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.
Let \(n\) numbers are given together with their product \(p\). The difference between \(p\) and each of these numbers is an odd number.
Prove that all \(n\) numbers are irrational.
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.
a) Could an additional \(6\) digits be added to any \(6\)-digit number starting with a \(5\), so that the \(12\)-digit number obtained is a complete square?
b) The same question but for a number starting with a \(1\).
c) Find for each \(n\) the smallest \(k = k (n)\) such that to each \(n\)-digit number you can assign \(k\) more digits so that the resulting \((n + k)\)-digit number is a complete square.
Initially, on each cell of a \(1 \times n\) board a checker is placed. The first move allows you to move any checker onto an adjacent cell (one of the two, if the checker is not on the edge), so that a column of two pieces is formed. Then one can move each column in any direction by as many cells as there are checkers in it (within the board); if the column is on a non-empty cell, it is placed on a column standing there and unites with it. Prove that in \(n - 1\) moves you can collect all of the checkers on one square.
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.