{"status": "success", "data": {"description_md": "A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16$. The set of all possible values of $r$ is an interval $[a,b)$. What is $a+b$?\n\n$\\textbf{(A)}\\ 5\\sqrt{2}+4 \\qquad\\textbf{(B)}\\ \\sqrt{17}+7 \\qquad\\textbf{(C)}\\ 6\\sqrt{2}+3 \\qquad\\textbf{(D)}\\ \\sqrt{15}+8 \\qquad\\textbf{(E)}\\ 12$\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>  passes through both foci of, and exactly four points on, the ellipse with equation  <span class=\"katex--inline\">x^2+16y^2=16</span> . The set of all possible values of  <span class=\"katex--inline\">r</span>  is an interval  <span class=\"katex--inline\">[a,b)</span> . What is  <span class=\"katex--inline\">a+b</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 5\\sqrt{2}+4 \\qquad\\textbf{(B)}\\ \\sqrt{17}+7 \\qquad\\textbf{(C)}\\ 6\\sqrt{2}+3 \\qquad\\textbf{(D)}\\ \\sqrt{15}+8 \\qquad\\textbf{(E)}\\ 12</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": "2015 AMC 12A Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/15_amc12A_p22", "prev": "/problem/15_amc12A_p20"}}