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.
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?
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?
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 \(M = K\).
The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?
Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?
Two people toss a coin: one tosses it 10 times, the other – 11 times. What is the probability that the second person’s coin showed heads more times than the first?
In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.
Let \(f (x)\) be a polynomial about which it is known that the equation \(f (x) = x\) has no roots. Prove that then the equation \(f (f (x)) = x\) does not have any roots.