Prove that in any group of 6 people there are either three pairs of people who know one another, or three pairs of people who do not know one another.
A warehouse contains 200 boots of each of the sizes 8, 9, and 10. Amongst these 600 boots, 300 are left boots and 300 are right boots. Prove that there are at least 100 usable pairs of boots in the warehouse.
The alphabet of the Ni-Boom-Boom tribe contains 22 consonants and 11 vowels. A word in this language is defined as any combination of letters in which there are no consecutive consonants and no letter is used more than once. The alphabet is divided into 6 non-empty groups. Prove that it is possible to construct a word from all the letters in the group in at least one of the groups.
Prove that in any group of 10 whole numbers there will be a few whose sum is divisible by 10.
You are given 11 different natural numbers that are less than or equal to 20. Prove that it is always possible to choose two numbers where one is divisible by the other.
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 \(B\).
Is it possible to place the numbers \(-1, 0, 1\) in a \(6\times 6\) square such that the sums of each row, column, and diagonal are unique?
Let \(p\) be a prime number, and \(a\) an integer number not divisible by \(p\). Prove that there is a positive integer \(b\) such that \(ab \equiv 1 \pmod p\).
On the plane, 10 points are marked so that no three of them lie on the same line. How many triangles are there with vertices at these points?
At a conference there are 50 scientists, each of whom knows at least 25 other scientists at the conference. Prove that is possible to seat four of them at a round table so that everyone is sitting next to people they know.