Two cities are connected by a highway. In the cities, there are cars that can refuel with just enough gasoline to cover half the distance between the two cities. Using helper cars, we want to get a car $A$ from one city to the other. The helper cars can transfer gasoline to each other and to $A$ at certain points along the route, but only enough so that they can return to their starting station. What is the minimum number of helper cars needed?
The task description does not exclude the possibility that the auxiliary cars (a finite number of times) return to their station, refuel, and return to $A$ or to the other auxiliary cars on the way. We will use this possibility in the solution. Let's choose as a unit of distance the distance that a car with a full tank can travel without further assistance, according to this, the distance between the two cities is 2 units. Let's denote the cities by $B$ and $C$, and for a given plan, let $B^{*}$ and $C^{*}$ denote the farthest point from $B$ and $C$, respectively, on the $BC$ segment, to which the auxiliary cars from $B$ and $C$ can reach according to the plan. It is easy to see that it is sufficient to examine only those plans according to which the tank of $A$ is full in $B^{*}$ and empty in $C^{*}$.

If only one auxiliary car is used in $B$, then $B B^{*} < 1 / 2$. Indeed, in the last round, the auxiliary car must reach $B^{*}$, give all its fuel to $A$, and still be able to return to $B$. This is only possible if twice the length of the $B B^{*}$ segment is less than 1, i.e., $B B^{*} < 1 / 2$. Similarly, the length of the $C^{*} C$ segment is less than 1/2 if only one auxiliary car is used in $C$, since the auxiliary car from $C$ must reach $C^{*}$, give all its fuel to $A$, and then be able to return to $C$. According to this, two auxiliary cars are not enough if one is used in $B$ and the other in $C$. (To prove this, it is only necessary to note, in addition to the above, that $B^{*} C^{*} = 1$.)

Two auxiliary cars are also not enough if both have $B$ as their station. Let $B_{1}$ be the point from which the auxiliary car that fills $A$ in $B^{*}$ starts with a full tank, and $B_{2}$ be the point where $A$'s tank is empty after being filled and heading back to $B$. According to the above, $B B_{1}$ and $B B_{2}$ are less than 1/2, and thus $B B^{*} < 1$, so two auxiliary cars from $B$ are indeed not enough. Similarly, it can be shown that two auxiliary cars from $C$ are not enough either.

![](https://cdn.mathpix.com/cropped/2024_05_02_9cfe242b1b3de1bad91eg-1.jpg?height=800&width=1244&top_left_y=939&top_left_x=430)

We will show that three auxiliary cars are sufficient to get $A$ from $B$ to $C$. Let's start $A$ from $B$ with two auxiliary cars, and let them travel together to $B_{0}$, where $B B_{0} = 1 / 3$. At this point, the $S_{1}$ auxiliary car should fill $A$'s tank and return to $B$, and in the second round, it should fill the tank of the $S_{2}$ auxiliary car at $B_{0}$. Let $A$ continue from $B_{0}$ with the $S_{2}$ auxiliary car, and let them travel another $1 / 3$ unit of distance together. Then $S_{2}$ should fill $A$ and return to $B_{0}$. In the third round, the $S_{1}$ auxiliary car should come from $B$ to meet them, so they can return home together. Thus, according to our plan, $B B^{*} = 2 / 3$. From the $B^{*}$ point, $A$ can reach the $C^{*}$ point, where $C^{*} C = 1 / 3$. Let the third, $S_{3}$ auxiliary car come from $C$ to meet them, and from there, they can travel together to $C$. (The small pie charts on the diagram show the tank contents.)