Let \(p\) be a prime number, and \(a\) an integer number not divisible by \(p\). Prove that there is a positive integer \(b\) such that \(ab \equiv 1 \pmod p\).
30 people vote on five proposals. In how many ways can the votes be distributed if everyone votes only for one proposal and only the number of votes cast for each proposal is taken into account?
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.
Prove that the number of US states with an odd number of neighbours is even.
a) What is the minimum number of pieces of wire needed in order to weld a cube’s frame?
b) What is the maximum length of a piece of wire that can be cut from this frame? (The length of the edge of the cube is 1 cm).
Prove there are no integer solutions for the equation \(x^2 + 1990 = y^2\).
Recall that a natural number \(x\) is called prime if \(x\) has no divisors except \(1\) and itself. Solve the equation with prime numbers \(pqr = 7(p + q + r)\).
Solve the equation with natural numbers \(1 + x + x^2 + x^3 = 2y\).