Neustroeva E.

In the relay race Moscow-Petushki, two teams of $\$ 20 \$$ people each participated. Each team divided the distance into $\$ 20 \$$ not necessarily equal segments and distributed them among the participants so that each ran exactly one segment (the speed of each participant is constant, but the speeds of different participants may vary). The first participants of both teams started simultaneously, and the baton handover happens instantaneously. What is the maximum number of overtakes that could occur in such a race? Overtaking at the segment boundaries is not counted as an overtake.
First, let's prove that no more than $38$ overtakes occurred. Note that between the start and the first overtake, and between two consecutive overtakes, at least one of the teams must have changed the runner. There were $19$ runner changes in each team, which means a total of $38$, and thus, there were no more than $38$ overtakes.

Let's prove that $38$ overtakes were possible. We will illustrate the trajectories of the teams on a graph with the horizontal axis representing time and the vertical axis representing the distance covered. Then, both trajectories are $20$-segmented broken lines with segments directed upwards to the right. The lower left end of each broken line coincides with the origin, and the upper right ends of the broken lines have the same ordinate. In this situation, an overtake is a point of intersection of the broken lines that is not a vertex.

So, let's provide an example. First, draw a broken line consisting of 40 consecutive sides of a regular 160-gon, such that the last side is horizontal. In other words, the first segment of the broken line starts from the point with coordinates $(0,0)$ at an angle of $9/4^\circ$ to the vertical, and each subsequent segment is rotated $9/4^\circ$ clockwise compared to the previous one. Thus, the last segment will be horizontal, i.e., rotated by $40 \cdot 9/4^\circ = 90^\circ$ relative to the vertical axis. Number the vertices sequentially from 1 to 41 and connect the vertices with numbers $1$, $2$, $4$, $2$, $40$, this will be the trajectory of the first team. The trajectory of the second team will be a broken line connecting the vertices with numbers $1$, $3$, $5$, ..., $41$. The segments $(i, i+2)$ and $(i+1, i+3)$ for $i=1,2, \ldots, 38$ will intersect, and their points of intersection will be the 38 overtakes.

Let's illustrate the example for the case of teams consisting of two participants (see the figure, the first team is a solid black broken line, the second - a dashed one).

distance

![](https://cdn.mathpix.com/cropped/2024_05_06_de7bb834a1e75b584fa6g-15.jpg?height=627&width=1020&top_left_y=1207&top_left_x=42)

## Answer

38 overtakes.

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