We solve problems

Problem #PRU-32987

Problems Algebra Algebraic equations and systems of equations

11–13 2.0

Prove there are no integer solutions for the equation \(3x^2 + 2 = y^2\).

Problem #PRU-32991

Problems Number Theory Divisibility Odd and even numbers

13–13 2.0

Can seven phones be connected with wires so that each phone is connected to exactly three others?

Problem #PRU-32992

Problems Geometry Solid geometry Regular polyhedra Main features and definitions of regular polyhedra Space transformations Projection Projection helps solve the problem Discrete Mathematics 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-32994

Problems Number Theory Divisibility Divisibility of a number. General properties

13–13 2.0

Write out in a row the numbers from \(1\) to \(9\) (every number once) so that every two consecutive numbers give a two-digit number that is divisible by \(7\) or by \(13\).

Problem #PRU-32996

Problems Discrete Mathematics Graph Theory

13–13 2.0

Several Top Secret Objects are connected by an underground railway in such a way that each Object is directly connected to no more than three others and from each Object one can reach any other Object by going and by changing no more than once. What is the maximum number of Top Secret Objects?

Problem #PRU-33134

Problems Number Theory Divisibility Odd and even numbers

11–13 2.0

Prove that the sum of

a) any number of even numbers is even;

b) an even number of odd numbers is even;

c) an odd number of odd numbers is odd.

Problem #PRU-33135

Problems Number Theory Divisibility Odd and even numbers

11–13 2.0

Prove that the product of

a) two odd numbers is odd;

b) an even number with any integer is even.

Problem #PRU-34834

Problems Algebra Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

12–14 3.0

A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?

Problem #PRU-34840

Problems Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

12–13 2.0

Several guests are sitting at a round table. Some of them are familiar with each other; mutually acquainted. All the acquaintances of any guest (counting himself) sit around the table at regular intervals. (For another person, these gaps may be different.) It is known that any two have at least one common acquaintance. Prove that all guests are familiar with each other.

Problem #PRU-34851

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

15 MPs take part in a debate. During the debate, each one criticises exactly \(k\) of the 14 other contributors. For what minimum value of \(k\) is it possible to definitively state that there will be two MPs who have criticised one another?