We solve problems

Problems Algebra Algebraic equations and systems of equations

Solve the equations \(x^2 = 14 + y^2\) in integers.

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

11–14 2.0

Solve the equation with integers \(x^2 + y^2 = 4z - 1\).

Problem #PRU-30682

Problems Number Theory Divisibility Divisibility of a number. General properties Division with remainders. Arithmetic of remainders Fermat's little theorem Methods Pigeonhole principle Pigeonhole principle (other)

14–15 3.0

Let \(p\) be a prime number, and \(a\) an integer number not divisible by \(p\). Prove that there is a positive integer \(b\) such that \(ab \equiv 1 \pmod p\).

Problem #PRU-30723

Problems Discrete Mathematics Combinatorics

13–14 3.0

30 people vote on five proposals. In how many ways can the votes be distributed if everyone votes only for one proposal and only the number of votes cast for each proposal is taken into account?

Problem #PRU-30782

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

a) they have 10 vertices, the degree of each of which is equal to 9?

b) they have 8 vertices, the degree of each of which is equal to 3?

c) are they connected, without cycles and contain 6 edges?

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-30811

Problems 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-30820

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 Problems Algebra

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-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.