$\underline{\text { Keshukhin } 0 . H .}$

Three cyclists are riding in the same direction on a circular track 300 meters long. Each of them moves at their own constant speed, and all speeds are different. A photographer can take a successful photo of the cyclists if all of them end up on some segment of the track $d$ meters long. For what smallest $d$ will the photographer definitely be able to take a successful photo sooner or later?
Without loss of generality, we assume that all cyclists are riding counterclockwise on the track, with the first one being the fastest and the third one being the slowest. Consider the motion of the cyclists in a reference frame attached to the second cyclist. Then the second cyclist is always at some point $A$, while the first and third cyclists move counterclockwise and clockwise, respectively. They periodically meet each other at equal intervals of time. Denote by $B_{1}, B_{2}, B_{3}, \ldots$ the sequence of their meeting points starting from the beginning of the observation of these athletes (see the left figure). Any two consecutive points $B_{n}$ and $B_{n+1}$ ($n$ - any natural number) are different, because the first cyclist cannot complete a full circle counterclockwise without meeting the third cyclist. Denote by $\beta_{n}$ the shorter of the two arcs of the track with endpoints $B_{n}, B_{n+1}$. Its length does not exceed 150 m. Since the meetings occur periodically with a shift in one direction, all arcs $\beta_{n}$ are equal to each other and the union of several of them covers the entire track. Therefore, there exists an arc $\beta_{m}$ that contains point $A$. In this case, the length of one of the arcs $B_{m} A$ or $A B_{m+1}$ does not exceed 75 m. Thus, at some moment of the meeting of the first and third cyclists, they will be no more than 75 m away from the second cyclist. Therefore, the photographer will definitely be able to take a successful photo if $d \geq$ 75.

![](https://cdn.mathpix.com/cropped/2024_05_06_411c51e953f1ebed484eg-42.jpg?height=434&width=1118&top_left_y=0&top_left_x=476)

Let's provide an example of the cyclists' motion in which they cannot be on any segment of the track shorter than $75 \mathrm{m}$ at any moment. Suppose that relative to point $A$, the first and third cyclists move with equal magnitudes but opposite directions of velocity, and at the initial moment, they are at point $B$, which is a quarter of the circle away from $A$ (see the right figure). Then they will meet alternately at diametrically opposite points $B$ and $C$ (which are 75 m away from $A$), and their positions at any moment will be symmetric with respect to the line $B C$. Therefore, at any moment, the distance from one of them to point $A$ will be no less than 75 m.

## Answer

75.