{"status": "success", "data": {"description_md": "Three circles of radius $s$ are drawn in the first quadrant of the $xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $x$-axis, and the third is tangent to the first circle and the $y$-axis. A circle of radius $r > s$ is tangent to both axes and to the second and third circles. What is $r/s$?<br><center><img class=\"problem-image\" alt='[asy] unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=3; pair O0=(9,9), O1=(1,1), O2=(3,1), O3=(1,3); pair P0=O0+9*dir(-45), P3=O3+dir(70); pair[] ps={O0,O1,O2,O3}; dot(ps); draw(Circle(O0,9)); draw(Circle(O1,1)); draw(Circle(O2,1)); draw(Circle(O3,1)); draw(O0--P0,linetype(\"3 3\")); draw(O3--P3,linetype(\"2 2\")); draw((0,0)--(18,0)); draw((0,0)--(0,18)); label(\"$r$\",midpoint(O0--P0),NE); label(\"$s$\",(-1.5,4)); draw((-1,4)--midpoint(O3--P3));[/asy]' class=\"latexcenter\" height=\"258\" src=\"https://latex.artofproblemsolving.com/6/5/7/657afd2eef0760abe2ccc3fe7453a83a0b22e681.png\" width=\"285\"/></center>\n\n$(\\mathrm {A}) \\ 5 \\qquad (\\mathrm {B}) \\ 6 \\qquad (\\mathrm {C})\\ 8 \\qquad (\\mathrm {D}) \\ 9 \\qquad (\\mathrm {E})\\ 10$\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 circles of radius  <span class=\"katex--inline\">s</span>  are drawn in the first quadrant of the  <span class=\"katex--inline\">xy</span> -plane. The first circle is tangent to both axes, the second is tangent to the first circle and the  <span class=\"katex--inline\">x</span> -axis, and the third is tangent to the first circle and the  <span class=\"katex--inline\">y</span> -axis. A circle of radius  <span class=\"katex--inline\">r &gt; s</span>  is tangent to both axes and to the second and third circles. What is  <span class=\"katex--inline\">r/s</span> ?<br/><center><img class=\"latexcenter\" alt=\"[asy] unitsize(3mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=3; pair O0=(9,9), O1=(1,1), O2=(3,1), O3=(1,3); pair P0=O0+9*dir(-45), P3=O3+dir(70); pair[] ps={O0,O1,O2,O3}; dot(ps); draw(Circle(O0,9)); draw(Circle(O1,1)); draw(Circle(O2,1)); draw(Circle(O3,1)); draw(O0--P0,linetype(&#34;3 3&#34;)); draw(O3--P3,linetype(&#34;2 2&#34;)); draw((0,0)--(18,0)); draw((0,0)--(0,18)); label(&#34;$r$&#34;,midpoint(O0--P0),NE); label(&#34;$s$&#34;,(-1.5,4)); draw((-1,4)--midpoint(O3--P3));[/asy]\" height=\"258\" src=\"https://latex.artofproblemsolving.com/6/5/7/657afd2eef0760abe2ccc3fe7453a83a0b22e681.png\" width=\"285\"/></center></p>&#10;<p> <span class=\"katex--inline\">(\\mathrm {A}) \\ 5 \\qquad (\\mathrm {B}) \\ 6 \\qquad (\\mathrm {C})\\ 8 \\qquad (\\mathrm {D}) \\ 9 \\qquad (\\mathrm {E})\\ 10</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": "2005 AMC 12A Problem 16", "can_next": true, "can_prev": true, "nxt": "/problem/05_amc12A_p17", "prev": "/problem/05_amc12A_p15"}}