2005 AMC 10A Problems/Problem 13

Problem

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 $125 \Rightarrow E$ .