In a chess tournament $ 2n\plus{}3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

There are given in a table numbers $1,2,...,18$. What is minimal number of numbers we should erase such that the sum of every two remaining numbers is not perfect square of a positive integer.

Danielle has an $m \times n$ board and wants to fill it with pieces composed of two or more diagonally connected squares as shown, without overlapping or leaving gaps: a) Find all values of $(m,n)$ for which it is possible to fill the board. b) If it is possible to fill an $m \times n$ board, find the minimum number of pieces Danielle can use to fill it. [i]Note: The pieces can be rotated[/i].

We say the point $i$ in the permutation $\sigma$ [b]ongoing[/b] if for every $j<i$ we have $\sigma (j)<\sigma (i)$. [b]a)[/b] prove that the number of permutations of the set $\{1,....,n\}$ with exactly $r$ ongoing points is $s(n,r)$. [b]b)[/b] prove that the number of $n$-letter words with letters $\{a_1,....,a_k\},a_1<.....<a_k$. with exactly $r$ ongoing points is $\sum_{m}\dbinom{k}{m} S(n,m) s(m,r)$.

On a table, there are $2025$ empty boxes numbered $1,2,\dots ,2025$, and $2025$ balls with weights $1,2,\dots ,2025$. Starting with Vadim, Vadim and Marian take turns selecting a ball from the table and placing it into an empty box. After all $2025$ turns, there is exactly one ball in each box. Denote the weight of the ball in box $i$ by $w_i$. Marian wins if $$\sum_{i=1}^{2025}i\cdot w_i\equiv 0 \pmod{23}.$$ If both players play optimally, can Marian guarantee a win? [i]Proposed by Pavel Ciurea[/i]

A certain Oxford professor, assigned to espionage cryptography services British, role played by Dirk Bogarde in a film, recruits his proposing small attention exercises, such as mentally reading a word the other way around. Frequently he does it with his own name: $SEBASTIAN$, what will there be to read $NAITSABES$. He wonders if there is any movement of the plane or of space that transforms one of these words in the other, just as they appear written. And if it had been called $AVITO$, like a certain Unamuno character? Give a reasoned explanation for each answer.

An integer $1$ is written on the blackboard. We are allowed to perform the following operations:to change the number $x$ to $3x+1$ of to $[\frac{x}{2}]$. Prove that we can get all positive integers using this operations.

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

In a lottery, a person must select six distinct numbers from $1, 2, 3,\dots, 36$ to put on a ticket. The lottery commitee will then draw six distinct numbers randomly from $1, 2, 3, \ldots, 36$. Any ticket with numbers not containing any of these $6$ numbers is a winning ticket. Show that there is a scheme of buying $9$ tickets guaranteeing at least one winning ticket, but $8$ tickets are not enough to guarantee a winning ticket in general.

Ana writes a three-digit code, and Beto has to guess it. To do so, he can ask about a sequence of three digits, and Ana will respond "warm" if the sequence Beto proposes has at least one correct digit in the correct position, and she will respond "cold" if none of the digits are correct. For example, if the correct code is $014$, then if Beto asks $099$ or $014$, he receives the answer "warm", and if he asks $140$ or $322$, he receives the answer "cold". Determine the minimum number of questions Beto needs to ask in order to know the correct code with certainty.

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection.

Let $X$ be an $n$-element set and let $A_1,\ldots ,A_m$ be subsets of $X$ such that i) $|A_i|=3$ for each $i=1,\ldots ,m$. ii) $|A_i\cap A_j|\le 1$ for any two distinct indices $i,j$. Show that there exists a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ elements which does not contain any of the $A_i$’s.

A regular $n$-gon is inscribed in a circle radius $1$. Let $X$ be the set of all arcs $PQ$, where $P, Q$ are distinct vertices of the $n$-gon. $5$ elements $L_1, L_2, ... , L_5$ of $X$ are chosen at random (so two or more of the $L_i$ can be the same). Show that the expected length of $L_1 \cap L_2 \cap L_3 \cap L_4 \cap L_5$ is independent of $n$.

Let $n > 2$ be an integer. The numbers $1, 2, . . . , n$ are written at the vertices of an $n$-gon in that order. A move consists of choosing two adjacent vertices and adding $1$ to the numbers written there. Determine all n for which it is possible to achieve that all numbers are identical after a finite number of moves.

Given a chessboard $n \times n$, where $n\geq 4$ and $p=n+1$ is a prime number. A set of $n$ unit squares is called [i]tactical[/i] if after putting down queens on these squares, no two queens are attacking each other. Prove that there exists a partition of the chessboard into $n-2$ tactical sets, not containing squares on the main diagonals. Queens are allowed to move horizontally, vertically and diagonally.

Let $k$ be a fixed natural number. In the infinite number of real line, each integer is colored with color ..., red, green, blue, red, green, blue, ... and so on. A number of flea settles at first at integer points. On each turn, a flea will jump over the other tick so that the distance $k$ is the original distance. Formally, we may choose $2$ tails $A, B$ that are spaced $n$ and move $A$ to the different side of $B$ so the current distance is $kn$. Some fleas may occupy the same point because we consider the size of fleas very small. Determine all the values of $k$ so that, whatever the initial position of the ticks, we always get a position where all ticks land on the same color.

The vertices of the regular $n$-gon are marked with some colours (each vertex -- with one colour) in such a way, that the vertices of one colour represent the right polygon. Prove that there are two equal ones among the smaller polygons.

Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card. Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$. [i]Proposed by Carl Schildkraut and Colin Tang[/i]

At a summer school there are $7$ courses. Each participant was a student in at least one course, and each course was taken by exactly $40$ students. It is known that for each $2$ courses there were at most $9$ students who took them both. Prove that at least $120$ students participated at this summer school.

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Consider all binary sequences (sequences consisting of 0’s and 1’s). In such a sequence the following four types of operation are allowed: (a) $010 \rightarrow 1$, (b) $1 \rightarrow 010$, (c) $110 \rightarrow 0$, and (d) $0 \rightarrow 110$. Determine if it is possible to obtain the sequence $100...0$ (with $2003$ zeroes) from the sequence $0...01$ (with $2003$ zeroes).

There are some gas stations on a circular highway. The total amount of gasoline in them is enough for two laps. Two drivers want to refuel at one station and starting from it, go in different directions, both of them completing an entire lap. Along the way, they can refuel at other stations, without necessarily taking all the gasoline. Prove that drivers will always be able to do this. [i]Proposed by I. Bogdanov[/i]

Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$.