Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 1989.
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?
A number is written on each edge of a cube. The sum of the 4 numbers on the adjacent edges is written on each face. Place the numbers \(1\) and \(-1\) on the edges so that the numbers written on the faces are all different.
It is known that among the members of the government of the Planet of Liars and truth tellers, consisting of 20 members, there is at least one honest one, and also that from any two at least one is a bribe taker. How many bribe takers are there in the government?
An adventurer is travelling to the planet of liars and truth tellers with an official guide and is introduced to a local. “Are you a truth teller?” asked the adventurer. The alien answers “Yrrg,” which means either “yes” or “no”. The adventurer asks the guide for a translation. The guide says “"yrrg" means "yes". I will add that the local is actully a liar.” Is the local alien liar or truth teller?