Prove that the midpoints of the sides of a regular polygon form a regular polygon.
Two perpendicular straight lines are drawn through the centre of the square. Prove that their intersection points with the sides of a square form a square.
A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.
There are 5 points inside an equilateral triangle with side of length 1. Prove that the distance between some two of them is less than 0.5.
A \(3\times 4\) rectangle contains 6 points. Prove that amongst them there will be two points, such that the distance between them is no greater than \(\sqrt5\).
There are 25 points on a plane, and among any three of them there can be found two points with a distance between them of less than 1. Prove that there is a circle of radius 1 containing at least 13 of these points.
A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius \(\frac{1}{7}\).
What is the minimum number of points necessary to mark inside a convex \(n\)-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?
A plane contains \(n\) straight lines, of which no two are parallel. Prove that some of the angles will be smaller than \(180^\circ/n\).
Several chords are drawn through a unit circle. Prove that if each diameter intersects with no more than \(k\) chords, then the total length of all the chords is less than \(\pi k\).