a) In a group of 4 people, who speak different languages, any three of them can communicate with one another; perhaps by one translating for two others. Prove that it is always possible to split them into pairs so that the two members of every pair have a common language.
b) The same, but for a group of 100 people.
c) The same, but for a group of 102 people.
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?
In order to glaze 15 windows of different shapes and sizes, 15 pieces of glass are prepared exactly for the size of the windows (windows are such that each window should have one glass). The glazier, not knowing that the glass is specifically selected for the size of each window, works like this: he approaches a certain window and sorts out the unused glass until he finds one that is large enough (that is, either an exactly suitable piece or one from which the right size can be cut), if there is no such glass, he goes to the next window, and so on, until he has assessed each window. It is impossible to make glass from several parts. What is the maximum number of windows which can be left unglazed?
On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.
What is the minimum number of lottery tickets for the Sport Lotto that it is necessary to buy in order to guarantee that at least one of the tickets will have one number correct. On any single ticket you can choose 6 of the available numbers 1 to 49.
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.
There were seven boxes. In some of them, seven more boxes were placed inside (not nested in each other), etc. As a result, there are 10 non-empty boxes. How many boxes are there now in total?
A class contains 25 pupils. It is known that within any group of 3 pupils there are two friends. Prove that there is a pupil who has no fewer than 12 friends.
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?