Consider a segment on a line of length \(3m\). Jack chose \(4\) random points on the segment and measured all the distances between points. Prove that at least one of the distances is less or equal than \(1m\).
Twelve lines are drawn on the plane, passing through a point \(A\). Prove that there are two of them with angle less than \(17^{\circ}\) between them.
Inside a square of area \(6\), there are three polygons, each of area \(3\). Show that some two of them overlap and the area of the overlap is at least \(1\).
A Wimbledon doubles court is 78 ft \(\times\) 36 ft. After a long practice match, there were 79 tennis balls in the court area. Show that some two of them were no further than \(6\sqrt{2}\) ft away.
There are \(n\) ambassadors sitting at a round table. Each ambassador has a flag of a country in front of them. Unfortunately flags have been mixed up and 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 a very large chessboard, consisting of white and black squares. We would like to place a stain of a specific shape on this chessboard and we know that the area of this stain is 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 less than \(\frac{180^{\circ}}{n}\).
There are thirteen boys and girls who met to play a football match. Eleven of them shook hands with everybody else in the group, but the last two shook hands with everybody else but not each other, because they were siblings and arrived together. How many handshakes took place?
The city of Konigsberg has seven bridges as depicted on the layout below. \[\includegraphics[scale=0.5]{WSP-000146.png}\] Is it possible for a great mathematician Leonard Euler to have an excursion in Konigsberg visiting all islands and land banks, but crossing each bridge only once?
There are \(6\) people at a party. Each two people either know each other or not, and the knowledge goes both ways: if \(A\) knows \(B\), then \(B\) knows \(A\). Show that there either is a trio of people who all know each other or a trio of people who all don’t know each other.