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.)
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?
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.
In chess, ‘check’ is when the king is under threat of capture from another piece. What is the largest number of kings that it is possible to place on a standard \(8\times 8\) chess board so that no two check one another.
Solve the equation \(2x^x = \sqrt {2}\) for positive numbers.
Find the first 99 decimal places in the number expansion of \((\sqrt{26} + 5)^{99}\).
Let \(M\) be a finite set of numbers. It is known that among any three of its elements there are two, the sum of which belongs to \(M\).
What is the largest number of elements in \(M\)?