Some two teams scored the same number of points in a volleyball tournament. Prove that there are teams \(A\), \(B\) and \(C\), in which \(A\) beat \(B\), \(B\) beat \(C\) and \(C\) beat \(A\).
In the country called Orientation a one-way traffic system was introduced on all the roads, and each city can be reached from any other one by driving on no more than two roads. One road was closed for repairs but from every city it remained possible to get to any other. Prove that for every two cities this can still be done whilst driving on no more than 3 roads.
In a circle, each member has one friend and one enemy. Prove that
a) the number of members is even.
b) the circle can be divided into two neutral circles.
In some country 89 roads emerge from the capital, from the city of Dalny – one road, from the remaining 1988 cities – 20 roads (in each).
Prove that from the capital you can drive to Dalny.
Out of a whole 100-vertex graph, 98 edges were removed. Prove that the remaining ones were connected.
In a graph there are 100 vertices, and the degree of each of them is not less than 50. Prove that the graph is connected.
The faces of a polyhedron are coloured in two colours so that the neighbouring faces are of different colours. It is known that all of the faces except for one have a number of edges that is a multiple of 3. Prove that this one face has a multiple of 3 edges.
In a country, each two cities are connected with a one-way road.
Prove that there is a city from which you can drive to any other whilst travelling along no more than two roads.
Prove that in a bipartite planar graph \(E \geq 2F\), if \(E \geq 2\) (\(E\) is the number of edges, \(F\) is the number of regions).
Solve the equation in integers \(2x + 5y = xy - 1\).