{"status": "success", "data": {"description_md": "Square $ABCD$ has side length $s$, a circle centered at $E$ has radius $r$, and $r$ and $s$ are both rational. The circle passes through $D$, and $D$ lies on $\\overline{BE}$. Point $F$ lies on the circle, on the same side of $\\overline{BE}$ as $A$. Segment $AF$ is tangent to the circle, and $AF=\\sqrt{9+5\\sqrt{2}}$. What is $r/s$?<br><!-- [[Image:AMC12_2006A_17.png|center]] --><br><center><img class=\"problem-image\" alt='[asy] real s = 90; real r = 50; pair e = (r/sqrt(2),r/sqrt(2)); pair f = (4.34, 74.58); draw((-s, 0) -- (-s,-s) -- (0, -s) -- (0,0) -- (-s, 0)); draw(circle(e,r)); draw((-s,0) -- f); dot(e); dot(f);  label(\"A\", (-s,0), W); label(\"B\", (-s,-s), W); label(\"C\", (0,-s), E); label(\"D\", (0,0), SW); label(\"E\", e, E); label(\"F\", f, N);  [/asy]' class=\"latexcenter\" height=\"245\" src=\"https://latex.artofproblemsolving.com/5/c/2/5c2fba73325243f6661e3a7593c657103c98be53.png\" width=\"252\"/></center>\n\n$\\mathrm{(A) \\ } \\frac{1}{2}\\qquad \\mathrm{(B) \\ } \\frac{5}{9}\\qquad \\mathrm{(C) \\ } \\frac{3}{5}\\qquad \\mathrm{(D) \\ } \\frac{5}{3}\\qquad \\mathrm{(E) \\ }  \\frac{9}{5}$\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\">ABCD</span>  has side length  <span class=\"katex--inline\">s</span> , a circle centered at  <span class=\"katex--inline\">E</span>  has radius  <span class=\"katex--inline\">r</span> , and  <span class=\"katex--inline\">r</span>  and  <span class=\"katex--inline\">s</span>  are both rational. The circle passes through  <span class=\"katex--inline\">D</span> , and  <span class=\"katex--inline\">D</span>  lies on  <span class=\"katex--inline\">\\overline{BE}</span> . Point  <span class=\"katex--inline\">F</span>  lies on the circle, on the same side of  <span class=\"katex--inline\">\\overline{BE}</span>  as  <span class=\"katex--inline\">A</span> . Segment  <span class=\"katex--inline\">AF</span>  is tangent to the circle, and  <span class=\"katex--inline\">AF=\\sqrt{9+5\\sqrt{2}}</span> . What is  <span class=\"katex--inline\">r/s</span> ?<br/><!-- [[Image:AMC12_2006A_17.png|center]] --><br/><center><img class=\"latexcenter\" alt=\"[asy] real s = 90; real r = 50; pair e = (r/sqrt(2),r/sqrt(2)); pair f = (4.34, 74.58); draw((-s, 0) -- (-s,-s) -- (0, -s) -- (0,0) -- (-s, 0)); draw(circle(e,r)); draw((-s,0) -- f); dot(e); dot(f);  label(&#34;A&#34;, (-s,0), W); label(&#34;B&#34;, (-s,-s), W); label(&#34;C&#34;, (0,-s), E); label(&#34;D&#34;, (0,0), SW); label(&#34;E&#34;, e, E); label(&#34;F&#34;, f, N);  [/asy]\" height=\"245\" src=\"https://latex.artofproblemsolving.com/5/c/2/5c2fba73325243f6661e3a7593c657103c98be53.png\" width=\"252\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A) \\ } \\frac{1}{2}\\qquad \\mathrm{(B) \\ } \\frac{5}{9}\\qquad \\mathrm{(C) \\ } \\frac{3}{5}\\qquad \\mathrm{(D) \\ } \\frac{5}{3}\\qquad \\mathrm{(E) \\ }  \\frac{9}{5}</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": "2006 AMC 12A Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/06_amc12A_p18", "prev": "/problem/06_amc12A_p16"}}