Difference between revisions of "2005 AMC 10A Problems/Problem 13"

Revision as of 19:32, 1 August 2006

How many positive integers $n$ satisfy the following condition:

$(130n)^{50} > n^{100} > 2^{200}$ ?

$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 7\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 65\qquad \mathrm{(E) \ } 125$

$(130n)^{50} > n^{100} > 2^{200}$

$\sqrt[50]{(130n)^{50}} > \sqrt[50]{n^{100}} > \sqrt[50]{2^{200}}$

$130n > n^2 > 2^4$

$130n < n^2 < 16$

Solving each part seperatly:

$n^2 > 16$

$n < 4$

$130n > n^2$

$130 < n$

So $4 < n < 130$ .

Therefore the answer is the number of positive integers over the interval $(4,130)$ which is $125 \Rightarrow E$ .

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 *[[2005 AMC 10A Problems]]
-*[[2005 AMC 10A Problems/Problem |Previous Problem]]
+*[[2005 AMC 10A Problems/Problem 12|Previous Problem]]
-*[[2005 AMC 10A Problems/Problem |Next Problem]]
+*[[2005 AMC 10A Problems/Problem 14|Next Problem]]