There are \(n\) cities in a country. Between each two cities an air service is established by one of two airlines. Prove that out of these two airlines at least one is such that from any city you can get to any other city whilst traveling on flights only of this airline.
Are there any irrational numbers \(a\) and \(b\) such that the degree of \(a^b\) is a rational number?
Three people play table tennis, and the player who lost the game gives way to the player who did not participate in it. As a result, it turned out that the first player played 10 games and the second played 21 games. How many games did the third player play?
In the secret service, there are \(n\) agents – 001, 002, ..., 007, ..., \(n\). The first agent monitors the one who monitors the second, the second monitors the one who monitors the third, etc., the nth monitors the one who monitors the first. Prove that \(n\) is an odd number.
Construct a function defined at all points on a real line which is continuous at exactly one point.
In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.
Every point in a plane, which has whole-number co-ordinates, is plotted in one of \(n\) colours. Prove that there will be a rectangle made out of 4 points of the same colour.
Prove that multiplying the polynomial \((x + 1)^{n-1}\) by any polynomial different from zero, we obtain a polynomial having at least \(n\) nonzero coefficients.
One of \(n\) prizes is embedded in each chewing gum pack, where each prize has probability \(1/n\) of being found.
How many packets of gum, on average, should I buy to collect the full collection prizes?
On a \(100 \times 100\) board 100 rooks are placed that cannot capturing one another.
Prove that an equal number of rooks is placed in the upper right and lower left cells of \(50 \times 50\) squares.