{"status": "success", "data": {"description_md": "Let $f_{1}(x)=\\sqrt{1-x}$, and for integers $n \\geq 2$, let $f_{n}(x)=f_{n-1}(\\sqrt{n^2 - x})$. If $N$ is the largest value of $n$ for which the domain of $f_{n}$ is nonempty, the domain of $f_{N}$ is $\\{ c\\}$. What is $N+c$?\n\n$\\textbf{(A)}\\ -226 \\qquad<br>\\textbf{(B)}\\ -144 \\qquad<br>\\textbf{(C)}\\ -20 \\qquad<br>\\textbf{(D)}\\ 20 \\qquad<br>\\textbf{(E)}\\ 144$\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>Let  <span class=\"katex--inline\">f_{1}(x)=\\sqrt{1-x}</span> , and for integers  <span class=\"katex--inline\">n \\geq 2</span> , let  <span class=\"katex--inline\">f_{n}(x)=f_{n-1}(\\sqrt{n^2 - x})</span> . If  <span class=\"katex--inline\">N</span>  is the largest value of  <span class=\"katex--inline\">n</span>  for which the domain of  <span class=\"katex--inline\">f_{n}</span>  is nonempty, the domain of  <span class=\"katex--inline\">f_{N}</span>  is  <span class=\"katex--inline\">\\{ c\\}</span> . What is  <span class=\"katex--inline\">N+c</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ -226 \\qquad\\textbf{(B)}\\ -144 \\qquad\\textbf{(C)}\\ -20 \\qquad\\textbf{(D)}\\ 20 \\qquad\\textbf{(E)}\\ 144</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": "2011 AMC 12A Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/11_amc12A_p22", "prev": "/problem/11_amc12A_p20"}}