Problems

Prove that the number of US states with an odd number of neighbours is even.

Find the last digit of the number $1\times 2+2\times 3+\cdots +999\times 1000$.

Is the number 12345678926 square?

Prove there are no integer solutions for the equation $x^2+1990=y^2$.

There are 100 notes of two types: $a$ and $b$ pounds, and $a\ne b(\mathrm{mod}101)$. Prove that you can select several bills so that the amount received (in pounds) is divisible by 101.

Solve the equation with natural numbers $1+x+x^2+x^3=2y$.

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?