We solve problems

Problems Geometry Solid geometry Convex solids Graph theory Polyhedra and polygons in space

Prove that for every convex polyhedron there are two faces with the same number of sides.

Problems Geometry Solid geometry Spheres Circle on a sphere Methods Pigeonhole principle Pigeonhole principle (area and volume) Extremal principle Smallest or largest distance (length)

14–16 4.0

A spherical sun is observed to have a finite number of circular sunspots, each of which covers less than half of the sun’s surface. These sunspots are said to be enclosed, that is no two sunspots can touch, and they do not overlap with one another. Prove that the sun will have two diametrically opposite points that are not covered by sunspots.

Problem #PRU-87973

Problems Algebra and arithmetic Arithmetics. Mental maths Geometry Solid geometry Volume Ratio of volumes Space transformations Similarity

10–12 2.0

When Gulliver came to Lilliput, he found that everything was exactly 12 times shorter than in his homeland. Can you say how many Lilliputian matchboxes fit into the matchbox of Gulliver?

Problem #PRU-88301

Problems Algebra and arithmetic Arithmetics. Mental maths Geometry Solid geometry Volume

10–12 2.0

A cube with a side of 1 m was sawn into cubes with a side of 1 cm and they were in a row (along a straight line). How long was the line?

Problem #PRU-102815

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Propose a method for measuring the diagonal of a conventional brick, which is easily realied in practice (without the Pythagorean theorem).

Problem #PRU-103008

Problems Algebra and arithmetic Arithmetics. Mental maths Geometry Solid geometry Volume Ratio of volumes

10–11 2.0

When Gulliver came to Lilliput, he found that there all things were exactly 12 times shorter than in his homeland. Can you say how many Lilliputian matchboxes fit into one of Gulliver’s matchboxes?

Problem #PRU-103824

Problems Geometry Solid geometry Spheres Circle on a sphere Visual geometry in space

11–13 2.0

A globe has 17 parallels and 24 meridians. How many parts is the globe’s surface divided into? The meridian is an arc connecting the North Pole with the South Pole. A parallel is a circle parallel to the equator (the equator is also a parallel).

Problem #PRU-103888

Problems Methods Algebraic methods Proof by exhaustion Geometry Solid geometry Visual geometry in space

11–12 2.0

A square napkin was folded in half, the resulting rectangle was then folded in half again (see the figure). The resulting square was then cut with scissors (in a straight line). Could the napkin have been broken up a) into 2 parts? b) into 3 parts? c) into 4 parts? d) into 5 parts? If yes – illustrate such a cut, if not – write the word “no”.

Problem #PRU-104021

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Algebra and arithmetic Number theory. Divisibility Odd and even numbers Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

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.

Problem #PRU-107702

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Combinatorics Dissections, partitions, covers and tilings Features of dissection pieces Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–16 3.0

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 a vertex or are directly adjacent to another marked square?