Problems

Peter has 28 classmates. Each 2 out of these 28 have a different number of friends in the class. How many friends does Peter have?

Prove that there is a vertex in the tree from which exactly one edge emerges (such a vertex is called a hanging top).

Given $n$ points that are connected by segments so that each point is connected to some other and there are no two points that would be connected in two different ways. Prove that the total number of segments is $n-1$.

A system of points connected by segments is called “connected” if from each point one can go to any other one along these segments. Is it possible to connect five points to a connected system so that when erasing any segment, exactly two connected points systems are formed that are not related to each other? (We assume that in the intersection of the segments, the transition from one of them to another is impossible).

A group of $2n$ people were gathered together. Each person knew at least $n$ of the other people present. Prove that it is possible to select 4 people and seat them around a table so that each person sits next to people they know. ($n\ge 2$)

Airlines connect pairs of cities. How can you connect 50 cities with the fewest number of airlines so that from every city you can get to any other city by taking at most two flights?

In the country of Mara there are several castles. Three roads lead from each castle. A knight left from one of the castles. Traveling along the roads, he turns from each castle standing in his way, either to the right or to the left depending on the road on which he came. The knight never turns to the side which he turned before it. Prove that one day he will return to the original castle.

Prove that for every convex polyhedron there are two faces with the same number of sides.

Prove that the following facts are true for any graph:

a) The sum of degrees of all vertices is equal to twice the number of edges (and therefore it is even);

b) The number of vertices of odd degree is even.

During a chess tournament, some of the players played an odd number of games. Prove that the number of such players is even.