10.1. Ivan was walking from Pakhomovo to Vorobyevo. At noon, when he had covered $4 / 9$ of the entire distance, Foma set off after him from Pakhomovo on a bicycle, and Erema set off towards him from Vorobyevo. Foma overtook Ivan at 1 PM, and met Erema at 1:30 PM. When will Ivan and Erema meet?
Answer: At 14:30

Solution. Let the distance from Pakhomo to Vorobyovo be denoted as P, and the speeds of the performance participants as I, F, and E km per hour, respectively, using the first letters of their names. Then, according to the problem, Foma and Ivan are moving in the same direction, while Foma and Yerema are moving towards each other, and at noon, they are separated by: the first pair by $4 / 9 \cdot P$, and the second pair by exactly P. We set up the equations $\frac{4 / 9 \cdot P}{F-I}=1$, $\frac{P}{F+E}=\frac{3}{2}$, from which $I+E=\frac{2 P}{3}-\frac{4 P}{9}=\frac{2 P}{9}$ and the time it takes for Ivan and Yerema to cover the $5 / 9 \cdot P$ separating them, moving towards each other, is $\frac{5 P / 9}{2 P / 9}=\frac{5}{2}$. Therefore, they will meet at $12+2.5=14.5$ hours $=14$ hours 30 minutes.

Grading criteria. Correct answer with verification: 1 point. Correct system of equations written (but not solved): 3 points.