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

## Problem

How many positive integers $n$ satisfy the following condition: $(130n)^{50} > n^{100} > 2^{200}$? $\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 7\qquad \mathrm{(C) \ } 12\qquad \mathrm{(D) \ } 65\qquad \mathrm{(E) \ } 125$

## Solution

We're given $(130n)^{50} > n^{100} > 2^{200}$, so $\sqrt{(130n)^{50}} > \sqrt{n^{100}} > \sqrt{2^{200}}$ (because all terms are positive) and thus $130n > n^2 > 2^4$ $130n > n^2 > 16$

Solving each part separately: $n^2 > 16 \Longrightarrow n > 4$ $130n > n^2 \Longrightarrow 130 > n$

So $4 < n < 130$.

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

## See also

 2005 AMC 10A (Problems • Answer Key • Resources) Preceded byProblem 12 Followed byProblem 14 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

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