A pawn stands on one of the squares of an endless in both directions chequered strip of paper. It can be shifted by \(m\) squares to the right or by \(n\) squares to the left. For which \(m\) and \(n\) can it move to the next cell to the right?
Let \(p\) be a prime number, and \(a\) an integer number not divisible by \(p\). Prove that there is a positive integer \(b\) such that \(ab \equiv 1 \pmod p\).
How many different four-digit numbers, divisible by 4, can be made up of the digits 1, 2, 3 and 4,
a) if each number can occur only once?
b) if each number can occur several times?
How many integers are there from 0 to 999999, in the decimal notation of which there are no two identical numbers next to each other?
Each of the 102 pupils of one school is friends with at least 68 others. Prove that among them there are four who have the same number of friends.
In what number system is the equality \(3 \times 4 = 10\) correct?
Prove that any axis of symmetry of a 45-gon passes through its vertex.
Is the number \(1 + 2 + 3 + \dots + 1990\) odd or even?
Every Martian has three hands. Can seven Martians join hands?
At the vertices of a \(n\)-gon are the numbers \(1\) and \(-1\). On each side is written the product of the numbers at its ends. It turns out that the sum of the numbers on the sides is zero. Prove that a) \(n\) is even; b) \(n\) is divisible by 4.