How many ways can you build a closed line whose vertices are the vertices of a regular hexagon (the line can be self-intersecting)?
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?
Find the number of rectangles made up of the cells of a board with \(m\) horizontals and \(n\) verticals that contain a cell with the coordinates \((p, q)\).
Prove that there is no graph with five vertices whose degrees are equal to 4, 4, 4, 4, 2.
Prove that a graph, in which every two vertices are connected by exactly one simple path, is a tree.
Prove that, in a tree, every two vertices are connected by exactly one simple path.
Prove that there is a vertex in the tree from which exactly one edge emerges (such a vertex is called a hanging top).
At a conference there are 50 scientists, each of whom knows at least 25 other scientists at the conference. Prove that is possible to seat four of them at a round table so that everyone is sitting next to people they know.
Each of the edges of a complete graph consisting of 6 vertices is coloured in one of two colours. Prove that there are three vertices, such that all the edges connecting them are the same colour.