In each square of a rectangular table of size \(M \times K\), a number is written. The sum of the numbers in each row and in each column, is 1. Prove that \(M = K\).
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.
The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from \(n\) books of different heights, if: a) \(n = 4\); b) \(n = 5\)?
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.
Lessons at the Evening Mathematical School take place in nine auditoriums. Amongst the class were 19 students from the same school.
a) Prove that no matter how these students are arranged at least one auditorium will contain no fewer than 3 of these students.
b) Is it true that one of the auditoriums must contain exactly 3 of these students?
12 straight lines passing through the origin are drawn on a plane. Prove that it is possible to choose two of these lines such that the angle between them is less than 17 degrees.
Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?