We solve problems

Problem #PRU-31078

Problems Algebra Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

In a graph there are 100 vertices, and the degree of each of them is not less than 50. Prove that the graph is connected.

Problem #PRU-31087

Problems Algebra Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

The faces of a polyhedron are coloured in two colours so that the neighbouring faces are of different colours. It is known that all of the faces except for one have a number of edges that is a multiple of 3. Prove that this one face has a multiple of 3 edges.

Problem #PRU-31282

Problems Algebra Algebraic equations and systems of equations Algebraic equations and systems of equations (other)

12–14 2.0

Solve the equation in integers \(2x + 5y = xy - 1\).

Problem #PRU-31287

Problems Number Theory Divisibility Equations in integer numbers

12–14 3.0

Prove there are no integer solutions for the equation \(x^2=y^2+1990\).

Problem #PRU-31302

Problems Algebra Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other)

12–14 3.0

Recall that a natural number \(x\) is called prime if \(x\) has no divisors except \(1\) and itself. Solve the equation with prime numbers \(pqr = 7(p + q + r)\).

Problem #PRU-31304

Problems Number Theory Divisibility Equations in integer numbers

12–14 3.0

Solve the equation with natural numbers \(1 + m + m^2 + m^3 = 2^n\).

Problem #PRU-31362

Problems Discrete Mathematics Algorithm Theory Theory of algorithms (other)

12–14 2.0

Some person \(A\) thought of a number from 1 to 15. Some person \(B\) asks some questions to which you can answer ‘yes’ or ‘no’. Can \(B\) guess the number by asking a) 4 questions; b) 3 questions.

Problem #PRU-31363

Problems Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

12–14 3.0

a) In a group of 4 people, who speak different languages, any three of them can communicate with one another; perhaps by one translating for two others. Prove that it is always possible to split them into pairs so that the two members of every pair have a common language.

b) The same, but for a group of 100 people.

c) The same, but for a group of 102 people.

Problem #PRU-31377

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–14 3.0

There are two identical gears with 14 teeth on a common shaft. They are aligned and four pairs of teeth are removed.

Prove that the gears can be rotated so that they form a complete gear (one containing no gaps).

Problem #PRU-32024

Problems Methods Pigeonhole principle Pigeonhole principle (other) Algebraic methods Processes and operations

12–14 3.0

In order to glaze 15 windows of different shapes and sizes, 15 pieces of glass are prepared exactly for the size of the windows (windows are such that each window should have one glass). The glazier, not knowing that the glass is specifically selected for the size of each window, works like this: he approaches a certain window and sorts out the unused glass until he finds one that is large enough (that is, either an exactly suitable piece or one from which the right size can be cut), if there is no such glass, he goes to the next window, and so on, until he has assessed each window. It is impossible to make glass from several parts. What is the maximum number of windows which can be left unglazed?