What are the sides of a rectangle, from which, by cutting off $3 \mathrm{~cm}$ from its length and $2 \mathrm{~cm}$ from its width, we get a square whose area is half of the rectangle's area?
If the side of the square is $x$, then according to the problem,

$$
(x+2)(x+3)=2 x^{2},
$$

or

$$
x^{2}-5 x-6=(x+1)(x-6)=0
$$

from which the positive value of $x$ is: $x=6$, so the length of the rectangle is $9 \mathrm{~cm}$, and the width is $8 \mathrm{~cm}$.

(Klein Adolf, Székesfehérvár.)

Number of solutions: 32.