For a roof of a given width, at what angle of inclination will the rainwater flow off in the shortest possible time?
If the width of the roof, i.e., the base of the slope, is $a$, the length of the slope is $h$, and the angle of inclination is $\alpha$, then the time it takes for the rainwater to flow down is:

$$
t=\sqrt{\frac{2 h}{g \sin \alpha}}
$$

but since

$$
h=\frac{a}{\cos \alpha}
$$

therefore

$$
t=\sqrt{\frac{2 a}{g \sin \alpha \cos \alpha}}=\sqrt{\frac{4 a}{g \sin 2 \alpha}}
$$

Thus, $t$ is the smallest when $\sin 2 \alpha$ is the largest, i.e., when

$$
2 \alpha=90^{\circ}
$$

or

$$
\alpha=45^{\circ}
$$

(Jeno Filkorn, Nyitra.)