2005 AMC 10A Problem 13

Below is the professionally curated solution for Problem 13 of the 2005 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 10A solutions, or check the answer key.

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Concepts:inequality exponent counting integers in a range

Difficulty rating: 1540

13.

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

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

$0$

$7$

$12$

$65$

$125$

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 $n \lt 130.$ So $n$ ranges over the integers $5, 6, \ldots, 129,$ which is $125$ values.

Thus, the correct answer is E.

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