8.4. Nikita and Egor are running on a circular road, having started from the same place in opposite directions. It is known that Nikita runs a lap 12 seconds faster than Egor, but still takes more than 30 seconds to do so. It turned out that for the seventh time after the start, they met at the same place where they began. How long does each of them take to run a lap?
Answer: Nikita in 36 seconds, Egor in 48 seconds.

Solution: Note that between any two meetings, the boys run a total of one lap. Therefore, by the 7th meeting, they have run 7 laps together. Also, note that each of them ran an integer number of laps, as they met at the starting line. Since Nikita ran faster, he ran more laps than Egor, i.e., Nikita ran 4, 5, or 6 laps (he couldn't have run 7, otherwise Egor would have been standing), and Egor ran 3, 2, or 1 lap, respectively.

Let's assume that Nikita takes $t$ seconds to run one lap, and Egor takes $t+12$ seconds. Let's consider all cases of how many laps Nikita ran by the 7th meeting.

1) 4 laps, and Egor - 3. Then the time it takes for Nikita to run 1 lap is $4 / 3$ times less than the time it takes for Egor (in the same amount of time, one ran 4 laps, and the other - 3). Therefore, $t=3 / 4(t+12)$, from which $\mathrm{t} / 4=9$, i.e., $\mathrm{t}=36$, and Egor's time is $\mathrm{t}+12=48$ seconds.
2) 5 laps, and Egor - 2. Similarly, we get that $t=2 / 5(t+12)$, from which $t=8$, which does not fit the condition.
3) 6 laps, and Egor - 1. We get that $t=1 / 6(t+12)$, from which $t=12 / 5$, which also does not fit.

Therefore, only the first case fits.

## Criteria:

Answer, answer with verification - 0 points.

Noted that the boys ran a total of 7 laps - 1 point.

Noted that Nikita ran 4-6 laps - 1 point, combined with the previous one.

Lost one case - no more than 5 points.

Lost more than one case - no more than 3 points.