In a room there are some chairs with 4 legs and some stools with 3 legs. When each chair and stool has one person sitting on it, then in the room there are a total of 39 legs. How many chairs and stools are there in the room?
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.
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?
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?
Two grandmasters in turn put rooks on a chessboard (one turn – one rook) so that they cannot capture each other. The person who cannot put a rook on the chessboard loses. Who will win with the game – the first or second grandmaster?