In the graph, each vertex is either blue or green. Each blue vertex is linked to five blue and ten green vertices, and each green vertex is linked to nine blue and six green vertices. Which vertices are there more of – blue or green ones?
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.
Find the last digit of the number \(1 \times 2 + 2 \times 3 + \dots + 999 \times 1000\).
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 \(a \neq b \pmod {101}\). 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.