The height of a square-based right pyramid is $m$ and the area of its base is $A$. Place a cube on the base of this pyramid so that its four corners are in the pyramid's side edges. What are the lengths of the cube's edges?
According to the pyramid, the areas of the base and a section parallel to the base are in the ratio of the squares of the corresponding heights, therefore, if $x$ is one edge of the smaller pyramid,

$$
A: x^{2}=m^{2}:(m-x)^{2}
$$

from which

$$
x=\frac{m \sqrt{A}}{m+\sqrt{A}}
$$

(Henrik Friedländer, Beregszász.)

Number of solutions: 30.