Let \(x_1, x_2, \dots , x_n\) be some numbers belonging to the interval \([0, 1]\). Prove that on this segment there is a number \(x\) such that \[\frac{1}{n} (|x - x_1| + |x - x_2| + \dots + |x - x_n|) = 1/2.\]
A moth makes \(51\) little holes on a square cloth that is \(1\) meter on each side. Think of the holes as just tiny dots with no size. Explain why you can always cover at least \(3\) of the holes with a square patch that is \(20\) centimeters on each side.
Prove that amongst numbers written only using the number 1, i.e.: 1, 11, 111, etc, there is a number that is divisible by 1987.
Prove that there is a power of \(3\) that ends in \(001\). You can take the following fact as given: if the product \(a\times b\) of two numbers is divisible by another number \(c\), but \(a\) and \(c\) share no prime factors (we say that \(a\) and \(c\) are coprime) then \(b\) must be divisible by \(c\).
The numbers \(1, 2, \dots , 9\) are divided into three groups. Prove that the product of the numbers in one of the groups will always be no less than 72.
Prove that in any group of \(10\) whole numbers you can always find some of them that add up to a multiple of \(10\).
11 scouts are working on 5 different badges. Prove that there will be two scouts \(A\) and \(B\), such that every badge that \(A\) is working towards is also being worked towards by \(B\).
Prove that \(n^2 + 1\) is not divisible by \(3\) for any natural \(n\).
Prove that there are no natural numbers \(a\) and \(b\) such that \(a^2 - 3b^2 = 8\).
The board has the form of a cross, which is obtained if corner boxes of a square board of \(4 \times 4\) are erased. Is it possible to go around it with the help of the knight chess piece and return to the original cell, having visited all the cells exactly once?