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All the sides of a triangle have integer length.  The perimeter of the triangle is $60$, and the triangle is isosceles.  How many such non-congruent triangles are there?  (A 3-4-5 triangleis is considered congruent to a 3-5-4 triangle because we can reflect and rotate the triangles until they match up.)

 Dec 21, 2021
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Let the sides of the triangle be x and y, where x is the repeated side length. It follows that 2x+y = 60. Also, by the triangle inequality, 2x>y and x+y>x (which is trivial, since this simplifies to y>0.) Defining pairs of (2x,y) over 2x>y and \(x, y \in \mathbb{Z}\), we have (32,28), (34,26), ... (58,2) which totals to 14 pairs.

 Dec 21, 2021

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