8.2. Matvey left from Totma, and at the same time, Petr ran towards him from Kaluga along the same road. Under identical conditions, the boys' speeds are in the ratio of $2: 3$ and are constant. At some point, they encounter a rough road (possibly at different times) that continues until the moment they meet. On the rough road, each boy's speed decreases by half. How do the distances traveled on the rough road by Petr and Matvey relate if they met exactly halfway between the cities, and the rough road constitutes $2 / 3$ of the distance between Kaluga and Totma?
Answer: The distances traveled off-road by Matvei and Petr are in the ratio of $1: 3$.

Solution: If the off-road terrain on Matvei's side were more or equal to that on Petr's side, Petr's speed would always be higher, and the meeting would not have occurred halfway. Therefore, Petr walked on off-road terrain for a longer time.

Let Petr walk $6a$ kilometers before reaching the off-road terrain, then Matvei would have walked $4a$ kilometers in the same time. After that, Petr walks $3b$ kilometers until the off-road terrain starts for Matvei. During this time, Matvei walks $4b$ kilometers. Then, until the meeting, Petr walks $3c$ kilometers, and Matvei walks $2c$ kilometers.

![](https://cdn.mathpix.com/cropped/2024_05_06_ee373ffab09e7a30d5bdg-1.jpg?height=254&width=1490&top_left_y=1552&top_left_x=183)

On one hand: $4a + 4b + 2c = 6a + 3b + 3c$, since the meeting occurred halfway. On the other hand, since the off-road terrain constitutes $2/3$ of the total distance, we have: $(5c + 3b) : (10a + 7b + 5c) = 2/3$.

From the first equation, we get $b = 2a + c$.

From the second equation:

$15c + 9b = 20a + 14b + 10c$

$5c = 5b + 20a$

$c = b + 4a$

Substituting the second result into the first, we get:

$b = 2a + c = 2a + b + 4a = 6a + b$

$0 = 6a$

Thus, for Petr, the off-road terrain started immediately, meaning he walked his entire part of the journey on off-road terrain, while Matvei had to walk only $2/3 - 1/2 = 1/6$ of the distance on off-road terrain. Therefore, the distances traveled off-road by Matvei and Petr are in the ratio of $1: 3$.

Criteria: Correctly formulated system of equations - 3 points. Only answer - 0 points. Answer with verification - 2 points.