Is the number 12345678926 square?
There are 100 notes of two types: \(a\) and \(b\) pounds, and \(a \neq b \pmod {101}\). Prove that you can select several bills so that the amount received (in pounds) is divisible by 101.
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.
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?
There are two identical gears with 14 teeth on a common shaft. They are aligned and four pairs of teeth are removed.
Prove that the gears can be rotated so that they form a complete gear (one containing no gaps).
If a class of 30 children is seated in the auditorium of a cinema there will always be at least one row containing no fewer than two classmates. If we do the same with a class of 26 children then at least three rows will be empty. How many rows are there in the cinema?
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?