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> | ||
+ | |||
<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== | + | == 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>. | ||
− | + | From the given that <math>2^{2013}<5^{867}<2^{2014}</math>, we know that these <math>867</math> intervals together have <math>2013</math> powers of <math>2</math>. Let <math>x</math> of them have <math>2</math> powers of <math>2</math> and <math>y</math> of them have <math>3</math> powers of <math>2</math>. Thus we have the system | |
− | + | <cmath>x+y=867</cmath><cmath>2x+3y=2013</cmath> | |
− | From the given that <math>2^{2013}<5^{867}<2^{2014}</math>, we know that these 867 intervals together have 2013 powers of 2. Let <math>x</math> of them have 2 powers of 2 and <math>y</math> of them have 3 powers of 2. Thus we have the system | ||
− | <cmath>x+y | ||
from which we get <math>y=279</math>, so the answer is <math>\boxed{\textbf{(B)}}</math>. | from which we get <math>y=279</math>, so the answer is <math>\boxed{\textbf{(B)}}</math>. | ||
− | + | == Video Solution by Richard Rusczyk == | |
− | + | https://artofproblemsolving.com/videos/amc/2014amc10a/379 | |
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+ | == See Also == | ||
{{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}} |
Latest revision as of 22:44, 24 May 2021
- 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
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 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.