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.
There are 30 people, among which some are friends. Prove that the number of people who have an odd number of friends is even.
25 cells were coloured in on a sheet of squared paper. Can each of them have an odd number of coloured in neighbouring cells?
Can the degrees of vertices in the graph be equal to:
a) 8, 6, 5, 4, 4, 3, 2, 2?
b) 7, 7, 6, 5, 4, 2, 2, 1?
c) 6, 6, 6, 5, 5, 3, 2, 2?