For which \(n > 3\), can a set of weights with masses of \(1, 2, 3, ..., n\) grams be divided into three groups of equal mass?
Prove that the infinite decimal \(0.1234567891011121314 \dots\) (after the decimal point, all of the natural numbers are written out in order) is an irrational number.
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.
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.
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.
Find the number of zeros in which the number \(11^{100} - 1\) ends.
You are given a table of size \(m \times n\) (\(m, n > 1\)). In it, the centers of all cells are marked. What is the largest number of marked centers that can be chosen so that no three of them are the vertices of a right triangle?
There are several cities (more than one) in a country; some pairs of cities are connected by roads. It is known that you can get from every city to any other city by driving along several roads. In addition, the roads do not form cycles, that is, if you leave a certain city on some road and then move so as not to pass along one road twice, it is impossible to return to the initial city. Prove that in this country there are at least two cities, each of which is connected by a road with exactly one city.
Solve the equation \(xy = x + y\) in integers.