1. (17 points) Three cyclists, Dima, Misha, and Petya, started a 1 km race simultaneously. At the moment Misha finished, Dima still had to cover one tenth of the distance, and at the moment Dima finished, Petya still had to cover one tenth of the distance. How far apart (in meters) were Petya and Dima when Misha finished? (The cyclists are riding at constant speeds).
Answer: 90.

Solution. Let $v_{1}$ be the speed of Misha, $v_{2}$ be the speed of Dima, and $v_{3}$ be the speed of Petya. Then $v_{2}=0.9 v_{1}$, since in the same amount of time Misha traveled 1000 m, while Dima traveled 900 m, and $v_{3}=0.9 v_{2}$, since in the same amount of time Dima traveled 1000 m, while Petya traveled 900 m. Since $v_{3}=$ $0.9 \cdot 0.9 v_{1}=0.81 v_{1}$, at the moment Misha finished, Petya had traveled 810 meters. The distance between Petya and Dima: $900-810=90$.