There are seven lakes in some country, connected by ten non-overlapping canals, and each lake can be reached from any other. How many islands are there in this country?
Prove that for a flat graph the inequality \(2E \geq 3F\) is valid.
Dan drew seven graphs on the board, each of which is a tree with six vertices. Prove that among them there are two which are isomorphic.
Several teams played a volleyball tournament amongst themselves. We will say that team \(A\) is better than team \(B\), if either \(A\) has either beaten team \(B\), or there exists such a team \(C\) that was beaten by \(A\), whilst \(C\) beat team \(B\).
a) Prove that there is a team that is better than all.
b) Prove that the team that won the tournament is the best.
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.
Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?
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.
Out of a whole 100-vertex graph, 98 edges were removed. Prove that the remaining ones were connected.
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.