The number of incongruent triangles with integer side lengths and perimeter 10100 is

208 3 . . .3︸ ︷︷ ︸ 196


How do we arrive at such a number? Let’s first solve some simpler problems. How many incongruent triangles have integer side lengths and perimeter 10? There are only two: (2, 4, 4) and (3, 3, 4). (We specify a triangle by giving the ordered triple of its side lengths in nondecreasing order. A triple (a, b, c) must satisfy the triangle inequality a+ b > c.)

Let t(n) be the number of integer triangles of perimeter n. Let’s generate some data. It is convenient to set t(0) = 0.