The area of a rhombus is $117 \mathrm{~cm}^{2}$. The perimeter of the rectangle determined by the midpoints of the sides of the rhombus is $31 \mathrm{~cm}$. What are the lengths of the diagonals of the rhombus?
If $x$ and $y$ are the diagonals of the rhombus, then the sides of the rectangle are $\frac{x}{2}$ and $\frac{y}{2}$, so

$$
\frac{x \cdot y}{2}=117
$$

and

$$
x+y=31
$$

From these two equations, we derive that

$$
x=18 \text{ cm} \quad \text{or} \quad 13 \text{ cm}
$$

and

$$
y=13 \text{ cm} \quad \text{or} \quad 18 \text{ cm}
$$

(Frigyes Frankel, Budapest.)

The problem was also solved by: Baján A., Basch R., Bauer El., Breuer P., Czigler I., Deschán Z., Dénes M., Engler J., Erdős V., Fried E., Grünwald Gy., Koffler B., Lengyel K., Schnabel L., Sichermann F., Spitzer L., Szántó L., Szilárd V., Szóbel I., Vilcsek A., Viola R., Weinberger Gy.