{"status": "success", "data": {"description_md": "A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$. What is $r$?\n\n$\\mathrm{(A) \\ } 2\\sqrt{2}+2\\sqrt{3}\\qquad \\mathrm{(B) \\ } 3\\sqrt{3}+\\sqrt{2}\\qquad \\mathrm{(C) \\ } 2\\sqrt{6}+\\sqrt{3} \\qquad \\mathrm{(D) \\ } 3\\sqrt{2}+\\sqrt{6}\\qquad \\mathrm{(E) \\ }  6\\sqrt{2}-\\sqrt{3}$\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 circle of radius  <span class=\"katex--inline\">r</span>  is concentric with and outside a regular hexagon of side length  <span class=\"katex--inline\">2</span> . The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is  <span class=\"katex--inline\">1/2</span> . What is  <span class=\"katex--inline\">r</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A) \\ } 2\\sqrt{2}+2\\sqrt{3}\\qquad \\mathrm{(B) \\ } 3\\sqrt{3}+\\sqrt{2}\\qquad \\mathrm{(C) \\ } 2\\sqrt{6}+\\sqrt{3} \\qquad \\mathrm{(D) \\ } 3\\sqrt{2}+\\sqrt{6}\\qquad \\mathrm{(E) \\ }  6\\sqrt{2}-\\sqrt{3}</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": "2006 AMC 12A Problem 22", "can_next": true, "can_prev": true, "nxt": "/problem/06_amc12A_p23", "prev": "/problem/06_amc12A_p21"}}