Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?
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.
In a circle, each member has one friend and one enemy. Prove that
a) the number of members is even.
b) the circle can be divided into two neutral circles.
Out of a whole 100-vertex graph, 98 edges were removed. Prove that the remaining ones were connected.
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?