How many different four-digit numbers, divisible by 4, can be made up of the digits 1, 2, 3 and 4,
a) if each number can occur only once?
b) if each number can occur several times?
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
Prove that there is no graph with five vertices whose degrees are equal to 4, 4, 4, 4, 2.
Prove that there exists a graph with 2n vertices whose degrees are \(1, 1, 2, 2, \dots , n, n\).
In a graph, all the vertices have degree of 3. Prove that there is a cycle in it.
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.
Eugenie, arriving from Big-island, said that there are several lakes connected by rivers. Three rivers flow from each lake, and four rivers flow into each lake. Prove that she is wrong.
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.