Question

Is it possible to locate 1000 segments on the plane so that each segment at both ends can be rested strictly inside other segments?

Answer 1

Let's consider that there are 1000 segments on the plane. Take an arbitrary straight line l, not perpendicular to any of them, and project of all these segments ends on it. It is clear that the end of the segment (projected on the left side of the obtained points) can not simply focus strictly inside the other segment.