Task 2. The lengths of the sides of an isosceles triangle are expressed in natural numbers in centimeters. How many different isosceles triangles can be constructed if the perimeter of the triangle is $22 \mathrm{~cm}$.
Solution. The sum of the lengths of the two sides of a triangle must be greater than the length of the third side. Furthermore,

$$
a+2 b=22
$$

since $2 b$ and 22 are even numbers, $a$ must also be an even number. From the inequality $2 b>a$ and from (1), the following table is derived

![](https://cdn.mathpix.com/cropped/2024_06_05_b353751bdd87cc544500g-05.jpg?height=300&width=200&top_left_y=965&top_left_x=1265)

| $a$ | 2 | 4 | 6 | 8 | 10 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| $b$ | 10 | 9 | 8 | 7 | 6 |

According to this, there are five different isosceles triangles that satisfy the condition of the problem.