In the Kingdom of Rabbitland some cities are connected with each other by a railroad. There are two ways of getting from Longears city to Pufftown, one of them passes through an even number of other cities, and the other passes through an odd number of other cities. Show that in the Kingdom of Rabbitland there is a cycle of an odd number of distinct cities, such that each is connected to the next by a railroad, and the last one is connected to the first one by a railroad.
During a chess tournament some people played 5 games and some people played 6. Prove that the number of people who played 5 games is even.
There are \(6\) points in three dimensional space. To connect some of the points we draw \(10\) segments. Show that there is at least one fully drawn triangle (with all vertices and sides).
There are ten islands in the Fantasia Archipelago. There used to be cruises connecting each island to each other island, each cruise served by a different ship. One day eight of those ships were taken over by pirates and do not serve cruises any more. Show that despite the efforts of pirates, it is still possible to get from each island to every other island, possibly changing ships on the way.
In Rabbitland you can get from any city to any other city. Bugs Bunny decided to find out what is the largest number of cities he can visit, while never visiting any city twice. He can start from any city and finish at any other. He realised that there are two longest paths he can take and they both involve the same number of cities. Show that those two paths have at least one city in common.
There are \(n\) people at a royal ball. Every time three of them meet and talk, it turns out some two of them didn’t know each other before. Show that the maximum number of connections between the guests is \(\frac14 n^2\). If any guest \(A\) knows guest \(B\), then \(B\) knows \(A\), and this counts as one connection.
An animal with \(n\) cells is a connected figure consisting of \(n\) equal-sized adjacent square cells. A dinosaur is an animal with at least 2007 cells. It is said to be primitive if its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
The kingdom of Rabbitland consists of a finite number of cities. No matter how you split the kingdom into two, there is always a train connection from a city in one part of the divide to a city in the other part of the divide. Show that one can in fact travel from any city to any other, possibly changing trains.
A poetry society has 33 members, and each person knows at least 16 people from the society. Show that you can get to know everyone in the society by a series of introductions if you already know someone from the society.
Some Star Trek fans and some Doctor Who fans met at a science fiction convention. It turned out that everyone knew exactly three people at the convention. However, none of the Star Trek fans knew each other and none of the Doctor Who fans knew each other. Show that there are the same number of Star Trek fans as the number of Doctor Who fans at the convention.