1.1. Around a circle, Petya and Vasya are riding bicycles at constant speeds. Petya's speed is 8 km/h, and Vasya's speed is 10 km/h. Initially, they were riding in opposite directions (Petya - clockwise, and Vasya - counterclockwise), and then Petya changed his direction (started moving counterclockwise) and simultaneously doubled his speed. After this, the cyclists started meeting $k$ times less frequently. Find $k$.
# Answer: 3

Solution. Let $d$ km be the length of the circumference. Then, initially, the time between two consecutive meetings of the cyclists is $t_{1}=d /(8+10)=d / 18$ hours. Then, Petya's speed increased to 16 km/h, and the cyclists started moving in the same direction, so the time between two consecutive meetings of the cyclists became $t_{2}=d /(16-10)=d / 6$ hours. From this, it follows that the frequency of meetings decreased by $t_{2} / t_{1}=3$ times.