Problems

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?

You are given 25 numbers. The sum of any 4 of these numbers is positive. Prove that the sum of all 25 numbers is also positive.

In a tournament by the Olympic system (the loser is eliminated), 50 boxers participate. What is the minimum number of matches needed to be conducted in order to identify the winner?

A square area of size \(100\times 100\) is covered in tiles of size \(1\times 1\) in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?

In each square of a rectangular table of size \(M \times K\), a number is written. The sum of the numbers in each row and in each column, is 1. Prove that \(M = K\).

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?