Integer Triangles of Perimeter Thirty

Worked solution

The answer is 19. Write the sides as $a \le b \le c$ with $a + b + c = 30$. The triangle inequality reduces to $a + b > c$, i.e. $c < 15$, and $c \ge 10$ since $c$ is the largest. For each $c$, count pairs $a \le b \le c$ with $a + b = 30 - c$: $b$ ranges from $\lceil (30-c)/2 \rceil$ to $c$. This gives $c = 10$: 1 (10,10,10); $c = 11$: 2; $c = 12$: 4; $c = 13$: 5; $c = 14$: 7. Total $1 + 2 + 4 + 5 + 7 = 19$.