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| − | {{duplicate|[[2014 AMC 12A Problems|2014 AMC 12A #22]] and [[2014 AMC 10A Problems|2014 AMC 10A #25]]}}
| + | #REDIRECT[[2014 AMC 10A Problems/Problem 25]] |
| − | ==Problem==
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| − | 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
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| − | \textbf{(B) }279\qquad
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| − | \textbf{(C) }280\qquad
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| − | \textbf{(D) }281\qquad
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| − | \textbf{(E) }282\qquad</math>
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| − | ==Solution==
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| − | Between any two powers of 5 there are either 2 or 3 powers of 2 (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 3 powers of 2.
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| − | 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
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| − | <cmath>x+y&=867</cmath><cmath>2x+3y&=2013</cmath>
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| − | from which we get <math>y=279</math>, so the answer is <math>\boxed{\textbf{(B)}}</math>.
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| − | (Solution by superpi83)
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| − | ==See Also==
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| − | {{AMC10 box|year=2014|ab=A|num-b=24|after=Last Problem}}
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| − | {{AMC12 box|year=2014|ab=A|num-b=21|num-a=23}}
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| − | {{MAA Notice}}
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