We solve problems

Problems Geometry Solid geometry Visual geometry in space

Out of a whole 100-vertex graph, 98 edges were removed. Prove that the remaining ones were connected.

Problems Algebra and arithmetic 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 and arithmetic 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-31088

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In a country, each two cities are connected with a one-way road.

Prove that there is a city from which you can drive to any other whilst travelling along no more than two roads.

Problem #PRU-31110

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

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

Problem #PRU-31282

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

12–14 2.0

Solve the equation in integers $2 x + 5 y = x y - 1$ .

Problem #PRU-31287

Problems Number Theory Number theory. Divisibility Equations in integer numbers

12–14 3.0

Prove there are no integer solutions for the equation $x^{2} + 1990 = y^{2}$ .

Problem #PRU-31288

Problems Algebra and arithmetic Algebraic equations and systems of equations Diophantine equations

11–13 2.0

Prove there are no integer solutions for the equation $4^{k} - 4^{l} = 10^{n}$ .

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-32088

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

13–16 4.0

It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.