{"status": "success", "data": {"description_md": "For every real number $x$, let $\\lfloor x\\rfloor$ denote the greatest integer not exceeding $x$, and let $$f(x)=\\lfloor x\\rfloor(2014^{x-\\lfloor x\\rfloor}-1).$$  The set of all numbers $x$ such that $1\\leq x<2014$ and $f(x)\\leq 1$ is a union of disjoint intervals.  What is the sum of the lengths of those intervals?\n\n$\\textbf{(A) }1\\qquad<br>\\textbf{(B) }\\dfrac{\\log 2015}{\\log 2014}\\qquad<br>\\textbf{(C) }\\dfrac{\\log 2014}{\\log 2013}\\qquad<br>\\textbf{(D) }\\dfrac{2014}{2013}\\qquad<br>\\textbf{(E) }2014^{\\frac1{2014}}\\qquad$\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>For every real number  <span class=\"katex--inline\">x</span> , let  <span class=\"katex--inline\">\\lfloor x\\rfloor</span>  denote the greatest integer not exceeding  <span class=\"katex--inline\">x</span> , and let  <span class=\"katex--display\">f(x)=\\lfloor x\\rfloor(2014^{x-\\lfloor x\\rfloor}-1).</span>   The set of all numbers  <span class=\"katex--inline\">x</span>  such that  <span class=\"katex--inline\">1\\leq x&lt;2014</span>  and  <span class=\"katex--inline\">f(x)\\leq 1</span>  is a union of disjoint intervals.  What is the sum of the lengths of those intervals?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }1\\qquad\\textbf{(B) }\\dfrac{\\log 2015}{\\log 2014}\\qquad\\textbf{(C) }\\dfrac{\\log 2014}{\\log 2013}\\qquad\\textbf{(D) }\\dfrac{2014}{2013}\\qquad\\textbf{(E) }2014^{\\frac1{2014}}\\qquad</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": "2014 AMC 12A Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/14_amc12A_p22", "prev": "/problem/14_amc12A_p20"}}