$11$ theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least $1$ performance of every other group. At least how many days did the festival last?

[b]O[/b]n a blackboard a stranger writes the values of $s_7(n)^2$ for $n=0,1,...,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.

In a checkered square, there is a closed door between any two cells adjacent by side. A beetle starts from some cell and travels through cells, passing through doors; she opens a closed door in the direction she is moving and leaves that door open. Through an open door, the beetle can only pass in the direction the door is opened. Prove that if at any moment the beetle wants to return to the starting cell, it is possible for her to do that.

Henry invited $2N$ guests to his birthday party. He has $N$ white hats and $N$ black hats. He wants to place hats on his guests and split his guests into one or several dancing circles so that in each circle there would be at least two people and the colors of hats of any two neighbours would be different. Prove that Henry can do this in exactly $(2N)!$ different ways. (All the hats with the same color are identical, all the guests are obviously distinct, $(2N)! = 1 \cdot 2 \cdot . . . \cdot (2N)$.) Gleb Pogudin

Prair has a box with some combination of red and green balls. If she randomly draws two balls out of the box (without replacement), the probability of drawing two balls of the same color is equal to the probability of drawing two balls of different colors! How many possible values between $200$ and $1000$ are there for the total number of balls in the box? [i]Proposed by Andrew Wu[/i]

$2023$ points are chosen on a circle. Determine the parity of the number of ways to color the chosen points blue and red (each in one color, not necessarily using both), such that among any $31$ consecutive points, there is at least one red point.

In a certain city, exactly 3 streets converge at any intersection. The streets are painted in three colors so that they converge at each intersection streets of three different colors. Three roads leave the city. Prove that they have different colors.

White and black checkers are put on the squares of an $n\times n$ chessboard $(n\ge2)$ according to the following rule. Initially, a black checker is put on an arbitrary square. In every consequent step, a white checker is put on a free square, whereby all checkers on the squares neighboring by side are replaced by checkers of the opposite colors. This process is continued until there is a checker on every square. Prove that in the final configuration there is at least one black checker.

Between the states of Alinaesquina and Berlinda, each road connects one city of Alinaesquina to one city of Berlinda. All the roads are in two-ways, and between any two cities, it is possible to travel from one to the other, using only these (possibly more than one) roads. Furthermore, it is known that, from any city of anyone of the two states, the same number of $k$ roads are going out. We know that $k\geq 2$. Prove that governments of the states can close anyone of the roads, and there will still be a route (possibly through several roads) between any two cities.

Each of the $16$ squares in a $4 \times 4$ table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to $1$. Determine the sum of all $16$ numbers in the table. (R Zhenodarov)

Consider a natural number $n\geqslant 3$ and a graph $G{}$ with a chromatic number $\chi(G)=n$ which has more than $n{}$ vertices. Prove that there exist two vertex-disjoint subgraphs $G_1{}$ and $G_2{}$ of $G{}$ such that $\chi(G_1)+\chi(G_2)\geqslant n+1.$ [i]Proposed by V. Dolnikov[/i]

Using one straight cut we partition a rectangular piece of paper into two pieces. We call this one "operation". Next, we cut one of the two pieces so obtained once again, to partition [i]this piece[/i] into two smaller pieces (i.e. we perform the operation on any [i]one[/i] of the pieces obtained). We continue this process, and so, after each operation we increase the number of pieces of paper by $1$. What is the minimum number of operations needed to get $47$ pieces of $46$-sided polygons? [obviously there will be other pieces too, but we will have at least $47$ (not necessarily [i]regular[/i]) $46$-gons.]

Let $A_1A_2...A_{101}$ be a regular $101$-gon, and colour every vertex red or blue. Let $N$ be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the $101$-gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour. $(1)$ Find the largest possible value of $N$. $(2)$ Find the number of ways to colour the vertices such that maximum $N$ is acheived. (Two colourings a different if for some $A_i$ the colours are different on the two colouring schemes).

Let be a board with $2007 \times 2007$ cells, we colour with black $P$ cells such that: $\bullet$ there are no 3 colored cells that form a L-trinomes in any of its 4 orientations Find the minimum value of $P$, such that when you colour one cell more, this configuration can't keep the condition above.

A point $O$ inside a convex $n$-gon $A_1A_2 . . .A_n$ is connected with segments to its vertices. The sides of this $n$-gon are numbered $1$ to $n$ (distinct sides have distinct numbers). The segments $OA_1,OA_2, . . . ,OA_n$ are similarly numbered. a) For $n = 9$ find a numeration such that the sum of the sides’ numbers is the same for all triangles $A_1OA_2, A_2OA_3, . . . , A_nOA_1$. b) Prove that for $n = 10$ there is no such numeration.

In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $ m$ and $ n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $ m$ and $ n$, lie along edges of the squares. Let $ S_1$ be the total area of the black part of the triangle and $ S_2$ be the total area of the white part. Let $ f(m,n) \equal{} | S_1 \minus{} S_2 |$. a) Calculate $ f(m,n)$ for all positive integers $ m$ and $ n$ which are either both even or both odd. b) Prove that $ f(m,n) \leq \frac 12 \max \{m,n \}$ for all $ m$ and $ n$. c) Show that there is no constant $ C\in\mathbb{R}$ such that $ f(m,n) < C$ for all $ m$ and $ n$.

Let $K$ be the set of all sides and diagonals of a convex $2010-gon$ in the plane. For a subset $A$ of $K,$ if every pair of line segments belonging to $A$ intersect, then we call $A$ as an [i]intersecting set.[/i] Find the maximum possible number of elements of union of two [i]intersecting sets.[/i]

On $5 \times 5$ grid of lattice points, every point is uniformly randomly colored blue, red, or green. Find the expected number of monochromatic triangles T with vertices chosen from the lattice grid, such that some two sides of $T$ are parallel to the axis.

On a circle $2018$ points are marked. Each of these points is labeled with an integer. Let each number be larger than the sum of the preceding two numbers in clockwise order. Determine the maximal number of positive integers that can occur in such a configuration of $2018$ integers. [i](Proposed by Walther Janous)[/i]

In a class of $12$ students, no two people are the same height. Compute the total number of ways for the students to arrange themselves in a line such that: [list] [*] for all $1 < i < 12$, the person in the $i$-th position (with the leftmost position being $1$) is taller than exactly $i\pmod 3$ of their adjacent neighbors, and [*] the students standing at positions which are multiples of $3$ are strictly increasing in height from left to right. [/list] [i]Proposed by Nancy Kuang[/i]

A company with $2004$ workers celebrated its anniversary by inviting everyone to a lunch served at a round table. When the $2004$ workers sat around this table, they formed a circle of people and soon discovered that they all had salaries. different and also that the difference between the salaries of any two neighbors, at the round table, was $2000$ or $3000$ pesos. Calculate the maximum difference that can exist between the wages of these workers.

Depei is imprisoned by an evil wizard and is coerced to play the following game. Every turn, Depei flips a fair coin. Then, the following events occur in this order: $\bullet$ The wizard computes the difference between the total number of heads and the total number of tails Depei has flipped. If that number is greater than or equal to $4$ or less than or equal to $-3$, then Depei is vaporized by the wizard. $\bullet$ The wizard determines if Depei has flipped at least $10$ heads or at least $10$ tails. If so, then the wizard releases Depei from the prison. The probability that Depei is released by the evil wizard equals $\frac{m}{2^k}$ , where $m, k$ are positive integers. Compute $m+k$.

Let $n \geq 3$ be a positive integer. In an election debate, we have $n$ seats arranged in a circle and these seats are numbered from $1$ to $n$, clockwise. In each of these chairs sits a politician, who can be a liar or an honest one. Lying politicians always tell lies, and honest politicians always tell the truth. At one heated moment in the debate, they accused each other of being liars, with the politician in chair $1$ saying that the politician immediately to his left is a liar, the politician in chair $2$ saying that all the $2$ politicians immediately to his left are liars, the politician in the char $3$ saying that all the $3$ politicians immediately to his left are liars, and so on. Note that the politician in chair $n$ accuses all $n$ politicians (including himself) of being liars. For what values of $n$ is this situation possible to happen?

A $7 \times 7$ square is made up of $16$ $1 \times 3$ tiles and $1$ $1 \times 1$ tile. Prove that the $1 \times 1$ tile lies either at the centre of the square or adjoins one of its boundaries .

Juty and Azgor plays the following game on a \((2n+1) \times (2n+1)\) board with Juty moving first. Initially all cells are colored white. On Juty's turn, she colors a white cell green and on Azgor's turn, he colors a white cell red. The game ends after they color all the cells of the board. Juty wins if all the green cells are connected, i.e. given any two green cells, there is at least one chain of neighbouring green cells connecting them (we call two cells [i]neighboring[/i] if they share at least one corner), otherwise Azgor wins. Determine which player has a winning strategy. [i]Proposed by Atonu Roy Chowdhury[/i]