1. Cyclists Petya, Vlad, and Timur simultaneously started a warm-up race on a circular cycling track. Their speeds are 27 km/h, 30 km/h, and 32 km/h, respectively. After what shortest time will they all be at the same point on the track again? (The length of the cycling track is 400 meters.)
Answer: 24 min.

Solution. Vlad rides 3 km/h faster than Petya. Therefore, he will overtake him by one lap (i.e., ride 400 m more) in $\frac{0.4}{3}$ h $=8$ min. Timur rides 2 km/h faster than Vlad. Therefore, he will overtake him by one lap in $\frac{0.4}{2}$ h $=12$ min. The least common multiple of the numbers 8 and 12 is 24.

Grading. 7 points for the correct solution.