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.
A conference was attended by a finite group of scientists, some of whom are friends. It turned out that every two scientists, who have an equal number of friends at the conference, do not have friends in common. Prove that there is a scientist who has exactly one friend among the conference attendees.
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?
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?
In the Land of Linguists live \(m\) people, who have opportunity to speak \(n\) languages. Each person knows exactly three languages, and the sets of known languages may be different for different people. It is known that \(k\) is the maximum number of people, any two of whom can talk without interpreters. It turned out that \(11n \leq k \leq m/2\). Prove that then there are at least \(mn\) pairs of people in the country who will not be able to talk without interpreters.
Let’s prove the following statement: every graph without isolated vertices is connected.
Proof We use the induction on the number of vertices. Clearly the statement is true for graphs with \(2\) vertices. Now, assume we have proven the statement for graphs with up to \(n\) vertices.
Take a graph with \(n\) vertices by induction hypothesis it must be connected. Let’s add a non-isolated vertex to it. As this vertex is not isolated, it is connected to one of the other \(n\) vertices. But then the whole graph of \(n+1\) vertices is connected!