We solve problems

Problem #PRU-98648

Problems Methods Examples and counterexamples. Constructive proofs Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

Elephants, rhinoceroses, giraffes. In all zoos where there are elephants and rhinoceroses, there are no giraffes. In all zoos where there are rhinoceroses and there are no giraffes, there are elephants. Finally, in all zoos where there are elephants and giraffes, there are also rhinoceroses. Could there be a zoo in which there are elephants, but there are no giraffes and no rhinoceroses?

Problem #PRU-98652

Problems Geometry Plane geometry Area Area ratio Area ratio of similar figures

11–12 1.0

The population of China is one billion people. It would seem that on a map of China with a scale of 1 : 1,000,000 (1 cm : 10 km), it would be possible to fit a million times fewer people than there is in the whole country. However, in fact, not only 1000, but even 100 people will not be able to be placed on this map. Can you explain this contradiction?

Problem #PRU-98665

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

10–12 2.0

Several natives of an island met up (each either a liar or a knight), and everyone said to everyone else: “You are all liars.” How many knights were there among them?

Problem #PRU-98670

Problems Methods Examples and counterexamples. Constructive proofs

11–12 2.0

Among 4 people there are no three with the same name, the same middle name and the same surname, but any two people have either the same first name, middle name or surname. Can this be so?

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-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

Problems Fun Problems Word Problems

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-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.