MathDB
Function is Less Than One

Source: 2014 AMC 12A #21

February 5, 2014
functionfloor functionlogarithmsalgebradomainAMC

Problem Statement

For every real number xx, let ⌊x⌋\lfloor x\rfloor denote the greatest integer not exceeding xx, and let f(x)=⌊x⌋(2014x−⌊x⌋−1).f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1). The set of all numbers xx such that 1≤x<20141\leq x<2014 and f(x)≤1f(x)\leq 1 is a union of disjoint intervals. What is the sum of the lengths of those intervals?
<spanclass=′latex−bold′>(A)</span>1<spanclass=′latex−bold′>(B)</span>log⁡2015log⁡2014<spanclass=′latex−bold′>(C)</span>log⁡2014log⁡2013<spanclass=′latex−bold′>(D)</span>20142013<spanclass=′latex−bold′>(E)</span>201412014<span class='latex-bold'>(A) </span>1\qquad <span class='latex-bold'>(B) </span>\dfrac{\log 2015}{\log 2014}\qquad <span class='latex-bold'>(C) </span>\dfrac{\log 2014}{\log 2013}\qquad <span class='latex-bold'>(D) </span>\dfrac{2014}{2013}\qquad <span class='latex-bold'>(E) </span>2014^{\frac1{2014}}\qquad
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