In a graph, three edges emerge from each vertex. Can there be a 1990 edges in this graph?
Prove that the number of US states with an odd number of neighbours is even.
a) What is the minimum number of pieces of wire needed in order to weld a cube’s frame?
b) What is the maximum length of a piece of wire that can be cut from this frame? (The length of the edge of the cube is 1 cm).
Find the last digit of the number \(1 \times 2 + 2 \times 3 + \dots + 999 \times 1000\).
Is the number 12345678926 square?
There are 100 notes of two types: \(a\) and \(b\) pounds, and \(a \neq b \pmod {101}\). Prove that you can select several bills so that the amount received (in pounds) is divisible by 101.
In a room there are some chairs with 4 legs and some stools with 3 legs. When each chair and stool has one person sitting on it, then in the room there are a total of 39 legs. How many chairs and stools are there in the room?
Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 1989.
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.
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?