Problems

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.

A bag contains balls of two different colours: black and white. What is the smallest number of balls that needs to be taken out of the bag blindly so that among them there are obviously two balls of the same colour?

Tom wants to see a movie with his girlfriend Katie. Unfortunately the ticket selling service is broken. 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 see a movie together?

In the classroom there are $38$ people. Prove that among them there are four who were born in the same month.

Is it possible to split $44$ balls into $9$ piles so that the number of balls in different piles is different? (Each pile has to have at least one ball)

$20$ 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 at least $5$ birds of one species and at least $3$ of another species remaining in the studio?

Jamie’s drawer is pretty big. It has infinitely many crayons. In fact, for every colour of crayon, there are infinitely many crayons of that colour.
In every group of $9$ randomly chosen crayons from Jamie’s drawer, some $3$ will have the same colour. Jamie chooses $25$ crayons at random. Prove that some $7$ of them will have the same colour.

Suppose $n\ge 2$ cricket teams play in a tournament. No two teams play each other more than once, and no team plays itself. 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 one of the available flavours at least 3 times, although you don’t mind which flavour it is. How many samples do you need to obtain in total to ensure that?

Prove that among $11$ different infinite decimal fractions, you can choose two fractions which coincide in an infinite number of digits.