Problem 4. On each of the two branches of the river, a kilometer before their confluence, there is a pier, and another pier is located two kilometers downstream from the confluence (see figure). A boat traveled from one of the piers to another (unknown which) in 30 minutes, and from another to the third in 18 minutes. How many minutes can it take to travel from the third pier to the first? (The speed of the river current is constant and the same in all parts of the river. The boat's own speed is also constant.)

[6 points] (V. M. Gurvich)

![](https://cdn.mathpix.com/cropped/2024_05_06_13e5d2ec23c25d878726g-2.jpg?height=169&width=487&top_left_y=1183&top_left_x=479)
Answer: Either 24 or 72 minutes.

Solution. Let's denote the piers in the order of visit as $A, B$, and $C$. Let's think about which of them could have been after the confluence.

- Pier $B$ could not have been after the confluence. Indeed, otherwise the boat would have traveled 3 km downstream to it, and the same 3 km upstream from it. But according to the condition, the time of the first part of the journey is greater than the time of the second.
- Suppose that after the confluence stands pier $A$. Then 1 km upstream the boat travels in $30 / 3=10$ minutes, and 1 km downstream in $18-10=8$ minutes. Accordingly, the journey from $C$ to $A$ (3 km downstream) takes $3 \cdot 8=24$ minutes.
- Finally, suppose that after the confluence stands pier $C$. In this case, 1 km downstream the boat travels in $18 / 3=6$ minutes, and 1 km upstream in $30-6=24$ minutes. Accordingly, the journey from $C$ to $A$ (3 km upstream) takes $3 \cdot 24=72$ minutes.