## Task $2 / 74$

All rectangles with the following property are to be found: The side lengths can be expressed by whole numbers, whose sum is numerically equal to the area of the rectangle.
If the side lengths of the rectangle are denoted by $x$ and $y$, the condition of the problem leads to the Diophantine equation $2x + 2y = xy$. From this, it follows that

$$
x = \frac{2y}{y-2} = 2 + \frac{4}{y-2}
$$

For $x$ to be an integer, $4 = k(y-2)$ must hold, where $k$ and $y-2$ are integers; furthermore, $y > 0$ must be true. Since $x > 0$, $y = 1$ is also excluded, and because $y - 2 > 0$, $y = 2$ is also excluded. Therefore, the remaining possibilities are:

1. $y = 3$ with $x = 6$
2. $y = 4$ with $x = 4$
3. $y = 6$ with $x = 3$

For $y > 6$, $x$ will not be an integer. Therefore, there are exactly two different rectangles with the required property:

1.) Side lengths 3 and 6 and 2.) Side lengths 4 and 4 (square)