10.1. Vikentiy walked from the city to the village, and at the same time Afanasiy walked from the village to the city. Find the distance between the village and the city, given that the distance between the pedestrians was 2 km twice: first, when Vikentiy had walked half the way to the village, and then, when Afanasiy had walked a third of the way to the city.
Answer: 6 km.

Solution. Let the distance between the village and the city be denoted as $S$ km, the speeds of Vikentiy and Afanasy as $x$ and $y$, and calculate the time spent by the travelers in the first and second cases. In the first case, we get: $\frac{S / 2}{x}=\frac{S / 2-2}{y}$, in the second case $\frac{2 S / 3+2}{x}=\frac{S / 3}{y}$. From here, eliminating $x$ and $y$, we have $S^{2}-2 S-24=0$, from which $S=6$ km.

Instructions. Answer with verification: 1 point. Formulation of equations: 3 points.