We solve problems

Problem #PRU-31097

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Algebra Number theory. Divisibility Odd and even numbers Cross-sections, nets, and skeletons Skeletons of multifaceted figures

11–13 3.0

a) What is the minimum number of pieces of wire needed in order to weld a cube’s frame?

b) What is the maximum length of a piece of wire that can be cut from this frame? (The length of the edge of the cube is 1 cm).

Problem #PRU-31110

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Prove that in a bipartite planar graph \(E \geq 2F\), if \(E \geq 2\) (\(E\) is the number of edges, \(F\) is the number of regions).

Problem #PRU-31364

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

12 teams played a volleyball tournament in one round. Two teams scored exactly 7 wins.

Prove that there are teams \(A\), \(B\), \(C\) where \(A\) won against \(B\), \(B\) won against \(C\), and \(C\) won against \(A\).

Problem #PRU-32124

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Geometry Solid geometry Tetrahedron and pyramid Particular cases of tetrahedra Regular tetrahedron

13–15 3.0

A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.

Problem #PRU-32901

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Find a natural number greater than one that occurs in the Pascal triangle a) more than three times; b) more than four times.

Problem #PRU-32992

Problems Geometry Solid geometry Regular polyhedra Main features and definitions of regular polyhedra Space transformations Projection Projection helps solve the problem Combinatorics Systems of points and line segments Systems of points and line segments (counterexamples)

13–13 2.0

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?

Problem #PRU-34972

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

For which \(n > 3\), can a set of weights with masses of \(1, 2, 3, ..., n\) grams be divided into three groups of equal mass?

Problem #PRU-34980

Problems Geometry Solid geometry Lines and planes in space Methods Pigeonhole principle

14–15 3.0

A spherical planet is surrounded by 25 point asteroids. Prove, that at any given moment there will be a point on the surface of the planet from which an astronomer will not be able to see more than 11 asteroids.

Problem #PRU-35089

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

There are \(n\) cities in a country. Between each two cities an air service is established by one of two airlines. Prove that out of these two airlines at least one is such that from any city you can get to any other city whilst traveling on flights only of this airline.

Problem #PRU-35146

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In the secret service, there are \(n\) agents – 001, 002, ..., 007, ..., \(n\). The first agent monitors the one who monitors the second, the second monitors the one who monitors the third, etc., the nth monitors the one who monitors the first. Prove that \(n\) is an odd number.