How many (non-degenerate) triangles with integer side lengths and a perimeter of 100 exist?

#
We need to find the number of triples $(a, b, c)$ of natural numbers where $a \leq b \leq c, a+b+c=100, a+b>c$.

It is clear that $c$ can take values from 34 to 49. For each of these values, $a+b=100-c$, so $b$ can take values from

$\frac{1}{2}(100-c)$ to $c$, more precisely, from $50-\frac{c}{2}$ to $c$ for even $c$ (a total of $\frac{3c}{2}-49$ options) and from $50-\frac{c-1}{2}$ to $c$ for odd $c$ (a total of $\frac{3c-1}{2}-49$ options).

Thus, for $c=34,36, \ldots, 48$ we get 2, 5, ..., 23 triangles, and for $c=35,37, \ldots, 49$ we get 3, 6, ..., 24 triangles. In total, $8 \cdot 26=208$ triangles.