Problems
Prove that out of \(n\) objects an even number of objects can be chosen in \(2^{n-1}\) ways.
Prove that every number \(a\) in Pascal’s triangle is equal to
a) the sum of the numbers of the previous right diagonal, starting from the leftmost number up until the one to the right above the number \(a\).
b) the sum of the numbers of the previous left diagonal, starting from the leftmost number to the one to left of the number which is above \(a\).
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.
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 \(A\).