{"status": "success", "data": {"description_md": "A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$?<br><center><img class=\"problem-image\" alt='[asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label(\"\\(x\\)\",(-1.55,2.1),E); label(\"\\(1\\)\",(-0.5,3.8),S);[/asy]' class=\"latexcenter\" height=\"155\" src=\"https://latex.artofproblemsolving.com/4/5/5/455ee9e9f5150b8651dd85c1adc2f62d3193a852.png\" width=\"155\"/></center>\n\n$\\mathrm{(A)}\\ 2\\sqrt{5}-\\sqrt{3}\\qquad\\mathrm{(B)}\\ 3\\qquad\\mathrm{(C)}\\ \\frac{3\\sqrt{7}-\\sqrt{3}}{2}\\qquad\\mathrm{(D)}\\ 2\\sqrt{3}\\qquad\\mathrm{(E)}\\ \\frac{5+2\\sqrt{3}}{2}$\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>A round table has radius  <span class=\"katex--inline\">4</span> . Six rectangular place mats are placed on the table. Each place mat has width  <span class=\"katex--inline\">1</span>  and length  <span class=\"katex--inline\">x</span>  as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length  <span class=\"katex--inline\">x</span> . Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is  <span class=\"katex--inline\">x</span> ?<br/><center><img class=\"latexcenter\" alt=\"[asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label(&#34;\\(x\\)&#34;,(-1.55,2.1),E); label(&#34;\\(1\\)&#34;,(-0.5,3.8),S);[/asy]\" height=\"155\" src=\"https://latex.artofproblemsolving.com/4/5/5/455ee9e9f5150b8651dd85c1adc2f62d3193a852.png\" width=\"155\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A)}\\ 2\\sqrt{5}-\\sqrt{3}\\qquad\\mathrm{(B)}\\ 3\\qquad\\mathrm{(C)}\\ \\frac{3\\sqrt{7}-\\sqrt{3}}{2}\\qquad\\mathrm{(D)}\\ 2\\sqrt{3}\\qquad\\mathrm{(E)}\\ \\frac{5+2\\sqrt{3}}{2}</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": 5, "problem_name": "2008 AMC 12A Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/08_amc12A_p23", "prev": "/problem/08_amc12A_p21"}}