# Problem #1838

 1838 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? $\textbf{(A) }1\qquad \textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad \textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad \textbf{(D) }\dfrac{2014}{2013}\qquad \textbf{(E) }2014^{\frac1{2014}}\qquad$ This problem is copyrighted by the American Mathematics Competitions.
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