{"status": "success", "data": {"description_md": "Let $f(x)=|2\\{x\\}-1|$ where $\\{x\\}$ denotes the fractional part of $x$.  The number $n$ is the smallest positive integer such that the equation $$nf(xf(x))=x$$ has at least $2012$ real solutions.  What is $n$?  '''Note:''' the fractional part of $x$ is a real number $y=\\{x\\}$ such that $0\\le y<1$ and $x-y$ is an integer.\n\n$\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ 31\\qquad\\textbf{(C)}\\ 32\\qquad\\textbf{(D)}\\ 62\\qquad\\textbf{(E)}\\ 64$\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(x)=|2\\{x\\}-1|</span>  where  <span class=\"katex--inline\">\\{x\\}</span>  denotes the fractional part of  <span class=\"katex--inline\">x</span> .  The number  <span class=\"katex--inline\">n</span>  is the smallest positive integer such that the equation  <span class=\"katex--display\">nf(xf(x))=x</span>  has at least  <span class=\"katex--inline\">2012</span>  real solutions.  What is  <span class=\"katex--inline\">n</span> ?  &#8216;&#8217;&#8216;Note:&#8217;&#8217;&#8217; the fractional part of  <span class=\"katex--inline\">x</span>  is a real number  <span class=\"katex--inline\">y=\\{x\\}</span>  such that  <span class=\"katex--inline\">0\\le y&lt;1</span>  and  <span class=\"katex--inline\">x-y</span>  is an integer.</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 30\\qquad\\textbf{(B)}\\ 31\\qquad\\textbf{(C)}\\ 32\\qquad\\textbf{(D)}\\ 62\\qquad\\textbf{(E)}\\ 64</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": "2012 AMC 12A Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/12_amc12A_p24"}}