2005 AMC 10A Problem 13

How many positive integers

n

satisfy the following condition:

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

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Solution

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.

n

ranges over the integers

5, 6, \ldots, 129,

which is

125

values. Thus, the correct answer is E.

Concepts: inequality · exponent · counting integers in a range

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.