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| − | == Problem ==
| + | #REDIRECT [[2006 AIME I Problems/Problem 2]] |
| − | 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]]:
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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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| − | Also:
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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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| − | Combining these two inequalities:
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| − | <math> 6.25 < n < 900 </math>
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| − | The number of possible integer values for <math>n</math> is the number of integers over the interval <math>(6.25 , 900)</math> which is <math>893</math>
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| − | == See also ==
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| − | *[[2006 AIME II Problems]]
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