There are 101 buttons of 11 different colours. Prove that amongst them there are either 11 buttons of the same colour, or 11 buttons of different colours.
Prove that the equation \[a_1 \sin x + b_1 \cos x + a_2 \sin 2x + b_2 \cos 2x + \dots + a_n \sin nx + b_n \cos nx = 0\] has at least one root for any values of \(a_1 , b_1, a_2, b_2, \dots, a_n, b_n\).
A journalist came to a company which had \(N\) people. He knows that this company has a person \(Z\), who knows all the other members of the company, but nobody knows him. A journalist can address each member of the company with the question: “Do you know such and such?” Find the smallest number of questions sufficient to surely find \(Z\). (Everyone answers the questions truthfully. One person can be asked more than one question.)
Let \(f (x)\) be a polynomial about which it is known that the equation \(f (x) = x\) has no roots. Prove that then the equation \(f (f (x)) = x\) does not have any roots.
At a round table, 10 boys and 15 girls were seated. It turned out that there are exactly 5 pairs of boys sitting next to each other.
How many pairs of girls are sitting next to each other?
Prove that there exist numbers, that can be presented in no fewer than 100 ways in the form of a summation of 20001 terms, each of which is the 2000th power of a whole number.
Arrows are placed on the sides of a polygon. Prove that the number of vertices in which two arrows converge is equal to the number of vertices from which two arrows emerge.
In the government of the planet of liars and truth tellers there are \(101\) ministers. In order to reduce the budget, it was decided to reduce the number of ministers by \(1.\) But each of the ministers said that if they were to be removed from the government, then the majority of the remaining ministers would be liars. How many truth tellers and how many liars are there in the government?
Is it possible to fill an \(n\times n\) table with the numbers \(-1\), \(0\), \(1\), such that the sums of all the rows, columns, and diagonals are unique?
Prove that in any group of friends there will be two people who have the same number of friends.