Each number of the set $\{1,2, 3,4,5,6, 7,8\}$ is colored red or blue, following the following rules: (a) Number $4$ is colored red, and there is at least one blue number, (b) if two numbers $x,y$ have different colors and $x + y \le 8$, so the number $x + y$ is colored blue, (c) if two numbers $x,y$ have different colors and $x \cdot y \le 8$, then the number $x \cdot y$ is colored red. Find all the possible ways to color this set.

$N\geq9$ distinct real numbers are written on a blackboard. All these numbers are nonnegative, and all are less than $1$. It happens that for very $8$ distinct numbers on the board, the board contains the ninth number distinct from eight such that the sum of all these nine numbers is integer. Find all values $N$ for which this is possible. [i](F. Nilov)[/i]

At a party, there are $2011$ people with a glass of fruit juice each sitting around a circular table. Once a second, they clink glasses obeying the following two rules: (a) They do not clink glasses crosswise. (b) At each point of time, everyone can clink glasses with at most one other person. How many seconds pass at least until everyone clinked glasses with everybody else? [i](Swiss Mathematical Olympiad 2011, Final round, problem 1)[/i]

A "labyrinth" is an $8 \times 8$ chessboard with walls between some neighboring squares. If a rook can traverse the entire board without jumping over the walls, the labyrinth is "good" ; otherwise it is "bad" . Are there more good labyrinths or bad labyrinths? (A Shapovalov)

Let $m, n, k > 1$ be positive integers. For a set $S$ of positive integers, define $S(i,j)$ for $i<j$ to be the number of elements in $S$ strictly between $i$ and $j$. We say two sets $(X,Y)$ are a [i]fat[/i] pair if \[ X(i,j)\equiv Y(i,j) \pmod{n} \] for every $i,j \in X \cap Y$. (In particular, if $\left\lvert X \cap Y \right\rvert < 2$ then $(X,Y)$ is fat.) If there are $m$ distinct sets of $k$ positive integers such that no two form a fat pair, show that $m<n^{k-1}$. [i]Proposed by Allen Liu[/i]

a) How many distinct ways are there are there of painting the faces of a cube six different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.) b)* How many distinct ways are there are there of painting the faces of a dodecahedron $12$ different colors? (Colorations are considered distinct if they do not coincide when the cube is rotated.)

Let $\mathbb X$ be the set of all bijective functions from the set $S=\{1,2,\cdots, n\}$ to itself. For each $f\in \mathbb X,$ define \[T_f(j)=\left\{\begin{aligned} 1, \ \ \ & \text{if} \ \ f^{(12)}(j)=j,\\ 0, \ \ \ & \text{otherwise}\end{aligned}\right.\] Determine $\sum_{f\in\mathbb X}\sum_{j=1}^nT_{f}(j).$ (Here $f^{(k)}(x)=f(f^{(k-1)}(x))$ for all $k\geq 2.$)

Each of $N$ people have chosen some $5$ elements from a $23$-element set so that any two people share at most $3$ chosen elements. Does this mean that $N \le 2020$? Answer the same question with $25$ instead of $23$.

Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that at least 85 dominoes(1×2 rectangle) can be removed from the remainder. Proposed by S. Berlov

Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop." [list][b](a)[/b] Can Andy guarantee that after he says "stop," no card is in its initial spot? [b](b)[/b] Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to the empty spot?[/list]

There is $207$ boxes on the table which numbered $1,2, \dots , 207$ respectively. Firstly Aslı puts a red ball in each of the $100$ boxes that she chooses and puts a white ball in each of the remaining ones. After that Zehra, writes a pair $(i,j)$ on the blackboard such that $1\leq i \leq j \leq 207$. Finally, Aslı tells Zehra that for every pair; whether the color of the balls which is inside the box which numbered by these numbers are the same or not. Find the least possible value of $N$ such that Zehra can guarantee finding all colors that has been painted to balls in each of the boxes with writing $N$ pairs on the blackboard.

A frog starts a journey at $(6,9).$ A skip is the act of traveling a positive integer number of units straight south or a positive integer number of units straight west. A jump is the act of traveling one unit straight west. A hop consists of any skip followed by a jump. How many different sequences of hops can the frog take so that the frog's final destination is $(0,0)$? [i]Proposed by Jack Ma[/i]

A square board with $8\text{cm}$ sides is divided into $64$ squares square with each side $1\text{cm}$. Each box can be painted white or black. Find the total number of ways to colour the board so that each square of side $2\text{cm}$ formed by four squares with a common vertex contains two white and two black squares.

3. Eight farmers have a checkered $8 \times 8$ field. There is a fence along the boundary of the field. The entire field is completely covered with berries (there is a berry in every point of the field, except the points of the fence). The farmers divided the field along the grid lines in 8 plots of equal area (every plot is a polygon), however they did not demarcate their boundaries. Each farmer takes care of berries only inside his own plot (not on its boundaries). A farmer will notice a loss only if at least two berries disappeared inside his plot. There is a crow which knows all of the above, except the location of boundaries of plots. Can the crow carry off 9 berries from the field so that for sure no farmer will notice this? Tatiana Kazitsyna

We have a row of 203 cells. Initially the leftmost cell contains 203 tokens, and the rest are empty. On each move we can do one of the following: 1)Take one token, and move it to an adjacent cell (left or right). 2)Take exactly 20 tokens from the same cell, and move them all to an adjacent cell (all left or all right). After 2023 moves each cell contains one token. Prove that there exists a token that moved left at least nine times.

There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ different starting points and $k$ different finishing points, and a cable car which starts higher also finishes higher. The same conditions hold for $B$. We say that two stations are linked by a company if one can start from the lower station and reach the higher one by using one or more cars of that company (no other movements between stations are allowed). Determine the smallest positive integer $k$ for which one can guarantee that there are two stations that are linked by both companies. [i]Proposed by Tejaswi Navilarekallu, India[/i]

Is it possible to partition the set $A = \{1, 2, 3, ... , 32, 33\}$ into eleven subsets that contain three integers each, such that for every one of these eleven subsets, one of the integers is equal to the sum of the other two? If so, give such a partition, if not, prove that such a partition cannot exist.

A $9\times9$ checkerboard is colored with 2 colors. If we choose any $3\times1$ region on the checkerboard we can paint all of the squares in that region with the color that is in the majority in that region. Show that with a finite number of these operations, we can paint the checkerboard all in the same color.

Let there be a $n\times n$ board. Write down $0$ or $1$ in all $n^2$ squares. For $1 \le k \le n$, let $A_k$ be the product of all numbers in the $k$th row. How many ways are there to write down the numbers so that $A_1 + A_2 + ... + A_n$ is even?

Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers. (Day 1, 3rd problem author: Michal Rolínek)

Consider a box of dimensions $10$ cm $\times 16$ cm $\times 1$ cm. Determine the maximum number of balls of diameter $ 1$ cm that the box can contain.

For a given set of points in space it is allowed to mirror a point from the set with respect to another point from the set, and to include the image in the set. Starting with a set of seven vertices of a cube, is it possible to include the eight vertex in the set after finitely many such steps?

a) A game is played on an infinite plane. There are fifty one pieces, one “wolf” and $50$ “sheep”. There are two players. The first commences by moving the wolf. Then the second player moves one of the sheep, the first player moves the wolf, the second player moves a sheep, and so on. The wolf and the sheep can move in any direction through a distance of up to one metre per move. Is it true that for any starting position the wolf will be able to capture at least one sheep? b) A game is played on an infinite plane. There are two players. One has a piece known as a “wolf”, while the other has $K$ pieces known as “sheep”. The first player moves the wolf, then the second player moves a sheep, the first player moves the wolf again, the second player moves a sheep, and so on. The wolf and the sheep can move in any direction, with a maximum distance of one metre per move. Is it true that for any value of $K$ there exists an initial position from which the wolf can not capture any sheep? PS. (a) was the junior version, (b) the senior one

Ria has $4$ green marbles and 8 red marbles. She arranges them in a circle randomly, if the probability that no two green marbles are adjacent is $\frac{p}{q}$ where the positive integers $p,q$ have no common factors other than $1$, what is $p+q$?

Let $ a_1 , a_2 , \cdots , a_k $ be numbers such that $ a_i \in \{ 0,1,2,3 \} ( i= 1, 2, \cdots ,k) $. Let $ z = ( x_k , x_{k-1} , \cdots , x_1 )_4 $ be a base 4 expansion of $ z \in \{ 0, 1, 2, \cdots , 4^k -1 \} $. Define $ A $ as follows: \[ A = \{ z | p(z)=z, z=0, 1, \cdots ,4^k-1 \}\] where \[ p(z) = \sum_{i=1}^{k} a_i x_i 4^{i-1} . \] Prove that the number of elements in $ X $ is a power of 2.