[ Systems of points and segments. Examples and counterexamples ] [ Classical combinatorics (miscellaneous). ]

On a plane, 10 equal segments were drawn, and all their points of intersection were marked. It turned out that each point of intersection divides any segment passing through it in the ratio $3: 4$. What is the maximum possible number of marked points?
On each segment, there are no more than two points. On the other hand, each intersection point belongs to at least two segments. Therefore, there are no more than $10 \cdot 2: 10=10$ points. An example with 10 points is shown in the figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_9bfea73962495faf8ed9g-25.jpg?height=370&width=1352&top_left_y=1278&top_left_x=364)

Answer

10 points.