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?

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).

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?

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.

A schoolboy told his friend Bob:

“We have thirty-five people in the class. And imagine, each of them is friends with exactly eleven classmates...”

“It cannot be,” Bob, the winner of the mathematical Olympiad, answered immediately. Why did he decide this?

A group of $2n$ people were gathered together, of whom each person knew no less than $n$ 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. ($n\ge 2$)

In the town of Ely, all the families have separate houses. On one fine day, each family moved into another, one of the houses house that used to be occupied by other families. They afterwards decided to paint all houses in red, blue or green colors in such a way that for each family the colour of the new and old houses would not match. Is this always possible to paint te houses in such a way, regardless of how families decided to move?

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