We solve problems

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-31081

Problems Algebra and arithmetic Number theory. Divisibility Divisibility of a number. General properties

11–13 2.0

In a graph, three edges emerge from each vertex. Can there be a 1990 edges in this graph?

Problem #PRU-31082

Problems Algebra and arithmetic Number theory. Divisibility Odd and even numbers

11–13 3.0

Prove that the number of US states with an odd number of neighbours is even.

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-31235

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders

11–13 2.0

Find the last digit of the number \(1 \times 2 + 2 \times 3 + \dots + 999 \times 1000\).

Problem #PRU-31276

Problems Algebra and arithmetic Number theory. Divisibility Divisibility rules Divisibility tests for 2 and 4

11–13 2.0

Is the number 12345678926 square?

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 \(2x + 5y = xy - 1\).

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-31295

Problems Algebra and arithmetic Number theory. Divisibility Equations in integer numbers Methods Pigeonhole principle Pigeonhole principle (other)

11–13 4.0

There are 100 notes of two types: \(a\) and \(b\) pounds, and \(a \neq b \pmod {101}\). Prove that you can select several bills so that the amount received (in pounds) is divisible by 101.

Problem #PRU-31302

Problems Algebra and arithmetic 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 \(pqr = 7(p + q + r)\).