{"status": "success", "data": {"description_md": "Square $ABCD$ has side length $2$. A semicircle with diameter $\\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\\overline{AD}$ at $E$. What is the length of $\\overline{CE}$?<br><center><img class=\"problem-image\" alt='[asy] size(100); defaultpen(fontsize(10)); pair A=(0,0), B=(2,0), C=(2,2), D=(0,2), E=(0,1/2); draw(A--B--C--D--cycle);draw(C--E); draw(Arc((1,0),1,0,180)); label(\"$A$\",A,(-1,-1)); label(\"$B$\",B,( 1,-1)); label(\"$C$\",C,( 1, 1)); label(\"$D$\",D,(-1, 1)); label(\"$E$\",E,(-1, 0)); [/asy]' class=\"latexcenter\" height=\"165\" src=\"https://latex.artofproblemsolving.com/d/a/e/daef6f3ebe3e1fc6cfacd8611db1d6d449c8d2d4.png\" width=\"168\"/></center>\n\n$\\text {(A) } \\frac {2 + \\sqrt5}{2} \\qquad \\text {(B) } \\sqrt 5 \\qquad \\text {(C) } \\sqrt 6 \\qquad \\text {(D) } \\frac52 \\qquad \\text {(E) }5 - \\sqrt5$\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\">2</span> . A semicircle with diameter  <span class=\"katex--inline\">\\overline{AB}</span>  is constructed inside the square, and the tangent to the semicircle from  <span class=\"katex--inline\">C</span>  intersects side  <span class=\"katex--inline\">\\overline{AD}</span>  at  <span class=\"katex--inline\">E</span> . What is the length of  <span class=\"katex--inline\">\\overline{CE}</span> ?<br/><center><img class=\"latexcenter\" alt=\"[asy] size(100); defaultpen(fontsize(10)); pair A=(0,0), B=(2,0), C=(2,2), D=(0,2), E=(0,1/2); draw(A--B--C--D--cycle);draw(C--E); draw(Arc((1,0),1,0,180)); label(&#34;$A$&#34;,A,(-1,-1)); label(&#34;$B$&#34;,B,( 1,-1)); label(&#34;$C$&#34;,C,( 1, 1)); label(&#34;$D$&#34;,D,(-1, 1)); label(&#34;$E$&#34;,E,(-1, 0)); [/asy]\" height=\"165\" src=\"https://latex.artofproblemsolving.com/d/a/e/daef6f3ebe3e1fc6cfacd8611db1d6d449c8d2d4.png\" width=\"168\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\text {(A) } \\frac {2 + \\sqrt5}{2} \\qquad \\text {(B) } \\sqrt 5 \\qquad \\text {(C) } \\sqrt 6 \\qquad \\text {(D) } \\frac52 \\qquad \\text {(E) }5 - \\sqrt5</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 18", "can_next": true, "can_prev": true, "nxt": "/problem/04_amc12A_p19", "prev": "/problem/04_amc12A_p17"}}