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 \(\frac{180^{\circ}}{n}\).
Jane’s birthday cake is square-shaped and has side length 25 cm. Suppose she makes 4 horizontal cuts perpendicular to the vertical edges of the cake and 4 vertical cuts perpendicular to the horizontal edges of the cake. Show that at least one of the pieces has an area of at most 25 cm\(^2\).
Show that given any nine points on a sphere, there is a closed hemisphere that contains six of them. A closed hemiphere is one that contains the equator with respect to the division.