Task B-3.4. How many isosceles triangles have side lengths that are integers, and a perimeter of $30 \mathrm{~cm}$?
Solution. Let $a$ be the length of the base, and $b$ the length of the legs of the isosceles triangle. From $a+2 b=30$ it follows that $a$ must be an even number, and from the triangle inequality $a<b+b=2 b$.

(2 points)

Therefore, $a \leq 14$ and is an even number, so there are only these 7 cases:

$$
(a, b) \in\{(2,14),(4,13),(6,12),(8,11),(10,10),(12,9),(14,8)\}
$$

If the student does not have the possibility that $a=b=10 \mathrm{~cm}$, one point should be deducted.