10.1. A motorist and a cyclist set off from points A and B, not at the same time, towards each other. Meeting at point C, they immediately turned around and continued with the same speeds. Upon reaching their respective points A and B, they turned around again and met for the second time at point D. Here they turned around once more and so on. At which point on the segment AB will their 2015th meeting occur?
Answer: At point C.

Solution. Let the speeds of the motorist and the cyclist be $x$ and $y$ respectively. Suppose the motorist left earlier than the cyclist by $t$ hours and, by the time the cyclist set off, was at point P, a distance of $xt$ from A. After meeting at C and turning around, when the cyclist returned to B, the motorist returned to P, and then drove to A, by which time the cyclist had turned around and was at point T, a distance of $yt$ from B. Then they continued forward, met at D, turned around, the cyclist returned to T, and the motorist to A. Next, the cyclist, having traveled $yt$ km, returned to B, while the motorist turned around and, in the same time, having traveled $xt$ km, was back at P. The cycle is closed, they are again in the same positions as they were $t$ hours after the start. Thus, they will meet at C on even-numbered occasions, and at D on odd-numbered occasions.

Criterion. Answer only: 0 points.