Solve the equation in integers \(2x + 5y = xy - 1\).
Prove there are no integer solutions for the equation \(4^k - 4^l = 10^n\).
If a salary is first increased by 20%, and then reduced by 20%, will the salary paid increase or decrease as a result?
Arrange in a row the numbers from 1 to 100 so that any two neighbouring ones differ by at least 50.
An \(8 \times 8\) square is painted in two colours. You can repaint any \(1 \times 3\) rectangle in its predominant colour. Prove that such operations can make the whole square monochrome.
Some person \(A\) thought of a number from 1 to 15. Some person \(B\) asks some questions to which you can answer ‘yes’ or ‘no’. Can \(B\) guess the number by asking a) 4 questions; b) 3 questions.
12 teams played a volleyball tournament in one round. Two teams scored exactly 7 wins.
Prove that there are teams \(A\), \(B\), \(C\) where \(A\) won against \(B\), \(B\) won against \(C\), and \(C\) won against \(A\).
The numbers from 1 to 9999 are written out in a row. How can I remove 100 digits from this row so that the remaining number is a) maximal b) minimal?
Is it possible to draw five lines from one point on a plane so that there are exactly four acute angles among the angles formed by them? Angles between not only neighboring rays, but between any two rays, can be considered.
A group of 20 tourists go on a trip. The oldest member of the group is 35, the youngest is 20. Is it true that there are members of the group that are the same age?