2003 AMC 12A Problems/Problem 7
Problem
How many non- congruent triangles with perimeter have integer side lengths?
Solution
By the triangle inequality, no one side may have a length greater than half the perimeter which is .
Since all sides must be integers, the largest possible length of a side is
Therefore, all such triangles must have all sides of length , , or .
Since , atleast one side must have a length of
Thus, the remaining two sides have a combined length of .
So, the remaining sides must be either and or and .
Therefore, the number of triangles is .