We solve problems

Problem #PRU-102797

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

Solve the equation \(\lfloor x^3\rfloor + \lfloor x^2\rfloor + \lfloor x\rfloor = \{x\} - 1\).

Problem #PRU-102844

Problems Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

12–14 3.0

The cells of a \(15 \times 15\) square table are painted red, blue and green. Prove that there are two lines which at least have the same number of cells of one colour.

Problem #PRU-102964

Problems Set theory and logic Theory of algotithms Game theory Game theory (other) Algebra and arithmetic Number theory. Divisibility Odd and even numbers

10–12 2.0

Petya and Misha play such a game. Petya takes in each hand a coin: one – 10 pence, and the other – 15. After that, the contents of the left hand are multiplied by 4, 10, 12 or 26, and the contents of the right hand – by 7, 13, 21 or 35. Then Petya adds the two results and tells Misha the result. Can Misha, knowing this result, determine which hand – the right or left – contains the 10 pence coin?

Problem #PRU-102989

Problems Algebra and arithmetic Arithmetics. Mental maths Set theory and logic Joke problem

10–12 2.0

In a bookcase, there are four volumes of the collected works of Astrid Lindgren, with each volume containing 200 pages. A worm who lives on this bookshelf has gnawed its way from the first page of the first volume to the last page of the fourth volume. Through how many pages has the worm gnawed its way through?

Problem #PRU-102998

Problems Algebra and arithmetic Number theory. Divisibility Divisibility of a number. General properties Set theory and logic Joke problem

10–12 2.0

Can the equality \(K \times O \times T = U \times W \times E \times N \times H \times Y\) be true if the numbers from 1 to 9 are substituted for the letters? Different letters correspond to different numbers.

Problem #PRU-103008

Problems Algebra and arithmetic Arithmetics. Mental maths Geometry Solid geometry Volume Ratio of volumes

10–11 2.0

When Gulliver came to Lilliput, he found that there all things were exactly 12 times shorter than in his homeland. Can you say how many Lilliputian matchboxes fit into one of Gulliver’s matchboxes?

Problem #PRU-103744

Problems Algebra and arithmetic Number systems Decimal number system

11–11 2.0

Find numbers equal to twice the sum of their digits.

Problem #PRU-103802

Problems Algebra and arithmetic 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 and arithmetic 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 and arithmetic 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.