{"status": "success", "data": {"description_md": "Circle $C_1$ and $C_2$ each have radius $1$, and the distance between their centers is $\\frac{1}{2}$. Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$. Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$. What is the radius of $C_4$?<br><center><img class=\"problem-image\" alt='[asy] import olympiad; size(10cm); draw(circle((0,0),0.75)); draw(circle((-0.25,0),1)); draw(circle((0.25,0),1)); draw(circle((0,6/7),3/28)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118); dot(B^^C); draw(B--E, dashed); draw(C--F, dashed); draw(B--C); label(\"$C_4$\", D); label(\"$C_1$\", (-1.375, 0)); label(\"$C_2$\", (1.375,0)); label(\"$\frac{1}{2}$\", (0, -.125)); label(\"$C_3$\", (-0.4, -0.4)); label(\"$1$\", (-.85, 0.70)); label(\"$1$\", (.85, -.7)); import olympiad; markscalefactor=0.005; [/asy]' class=\"latexcenter\" height=\"325\" src=\"https://latex.artofproblemsolving.com/4/0/6/406135ba18fccc6f68ffce10d698ac01c265ec90.png\" width=\"475\"/></center>\n\n$\\textbf{(A) } \\frac{1}{14} \\qquad \\textbf{(B) } \\frac{1}{12} \\qquad \\textbf{(C) } \\frac{1}{10} \\qquad \\textbf{(D) } \\frac{3}{28} \\qquad \\textbf{(E) } \\frac{1}{9}$\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>Circle  <span class=\"katex--inline\">C_1</span>  and  <span class=\"katex--inline\">C_2</span>  each have radius  <span class=\"katex--inline\">1</span> , and the distance between their centers is  <span class=\"katex--inline\">\\frac{1}{2}</span> . Circle  <span class=\"katex--inline\">C_3</span>  is the largest circle internally tangent to both  <span class=\"katex--inline\">C_1</span>  and  <span class=\"katex--inline\">C_2</span> . Circle  <span class=\"katex--inline\">C_4</span>  is internally tangent to both  <span class=\"katex--inline\">C_1</span>  and  <span class=\"katex--inline\">C_2</span>  and externally tangent to  <span class=\"katex--inline\">C_3</span> . What is the radius of  <span class=\"katex--inline\">C_4</span> ?<br/><center><img class=\"latexcenter\" alt=\"[asy] import olympiad; size(10cm); draw(circle((0,0),0.75)); draw(circle((-0.25,0),1)); draw(circle((0.25,0),1)); draw(circle((0,6/7),3/28)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118); dot(B^^C); draw(B--E, dashed); draw(C--F, dashed); draw(B--C); label(&#34;$C_4$&#34;, D); label(&#34;$C_1$&#34;, (-1.375, 0)); label(&#34;$C_2$&#34;, (1.375,0)); label(&#34;$&#12;rac{1}{2}$&#34;, (0, -.125)); label(&#34;$C_3$&#34;, (-0.4, -0.4)); label(&#34;$1$&#34;, (-.85, 0.70)); label(&#34;$1$&#34;, (.85, -.7)); import olympiad; markscalefactor=0.005; [/asy]\" height=\"325\" src=\"https://latex.artofproblemsolving.com/4/0/6/406135ba18fccc6f68ffce10d698ac01c265ec90.png\" width=\"475\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) } \\frac{1}{14} \\qquad \\textbf{(B) } \\frac{1}{12} \\qquad \\textbf{(C) } \\frac{1}{10} \\qquad \\textbf{(D) } \\frac{3}{28} \\qquad \\textbf{(E) } \\frac{1}{9}</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": "2023 AMC 12A Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12A_p19", "prev": "/problem/23_amc12A_p17"}}