You are given 25 numbers. The sum of any 4 of these numbers is positive. Prove that the sum of all 25 numbers is also positive.
A class contains 25 pupils. It is known that within any group of 3 pupils there are two friends. Prove that there is a pupil who has no fewer than 12 friends.
A square area of size \(100\times 100\) is covered in tiles of size \(1\times 1\) in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?
One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?
It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 1989.
A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.
101 points are marked on a plane; not all of the points lie on the same straight line. A red pencil is used to draw a straight line passing through each possible pair of points. Prove that there will always be a marked point on the plane through which at least 11 red lines pass.
33 representatives of four different races – humans, elves, gnomes, and goblins – sit around a round table.
It is known that humans do not sit next to goblins, and that elves do not sit next to gnomes. Prove that some two representatives of the same peoples must be sitting next to one another.
What is the maximum number of rooks – also known as castles – you could place on an 8 by 8 chess board such that no two could take one another? Rooks can attack any number of squares horizontally and vertically, but not diagonally.