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This problem trains for: AMC-10, AMC-12, AIME.

How many non-degenerate triangles have integer length sides, an area of 108 square units and a circle of radius 12 can be inscribed in them?

The area of a triangle is related to the radius of the inscribed circle and to the perimeter of the triangle:

equation

The conditions set by the problem are equivalent to finding all triangles with integer length sides and perimeter equal to:

equation

Therefore, if the lengths of the sides are denoted by a, b and c, then:

equation

Since the minimum length of a side must be 1 (the triangle is non-degenerate) we make the change of variable:

equation

equation

equation

in order to have all integer values start at 0.

Now, the Diophantine equation looks like:

equation

and, by the "stars and bars" method, its number of distinct solutions is:

equation