Determine all integer solutions of the equation \(3x - 12y = 7\).
Determine all the integer solutions for the equation \(21x + 48y = 6\).
Solve the equation with integers \(x^2 + y^2 = 4z - 1\).
a) they have 10 vertices, the degree of each of which is equal to 9?
b) they have 8 vertices, the degree of each of which is equal to 3?
c) are they connected, without cycles and contain 6 edges?
On the plane 100 circles are given, which make up a connected figure (that is, not falling apart into pieces). Prove that this figure can be drawn without taking the pencil off of the paper and going over any line twice.
In some country there is a capital and another 100 cities. Some cities (including the capital) are connected by one-way roads. From each non-capital city 20 roads emerge, and 21 roads enter each such city. Prove that you cannot travel to the capital from any city.
Prove that on the edges of a connected graph one can arrange arrows so that from some vertex one can reach any other vertex along the arrows.
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.
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.