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

(Created page with "==Problem== For every real number <math>x</math>, let <math>\lfloor x\rfloor</math> denote the greatest integer not exceeding <math>x</math>, and let <cmath>f(x)=\lfloor x\rfloor...")
 
(→‎See Also)
 
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==Solution==
 
==Solution==
 
Let <math>\lfloor x\rfloor=k</math> for some integer <math>1\leq k\leq 2013</math>.  Then we can rewrite <math>f(x)</math> as <math>k(2014^{x-k}-1)</math>.  In order for this to be less than or equal to <math>1</math>, we need <math>2014^{x-k}-1\leq\dfrac1k\implies x\leq k+\log_{2014}\left(\dfrac{k+1}k\right)</math>.  Combining this with the fact that <math>\lfloor x\rfloor =k</math> gives that <math>x\in\left[k,k+\log_{2014}\left(\dfrac{k+1}k\right)\right]</math>, and so the length of the interval is <math>\log_{2014}\left(\dfrac{k+1}k\right)</math>.  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 <math>k=1</math> to <math>k=2013</math> to get that the desired sum is <cmath>\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)}}.</cmath>
 
Let <math>\lfloor x\rfloor=k</math> for some integer <math>1\leq k\leq 2013</math>.  Then we can rewrite <math>f(x)</math> as <math>k(2014^{x-k}-1)</math>.  In order for this to be less than or equal to <math>1</math>, we need <math>2014^{x-k}-1\leq\dfrac1k\implies x\leq k+\log_{2014}\left(\dfrac{k+1}k\right)</math>.  Combining this with the fact that <math>\lfloor x\rfloor =k</math> gives that <math>x\in\left[k,k+\log_{2014}\left(\dfrac{k+1}k\right)\right]</math>, and so the length of the interval is <math>\log_{2014}\left(\dfrac{k+1}k\right)</math>.  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 <math>k=1</math> to <math>k=2013</math> to get that the desired sum is <cmath>\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)}}.</cmath>
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 +
=== Video Solution by Richard Rusczyk ===
 +
 +
https://artofproblemsolving.com/videos/amc/2014amc12a/380
 +
 +
~ dolphin7
 +
 +
==Video Solution by Punxsutawney Phil==
 +
https://youtube.com/watch?v=MI3ax4WJBZA
 +
 +
(The video is no longer available on YouTube)
 +
 +
==See Also==
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{{AMC12 box|year=2014|ab=A|num-b=20|num-a=22}}
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{{MAA Notice}}
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[[Category:Intermediate Algebra Problems]]

Latest revision as of 14:38, 12 June 2024

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

(The video is no longer available on YouTube)

See Also

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

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