10.4. Two athletes are swimming along the pool lane at constant but different speeds. Meeting for the first time at point $C$ at some moment, they turn around, each swims to their end of the lane, immediately turn around and swim back, meet for the second time at point $D$, and so on. Where will their 20th meeting occur?
# 10.4. Answer: At point $D$.

Let's use a graphical representation. Along the x-axis, we plot time, and along the y-axis, we plot distance. On the y-axis, we mark the edges of the pool as points $A$ and $B$, and the meeting points as $C$ and $D$. The movement of one swimmer is represented by a solid line, and the other by a dashed line. Movement at a constant speed is depicted on the graph as a straight line segment. When the direction of movement changes after meeting the edge of the pool or another swimmer, the segment changes direction, with the angle to the x-axis changing sign but not in absolute value.

![](https://cdn.mathpix.com/cropped/2024_05_06_419155ca21d2b5501595g-2.jpg?height=685&width=1427&top_left_y=214&top_left_x=383)

Let point $M$ on the graph correspond to the moment of the second meeting (at point $D$ in the pool). The graphs of the movements of both athletes are symmetric relative to the vertical line passing through point $M$. Therefore, the third meeting will occur at the same point as the first, that is, at point $C$. The same situation applies to any other meeting point. Therefore, the meeting points alternate: odd meetings occur at point $C$, and even meetings at point $D$.