8.2. On a line, there are blue and red points, with no fewer than 5 red points. It is known that on any segment with endpoints at red points, containing a red point inside, there are at least 3 blue points. And on any segment with endpoints at blue points, containing 2 blue points inside, there are at least 2 red points. What is the maximum number of blue points that can be on a segment with endpoints at red points, not containing other red points?
# Answer. 3.

Solution. Note that on a segment with endpoints at red points, not containing other red points, there cannot be 4 blue points. Indeed, in this case, between the outermost blue points there will be 2 blue points, which means there will be at least 2 more red points. Therefore, there are no more than 3 blue points on such a segment.

We will show that 3 blue points can lie on a segment with endpoints at red points, not containing other red points. Suppose the points are arranged on a line in the following order: 2 red - 3 blue - 2 red - 3 blue - 2 red. Then all conditions are satisfied, and there is a segment with 3 blue points.

Comment. It is proven that there are no more than 3 blue points between neighboring red points - 4 points.

A correct example with 3 blue points is provided - 3 points.