We solve problems

Problem #PRU-30808

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

On the plane 100 circles are given, which make up a connected figure (that is, not falling apart into pieces). Prove that this figure can be drawn without taking the pencil off of the paper and going over any line twice.

Problem #PRU-30810

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–13 3.0

At a conference there are 50 scientists, each of whom knows at least 25 other scientists at the conference. Prove that is possible to seat four of them at a round table so that everyone is sitting next to people they know.

Problem #PRU-30811

Problems Algebra Number theory. Divisibility Odd and even numbers Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

13–15 4.0

Each of the 102 pupils of one school is friends with at least 68 others. Prove that among them there are four who have the same number of friends.

Problem #PRU-30813

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Dan drew seven graphs on the board, each of which is a tree with six vertices. Prove that among them there are two which are isomorphic.

Problem #PRU-30815

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other)

12–13 3.0

Each of the edges of a complete graph consisting of 6 vertices is coloured in one of two colours. Prove that there are three vertices, such that all the edges connecting them are the same colour.

Problem #PRU-30820

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 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 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-30830

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–14 4.0

In some state, there are 101 cities.

a) Each city is connected to each of the other cities by one-way roads, and 50 roads lead into each city and 50 roads lead out of each city. Prove that you can get from each city to any other, having travelled on no more than on two roads.

b) Some cities are connected by one-way roads, and 40 roads lead into each city and 40 roads lead out of each. Prove that you can get form each city to any other, having travelled on no more than on three roads.