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?
A professional tennis player plays at least one match each day for training purposes. However in order to ensure he does not over-exert himself he plays no more than 12 matches a week. Prove that it is possible to find a group of consecutive days during which the player plays a total of 20 matches.
During the year, the price for a strudel were twice raised by 50%, and before the New Year they were sold at half price. How much does one strudel cost now, if at the beginning of the year it cost 80 pence?
Harry and Matt came down from a mountain. Harry walked on foot, and Matt went skiing, which was seven times faster than Harry. Halfway down, Matt fell, broke his skis and his leg, and hence travelled twice as slow as Harry. Who will descend first from the mountain?
A country is called a Fiver if, in it, each city is connected by airlines with exactly with five other cities (there are no international flights).
a) Draw a scheme of airlines for a country that is made up of 10 cities.
b) How many airlines are there in a country of 50 cities?
c) Can there be a Fiver country, in which there are exactly 46 airlines?