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.)

There are two countries $A$ and $B$, where each countries have $n(\ge 2)$ airports. There are some two-way flights among airports of $A$ and $B$, so that each airport has exactly $3$ flights. There might be multiple flights among two airports; and there are no flights among airports of the same country. A travel agency wants to plan an [i]exotic traveling course[/i] which travels through all $2n$ airports exactly once, and returns to the initial airport. If $N$ denotes the number of all exotic traveling courses, then prove that $\frac{N}{4n}$ is an even integer. (Here, note that two exotic traveling courses are different if their starting place are different.)

The road infrastructure in a country consists of an even number of direct roads, each of which is bidirectional. Moreover, for any two cities $X$ and $Y$, there is at most one direct road between the two of them and there exists a sequence $X = X_0, X_1, ..., X_{n - 1}, X_n = Y$ of cities such that for any $i = 0, ..., n - 1$, there exists a direct road between $X_i$ and $X_{i + 1}$. Prove that all direct roads in this country can be oriented (i.e. each road can become a one-way road) such that each city $X$ is the starting point for an even number of direct roads. Proposed by [i]Mirko Petrushevski[/i]

$n \geq 4$ players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie game. We call a company of four players $bad$ if one player was defeated by the other three players, and each of these three players won a game and lost another game among themselves. Suppose that there is no bad company in this tournament. Let $w_i$ and $l_i$ be respectively the number of wins and losses of the $i$-th player. Prove that \[\sum^n_{i=1} \left(w_i - l_i\right)^3 \geq 0.\] [i]Proposed by Sung Yun Kim, South Korea[/i]

In the plane, $10^6$ points are marked, no three of which are collinear. All possible segments between them are drawn. Grisha assigned to each drawn segment a real number with absolute value no greater than $1$. For every group of $6$ marked points, he calculated the sum of the numbers on all $15$ connecting segments. It turned out that the absolute value of each such sum is at least \(C\), and there are both positive and negative such sums. What is the maximum possible value of \(C\)?

There is a queue of $n{}$ girls on one side of a tennis table, and a queue of $n{}$ boys on the other side. Both the girls and the boys are numbered from $1{}$ to $n{}$ in the order they stand. The first game is played by the girl and the boy with the number $1{}$ and then, after each game, the loser goes to the end of their queue, and the winner remains at the table. After a while, it turned out that each girl played exactly one game with each boy. Prove that if $n{}$ is odd, then a girl and a boy with odd numbers played in the last game. [i]Proposed by A. Gribalko[/i]

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]

At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two. [color=#BF0000]Rewording of the last line for clarification:[/color] Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.

On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.

A tree $T$ is given. Starting with the complete graph on $n$ vertices, subgraphs isomorphic to $T$ are erased at random until no such subgraph remains. For what trees does there exist a positive constant $c$ such that the expected number of edges remaining is at least $cn^2$ for all positive integers $n$? [i]David Yang.[/i]

In a company of $6$ sousliks each souslik has $4$ friends. Is it always possible to divide this company into two groups of $3$ sousliks such that in both groups all sousliks are friends?

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]

There are $N$ cities in a country, some of which are connected by two-way flights. No city is directly connected with every other city. For each pair $(A, B)$ of cities there is exactly one route using at most two flights between them. Prove that $N - 1$ is a square of an integer.

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 $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

Let $\{x_n\}$ be a Van Der Corput series,that is,if the binary representation of $n$ is $\sum a_{i}2^{i}$ then $x_n=\sum a_i2^{-i-1}$.Let $V$ be the set of points on the plane that have the form $(n,x_n)$.Let $G$ be the graph with vertex set $V$ that is connecting any two points $(p,q)$ if there is a rectangle $R$ which lies in parallel position with the axes and $R\cap V= \{p,q\}$.Prove that the chromatic number of $G$ is finite.

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 $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

A weighted complete graph with distinct positive wights is given such that in every triangle is [i]degenerate [/i] that is wight of an edge is equal to sum of two other. Prove that one can assign values to the vertexes of this graph such that the wight of each edge is the difference between two assigned values of the endpoints. [i]Proposed by Morteza Saghafian [/i]

In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo). What is the maximal possible number of edges in such a graph?

In every acyclic graph with 2022 vertices we can choose $k$ of the vertices such that every chosen vertex has at most 2 edges to chosen vertices. Find the maximum possible value of $k$.

In the space there are 8 points that no four of them are in the plane. 17 of the connecting segments are coloured blue and the other segments are to be coloured red. Prove that this colouring will create at least four triangles. Prove also that four cannot be subsituted by five. Remark: Blue triangles are those triangles whose three edges are coloured blue.

In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game. If after ending of tournament participant have at least $ 75 % $ of maximum possible points he called $winner$ $of$ $tournament$. Find maximum possible numbers of $winners$ $of$ $tournament$.

There are $N$ cities in a country. Any two of them are connected either by a road or by an airway. A tourist wants to visit every city exactly once and return to the city at which he started the trip. Prove that he can choose a starting city and make a path, changing means of transportation at most once.

In ancient times there was a Celtic tribe consisting of several families. Many of these families were at odds with each other, so that their chiefs would not shake hands. At some point at the annual meeting of the chiefs they found it even impossible to assemble four or more of them in a circle with each of them being willing to shake his neighbour's hand. To emphasize the gravity of the situation, the Druid collected three pieces of gold from each family. The Druid then let all those chiefs shake hands who were willing to. For each handshake of two chiefs he paid each of them a piece of gold as a reward. Show that the number of pieces of gold collected by the Druid exceeds the number of pieces paid out by at least three.

Let $n \geq 1$ be an integer, and let $S$ be a set of integer pairs $(a,b)$ with $1 \leq a < b \leq 2^n$. Assume $|S| > n \cdot 2^{n+1}$. Prove that there exists four integers $a < b < c < d$ such that $S$ contains all three pairs $(a,c)$, $(b,d)$ and $(a,d)$.

Ten localities are served by two international airlines such that there exists a direct service (without stops) between any two of these localities and all airline schedules offer round-trip service between the cities they serve. Prove that at least one of the airlines can offer two disjoint round trips each containing an odd number of landings.