4. $n$ trains are circulating in one direction at equal distances around a circle. On this road, stations $A$, $B$, and $C$ are located at the vertices of an equilateral triangle (labeled in the direction of travel). Ira enters station $A$ and at the same time, Lesha enters station $B$ to catch the nearest trains. It is known that if they enter the stations at the moment when the driver Roma passes through the forest, Ira will board the train before Lesha, and in all other cases, Lesha will board the train before Ira or at the same time as her. What fraction of the road passes through the forest?
4. For clarity, we will consider trains and stations as points.

It is clear that if $n$ is divisible by 3, then Lёsha and Ira always leave at the same time. Therefore, in this case, there is no forest. Let $n$ not be divisible by 3. Then, whenever they arrive at the station, either Ira leaves before Lёsha, or Lёsha leaves before Ira.

Let the distance between neighboring trains be denoted by $l$. If there is a forest at some point $X$, then at point $Y$, which is at a distance that is a multiple of $l$ from $X$, there is also a forest. Indeed, if Ira arrives at the station when Roma is at point $X$, then she will leave first. But when Roma is at point $Y$, the arrangement of the trains is the same as when he is at point $X$, so in this case, Ira will also leave first, which means there is a forest at point $Y$ as well.

Thus, the "structure" of the forest is periodic, so it is sufficient to determine the location of the forest on an interval of length $l$.

Consider the moment when a certain train departs from station $B$ (Fig. 85). Let the train that Ira will board

![](https://cdn.mathpix.com/cropped/2024_05_21_6455f48598646f41a890g-196.jpg?height=357&width=374&top_left_y=1049&top_left_x=128)

Fig. 85 (i.e., the nearest train against the direction of movement to station $A$) be at a distance $x$ from $A$ at this moment. Then the entire interval between this train and point $A$ is covered by the forest. Indeed, if driver Roma is on this interval, he will take Ira because Lёsha "missed" his train (strictly speaking, this follows from the fact that $x < l$).

We will show that the interval of length $l-x$, following $A$ in the direction of movement, is not covered by the forest. Indeed, when the train arrives at station $A$, the nearest train to $B$ against the direction of movement will be at a distance $l-x$. So if Roma is on the specified interval of length $l-x$, then Lёsha will board the train first, because Ira "missed" the train that Roma is driving.

Thus, on a segment of length $l$, the forest is $-x$, and the field is $-l-x$. Since the structure of the forest is periodic, the amount of forest relative to the amount of field is $x$ to $l-x$.

It remains to find $x$. The circumference is $n l$, so the length of the larger arc $B A$ is $\frac{2}{3} n l$. It is clear that the value of $x$ is the remainder of the division of the length of arc $B A$ by $l$. Therefore, if the remainder of the division of $n$ by 3 is 1, then $x = \frac{2 l}{3}$, i.e., the proportion of the forest is $\frac{2}{3}$.

Similarly, if the remainder of the division of $n$ by 3 is 2, then $x = \frac{l}{3}$, and the forest constitutes $\frac{1}{3}$.