51 points were thrown into a square of side 1 m. Prove that it is possible to cover some set of 3 points with a square of side 20 cm.
A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.
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.
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.
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.
A standard chessboard has more than a quarter of its squares filled with chess pieces. Prove that at least two adjacent squares, either horizontally, vertically, or diagonally, are occupied somewhere on the board.
2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.
All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.
Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.