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

Revision as of 19:11, 25 September 2007

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 $893$ .

@@ Line 29: / Line 29: @@
 == See also ==
-{{AIME box|year=2006|n=I|num-b=1|num-a=3}}
+{{AIME box|year=2006|n=II|num-b=1|num-a=3}}
 [[Category:Intermediate Geometry Problems]]
 [[Category:Intermediate Algebra Problems]]

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

Revision as of 19:11, 25 September 2007

Problem

Solution

See also