4. Petya observes two ants crawling along a straight path at constant speeds. After 3 minutes of observation, the distance between the ants was 9 meters, after 5 minutes - 5 meters, after 9 minutes - 3 meters. What was the distance between the ants after 8 minutes of observation?

$[5$ points]
![](https://cdn.mathpix.com/cropped/2024_05_06_053bf478cc99a970829eg-1.jpg?height=686&width=356&top_left_y=1666&top_left_x=1620)

| 1a | 1b | 2a | 2b | 2c | 2d | 3a | 3b | 3c | 4 | $\Sigma$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  |  |  |  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |  |  |  |
4. The faster ant is ahead of the slower one and is moving away from it.

It is clear that situation 1 can over time turn into situation 2, and situation 3 can turn into situation 4. But if the ants at some point find themselves in situation 2 or 4, then from that moment on, they will always be moving away from each other.

We know the distance between the ants at three moments in time. Note that between the first and second moment, 2 minutes have passed, and the distance between the ants has decreased by 4 meters. Between the second and third moment, 4 minutes have passed, and the distance has decreased by 2 meters. Since the distance decreased both times, we can be sure that 3 minutes and 5 minutes after the start of the observation, the ants were not in a "moving away" state, meaning they could only be in states 1 or 3.

If the meeting between them had not occurred in the next 4 minutes (from the 5th to the 9th minute), they would have remained in the same state (1 or 3) and continued to approach each other at a rate of 4 meters in 2 minutes, meaning they should have closed the distance by 8 meters in 4 minutes. However, the distance between them decreased by only 2 meters, so at some point between the 2nd and 3rd moments, a meeting must have occurred.

After the meeting, the ants move away from each other at the same speed they approached (in situations 1 and 2, the speeds of approaching and moving away are equal to the sum of the ants' speeds, while in situations 3 and 4, they are equal to the difference in their speeds). Therefore, in 4 minutes, the distance between the ants decreased from 5 meters to zero and then increased from zero to 3 meters. In total, 8 meters in 4 minutes, which matches the previously found rate of approaching. One minute earlier, the distance was 2 meters less. Therefore, 8 minutes after the start of the movement, the distance between the ants was 1 meter.