On a plane, there are given 2004 points. The distances between every pair of points is noted. Prove that among these noted distances at least thirty numbers are different.
Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.
Ten straight lines are drawn through a point on a plane cutting the plane into angles.
Prove that at least one of these angles is less than \(20^{\circ}\).
You are given \(7\) straight lines on a plane, no two of which are parallel. Prove that there will be two lines such that the angle between them is less than \(26^{\circ}\).