Inside a square of area \(6\), there are three polygons, each of area \(3\). Show that some two of these polygons overlap and the area of the overlap is at least \(1\).
A Wimbledon doubles court is \(78\)ft\(\times36\)ft. After a long practice match, there were \(79\) tennis balls in the court area. Show that some two of the balls were at most \(6\sqrt{2}\)ft away from each other.
There are \(n\) ambassadors, each from a different country, sitting at a round table. The flag for each country is on the table, but unfortunately the flags have been mixed up. As a result each ambassador has a wrong flag in front of them, while their flag is in front of some other ambassador. Show that you can rotate the table with the flags on it, in such a way that at least two ambassadors will have correct flags in front of them.
We have an infinitely large chessboard, consisting of white and black squares. We would like to place a stain of a specific shape on this chessboard. The stain is a bounded and connected shape with an area strictly less than the area of one square of the chessboard. Show that it is always possible to place the stain in such a way that it does not cover a vertex of any square.
There are \(n\) straight lines on a plane, no two among them are parallel to each other. Show that some two of them cross at an angle no more than \(\frac{180^{\circ}}{n}\).
Picasso got a new set of crayons. He started colouring various things. First, Picasso coloured a line under the following condition: each point on a line is coloured either red or blue. Show that there are three different points \(A,B,C\) on the line of the same colour such that \(AB = BC\).
Monet has coloured a plane. He wants to colour the entire plane in such a way that each straight line can only have points of three or fewer different colours. Show that he can use however many different colours he wants and still be able to achieve this goal.
Frida colours the plane. She decided to use \(9\) different colours, but she wants to colour the entire infinite plane in such a way that if two points are distance \(1\) apart, then they must be different colours. Show that this is always possible.
Mondrian coloured the vertices of a regular pentagon. He used only red and blue. Show that there are three vertices of the pentagon which are all the same colour and form an isosceles triangle.
Artemisia coloured the plane with red and blue crayons. Show that there is an equilateral triangle with all vertices of the same colour somewhere on the plane.