2014 AMC 12A Problems/Problem 22
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- 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)
Solution
Between any two powers of 5 there are either 2 or 3 powers of 2 (because 
). Consider the intervals 
. We want the number of intervals with 3 powers of 2.
From the given that 
, we know that these 867 intervals together have 2013 powers of 2. Let 
of them have 2 powers of 2 and 
of them have 3 powers of 2. Thus we have the system
\[x+y&=867\] (Error compiling LaTeX. Unknown error_msg)
\[2x+3y&=2013\] (Error compiling LaTeX. Unknown error_msg)
from which we get 
, so the answer is 
.
(Solution by superpi83)
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. 
