On the plane 100 circles are given, which make up a connected figure (that is, not falling apart into pieces). Prove that this figure can be drawn without taking the pencil off of the paper and going over any line twice.
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 some country there is a capital and another 100 cities. Some cities (including the capital) are connected by one-way roads. From each non-capital city 20 roads emerge, and 21 roads enter each such city. Prove that you cannot travel to the capital from any city.
Prove that on the edges of a connected graph one can arrange arrows so that from some vertex one can reach any other vertex along the arrows.
In what number system is the equality \(3 \times 4 = 10\) correct?
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.