How great is the distance between Budapest and Rozsnyo, if the latitude of Budapest is $\varphi=47^{\circ} 29^{\prime} 15^{\prime \prime}$, its longitude is $\lambda=36^{\circ} 42^{\prime} 17^{\prime \prime}$; the latitude of Rozsnyo is $\varphi^{\prime}=40^{\circ} 39^{\prime} 2^{\prime \prime}$, its longitude is $\lambda^{\prime}=38^{\circ} 12^{\prime} 28^{\prime \prime}$; by how much longer is the longest day in Rozsnyo compared to the longest day in Budapest, if the declination $\delta=23^{\circ} 27^{\prime} 55^{\prime \prime} ?$
Let's denote the center of the Earth as $C$, the North Pole as $P$, Budapest as $B$, and Rozsnyo as $R$. Draw meridians $CPB$ and $CPR$ through $B$ and $R$, and a great circle through $CBR$. Then the spherical distance $BR$ in the spherical triangle $BPR$ can be determined from the angles $BP = 90 - \varphi$, $RP = 90 - \varphi'$, and $BPR = \lambda' - \lambda$ using the second main theorem of spherical trigonometry. Thus, we have:

$$
\cos BR = \sin \varphi \sin \varphi' + \cos \varphi \cos \varphi' \cos (\lambda - \lambda')
$$

Furthermore,

$$
\cos BR = \sin \varphi \left( \sin \varphi' + \cos \varphi' \cot \varphi \cos (\lambda - \lambda') \right)
$$

Let's assume that

$$
\tan \omega = \cot \varphi \cos (\lambda - \lambda')
$$

Then,

$$
\begin{gathered}
\cos BR = \frac{\sin \varphi \left( \cos \omega \sin \varphi' + \cos \varphi' \sin \omega \right)}{\cos \omega} \\
\cos BR = \frac{\sin \varphi \sin (\omega + \varphi')}{\cos \omega}
\end{gathered}
$$

$BR$ is completely determined by equations 1) and 2). We get:

$$
\begin{gathered}
\log \tan \omega = \log \cot \varphi + \log \cos (\lambda - \lambda') = \\
= 9.96224 + 9.99985 = \\
= 9.96209 \\
\omega = 42^\circ 30' 10'' \\
\omega + \varphi' = 83^\circ 9' 12''
\end{gathered}
$$

$$
\log \cos BR = \log \sin \varphi + \log \sin (\omega + \varphi') - \log \cos \omega = \\
= 9.86754 + 9.99689 - 9.86761 = 9.99682
$$

$$
BR = 6^\circ 55' 30'' = \\
= 103 geographic miles.
$$

II.

Using the formula $\cos \omega = \tan \varphi \tan \delta$, we can determine the longest day for both locations.

$$
\begin{aligned}
1) \cos (180 - \omega') & = \tan \varphi' \tan \delta \\
2) \cos (180 - \omega_2) & = \tan \varphi_2 \tan \delta
\end{aligned}
$$

$$
\begin{gathered}
\log \cos (180 - \omega') = \log \tan \varphi' + \log \tan \delta \\
= 0.03775 + 9.63758 \\
= 9.67533 \\
180 - \omega' = 61^\circ 44' 15'' \\
\omega' = 118^\circ 15' 45''
\end{gathered}
$$

or expressed in time,

$$
\omega' = 7^\circ 53' 3''
$$

Thus, the duration of the longest day in Budapest is $15^\circ 46' 6''$;

$$
\begin{gathered}
\log \cos (180 - \omega_2) = \log \tan \varphi_2 + \log \tan \delta = \\
= 9.93381 + 9.63758 = \\
= 9.57139 \\
(180 - \omega_2) = 68^\circ 6' 58'' \\
(\omega_2) = 111^\circ 53' 02''
\end{gathered}
$$

expressed in time,

$$
7^\circ 27' 32''
$$

Thus, the duration of the longest day in Rozsnyo is $14^\circ 55' 4''$, and the longest day in Budapest is $51' 2''$ longer than in Rozsnyo.

(Gergely Jorga, student in the 8th grade of the main real school, Arad).