Problems

The bank of the Nile was approached by a group of six people: three Bedouins, each with his wife. At the shore is a boat with oars, which can withstand only two people at a time. A Bedouin can not allow his wife to be without him whilst in the company of another man. Can the whole group cross to the other side?

Cut the interval \([-1, 1]\) into black and white segments so that the integrals of any a) linear function; b) a square trinomial in white and black segments are equal.

Solve problem number 108736 for the inscription \(A\), \(BC\), \(DEF\), \(CGH\), \(CBE\), \(EKG\).

\(x_1\) is the real root of the equation \(x^2 + ax + b = 0\), \(x_2\) is the real root of the equation \(x^2 - ax - b = 0\).

Prove that the equation \(x^2 + 2ax + 2b = 0\) has a real root, enclosed between \(x_1\) and \(x_2\). (\(a\) and \(b\) are real numbers).

With a non-zero number, the following operations are allowed: \(x \rightarrow \frac{1+x}{x}\), \(x \rightarrow \frac{1-x}{x}\). Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

Members of the State parliament formed factions in such a way that for any two factions \(A\) and \(B\) (not necessarily different)

– also a faction (through

the set of all parliament members not included in \(C\) is denoted). Prove that for any two factions \(A\) and \(B\), \(A \cup % \includegraphics{https://problems-static.s3.eu-west-2.amazonaws.com/static/test/task_images/82/109909-3.png} B\) is also a faction.

A magician with a blindfold gives a spectator five cards with the numbers from 1 to 5 written on them. The spectator hides two cards, and gives the other three to the assistant magician. The assistant indicates to the spectator two of them, and the spectator then calls out the numbers of these cards to the magician (in the order in which he wants). After that, the magician guesses the numbers of the cards hidden by the spectator. How can the magician and the assistant make sure that the trick always works?

In 10 boxes there are pencils (there are no empty boxes). It is known that in different boxes there is a different number of pencils, and in each box, all pencils are of different colors. Prove that from each box you can choose a pencil so that they will all be of different colors.

Given a square trinomial \(f (x) = x^2 + ax + b\). It is known that for any real \(x\) there exists a real number \(y\) such that \(f (y) = f (x) + y\). Find the greatest possible value of \(a\).

Which numbers can stand in place of the letters in the equality \(AB \times C = DE\), if different letters denote different numbers and from left to right the numbers are written in ascending order?