The length of every edge of a rectangular cuboid is an integer. By adding the volume of the cuboid, half of its surface area, and the lengths of the edges meeting at one vertex, we get 2000. What are the lengths of the cuboid's edges?
The edges of the rectangular parallelepiped are: $a, b$ and $c$, that is, according to the condition,

$$
a b c + a b + b c + c a + a + b + c = 2000
$$

The expression on the left side is nothing more than $(a+1)(b+1)(c+1)-1$. (This can be easily verified by performing the multiplications. It is worth remembering this relationship, as it can be useful in other contexts.)

After transforming equation (1):

$$
(a+1)(b+1)(c+1) = 2001 = 3 \cdot 23 \cdot 29
$$

Since 3, 23, and 29 are prime numbers and the left side consists of the product of three integers greater than 1, the factors on the left side can be the factors on the right side. (The order depends only on which edge we labeled as $a$, $b$, and $c$.)

Let, for example, $a+1=29, b+1=23, c+1=3$, from which $a=28, b=22, c=2$.

Thus, the edges of the rectangular parallelepiped are 28, 22, and 2 units long.

Koch Dénes (Linz, Akad. Gymn., 11. o.t.)