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Two play a game on a chessboard $8 × 8.$ The player who makes the first move puts a knight on the board. Then they take turns moving it $($ according to the usual rules $)$, whilst you can not put the knight on a cell which he already visited. The loser is one who has nowhere to go. Who wins with the right strategy – the first player or his partner?

The function F is given on the whole real axis, and for each $x$ the equality holds: $F (x + 1) F (x) + F (x + 1) + 1 = 0.$

Prove that the function F can not be continuous.

For which natural $n$ does the number $\frac{n^2}{1,001^n}$ reach its maximum value?

A cat tries to catch a mouse in labyrinths A, B, and C. The cat walks first, beginning with the node marked with the letter “K”. Then the mouse $($ from the node “M”$)$ moves, then again the cat moves, etc. From any node the cat and mouse go to any adjacent node. If at some point the cat and mouse are in the same node, then the cat eats the mouse.

Can the cat catch the mouse in each of the cases A, B, C?

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A B C

a$)$ Give an example of a positive number a such that ${a} + {1 / a} = 1.$

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b$)$ Can such an a be a rational number?

There are 68 coins, and it is known that any two coins differ in weight. With 100 weighings on a two-scales balance without weights, find the heaviest and lightest coin.

A group of numbers $A_1, A_2, …, A_{100}$ is created by somehow re-arranging the numbers $1, 2, …, 100$.

100 numbers are created as follows:

$$B_1=A_1, B_2=A_1+A_2, B_3=A_1+A_2+A_3, …, B_{100} = A_1+A_2+A_3…+A_{100}$$

Prove that there will always be at least 11 different remainders when dividing the numbers $B_1, B_2, …, B_{100}$ by 100.

f$(x)$ is an increasing function defined on the interval [0, 1]. It is known that the range of its values belongs to the interval [0, 1]. Prove that, for any natural N, the graph of the function can be covered by N rectangles whose sides are parallel to the coordinate axes so that the area of each is $1/N^2$. $($In a rectangle we include its interior points and the points of its boundary$)$.

Given an endless piece of chequered paper with a cell side equal to one. The distance between two cells is the length of the shortest path parallel to cell lines from one cell to the other $($it is considered the path of the center of a rook$)$. What is the smallest number of colors to paint the board $($each cell is painted with one color$)$, so that two cells, located at a distance of 6, are always painted with different colors?

$a_1$, $a_2$, $a_3$, … is an increasing sequence of natural numbers. It is known that $a_{a_k} = 3k$ for any $k.$ Find a$)$ $a_{100}$; b$)$ $a_{2022}$.

Prove that for every natural number $n > 1$ the equality: $[n^{1 / 2}] + [n^{1/ 3}] + … + [n^{1 / n}] = [log_{2}n] + [log_{3}n] + … + [log_{n}n]$ is satisfied.

Determine all integer solutions of the equation $yk = x² + x$. Where $k$ is an integer greater than $1.$

The angle at the top of a crane is 20$^{\circ}$. How will the magnitude of this angle change when looking at the crane with binoculars which triple the size of everything?

In a vase, there is a bouquet of 7 white and blue lilac branches. It is known that 1$)$ at least one branch is white, 2$)$ out of any two branches, at least one is blue. How many white branches and how many blue are there in the bouquet?

The smell of a flowering lavender plant diffuses through a radius of 20m around it. How many lavender plants must be planted along a straight 400m path so that the smell of the lavender reaches every point on the path.

Decipher the following puzzle. All the numbers indicated by the letter E, are even (not necessarily equal); all the numbers indicated by the letter O are odd (also not necessarily equal).

Along two linear park alleys are planted five oaks – three along each alley. Where should the sixth oak be planted so that it is possible to lay two more linear alleys, along each of which there would also be three oak trees growing?

On an island live knights who always tell the truth, and liars who always lie. A traveler met three islanders and asked each of them: “How many knights are among your companions?”. The first one answered: “Not one.” The second one said: “One.” What did the third man say?

What is there a greater number of: cats, except for those cats that are not named Fluffy, or animals named Fluffy, except for those that are not cats?

True or false? Prince Charming went to find Cinderella. He reached the crossroads and started to daydream. Suddenly he sees the Big Bad Wolf. And everyone knows that this Big Bad Wolf on one day answers every question truthfully, and a day later he lies, he proceeds in such a manner on alternate days. Prince Charming can ask the Big Bad Wolf exactly one question, after which it is necessary for him to choose which of the two roads to go on. What question can Prince Charming ask the Big Bad Wolf to find out for sure which of the roads leads to the Magic kingdom?