2005 AMC 10A Problems/Problem 13

Problem

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

$\left(130n\right)^{50} > n^{100} > 2^{200} \ \text{?}$

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

Solution

Since $n > 0$ , all $3$ terms of the inequality are positive, so we may take the $50$ th root, yielding

$\begin{align*}&\sqrt[50]{(130n)^{50}} > \sqrt[50]{n^{100}} > \sqrt[50]{2^{200}} \\ \iff &130n > n^2 > 2^4 \\ \iff &130n > n^2 > 16.\end{align*}$

Solving each part separately, while noting that $n > 0$ , therefore gives $n^2 > 16 \iff n > 4$ and $130n > n^2 \iff n < 130$ .

Hence the solution is $4 < n < 130$ , and therefore the answer is the number of positive integers in the open interval $(4,130)$ , which is $129-5+1 = \boxed{\textbf{(E) } 125}$ .

2005 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 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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2005 AMC 10A Problems/Problem 13

Problem

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