Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled: [b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label. [b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side. [b](a)[/b] Find the maximal $r$ for which such a labelling is possible. [b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there? [hide="Easier version (5th German TST 2006) - contains answer to the harder version"] [i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide] [i]Proposed by Federico Ardila, Colombia[/i]

$3n - 2$ participants took part in the chess festival, some of them played one game of chess with each other. Prove that at least one of the following statements holds: (A) One can find $n$ chess players $A_1 , A_2 , . . . , A_n$ suchthat Ai has played a game with $A_{i+1}$ for all $i = 1, ...,n -1$. (B) Seven chess players can be found in $B_1 , . . . , B_7$, who have not played with each other, except perhaps three pairs $(B_1, B_2)$, $(B_3, B_4)$ and $(B_5, B_6)$, each of whom may or may not have played a game of chess.

Let $ A$ denote a subset of the set $ \{ 1,11,21,31, \dots ,541,551 \}$ having the property that no two elements of $ A$ add up to $ 552$. Prove that $ A$ can't have more than $ 28$ elements.

A binary sequence is a sequence in which each term is equal to $0$ or $1$. A binary sequence is called $\text{friendly}$ if each term is adjacent to at least on term that is equal to $1$. For example , the sequence $0,1,1,0,0,1,1,1$ is $\text{friendly}$. Let $F_{n}$ denote the number of $\text{friendly}$ binary sequences with $n$ terms. Find the smallest positive integer $n\ge 2$ such that $F_{n}>100$

Find all positive integer $n\ge 3$ such that it is possible to mark the vertices of a regular $n$- gon with the number from 1 to n so that for any three vertices $A, B$ and $C$ with $AB = AC$, the number in $A$ is greater or smaller than both numbers in $B, C$.

Find the smallest natural number $n$ for which it is possible to partition the set $M = \{1,2, ... ,40\}$ into n subsets $M_1, . . . ,M_n$ so that none of the $M_i$ contains elements $a,b,c$ (not necessarily distinct) with $a+b = c$.

There are $11$ quadratic equations on the board, where each coefficient is replaced by a star. Initially, each of them looks like this $$\star x^2 + \star x + \star= 0.$$ Two players are playing a game making alternating moves. In one move each ofthem replaces one star with a real nonzero number. The first player tries to make as many equations as possible without roots and the second player tries to make the number of equations without roots as small as possible. What is the maximal number of equations without roots that the first player can achieve if the second player plays to her best? Describe the strategies of both players.

The vigilantes are a group of five superheroes, such that each one has one and only one of the following powers: hypnosis, super speed, shadow manipulation, immortality and super strength (each has a different power). On an adventure to the island of Philippines, they meet the sorcerer Vicencio, an old wise man who offers them the following ritual to help them: The ritual consists of a superhero $A$ acquiring the gift (s) of $B$ without $B$ acquiring the gift (s) of $A$. Determine the fewest number of rituals to be performed by the sorcerer Vicencio so that each superhero controls each of the five gifts. Clarification: At the end of the ritual, a superhero $A$ will have his gifts and those of a superhero $B$, but $B$ does not acquire those of $A$, but it does keep its own.

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

The expressions $ a \plus{} b \plus{} c, ab \plus{} ac \plus{} bc,$ and $ abc$ are called the elementary symmetric expressions on the three letters $ a, b, c;$ symmetric because if we interchange any two letters, say $ a$ and $ c,$ the expressions remain algebraically the same. The common degree of its terms is called the order of the expression. Let $ S_k(n)$ denote the elementary expression on $ k$ different letters of order $ n;$ for example $ S_4(3) \equal{} abc \plus{} abd \plus{} acd \plus{} bcd.$ There are four terms in $ S_4(3).$ How many terms are there in $ S_{9891}(1989)?$ (Assume that we have $ 9891$ different letters.)

Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$, and $m$ distinct airports $c_1, \dots, c_m$, where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$; $c_2$ and $c_3$; $\dots$ ; $c_{m-1}$ and $c_m$; $c_m$ and $c_1$. Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.

Given are the lines $l_1,l_2,\ldots ,l_k$ in the plane, no two of which are parallel and no three of which are concurrent. For which $k$ can one label the intersection points of these lines by $1, 2,\ldots , k-1$ so that in each of the given lines all the labels appear exactly once?

A frog is placed on each cell of a $n \times n$ square inside an infinite chessboard (so initially there are a total of $n \times n$ frogs). Each move consists of a frog $A$ jumping over a frog $B$ adjacent to it with $A$ landing in the next cell and $B$ disappearing (adjacent means two cells sharing a side). Prove that at least $ \left[\frac{n^2}{3}\right]$ moves are needed to reach a configuration where no more moves are possible.

The positive integer $n$ is given and for all positive integers $k$, $1\leq k\leq n$, denote by $a_{kn}$ the number of all ordered sequences $(i_1,i_2,\ldots,i_k)$ of positive integers which verify the following two conditions: a) $1\leq i_1<i_2< \cdots i_k \leq n$; b) $i_{r+1}-i_r \equiv 1 \pmod 2$, for all $r \in\{1,2,\ldots,k-1\}$. Compute the number $a(n) = \sum\limits_{k=1}^n a_{kn}$. [i]Ioan Tomescu[/i]

In a simple graph, there are two disjoint set of vertices $A$ and $B$ where $A$ has $k$ and $B$ has $2016$ vertices. Four numbers are written to each vertex using the colors red, green, blue and black. There is no any edge at the beginning. For each vertex in $A$, we first choose a color and then draw all edges from this vertex to the vertices in $B$ having a larger number with the chosen color. It is known that for each vertex in $B$, the set of vertices in $A$ connected to this vertex are different. Find the minimal possible value of $k$.

An $n \times n$ board is given. In the top left corner cell there is a fox, whereas in the bottom left corner cell there is a rabbit. Every minute, the fox and the rabbit jump to a neighbouring cell at the same time. The fox can jump only to neighbouring cells that are below it or on its right, whereas the rabbit can only jump to the cells above it or in its right. They continue like this until they have no possible moves. The fox catches the rabbit if at a certain moment they are in the same cell, otherwise the rabbit gets away. Find all natural numbers $n$ for which the fox has a winning strategy to catch the rabbit. [i](Note: Two squares are considered neighbours if they have a common side.)[/i]

Let $n \ge k \ge 3$ be integers. Show that for every integer sequence $1 \le a_1 < a_2 < . . . < a_k \le n$ one can choose non-negative integers $b_1, b_2, . . . , b_k$, satisfying the following conditions: [list=i] [*] $0 \le b_i \le n$ for each $1 \le i \le k$, [*] all the positive $b_i$ are distinct, [*] the sums $a_i + b_i$, $1 \le i \le k$, form a permutation of the first $k$ terms of a non-constant arithmetic progression. [/list]

A rectangle with sides $ n$ and $p$ is divided into $np$ unit squares. Initially there are m unitary squares painted black and the remaining painted white. The following processoccurs repeatedly: if a unit square painted white has at minus two sides in common with squares painted black then Its color also turns black. Find the smallest integer $m$ that satisfies the property: there exists an initial position of $m$ black unit squares such that the entire $ n \times p$ rectangle is painted black when repeat the process a finite number of times.

A complete graph of $n$ vertices is a set of $n$ vertices and those vertices are connected in pairs by edges. Suppose the graph has $n$ vertices $A_1, A_2, ..., A_n$, the cycle is a set of edges of the form $A_{i_1}A_{i_2}, A_{i_2}A_{i_3},..., A_{i_m}A_{i_1}$ with $i_1, i_2, ..., i_m \in {1, 2, ..., n}$ double one different. We call $m$ the length of this cycle. Find the smallest positive integer$ n$ such that for every way of coloring all edges of a complete graph of $n$ vertices, each edge filled with one of three different colors, there is always a cycle of even length with the same color. PS. The same problem with another wording [url=https://artofproblemsolving.com/community/c6h151391p852296]here [/url].

An $L$ is an arrangement of $3$ adjacent unit squares formed by deleting one unit square from a $2 \times 2$ square. a) How many $L$s can be placed on an $8 \times 8$ board (with no interior points overlapping)? b) Show that if any one square is deleted from a $1987 \times 1987$ board, then the remaining squares can be covered with $L$s (with no interior points overlapping).

Suppose three direction on the plane . We draw $ 11$ lines in each direction . Find maximum number of the points on the plane which are on three lines .

A graph has $N$ vertices. An invisible hare sits in one of the vertices. A group of hunters tries to kill the hare. In each move all of them shoot simultaneously: each hunter shoots at a single vertex, they choose the target vertices cooperatively. If the hare was in one of the target vertices during a shoot, the hunt is finished. Otherwise the hare can stay in its vertex or jump to one of the neighboring vertices. The hunters know an algorithm that allows them to kill the hare in at most $N!$ moves. Prove that then there exists an algorithm that allows them to kill the hare in at most $2^N$ moves.

Fred and George are playing a game, in which Fred flips $2014$ coins and George flips $2015$ coins. Fred wins if he flips at least as many heads as George does, and George wins if he flips more heads than Fred does. Determine the probability that Fred wins.

2500 chess kings have to be placed on a $100 \times 100$ chessboard so that [b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex); [b](ii)[/b] each row and each column contains exactly 25 kings. Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.) [i]Proposed by Sergei Berlov, Russia[/i]

Let $n$ be a positive integer, $n\geq 4$. $n$ cards are arranged on a circle and the numbers $1$ or $-1$ are written on each of the cards. in a $question$ we may find out the product of the numbers on any $3$ cards. What is the minimum numbers if questions needed to find out the product of all $n$ numbers?