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−  == Problem ==
 +  #REDIRECT[[2003 AMC 12A Problems/Problem 7]] 
−  How many noncongruent triangles with perimeter <math>7</math> have integer side lengths?
 
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−  <math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 5 </math>
 
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−  == Solution ==
 
−  By the [[triangle inequality]], no one side may have a length greater than half the perimeter, which is <math>\frac{1}{2}\cdot7=3.5</math>
 
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−  Since all sides must be integers, the largest possible length of a side is <math>3</math>
 
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−  Therefore, all such triangles must have all sides of length <math>1</math>, <math>2</math>, or <math>3</math>.
 
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−  Since <math>2+2+2=6<7</math>, at least one side must have a length of <math>3</math>
 
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−  Thus, the remaining two sides have a combined length of <math>73=4</math>.
 
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−  So, the remaining sides must be either <math>3</math> and <math>1</math> or <math>2</math> and <math>2</math>.
 
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−  Therefore, the number of triangles is <math>2 \Rightarrow B</math>.
 
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−  == See Also ==
 
−  {{AMC10 boxyear=2003ab=Anumb=6numa=8}}
 
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−  [[Category:Introductory Geometry Problems]]
 