Difference between revisions of "2014 AMC 10A Problems/Problem 25"
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{{duplicate|[[2014 AMC 12A Problems|2014 AMC 12A #22]] and [[2014 AMC 10A Problems|2014 AMC 10A #25]]}} | {{duplicate|[[2014 AMC 12A Problems|2014 AMC 12A #22]] and [[2014 AMC 10A Problems|2014 AMC 10A #25]]}} | ||
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| + | == Problem == | ||
The number <math>5^{867}</math> is between <math>2^{2013}</math> and <math>2^{2014}</math>. How many pairs of integers <math>(m,n)</math> are there such that <math>1\leq m\leq 2012</math> and <cmath>5^n<2^m<2^{m+2}<5^{n+1}?</cmath> | The number <math>5^{867}</math> is between <math>2^{2013}</math> and <math>2^{2014}</math>. How many pairs of integers <math>(m,n)</math> are there such that <math>1\leq m\leq 2012</math> and <cmath>5^n<2^m<2^{m+2}<5^{n+1}?</cmath> | ||
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<math>\textbf{(A) }278\qquad | <math>\textbf{(A) }278\qquad | ||
\textbf{(B) }279\qquad | \textbf{(B) }279\qquad | ||
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\textbf{(E) }282\qquad</math> | \textbf{(E) }282\qquad</math> | ||
| − | ==Solution== | + | == Solutions == |
| − | + | === Solution 1 === | |
Between any two consecutive powers of <math>5</math> there are either <math>2</math> or <math>3</math> powers of <math>2</math> (because <math>2^2<5^1<2^3</math>). Consider the intervals <math>(5^0,5^1),(5^1,5^2),\dots (5^{866},5^{867})</math>. We want the number of intervals with <math>3</math> powers of <math>2</math>. | Between any two consecutive powers of <math>5</math> there are either <math>2</math> or <math>3</math> powers of <math>2</math> (because <math>2^2<5^1<2^3</math>). Consider the intervals <math>(5^0,5^1),(5^1,5^2),\dots (5^{866},5^{867})</math>. We want the number of intervals with <math>3</math> powers of <math>2</math>. | ||
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=== Video Solution by Richard Rusczyk === | === Video Solution by Richard Rusczyk === | ||
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https://artofproblemsolving.com/videos/amc/2014amc10a/379 | https://artofproblemsolving.com/videos/amc/2014amc10a/379 | ||
| − | + | == See Also == | |
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| − | ==See Also== | ||
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{{AMC10 box|year=2014|ab=A|num-b=24|after=Last Problem}} | {{AMC10 box|year=2014|ab=A|num-b=24|after=Last Problem}} | ||
{{AMC12 box|year=2014|ab=A|num-b=21|num-a=23}} | {{AMC12 box|year=2014|ab=A|num-b=21|num-a=23}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 00:12, 19 October 2020
- The following problem is from both the 2014 AMC 12A #22 and 2014 AMC 10A #25, so both problems redirect to this page.
Problem
The number 
is between 
and 
. How many pairs of integers 
are there such that 
and ![\[5^n<2^m<2^{m+2}<5^{n+1}?\]](http://latex.artofproblemsolving.com/c/0/d/c0dd87b90d59f4e58a09e5b2e78b8af6511482d5.png)
Solutions
Solution 1
Between any two consecutive powers of 
there are either 
or 
powers of (because 
). Consider the intervals 
. We want the number of intervals with powers of .
From the given that 
, we know that these 
intervals together have 
powers of . Let 
of them have powers of and 
of them have powers of . Thus we have the system
![\[x+y=867\]](http://latex.artofproblemsolving.com/1/5/8/1580c31f4d58c540f94347ac313ba29d0a32fcda.png)
![\[2x+3y=2013\]](http://latex.artofproblemsolving.com/3/9/b/39b29d4cbbbad5954993ebfc4f5348eb1a5ad481.png)
from which we get 
, so the answer is 
.
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2014amc10a/379
See Also
| 2014 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Last Problem | |
| 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 10 Problems and Solutions | ||
| 2014 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 21 |
Followed by Problem 23 |
| 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. 
