$B$ and $C$ cities are $20 \mathrm{~km}$ apart along the $D$ river. Trains run on a railway along the bank at a speed of $30 \mathrm{~km} /$ hour. Trains depart from $B$ to $C$ at 8:20 and 10:20, and from $C$ to $B$ at 10:00, arriving at 11:50. There is also an afternoon ferry service from $B$ to $C$ and back, and the boat is "at the same level" as the train in the first and third quarters of the $BC$ segment on both trips. - In which direction does $D$ flow, and what is its flow speed, if the boat's engine works the same on both trips? Also provide the departure and arrival times of the boat in $B$ and $C$.
The train's travel time is 40 minutes, so it arrives in $C$ at 9 and 11 o'clock, and departs from $C$ at 9:20 and 11:10 (the traffic can be handled by a single train). It is not possible for the boat to meet the train on the same route, because then they would travel the entire route together, the boat's speed would be the same in both directions, and $D$ would be still water.

![](https://cdn.mathpix.com/cropped/2024_05_02_66690cd5c2f4ee8248c8g-1.jpg?height=447&width=590&top_left_y=308&top_left_x=754)

According to this, during the train's first trip, it overtakes the boat at $E$ (first quarter) at 8:30, on the second trip they meet head-on at $H$ (third quarter) at 9:30, similarly on the third trip at 10:50, and finally on the fourth trip, it overtakes the boat again at $E$ at 11:40. Thus, the boat travels the 10 km $E H$ route in 60 minutes, and the $H E$ route in 50 minutes, with speeds of 10 km/h and 12 km/h, respectively. According to this, the river $D$ flows from $C$ towards $B$, with the boat's speed denoted by $V$ and the river's speed by $v$. From the system $V+v=12, V-v=10$, we get $V=11 \text{ km/h}$ and $v=1 \text{ km/h}$. The boat is on the route from $B$ to $C$ from 8 to 10 o'clock, and on the return trip from 10:25 to 12:05.

Maria Molnár (Sztálinváros, Kerpely A. kohóip. t. I. o. t.)

Note. It is noteworthy that only 14 solutions contain a graphical schedule similar to the one presented; many solutions tried to assist their thought process with other, more complex diagrams.