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Ladybirds gathered in a clearing. If the ladybug has on its back 6 spots, then it always speaks the truth, and if it has 4 spots then it always lies, and there are no other types of ladybirds in the meadow. The first ladybug said: “We each have the same number of spots on our backs.” The second one said: “Everyone has 30 spots on their backs in total.” “No, they all have 26 spots on their backs in total,” the third objected. “Of these three, exactly one told the truth,” – said each of the other ladybirds. How many ladybirds were gathered in the meadow?

In front of a gnome there lie three piles of diamonds: one with 17, one with 21 and one with 27 diamonds. In one of the piles lies one fake diamond. All the diamonds have the same appearance, and all real diamonds weigh the same, and the fake one differs in its weight. The gnome has a cup weighing scale without weights. The dwarf must find with one weighing a pile, in which all the diamonds are real. How should he do it?

2012 pine cones lay under the fir-tree. Winnie the Pooh and the donkey Eeyore play a game: they take turns picking up these pine cones. Winnie-the-Pooh takes either one or four cones in each of his turns, and Eeyore – either one or three. Winnie the Pooh goes first. The player who cannot make a move loses. Which of the players can be guaranteed to win, no matter how their opponent plays?

We are given a polynomial P $(x)$ and numbers $a_1$, $a_2$, $a_3$, $b_1$, $b_2$, $b_3$ such that $a_1a_2a_3$ ≠ 0. It turned out that P $(a_1x + b_1)$ + P $(a_2x + b_2)$ = P $(a_3x + b_3)$ for any real x. Prove that P $(x)$ has at least one real root.

On a board there are written four three-digit numbers, totaling 2012. To write them all, only two different digits were used. $\\$

Give an example of such numbers.

In the equality TIME + TICK = SPIT, replace the same letters with the same numbers, and different letters with different digits so that the word TICK is as small as possible $($ there are no zeros among the digits $)$.

At a round table, 30 people are sitting – knights and liars $($ knights always tell the truth, and liars always lie $)$. It is known that each of them at that table has exactly one friend, and for each knight this friend is a liar, and for a liar this friend is a knight $($ friendship is always mutual $)$. To the question “Does your friend sit next to you?” those in every other seat answered “yes”. How many of the others could also have said “Yes”?

There are 100 boxes numbered from 1 to 100. In one box there is a prize and the presenter knows where the prize is. The spectator can send the presented a pack of notes with questions that require a “yes” or “no” answer. The presenter mixes the notes in a bag and, without reading out the questions aloud, honestly answers all of them. What is the smallest number of notes you need to send to know for sure where the prize is?

There are 40 weights of weights of 1 g, 2 g, …, 40 grams. Of these, 10 weights of even weight were chosen and placed on the left hand side of the scales. Then we selected 10 weights of odd weight and put it on the right hand side of the scales. The scales were balanced. Prove that on one of the bowls of the scales there are two weights with a mass difference of 20 g.

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?

In the king’s prison, there are five cells numbered from 1 to 5. In each cell, there is one prisoner. Kristen persuaded the king to conduct an experiment: on the wall of each cell she writes at one point a number and at midnight, each prisoner will go to the cell with the indicated number $($if the number on the wall coincides with the cell number, the prisoner does not go anywhere$)$. On the following night at midnight, the prisoners again must move from their cell to another cell according to the instructions on the wall, and they do this for five nights. If the location of prisoners in the cells for all six days $($including the first$)$ is never repeated, then Kristen will be given the title of Wisdom, and the prisoners will be released. Help Kristen write numbers in the cells.

Hannah has a calculator that allows you to multiply a number by 3, add 3 to the number or $($4 if the number is divisible by 3 to make a whole number$)$ divide by 3. How can the number 11 be made on this calculator from the number 1?

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.

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.

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?

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 B is also a faction.

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?

$x_1$ 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 $)$.

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.