Let $X$ and $Y$ be two sets of points in the plane and $M$ be a set of segments connecting points from $X$ and $Y$ . Let $k$ be a natural number. Prove that the segments from $M$ can be painted using $k$ colors in such a way that for any point $x \in X \cup Y$ and two colors $\alpha$ and $\beta$ $(\alpha \neq \beta)$, the difference between the number of $\alpha$-colored segments and the number of $\beta$-colored segments originating in $X$ is less than or equal to $1$.

There are $n\geq 3$ cities in a country and between any two cities $A$ and $B$, there is either a one way road from $A$ to $B$, or a one way road from $B$ to $A$ (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define $d(A,B)$ to be the minimum number of roads you must go through to go from city $A$ to $B$. Consider all possible ways to build the roads. Find the minimum possible average value of $d(A,B)$ over all possible ordered pairs of distinct cities in the country.

Let $A$ be a subset of the edges of an $n\times n $ table. Let $V(A)$ be the set of vertices from the table which are connected to at least on edge from $A$ and $j(A)$ be the number of the connected components of graph $G$ which it's vertices are the set $V(A)$ and it's edges are the set $A$. Prove that for every natural number $l$: $$\frac{l}{2}\leq min_{|A|\geq l}(|V(A)|-j(A)) \leq \frac{l}{2}+\sqrt{\frac{l}{2}}+1$$

$2n$ real numbers with a positive sum are aligned in a circle. For each of the numbers, we can see there are two sets of $n$ numbers such that this number is on the end. Prove that at least one of the numbers has a positive sum for both of these two sets.

Define a [i]domino[/i] to be an ordered pair of [i]distinct[/i] positive integers. A [i]proper sequence[/i] of dominoes is a list of distinct dominoes 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 [i]both[/i] appear for any $i$ and $j$. Let $D_n$ be the set of all dominoes whose coordinates are no larger than $n$. Find the length of the longest proper sequence of dominoes that can be formed using the dominoes of $D_n$.

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

We fill in an $n\times n$ table with real numbers such that the sum of the numbers in each row and each coloumn equals $1$. For which values of $K$ is the following statement true: if the sum of the absolute values of the negative entries in the table is at most $K$, then it's always possible to choose $n$ positive entries of the table such that each row and each coloumn contains exactly one of the chosen entries. [i]Proposed by Dávid Bencsik, Budapest[/i]

There are $300$ participants to a mathematics competition. After the competition some of the contestants play some games of chess. Each two contestants play at most one game against each other. There are no three contestants, such that each of them plays against each other. Determine the maximum value of $n$ for which it is possible to satisfy the following conditions at the same time: each contestant plays at most $n$ games of chess, and for each $m$ with $1 \le m \le n$, there is a contestant playing exactly $m$ games of chess.

In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is[i] color-identifiable[/i] if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$. For all positive integers $n$ and $t$, determine the maximum integer $g(n, t)$ such that: In any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.

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]

A communications network consisting of some terminals is called a [i]$3$-connector[/i] if among any three terminals, some two of them can directly communicate with each other. A communications network contains a [i]windmill[/i] with $n$ blades if there exist $n$ pairs of terminals $\{x_{1},y_{1}\},\{x_{2},y_{2}\},\ldots,\{x_{n},y_{n}\}$ such that each $x_{i}$ can directly communicate with the corresponding $y_{i}$ and there is a [i]hub[/i] terminal that can directly communicate with each of the $2n$ terminals $x_{1}, y_{1},\ldots,x_{n}, y_{n}$ . Determine the minimum value of $f (n)$, in terms of $n$, such that a $3$ -connector with $f (n)$ terminals always contains a windmill with $n$ blades.

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]

Let $X$ be a set of $n + 1$ elements, $n\geq 2$. Ordered $n$-tuples $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ formed from distinct elements of $X$ are called[i] disjoint [/i]if there exist distinct indices $1\leq i \neq j\leq n$ such that $a_i = b_j$. Find the maximal number of pairwise disjoint $n$-tuples.

We are given a positive integer $ r$ and a rectangular board $ ABCD$ with dimensions $ AB \equal{} 20, BC \equal{} 12$. The rectangle is divided into a grid of $ 20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $ \sqrt {r}$. The task is to find a sequence of moves leading from the square with $ A$ as a vertex to the square with $ B$ as a vertex. (a) Show that the task cannot be done if $ r$ is divisible by 2 or 3. (b) Prove that the task is possible when $ r \equal{} 73$. (c) Can the task be done when $ r \equal{} 97$?

We call a graph with n vertices $k-flowing-chromatic$ if: 1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors. 2. we can choose a hamilton cycle $v_1,v_2,\cdots , v_n$, and move the chess on $v_i$ to $v_{i+1}$ with $i=1,2,\cdots ,n$ and $v_{n+1}=v_1$, such that any two neighboring chess also have different colors. 3. after some action of step 2 we can make all the chess reach each of the n vertices. Let T(G) denote the least number k such that G is k-flowing-chromatic. If such k does not exist, denote T(G)=0. denote $\chi (G)$ the chromatic number of G. Find all the positive number m such that there is a graph G with $\chi (G)\le m$ and $T(G)\ge 2^m$ without a cycle of length small than 2017.

In a school there are $n$ classes and $k$ student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than $k$ students. If $k-1$ is not a perfect square, prove that there exist a student that has attended in at least $\sqrt k$ classes. [i]Proposed by Mohammad Moshtaghi Far, Kian Shamsaie[/i] [b]Rated 4[/b]

Show that the length of a cycle that contains every edge of a connected graph is at most the sum between the vertices and nodes of the graph, minus $ 1. $

There are 1000 towns $A_{1},A_{2},\ldots ,A_{1000}$ with airports in a country and some of them are connected via flights. It's known that the $i$-th town is connected with $d_{i}$ other towns where $d_{1}\leq d_{2}\leq \ldots \leq d_{1000}$ and $d_{j}\geq j+1$ for every $j=1,2,\ldots 999-d_{999}$. Prove that if the airport of any town $A_{k}$ is closed, then we'd still be able to get from any town $A_{i}$ to any $A_{j}$ for $i,j\neq k$ (possibly by more than one flight).

A sequence of vertices $v_1,v_2,\ldots,v_k$ in a graph, where $v_i=v_j$ only if $i=j$ and $k$ can be any positive integer, is called a $\textit{cycle}$ if $v_1$ is attached by an edge to $v_2$, $v_2$ to $v_3$, and so on to $v_k$ connected to $v_1$. Rotations and reflections are distinct: $A,B,C$ is distinct from $A,C,B$ and $B,C,A$. Supposed a simple graph $G$ has $2013$ vertices and $3013$ edges. What is the minimal number of cycles possible in $G$?

A space figure is consisted of $n$ vertexes and $l$ lines connecting these vertices, where $n=q^2+q+1, l\geq\frac{1}{2}q(q+1)^2+1,q\geq2,q\in\mathbb{Z}_+$. The figure satisfies: every four vertices are not coplane, every vertex is connected by at least one line, and there is a vertex connected by at least $q+2$ lines. Prove that there exists a space quadrilateral in the figure. Note: a space quadrilateral is figure with four vertices $A, B, C, D$ and four lines $ AB, BC, CD, DA$.

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

What is the largest possible number of vertices one can have in a graph that satisfies the following conditions: each vertex is connected to exactly $3$ other vertices, and there always exists a path of length less than or equal to $2$ between any two vertices?

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

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?

Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.