Difference between revisions of "2014 AMC 12A Problems/Problem 21"

Revision as of 23:37, 28 January 2021

Problem

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?

$\textbf{(A) }1\qquad \textbf{(B) }\dfrac{\log 2015}{\log 2014}\qquad \textbf{(C) }\dfrac{\log 2014}{\log 2013}\qquad \textbf{(D) }\dfrac{2014}{2013}\qquad \textbf{(E) }2014^{\frac1{2014}}\qquad$

Solution

Let $\lfloor x\rfloor=k$ for some integer $1\leq k\leq 2013$ . Then we can rewrite $f(x)$ as $k(2014^{x-k}-1)$ . In order for this to be less than or equal to $1$ , we need $2014^{x-k}-1\leq\dfrac1k\implies x\leq k+\log_{2014}\left(\dfrac{k+1}k\right)$ . Combining this with the fact that $\lfloor x\rfloor =k$ gives that $x\in\left[k,k+\log_{2014}\left(\dfrac{k+1}k\right)\right]$ , and so the length of the interval is $\log_{2014}\left(\dfrac{k+1}k\right)$ . We want the sum of all possible intervals such that the inequality holds true; since all of these intervals must be disjoint, we can sum from $k=1$ to $k=2013$ to get that the desired sum is $\sum_{i=1}^{2013}\log_{2014}\left(\dfrac{i+1}i\right)=\log_{2014}\left(\prod_{i=1}^{2013}\dfrac{i+1}i\right)=\log_{2014}\left(\dfrac{2014}1\right)=\boxed{1\textbf{ (A)}}.$

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2014amc12a/380

~ dolphin7

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=MI3ax4WJBZA

2014 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 16: / Line 16: @@
 ~ dolphin7
+==Video Solution by Punxsutawney Phil==
+https://youtube.com/watch?v=MI3ax4WJBZA
 ==See Also==
 {{AMC12 box|year=2014|ab=A|num-b=20|num-a=22}}
 {{MAA Notice}}

Page

Toolbox

Search