There are \(6\) people playing a game online together. Among any \(3\) people at least \(2\) people know each other. Show that there is a group of \(3\) people that all know each other.
On a certain planet the time zones can only differ by a multiple of \(1\) hour and their day is divided into hours in the same way Earth’s day is divided into hours. Show that if we pick \(25\) cities on that planet, some two cities will have the same local time.
The bag contains balls of two different colours: black and white. What is the smallest number of balls that need to be taken out of the bag blindly so that among them there are obviously two balls of the same colour?
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?
\(20\) birds fly into a photographer’s studio: \(8\) starlings, \(7\) wagtails and \(5\) woodpeckers. Each time the photographer presses the shutter to take a photograph, one of the birds flies away and does not come back. How many photographs can the photographer take to be sure that at the end there will be no fewer than \(5\) birds of one species and no less than 3 of another species remaining in the studio.
In every group of \(9\) randomly chosen crayons from Jamie’s drawer, some \(3\) will have the same colour. Show that if Jamie chooses \(25\) crayons at random, some \(7\) will have the same colour.
Suppose \(n \ge 2\) cricket teams play in a tournament. If no two teams play each other more than once, prove that some two teams have to play the same number of games.
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?