11 scouts are working on 5 different badges. Prove that there will be two scouts \(A\) and \(B\), such that every badge that \(A\) is working towards is also being worked towards by \(B\).
Let \(p\) be a prime number, and \(a\) an integer number not divisible by \(p\). Prove that there is a positive integer \(b\) such that \(ab \equiv 1 \pmod p\).
30 people vote on five proposals. In how many ways can the votes be distributed if everyone votes only for one proposal and only the number of votes cast for each proposal is taken into account?
Each of the 102 pupils of one school is friends with at least 68 others. Prove that among them there are four who have the same number of friends.
In some state, there are 101 cities.
a) Each city is connected to each of the other cities by one-way roads, and 50 roads lead into each city and 50 roads lead out of each city. Prove that you can get from each city to any other, having travelled on no more than on two roads.
b) Some cities are connected by one-way roads, and 40 roads lead into each city and 40 roads lead out of each. Prove that you can get form each city to any other, having travelled on no more than on three roads.
Some person \(A\) thought of a number from 1 to 15. Some person \(B\) asks some questions to which you can answer ‘yes’ or ‘no’. Can \(B\) guess the number by asking a) 4 questions; b) 3 questions.
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).
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.