Let \( n \) be a positive integer. A graph on \( 2n - 1 \) vertices is given such that the size of the largest clique in the graph is \( n \). Prove that there exists a vertex that is present in every clique of size \( n\)

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.

A fighting game club has $2024$ members. One day, a game of Smash is played between some pairs of members so that every member has played against exactly $3$ other members. Each match has a winner and a loser. A member will be [i]happy[/i] if they won in at least $2$ of the matches. What is the maximum number of happy members over all possible match-ups and all possible outcomes?

There are 25 people. Every two of them are use some language to speak between. They use only one language even if they both know another one. Among every three of them there is one who speaking with two other on the same language. Prove that there exist one who speaking with 10 other on the same language.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

$2017$ engineers attend a conference. Any two engineers if they converse, converse with each other in either Chinese or English. No two engineers converse with each other more than once. It is known that within any four engineers, there was an even number of conversations and furthermore within this even number of conversations: i) At least one conversation is in Chinese. ii) Either no conversations are in English or the number of English conversations is at least that of Chinese conversations. Show that there exists $673$ engineers such that any two of them conversed with each other in Chinese.

1990 mathematicians attend a meeting, every mathematician has at least 1327 friends (the relation of friend is reciprocal). Prove that there exist four mathematicians among them such that any two of them are friends.

In a party among any four persons there are three people who are mutual acquaintances or mutual strangers. Prove that all the people can be separated into two groups $A$ and $B$ such that in $A$ everybody knows everybody else and in $B$ nobody knows anybody else.

Can you make $2015$ positive integers $1,2, \ldots , 2015$ to be a certain permutation which can be ordered in the circle such that the sum of any two adjacent numbers is a multiple of $4$ or a multiple of $7$?

Let $N = \{1, 2, \ldots, n\}$, $n \geq 3$. To each pair $i \neq j $ of elements of $N$ there is assigned a number $f_{ij} \in \{0, 1\}$ such that $f_{ij} + f_{ji} = 1$. Let $r(i)=\sum_{i \neq j} f_{ij}$, and write $M = \max_{i\in N} r(i)$, $m = \min_{i\in N} r(i)$. Prove that for any $w \in N$ with $r(w) = m$ there exist $u, v \in N$ such that $r(u) = M$ and $f_{uv}f_{vw} = 1$.

On some planet, there are $2^N$ countries $(N \geq 4).$ Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1,$ each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set. [i]Proposed by Tonći Kokan, Croatia[/i]

Let $V$ be a set of $2019$ points in space where any of the four points are not on the same plane, and $E$ be the set of edges connected between them. Find the smallest positive integer $n$ satisfying the following condition: if $E$ has at least $n$ elements, then there exists $908$ two-element subsets of $E$ such that [list][*]The two edges in each subset share a common vertice, [*]Any of the two subsets do not intersect.[/list]

How many graphs are there on $6$ vertices with degrees $1,1,2,3,4,5$?

Some $n \geq 3$ cities are connected with railways, so that you can travel from one city to every other, not necessarily directly. However, the railways are structured in such a way that there is only one way to get from one city to another, assuming you don't pass through the same city again. Let $A$ be the set of these cities and railways. Show that there exists a Subset of $A$, let's say $C$, such that (1) $C$ has at least $[(n+1)/2]$ cities as its element. (2) No two elements of $C$ are directly connected with railways.

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

The Imomi archipelago consists of $n\geq 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the traveller starts at, to visit all the islands exactly once (in particular, not returning to the island the traveller started at). Determine the maximal possible value of $k$ in terms of $n$. [i]Anton Trygub, Ukraine[/i]

Alice and Bob play a game on a connected graph with $2n$ vertices, where $n\in \mathbb{N}$ and $n>1$.. Alice and Bob have tokens named A and B respectively. They alternate their turns with Alice going first. Alice gets to decide the starting positions of A and B. Every move, the player with the turn moves their token to an adjacent vertex. Bob's goal is to catch Alice, and Alice's goal is to prevent this. Note that positions of A, B are visible to both Alice and Bob at every moment. Provided they both play optimally, what is the maximum possible number of edges in the graph if Alice is able to evade Bob indefinitely? [i]Proposed by Shashank Ingalagavi and Vighnesh Sangle[/i]

A simple graph is called "divisibility", if it's possible to put distinct numbers on its vertices such that there is an edge between two vertices if and only if number of one of its vertices is divisible by another one. A simple graph is called "permutationary", if it's possible to put numbers $1,2,...,n$ on its vertices and there is a permutation $ \pi $ such that there is an edge between vertices $i,j$ if and only if $i>j$ and $\pi(i)< \pi(j)$ (it's not directed!) Prove that a simple graph is permutationary if and only if its complement and itself are divisibility. [i]Proposed by Morteza Saghafian[/i] .

A badminton club consists of $2n$ members who are n couples. The club plans to arrange a round of mixed doubles matches where spouses neither play together nor against each other. Requirements are: $\cdot$ Each pair of members of the same gender meets exactly once as opponents in a mixed doubles match. $\cdot$ Any two members of the opposite gender who are not spouses meet exactly once as partners and also as opponents in a mixed doubles match. Given that $(n,6)=1$, can you arrange a round of mixed doubles matches that meets the above specifications and requirements?

In a competition there are $18$ teams and in each round $18$ teams are divided into $9$ pairs where the $9$ matches are played coincidentally. There are $17$ rounds, so that each pair of teams play each other exactly once. After $n$ rounds, there always exists $4$ teams such that there was exactly one match played between these teams in those $n$ rounds. Find the maximum value of $n$.

Adithya and Bill are playing a game on a connected graph with $n > 2$ vertices, two of which are labeled $A$ and $B$, so that $A$ and $B$ are distinct and non-adjacent and known to both players. Adithya starts on vertex $A$ and Bill starts on $B$. Each turn, both players move simultaneously: Bill moves to an adjacent vertex, while Adithya may either move to an adjacent vertex or stay at his current vertex. Adithya loses if he is on the same vertex as Bill, and wins if he reaches $B$ alone. Adithya cannot see where Bill is, but Bill can see where Adithya is. Given that Adithya has a winning strategy, what is the maximum possible number of edges the graph may have? (Your answer may be in terms of $n$.) [i]Proposed by Steven Liu[/i]

Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $j.$ Let $D_{40}$ be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$

In a party, there are $2n + 1$ people. It's well known that for every group of $n$ people, there exist a person(out of the group) who knows all them(the $n$ people of the group). Show that there exist a person who knows all the people in the party.

Let $G$ be a $2n-$vertices simple graph such that in any partition of the set of vertices of $G$ into two $n-$vertices sets $V_1$ and $V_2$, the number of edges from a vertex in $V_1$ to another vertex in $V_1$ is equal to the number of edges from a vertex in $V_2$ to another vertex in $V_2$. Prove that all the vertices have equal degrees.

A graph with $10^{100}$ vertices satisfies the following condition: Any simple odd cycle has length > 100. Prove there is an independent set in the graph of size at least $\frac{10^{100}}{102}$