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.
101 points are marked on a plane; not all of the points lie on the same straight line. A red pencil is used to draw a straight line passing through each possible pair of points. Prove that there will always be a marked point on the plane through which at least 11 red lines pass.
It is known that in a convex \(n\)-gon (\(n > 3\)) no three diagonals pass through one point. Find the number of points (other than the vertex) where pairs of diagonals intersect.
On two parallel lines \(a\) and \(b\), the points \(A_1, A_2, \dots , A_m\) and \(B_1, B_2, \dots , B_n\) are chosen, respectively, and all of the segments of the form \(A_iB_j\), where \(1 \leq i \leq m\), \(1 \leq j \leq n\). How many intersection points will there be if it is known that no three of these segments intersect at one point?