Difference between revisions of "2005 AMC 10A Problems/Problem 13"
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==Solution== | ==Solution== | ||
− | <math> (130n)^{50} > n^{100} > 2^{200} </math> | + | We're given <math> (130n)^{50} > n^{100} > 2^{200} </math>, so |
− | <math> \sqrt[50]{(130n)^{50}} > \sqrt[50]{n^{100}} > \sqrt[50]{2^{200}} </math> | + | <math> \sqrt[50]{(130n)^{50}} > \sqrt[50]{n^{100}} > \sqrt[50]{2^{200}} </math> (because all terms are positive) and thus |
<math> 130n > n^2 > 2^4 </math> | <math> 130n > n^2 > 2^4 </math> | ||
− | <math> 130n | + | <math> 130n > n^2 > 16 </math> |
Solving each part seperatly: | Solving each part seperatly: | ||
− | <math> n^2 > 16 </math> | + | <math> n^2 > 16 \Longrightarrow n > 4 </math> |
− | + | <math> 130n > n^2 \Longrightarrow 130 > n </math> | |
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− | <math> 130n > n^2 | ||
− | |||
− | |||
So <math> 4 < n < 130 </math>. | So <math> 4 < n < 130 </math>. | ||
− | Therefore the answer is the number of positive | + | Therefore the answer is the number of [[positive integer]]s over the interval <math> (4,130) </math> which is <math> 125 \Longrightarrow \mathrm{(E)} </math>. |
==See Also== | ==See Also== | ||
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*[[2005 AMC 10A Problems/Problem 14|Next Problem]] | *[[2005 AMC 10A Problems/Problem 14|Next Problem]] | ||
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+ | [[Category:Introductory Algebra Problems]] |
Revision as of 10:59, 2 August 2006
Problem
How many positive integers satisfy the following condition:
?
Solution
We're given , so
(because all terms are positive) and thus
Solving each part seperatly:
So .
Therefore the answer is the number of positive integers over the interval which is
.