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 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.
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?
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.