A gang contains 101 gangsters. The whole gang has never taken part in a raid together, but every possible pair of gangsters have taken part in a raid together exactly once. Prove that one of the gangsters has taken part in no less than 11 different raids.
Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?
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.
4 points \(a, b, c, d\) lie on the segment \([0, 1]\) of the number line. Prove that there will be a point \(x\), lying in the segment \([0, 1]\), that satisfies \[\frac{1}{ | x-a |}+\frac{1}{ | x-b |}+\frac{1}{ | x-c |}+\frac{1}{ | x-d |} < 40.\]
Some points with integer co-ordinates are marked on a Cartesian plane. It is known that no four points lie on the same circle. Prove that there will be a circle of radius 1995 in the plane, which does not contain a single marked point.
Some real numbers \(a_1, a_2, a_3,\dots ,a _{2022}\) are written in a row. Prove that it is possible to pick one or several adjacent numbers, so that their sum is less than 0.001 away from a whole number.
In a regular shape with 25 vertices, all the diagonals are drawn.
Prove that there are no nine diagonals passing through one interior point of the shape.
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?