We solve problems

Problem #PRU-30820

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 some country there is a capital and another 100 cities. Some cities (including the capital) are connected by one-way roads. From each non-capital city 20 roads emerge, and 21 roads enter each such city. Prove that you cannot travel to the capital from any city.

Problem #PRU-30822

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

Prove that on the edges of a connected graph one can arrange arrows so that from some vertex one can reach any other vertex along the arrows.

Problem #PRU-30826

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Several teams played a volleyball tournament amongst themselves. We will say that team \(A\) is better than team \(B\), if either \(A\) has either beaten team \(B\), or there exists such a team \(C\) that was beaten by \(A\), whilst \(C\) beat team \(B\).

a) Prove that there is a team that is better than all.

b) Prove that the team that won the tournament is the best.

Problem #PRU-30829

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Some two teams scored the same number of points in a volleyball tournament. Prove that there are teams \(A\), \(B\) and \(C\), in which \(A\) beat \(B\), \(B\) beat \(C\) and \(C\) beat \(A\).

Problem #PRU-30831

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In the country called Orientation a one-way traffic system was introduced on all the roads, and each city can be reached from any other one by driving on no more than two roads. One road was closed for repairs but from every city it remained possible to get to any other. Prove that for every two cities this can still be done whilst driving on no more than 3 roads.

Problem #PRU-31070

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In a circle, each member has one friend and one enemy. Prove that

a) the number of members is even.

b) the circle can be divided into two neutral circles.

Problem #PRU-31072

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 some country 89 roads emerge from the capital, from the city of Dalny – one road, from the remaining 1988 cities – 20 roads (in each).

Prove that from the capital you can drive to Dalny.

Problem #PRU-31074

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

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

Problem #PRU-31078

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.