An after school club was attended by 60 pupils. It turns out that in any group of 10 there will always be 3 classmates. Prove that within the group of 60 who attended there will always be at least 15 pupils from the same class.
A village infant school has 20 pupils. If we pick any two pupils they will have a shared granddad.
Prove that one of the granddads has no fewer than 14 grandchildren who are pupils at this school.
2022 dollars were placed into some wallets and the wallets were placed in some pockets. It is known that there are more wallets in total than there are dollars in any pocket. Is it true that there are more pockets than there are dollars in one of the wallets? You are not allowed to place wallets one inside the other.
In a basket, there are 30 mushrooms. Among any 12 of them there is at least one brown one, and among any 20 mushrooms, there is at least one chanterelle. How many brown mushrooms and how many chanterelles are there in the basket?
100 cars are parked along the right hand side of a road. Among them there are 30 red, 20 yellow, and 20 pink Mercedes. It is known that no two Mercedes of different colours are parked next to one another. Prove that there must be three Mercedes cars parked next to one another of the same colour somewhere along the road.
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 doesn’t come back. How many photographs can the photographer take to be sure that at the end there will be no fewer than 4 birds of one species and no less than 3 of another species remaining in the studio.