\"katex--inline\">A</span> at one vertex of <span class=\"katex--inline\">T</span>. Circles <span class=\"katex--inline\">C</span> and <span class=\"katex--inline\">D</span>, both with radius <span class=\"katex--inline\">2</span>, are internally tangent to circle <span class=\"katex--inline\">A</span> at the other two vertices of <span class=\"katex--inline\">T</span>. Circles <span class=\"katex--inline\">B</span>, <span class=\"katex--inline\">C</span>, and <span class=\"katex--inline\">D</span> are all externally tangent to circle <span class=\"katex--inline\">E</span>, which has radius <span class=\"katex--inline\">\\frac {m}{n}</span>, where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are relatively prime positive integers. Find <span class=\"katex--inline\">m + n</span>.<br/><br/>&#10;<img src=\"https://latex.artofproblemsolving.com/2/c/9/2c9d04c4ba8f11721bf58a4058fa3c4a849369c4.png\" alt=\"[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=4;  pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90); pair Ep=(0,4-27/5); pair[] dotted={A,B,C,D,Ep};  draw(Circle(A,10)); draw(Circle(B,3)); draw(Circle(C,2)); draw(Circle(D,2)); draw(Circle(Ep,27/5));  dot(dotted); label(&#34;&#34;,Ep,E); label(&#34;&#34;,A,W); label(&#34;&#34;,B,W); label(&#34;&#34;,C,W); label(&#34;&#34;,D,E); [/asy]\"/></p>&#10;<hr/>&#10;<p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. 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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2009 AIME II Problem 5", "can_next": true, "can_prev": true, "nxt": "/problem/09_aime_II_p06", "prev": "/problem/09_aime_II_p04"}}