A morning passenger train departing from Budapest to Érsekújvár travels at a speed of $v_{1}$; this train is to be overtaken by a later departing express train, which travels at a speed of $v_{2}$. After the passenger train has already covered $\frac{2}{3}$ of its journey, it continues at half its original speed, as a result of which the express train overtakes it $d$ kilometers before the Érsekújvár station. What is the distance between the two stations? $\left(d=27 \frac{1}{9} \mathrm{~km}\right.$, the passenger train covers $\frac{1}{2} \mathrm{~km}$ per minute, and the express train covers $1 \mathrm{~km}$ per minute.)
If the distance between the two stations is $x$ and the passenger train has already traveled $y$ when the express train starts, then according to the problem,

$$
\frac{x-y}{v_{1}}=\frac{x}{v_{2}}
$$

and

$$
\frac{\frac{2}{3} x-y}{v_{1}}+\frac{\frac{x}{3}-d}{\frac{v_{1}}{2}}=\frac{x-d}{v_{2}}
$$

from which

$$
x=\frac{3 d}{v_{2}}\left(2 v_{2}-v_{1}\right)=122 \mathrm{~km}
$$

(Filkorn Jenő, Nyitra.)

Number of solutions: 34.