Problem 2. Given 10 equal segments in a plane. All intersection points are colored. Each colored point divides each of the given segments it lies on in the ratio $3: 4$. What is the maximum number of colored points?
Solution. We will prove that the maximum number of colored points is 10.

The points that divide a given segment in the ratio $3: 4$ (or $4: 3$) are two; hence, on one segment, there are at most two colored points. On the other hand, each colored point lies on at least two segments. Therefore, the number of colored points is at most $\frac{10 \cdot 2}{2} = 10$.

The diagram below shows one of the possible

![](https://cdn.mathpix.com/cropped/2024_06_03_a96bf614c78521f03e19g-1.jpg?height=394&width=405&top_left_y=1473&top_left_x=1245)
examples with 10 colored points.