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21

Part of 2014 AMC 12/AHSME

Problems(2)

Function is Less Than One

Source: 2014 AMC 12A #21

2/5/2014
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
functionfloor functionlogarithmsalgebradomainAMC
Square and Rectangles

Source: 2014 amc 12b #21

2/20/2014
In the figure, ABCDABCD is a square of side length 1. The rectangles JKHGJKHG and EBCFEBCF are congruent. What is BEBE? [asy] unitsize(150); pair A,B,C,D,E,F,G,H,J,K; A=(1,0); B=(0,0); C=(0,1); D=(1,1); draw(A--B--C--D--A);
E=(2-sqrt(3),0); F=(2-sqrt(3),1); draw(E--F);
G=(1,sqrt(3)/2); H=(2.5-sqrt(3),1); K=(2-sqrt(3),1-sqrt(3)/2); J=(0.5,0); draw(G--H--K--J--G);
label("AA",A,SE); label("BB",B,SW); label("CC",C,NW); label("DD",D,NE); label("EE",E,S); label("FF",F,N); label("GG",G,E); label("HH",H,N); label("KK",K,W); label("JJ",J,S); [/asy]
<spanclass=′latex−bold′>(A)</span>12(6−2)<spanclass=′latex−bold′>(B)</span>14<spanclass=′latex−bold′>(C)</span>2−3<spanclass=′latex−bold′>(D)</span>36<spanclass=′latex−bold′>(E)</span>1−22 <span class='latex-bold'>(A) </span>\dfrac{1}{2}(\sqrt{6}-2)\qquad<span class='latex-bold'>(B) </span>\dfrac{1}{4}\qquad<span class='latex-bold'>(C) </span>2-\sqrt{3}\qquad<span class='latex-bold'>(D) </span>\dfrac{\sqrt{3}}{6}\qquad<span class='latex-bold'>(E) </span>1-\dfrac{\sqrt{2}}{2}
geometryrectangletrigonometrysimilar trianglesPythagorean TheoremAMC