In a simple graph $G$, an operation is defined as taking two neighbor vertices $u,v$ which have a common neighbor, deleting the edge between $u,v$ and adding a new vertex $w$ whose neighbors are exactly the common neighbors of $u$ and $v$. Starting with the complete graph $G=K_n$ where $n\ge 3$ is a positive integer, find the maximum number of operations that can be applied. Proposed by[i] Deniz Can Karaçelebi[/i]

Let $N$ be a positive integer. Kingdom Wierdo has $N$ castles, with at most one road between each pair of cities. There are at most four guards on each road. To cost down, the King of Wierdos makes the following policy: (1) For any three castles, if there are roads between any two of them, then any of these roads cannot have four guards. (2) For any four castles, if there are roads between any two of them, then for any one castle among them, the roads from it toward the other three castles cannot all have three guards. Prove that, under this policy, the total number of guards on roads in Kingdom Wierdo is smaller than or equal to $N^2$. [i]Remark[/i]: Proving that the number of guards does not exceed $cN^2$ for some $c > 1$ independent of $N$ will be scored based on the value of $c$. [i]Proposed by usjl[/i]

At a table tennis tournament, each one of the $ n\ge 2 $ participants play with all the others exactly once. Show that, at the end of the tournament, one and only one of these propositions will be true: $ \text{(1)} $ The players can be labeled with the numbers $ 1,2,...,n, $ such that $ 1 $ won $ 2, 2 $ won $ 3,...,n-1 $ won $ n $ and $ n $ won $ 1. $ $ \text{(2)} $ The players can be partitioned in two nonempty subsets $ A,B, $ such that whichever one from $ A $ won all that are in $ B. $

For each natural number $k$ find the least number $n$ such that in every tournament with $n$ vertices, there exists a vertex with in-degree and out-degree at least $k$. (Tournament is directed complete graph.)

In the unit squre For the given natural number $n \geq 2$ find the smallest number $k$ that from each set of $k$ unit squares of the $n$x$n$ chessboard one can achoose a subset such that the number of the unit squares contained in this subset an lying in a row or column of the chessboard is even

Is it possible to put $\binom{n}{2}$ consecutive natural numbers on the edges of a complete graph with $n$ vertices in a way that for every path (or cycle) of length $3$ where the numbers $a,b$ and $c$ are written on its edges (edge $b$ is between edges $c$ and $a$), $b$ is divisible by the greatest common divisor of the numbers $a$ and $c$? [i]Proposed by Morteza Saghafian[/i]

An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that: $ (i)$ Each player plays in each round, and every two players meet at most once. $ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$. Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament. [i]Proposed by Carlos di Fiore, Argentina[/i]

Given a simple, connected graph with $n$ vertices and $m$ edges. Prove that one can find at least $m$ ways separating the set of vertices into two parts, such that the induced subgraphs on both parts are connected.

There is a complete graph $G$ with $4027$ vertices drawn on the whiteboard. Ivan paints all the edges by red or blue colour. Find all ordered pairs $(r, b)$ such that Ivan can paint the edges so that every vertex is connected to exactly $r$ red edges and $b$ blue edges.

A kingdom has $2021$ towns. All of the towns lie on a circle, and there is a one-way road going from every town to the next $101$ towns in a clockwise order. Each road is colored in one color. Additionally, it is known that for any ordered pair of towns $A$ and $B$ it is possible to find a path from $A$ to $B$ so that no two roads of the path would have the same color. Find the minimal number of road colors in the kingdom.

There are $2^n$ words of length $n$ over the alphabet $\{0, 1\}$. Prove that the following algorithm generates the sequence $w_0, w_1, \ldots, w_{2^n-1}$ of all these words such that any two consecutive words differ in exactly one digit. (1) $w_0 = 00 \ldots 0$ ($n$ zeros). (2) Suppose $w_{m-1} = a_1a_2 \ldots a_n,\quad a_i \in \{0, 1\}$. Let $e(m)$ be the exponent of $2$ in the representation of $n$ as a product of primes, and let $j = 1 + e(m)$. Replace the digit $a_j$ in the word $w_{m-1}$ by $1 - a_j$. The obtained word is $w_m$.

Let $ n$ and $ k$ be positive integers such that $ \frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares.

Numbers 1 and −1 are written in the cells of a board 2000×2000. It is known that the sum of all the numbers in the board is positive. Show that one can select 1000 rows and 1000 columns such that the sum of numbers written in their intersection cells is at least 1000. [i]D. Karpov[/i]

The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\ge 4}$, find the least number of edges of a graph that can be obtained by repeated applications of this operation from the complete graph on $n$ vertices (where each pair of vertices are joined by an edge). [i]Proposed by Norman Do, Australia[/i]

In an election, there are $1395$ candidates and some voters. Each voter, arranges all the candidates by the priority order. We form a directed graph with $1395$ vertices, an arrow is directed from $U$ to $V$ when the candidate $U$ is at a higher level of priority than $V$ in more than half of the votes. (otherwise, there's no edge between $U,V$) Is it possible to generate all complete directed graphs with $1395$ vertices?

Given a set $S$ of $n$ variables, a binary operation $\times$ on $S$ is called [i]simple[/i] if it satisfies $(x \times y) \times z = x \times (y \times z)$ for all $x,y,z \in S$ and $x \times y \in \{x,y\}$ for all $x,y \in S$. Given a simple operation $\times$ on $S$, any string of elements in $S$ can be reduced to a single element, such as $xyz \to x \times (y \times z)$. A string of variables in $S$ is called[i] full [/i]if it contains each variable in $S$ at least once, and two strings are [i]equivalent[/i] if they evaluate to the same variable regardless of which simple $\times$ is chosen. For example $xxx$, $xx$, and $x$ are equivalent, but these are only full if $n=1$. Suppose $T$ is a set of strings such that any full string is equivalent to exactly one element of $T$. Determine the number of elements of $T$.

Given two positive integers $m$ and $n$, find the smallest positive integer $k$ such that among any $k$ people, either there are $2m$ of them who form $m$ pairs of mutually acquainted people or there are $2n$ of them forming $n$ pairs of mutually unacquainted people.

A connected graph has $100$ vertices, the degrees of all the vertices do not exceed $4$ and no two vertices of degree $4$ are adjacent. Prove that it is possible to remove several edges that have no common vertices from this graph such that there would be no triangles in the resulting graph.

For a competition a school wants to nominate a team of $k$ students, where $k$ is a given positive integer. Each member of the team has to compete in the three disciplines juggling, singing and mental arithmetic. To qualify for the team, the $n \ge 2$ students of the school compete in qualifying competitions, determining a unique ranking in each of the three disciplines. The school now wants to nominate a team satisfying the following condition: $(*)$ [i]If a student $X$ is not nominated for the team, there is a student $Y$ on the team who defeated $X$ in at least two disciplines.[/i] Determine all positive integers $n \ge 2$ such that for any combination of rankings, a team can be chosen to satisfy the condition $(*)$, when a) $k=2$, b) $k=3$.

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

There are $m \ge 2$ blue points and $n \ge 2$ red points in three-dimensional space, and no four points are coplanar. Geoff and Nazar take turns, picking one blue point and one red point and connecting the two with a straight-line segment. Assume that Geoff starts first and the one who first makes a cycle wins. Who has the winning strategy?

There are $1000$ students in a school. Every student has exactly $4$ friends. A group of three students $ \left \{A,B,C \right \}$ is said to be a [i]friendly triplet[/i] if any two students in the group are friends. Determine the maximal possible number of friendly triplets. [i]Proposed by Nikola Velov[/i]

At a two-round volleyball tournament participated 99 teams. Each played a match at home and a match away. Each team won exactly half of their home matches and exactly half of their away matches. Prove that one of the teams beat another team twice. [i]Proposed by M. Antipov[/i]

Let $k$ be a given integer, $3 < k \leq n$. Consider a graph $G$ with $n$ vertices satisfying the condition: for any two non-adjacent vertices $x$ and $y$ in graph $G$, the sum of their degrees must satisfy $d(x) + d(y) \geq k$. Please answer the following questions and prove your conclusions. (1) Suppose the length of the longest path in graph $G$ is $l$ satisfying the inequality $3 \leq l < k$, does graph $G$ necessarily contain a cycle of length $l+1$? (The length of a path or cycle refers to the number of edges that make up the path or cycle.) (2) For the case where $3 < k \leq n-1$ and graph $G$ is connected, can we determine that the length of the longest path in graph $G$, $l \geq k$? (3) For the case where $3 < k = n-1$, is it necessary for graph $G$ to have a path of length $n-1$ (i.e., a Hamiltonian path)?

Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than $180^{\circ}$. Let $C$ be a convex polygon with $s$ sides. The interior region of $C$ is the union of $q$ quadrilaterals, none of whose interiors overlap each other. $b$ of these quadrilaterals are boomerangs. Show that $q\ge b+\frac{s-2}{2}$.