The Vaskapu rising above the city of Esztergom is seen from point $A$ near the railway station at an angle of $6^{\circ} 50^{\prime} 33^{\prime \prime}$, and from 200 meters closer at an angle of $7^{\circ} 25^{\prime} 52^{\prime \prime}$; from the same point $A$, the Sashegy is seen at an angle of $5^{\circ} 24^{\prime} 52^{\prime \prime}$, and from 200 meters closer at an angle of $5^{\circ} 55^{\prime} 36^{\prime \prime}$; the angle formed by the horizontal direction lines starting from point $A$ is $31^{\circ} 4^{\prime} 34^{\prime \prime}$. How far apart are the two mountain peaks from each other?
Let the angles be given in their order as $\alpha, \beta, \gamma, \delta$ and $\varepsilon$. The footpoints of the two mountain peaks are $M$ and $M_{1}$, and their peaks are $O$ and $O_{1}$. It can be written that:

$$
O M = M A \operatorname{tg} \alpha = (M A - 200) \operatorname{tg} \beta
$$

from which

$$
M A = \frac{200 \cos \alpha \sin \beta}{\sin (\beta - \alpha)}
$$

Similarly,

$$
M_{1} A = \frac{200 \cos \gamma \sin \delta}{\sin (\delta - \gamma)}
$$

The detailed calculation gives:

$$
M A = 2502 \text{ m}, M_{1} A = 2302 \text{ m}
$$

From the triangle $A M M_{1}$, we obtain:

$$
M M_{1} = 1301 \text{ m}
$$

From the known parts of the trapezoid $M M_{1} O_{1} O$:

$$
\overline{O O_{1}} = \sqrt{\overline{M M_{1}}^{2} + (O M - O_{1} M_{1})^{2}}
$$

where

$$
O M = M A \operatorname{tg} \alpha, O_{1} M_{1} = M_{1} A \operatorname{tg} \gamma
$$

After substituting the values, we obtain that the distance between the mountain peaks is:

$$
O O_{1} = 1303 \text{ m}
$$

(Odön Kornis.)

The problem was also solved by. D. Barna, S. Bobál, F. Bojedain, D. Brandt, M. Devecis, E. Horovitz, J. Kárf, L. Kertész, Gy. Lukhaub, M. Roth, G. Sasvári, H. Schiffer, B. Schmidt, I. Szabó, B. Szitkey.