There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.
9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of \(2:3\). Prove that at least three of the nine lines pass through the same point.
In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.
Two points are placed inside a convex pentagon. Prove that it is always possible to choose a quadrilateral that shares four of the five vertices on the pentagon, such that both of the points lie inside or on the boundary the quadrilateral.
A pentagon is inscribed in a circle of radius 1. Prove that the sum of the lengths of its sides and diagonals is less than 17.
A Cartesian plane is coloured in in two colours. Prove that there will be two points on the plane that are a distance of 1 apart and are the same colour.
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.
Arrows are placed on the sides of a polygon. Prove that the number of vertices in which two arrows converge is equal to the number of vertices from which two arrows emerge.
A convex figure and point \(A\) inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point \(A\) and dividing it in half at point \(A\).
2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.