Tom wants to go see a movie with his girlfriend Katie. Unfortunately the ticket selling service is broken and when you buy a ticket, you receive a ticket to a random movie that is being played that day. There are \(7\) movies being played on Friday. How many random tickets does Tom need to buy to guarantee that he and Katie will be able to go see a movie together?
In the classroom there are \(38\) people. Prove that among them there are four who were born in one month.
Is it possible to split \(44\) balls into \(9\) piles so that the number of balls in different piles is different?
An ice cream machine distributes ice cream randomly. There are 5 flavours in the machine and you would like to have any one available flavour at least 3 times. How many total samples do you need to obtain to ensure that?
Determine all prime numbers \(p\) such that \(5p+1\) is also prime.
A graph is called Bipartite if it is possible to split all its vertices into two groups in such a way that there are no edges connecting vertices from the same group. Find out whic of the following graphs are bipartite and which are not:
The next day you have even harder situation: to the hotel, where all the rooms are occupied arrives a bus with infinitely many new customers. In the bus all the seats have numbers \(1,2,3...\) corresponding to all natural numbers. How to deal with this one?
What would you do about \(10000\) new guests arriving to the full hotel?
Imagine you have now a general finite number of new guests arriving to the full hotel. What do you do?
Today you saw two infinitely long buses with seats numbered as \(1,2,3,...\) carrying infinitely many guests each arriving at the full hotel. How do you accommodate everyone?