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

 
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#REDIRECT [[2006 AIME A Problems/Problem 2]]
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== Problem ==
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The lengths of the sides of a [[triangle]] with positive area are <math>\log_{10} 12</math>, <math>\log_{10} 75</math>, and <math>\log_{10} n</math>, where <math>n</math> is a positive integer. Find the number of possible values for <math>n</math>.
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== Solution ==
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By the [[Triangle Inequality]] and applying the well-known logarithmic property <math>\log_{c} a + \log_{c} b = \log_{c} ab</math>, we have that
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<div style="text-align:center;">
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<math>\log_{10} 12 + \log_{10} n > \log_{10} 75 </math>
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<math>\log_{10} 12n > \log_{10} 75 </math>
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<math> 12n > 75 </math>
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<math> n > \frac{75}{12} = \frac{25}{4} = 6.25 </math>
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</div>
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Also,
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<div style="text-align:center;">
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<math>\log_{10} 12 + \log_{10} 75 > \log_{10} n </math>
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<math>\log_{10} 12\cdot75 > \log_{10} n </math>
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<math> n < 900 </math>
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</div>
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Combining these two inequalities:
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<cmath> 6.25 < n < 900 </cmath>
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Thus <math>n</math> is in the set <math>(6.25 , 900)</math>; the number of positive integer <math>n</math> which satisfies this requirement is <math>\boxed{893}</math>.
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== See also ==
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{{AIME box|year=2006|n=II|num-b=1|num-a=3}}
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[[Category:Intermediate Geometry Problems]]
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[[Category:Intermediate Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 14:58, 14 April 2020

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 and applying the well-known logarithmic property $\log_{c} a + \log_{c} b = \log_{c} ab$, we have that

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

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

$12n > 75$

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

Also,

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

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

$n < 900$

Combining these two inequalities:

\[6.25 < n < 900\]

Thus $n$ is in the set $(6.25 , 900)$; the number of positive integer $n$ which satisfies this requirement is $\boxed{893}$.

See also

2006 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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