We solve problems

Problem #PRU-100705

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

Show that the equation \(x^2 +6x-1 = y^2\) has no solutions in integer \(x\) and \(y\).

Problem #PRU-102796

Problems Combinatorics Integer lattices Integer lattices (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.

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-102815

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Propose a method for measuring the diagonal of a conventional brick, which is easily realied in practice (without the Pythagorean theorem).

Problem #PRU-102818

Problems Set theory and logic Algorithm Theory Game Theory Symmetric strategies

12–13 3.0

A two-player game with matches. There are 37 matches on the table. In each turn, a player is allowed to take no more than 5 matches. The winner of the game is the player who takes the final match. Which player wins, if the right strategy is used?

Problem #PRU-102822

Problems Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

The seller with weights. With four weights the seller can weigh any integer number of kilograms, from 1 to 40 inclusive. The total mass of the weights is 40 kg. What are the weights available to the seller?

Problem #PRU-102839

12–14 2.0

In the US, it is customary to record the date as follows: the number of the month, then the number of the day and then the year. In Europe, the number comes first, then the month and then the year. How many days are there in the year, the date of which can be read definitively, without knowing how it was written?

Problem #PRU-102841

Problems Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

Two weighings. There are 7 coins which are identical on the surface, including 5 real ones (all of the same weight) and 2 counterfeit coins (both of the same weight, but lighter than the real ones). How can you find the 3 real coins with the help of two weighings on scales without weights?

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-102845

Problems Combinatorics Partitions Dissections with certain properties

11–12 2.0

Cutting into four parts. Cut each of the figures below into four equal parts (you can cut along the sides and diagonals of cells).