Prove that, in a circle of radius 10, you cannot place 400 points so that the distance between each two points is greater than 1.
On a line, there are 50 segments. Prove that either it is possible to find some 8 segments all of which have a shared intersection, or there can be found 8 segments, no two of which intersect.
On a plane \(n\) randomly placed lines are given. What is the number of triangles formed by them?