Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?
On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.
a) Can 4 points be placed on a plane so that each of them is connected by segments with three points (without intersections)?
b) Can 6 points be placed on a plane and connected by non-intersecting segments so that exactly 4 segments emerge from each point?
A Cartesian plane is coloured in in two colours. Prove that there will be two points on the plane that are a distance of 1 apart and are the same colour.
Prove that there is no polyhedron that has exactly seven edges.
Prove that the bisectors of a triangle intersect at one point.
Prove that a convex quadrilateral \(ABCD\) can be inscribed in a circle if and only if \(\angle ABC + \angle CDA = 180^{\circ}\).
Prove that a convex quadrilateral \(ICEF\) can have a circle inscribed into it if and only if \(IC+EH = CE+IF\).
The triangle \(ABC\) is given. Find the locus of the point \(X\) satisfying the inequalities \(AX \leq CX \leq BX\).
Three segments whose lengths are equal to \(a, b\) and \(c\) are given. Using only straightedge and compass construct a segment of length: a) \(ab/c\); b) \(\sqrt {ab}\).