3.1 Two ants are running along a circle towards each other at constant speeds. While one of them runs 9 laps, the other runs 6. At the point where the ants meet, a red dot appears. How many red dots will there be on the circle?

![](https://cdn.mathpix.com/cropped/2024_05_06_6a9b962ccc5e03d35f9eg-05.jpg?height=449&width=531&top_left_y=495&top_left_x=822)

#
# Answer: 5

Solution. The ratio of the speeds of the ants is $\frac{9}{6}=\frac{3}{2}$. Without loss of generality, assume the circumference of the circle is 5, and let the ants meet at some moment at point 0. Place points $1, 2, 3,$ and 4 clockwise so that they, together with point 0, divide the circle into 5 equal parts. If the slower ant runs counterclockwise, they will meet again at point 2. Now we can consider this point as the starting point, and thus they will meet next at point 4, and then at point 1 (since $4+2=6$, but the circumference is 5). The next meeting will be at point 3, then 0, after which the process will repeat, and there will be a total of 5 points.