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.
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 \(A\).
In the country called Orientation a one-way traffic system was introduced on all the roads, and each city can be reached from any other one by driving on no more than two roads. One road was closed for repairs but from every city it remained possible to get to any other. Prove that for every two cities this can still be done whilst driving on no more than 3 roads.
In what number system is the equality \(3 \times 4 = 10\) correct?
Prove that \(\frac {1}{2} (x^2 + y^2) \geq xy\) for any \(x\) and \(y\).
Prove that for \(a, b, c > 0\), the following inequality is valid: \(\left(\frac{a+b+c}{3}\right)^2 \ge \frac{ab+bc+ca}{3}\).
Prove that for \(x \geq 0\) the inequality is valid: \(2x + \frac {3}{8} \ge \sqrt[4]{x}\).
Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?
Prove that any axis of symmetry of a 45-gon passes through its vertex.