2005 AMC 10A Problem 13

Question

How many positive integers

n

satisfy the following condition:

(130n)^{50} \gt n^{100} \gt 2^{200}?

0

7

12

65

125

All quantities are positive, so take the

50

th root of each part of the inequality

The conditions become

130n \gt n^2

and

n^2 \gt 16,

which together give

4 \lt n \lt 130

Taking

50

th roots, the condition becomes

130n \gt n^2 \gt 2^4 = 16.

From

n^2 \gt 16

we get

n \gt 4,

and from

130n \gt n^2

we get

n \lt 130.

So

n

ranges over the integers

5, 6, \ldots, 129,

which is

125

values.

Thus, the correct answer is E.

Problem 13 in other years