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Difference between revisions of "2006 AIME A Problems/Problem 2"

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== See also ==
 
== See also ==
 
*[[2006 AIME II Problems]]
 
*[[2006 AIME II Problems]]
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[[Category:Intermediate Geometry Problems]]
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[[Category:Intermediate Algebra Problems]]

Revision as of 09:34, 16 July 2006

Problem

The lengths of the sides of a triangle with positive area are $\log_{10} 12$, $\log_{10} 75$, and $\log_{10} n$, where $n$ is a positive integer. Find the number of possible values for $n$.

Solution

By the Triangle Inequality:

$\log_{10} 12 + \log_{10} n > \log_{10} 75$

$\log_{10} 12n > \log_{10} 75$

$12n > 75$

$n > \frac{75}{12} = \frac{25}{4} = 6.25$

Also:

$\log_{10} 12 + \log_{10} 75 > \log_{10} n$

$\log_{10} 12\cdot75 > \log_{10} n$

$n < 900$

Combining these two inequalities:

$6.25 < n < 900$

The number of possible integer values for $n$ is the number of integers over the interval $(6.25 , 900)$ which is $893$

See also

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