We have a cuboid, one of whose edges is five times longer than another of its edges. If we increase the height of the cuboid by $2 \mathrm{~cm}$, its volume would increase by $90 \mathrm{~cm}^{3}$. If we further change the height of the thus increased cuboid to half, the volume of the new cuboid would be equal to three fifths of the volume of the original cuboid.

What could be the dimensions of the original cuboid? Determine all possibilities.

(E. Semerádová)
The volume of a cuboid is equal to the product of the area of the base and the height. A change in height by $2 \mathrm{~cm}$ causes a change in volume by $90 \mathrm{~cm}^{3}$; the corresponding base therefore has an area of $90: 2=45\left(\mathrm{~cm}^{2}\right)$.

When the height changes further, the base does not change, so a change in the volume of the cuboid to three fifths corresponds to a change in height also to three fifths. If we denote the height of the original cuboid in centimeters by $v$, we have

$$
\frac{v+2}{2}=\frac{3}{5} v
$$

From this, it is easy to express $v=10 \mathrm{~cm}$.

The last unused information from the problem is that one edge of the cuboid is five times longer than another of its edges. There can be three possibilities:

a) One edge of the base is five times longer than the height, i.e., $50 \mathrm{~cm}$. From this, the other edge of the base measures $45: 50=0.9(\mathrm{~cm})$.

b) One edge of the base is five times shorter than the height, i.e., $2 \mathrm{~cm}$. From this, the other edge of the base measures $45: 2=22.5(\mathrm{~cm})$.

c) One edge of the base is five times longer than the other edge of the base. If we denote the length of the shorter of these edges in centimeters by $a$, we have $a(5 a)=45$, so $a^{2}=9$ and $a=3(\mathrm{~cm})$. From this, the longer edge of the base measures $5 \cdot 3=15(\mathrm{~cm})$.

The dimensions of the original cuboid in centimeters can be $0.9 \times 50 \times 10$, or $2 \times 22.5 \times 10$, or $3 \times 15 \times 10$.

Scoring. 1 point for the area of the base; 2 points for the height of the original cuboid; 1 point for each correct solution.