We solve problems

Problem #PRU-103757

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 resident of one foreign intelligence agency informed the centre about the forthcoming signing of a number of bilateral agreements between the fifteen former republics of the USSR. According to his report, each of them will conclude an agreement exactly with three others. Should this resident be trusted?

Problem #PRU-103791

Problems Algebra Number systems Decimal number system Set theory and logic Mathematical logic Puzzles

12–12 3.0

In the rebus below, replace the letters with numbers such that the same numbers are represented with the same letter. The asterisks can be replaced with any numbers such that the equations hold.

An explanation of the notation used: the unknown numbers in the third and fourth rows are the results of multiplying 1995 by each digit of the number in the second row, respectively. These third and fourth rows are added together to get the total result of the multiplication \(1995 \times ***\), which is the number in the fifth row. This is an example of a “long multiplication table”.

Problem #PRU-103802

Problems Algebra Number systems Decimal number system Number theory. Divisibility Divisibility of a number. General properties Set theory and logic Set theory Set cardinality. One-to-one correspondence

12–13 2.0

Which five-digit numbers are there more of: ones that are not divisible by 5 or those with neither the first nor the second digit on the left being a five?

Problem #PRU-103812

Problems Algebra Arithmetic operations. Number identities Set theory and logic Mathematical logic Puzzles

11–11 2.0

Alex laid out an example of an addition of numbers from cards with numbers on them and then swapped two cards. As you can see, the equality has been violated. Which cards did Alex rearrange?

Problem #PRU-103815

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Algebra Word Problems Tables and tournaments Tables and tournaments (other)

11–11 2.0

Cut the board shown in the figure into four congruent parts so that each of them contains three shaded cells. Where the shaded cells are placed in each part need not be the same.

Problem #PRU-103825

Problems Algebra Arithmetics. Mental maths Methods Examples and counterexamples. Constructive proofs Set theory and logic Algorithm Theory Algorithm Theory

10–12 2.0

Three hedgehogs divided three pieces of cheese of mass of 5g, 8g and 11g. The fox began to help them. It can cut off and eat 1 gram of cheese from any two pieces at the same time. Can the fox leave the hedgehogs equal pieces of cheese?

Problem #PRU-103858

Problems Algebra Number theory. Divisibility The fundamental theorm of arithmetic. Prime factorisation.

12–13 2.0

Solve the rebus: \(AX \times UX = 2001\).

Problem #PRU-103884

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Methods Examples and counterexamples. Constructive proofs Algebra Fractions Simple fractions

11–12 2.0

A rectangle is cut into several smaller rectangles, the perimeter of each of which is an integer number of meters. Is it true that the perimeter of the original rectangle is also an integer number of meters?

Problem #PRU-103887

Problems Set theory and logic Mathematical logic Puzzles Algebra Fractions Simple fractions

12–12 2.0

Arrange brackets and arithmetic signs around these numbers so that the correct equality is obtained: \[\frac{1}{2}\quad \frac{1}{6}\quad \frac{1}{6009} \ = \ 2003.\]

Problem #PRU-103946

Problems Algebra Arithmetic operations. Number identities

11–12 2.0

The number \(A\) is positive, \(B\) is negative, and \(C\) is zero. What is the sign of the number \(AB + AC + BC\)?