Problem 2. Dodson, Williams, and their horse Bolivar want to get from city A to city B as quickly as possible. Along the road, there are 27 telegraph poles that divide the entire journey into 28 equal segments. Dodson takes 9 minutes to walk one segment, Williams takes 11 minutes, and either of them can ride Bolivar to cover the distance in 3 minutes (Bolivar cannot carry both). They set out from city A simultaneously; the journey is considered complete when all of them arrive in city B.

They agreed that part of the way Dodson will ride, then tie Bolivar to one of the telegraph poles and continue on foot, while Williams will initially walk and then ride Bolivar. At which pole should Dodson tie Bolivar to minimize the travel time to city B?

Answer: at the 12th, counting from A.
Solution. Let the distance from A to B be taken as a unit, and time will be measured in minutes. Then Dodson's speed is $1 / 9$, Williams' speed is $1 / 11$, and Bolivar's speed is $-1 / 3$.

Let the desired post have the number $k$ (i.e., the distance from city A is $k / 28$). Then Dodson will arrive in time

$$
\frac{k}{28}: \frac{1}{3}+\frac{28-k}{28}: \frac{1}{9}=9-\frac{6 k}{28}
$$

while Williams will take

$$
\frac{k}{28}: \frac{1}{11}+\frac{28-k}{28}: \frac{1}{3}=3+\frac{8 k}{28}
$$

We need to find $k$ for which these values coincide:

$$
9-\frac{6 k}{28}=3+\frac{8 k}{28} \quad \Leftrightarrow \quad 6 \cdot 28=14 k \quad \Leftrightarrow \quad k=12
$$

It remains to understand why $k=12$ gives the best time. Indeed, for smaller $k$, Dodson's time will increase, and for larger $k$, Williams' time will increase. Thus, for all other $k$, the time of at least one of the characters will be greater than when $k=12$.

## Criteria

The highest suitable criterion is used:

7 p. Any complete solution to the problem.

4 p. The position of the post is found where the time of the characters coincides, but it is not proven that this is the best time.

2 p. The idea of equating the time of the characters is present, but an error was made in the formulation or solution of the equation.

1 p. The correct answer is given.