The edges of a cube are assigned with integer values. For each vertex we look at the numbers corresponding to the three edges coming from this vertex and add them up. In case we get 8 equal results we call such cube “cute”. Are there any “cute” cubes with the following numbers corresponding to the edges:
(a) \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\);
(b) \(-6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6\)?
40% of adherents of some political party are women. 70% of the adherents of this party are townspeople. At the same time, 60% of the townspeople who support the party are men. Are the events “the adherent of the party is a townsperson” and “the adherent of party is a woman” independent?
Out of the given numbers 1, 2, 3, ..., 1000, find the largest number \(m\) that has this property: no matter which \(m\) of these numbers you delete, among the remaining \(1000 - m\) numbers there are two, of which one is divisible by the other.
a) A 1 or a 0 is placed on each vertex of a cube. The sum of the 4 adjacent vertices is written on each face of the cube. Is it possible for each of the numbers written on the faces to be different?
b) The same question, but if 1 and \(-1\) are used instead.
The surface of a \(3\times 3\times 3\) Rubik’s Cube contains \(54\) squares. What is the maximum number of squares we can mark so that no marked squares share at least one vertex?
Make sure you show that both (a) you can achieve this maximum and (b) that you can’t do better than this maximum.
Will the entire population of the Earth, all buildings and structures on it, fit into a cube with a side length of 3 kilometres?
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?
Prove that there is no polyhedron that has exactly seven edges.
The Great Pyramid of Giza is the largest pyramid in Egypt. For the purposes of this problem, assume that it’s a perfect square-based pyramid, with perpendicular height \(140\)m and the square has side length \(230\)m.
What is its volume in cubic metres?
The volume of a pyramid is \(\frac{1}{3}Bh\), where \(B\) is the area of the base and \(h\) is the perpendicular height. What’s the volume of a regular tetrahedron with side length \(1\)?