Let's calculate how high above the water level a straight line traveling horizontally from one end of Lake Balaton will pass at the other end of the lake. (The length of Lake Balaton is $76 \mathrm{~km}$; the average radius of the Earth: $6370430 \mathrm{~m}$).
If the center of the Earth is $O$, the length of Lake Balaton is $A C$, the sought height is $B C=x$, and the angle enclosed by the radii $A O$ and $C O$ is $\alpha$, then

$$
2 r \pi: 360^{\circ}=A C: \alpha
$$

from which

$$
\alpha=41^{\prime}
$$

From the right-angled triangle $O A B$:

$$
\cos \alpha=\frac{r}{r+x}
$$

from which

$$
x=r\left(\frac{1}{\cos \alpha-1}\right)=r \frac{2 \sin ^{2} \frac{\alpha}{2}}{\cos \alpha}
$$

Substituting the given values: $x=430 \mathrm{~m}$.

(Alfréd Haar, Budapest.)

The problem was also solved by: Déri Zs., Deutsch I., Erdélyi I., Kelemen M., Kürti I., Popoviciu M., Raab R., Söpkéz Gy.