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.
Eugenie, arriving from Big-island, said that there are several lakes connected by rivers. Three rivers flow from each lake, and four rivers flow into each lake. Prove that she is wrong.
In some state, there are 101 cities.
a) Each city is connected to each of the other cities by one-way roads, and 50 roads lead into each city and 50 roads lead out of each city. Prove that you can get from each city to any other, having travelled on no more than on two roads.
b) Some cities are connected by one-way roads, and 40 roads lead into each city and 40 roads lead out of each. Prove that you can get form each city to any other, having travelled on no more than on three 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.
Is the number \(1 + 2 + 3 + \dots + 1990\) odd or even?