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how many different (non-conguerent) triangles have integer side lengths and perimeter 2014
Take all the possible triangles. That is, all triplets of numbers a,b,c that have a total sum of 2014 and obey the conditions: a<b+c;b<c+a;c<a+b. All triangles have been taken 3 times, so divide the count by 3.
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