A grid consists of $n\times n$ points, with $n\in\mathbb{Z}^+$. In some of these points is a sentry. Every sentry chooses two directions, one perpendicular to the other (one vertical and the other horizontal) and watches over all the points that are found in the two chosen directions. Each sentry watches over her own point as well and the sentries on the edge of the grid can also watch the vacuum, depending on the directions they have chosen. For instance, in the figure below, representing a disposition of $5$ sentries in a $4\times 4$ grid, the sentries in $A,\,B,\,C,\,D,\,E$ watch over $1,\,3,\,4,\,5,\,7$ points, respectively; points $D$ and $E$ are watched by one sentry, point $C$ is watched by two sentries, points $A,\,B$ and $F$ are watched by three sentries. (a) Prove that we can place $12$ sentries in a $4\times 4$ grid in such a way that every point of the grid is watched by at most two sentries. (b) Let $S(n)$ be the maximum number of sentries we can place on an $n\times n$ grid in such a way that every point of the grid is watched by at most two sentries. Prove that $3n\leq S(n)\leq 4n$ for all $n\geq 3$.

There is a set of $17$ consecutive positive integers. Let $m$ be the smallest of these numbers. Determine for which values of $m$ the set can be divided into three subsets disjoint, such that the sum of the elements of each subset is the same.

Anna and Brian play a game where they put the domino tiles (of size $2 \times 1$) in a boards composed of $n \times 1$ boxes. Tiles must be placed so that they cover exactly two boxes. Players take turnslaying each tile and the one laying last tile wins. They play once for each $n$, where $n = 2, 3,\dots,2007$. Show that Anna wins at least $1505$ of the games if she always starts first and they both always play optimally, ie if they do their best to win in every move.

$2023$ wise men are located in a circle. Each of them thinks either that the earth is the center of the universe, or that it is not. Once a minute, all the wise men express their opinion at the same time. Every wise man who is between two wise men with an opinion different from his will change his mind at that moment. The others don't change their minds. The others don't change their minds. Determine the smallest necessary time for all the wise men to have the same opinion, without regardless of initial opinions or your location.

A king decides to reward one of his knights by making a game. He sits the knights at a round table and has them call out $1,2,3,1,2,3,\dots$ around the circle (that is, clockwise, and each person says a number). The people who say $2$ or $3$ immediately lose, and this continues until the last knight is left, the winner. Numbering the knights initially as $1,2,\dots,n$, find all values of $n$ such that knight $2008$ is the winner.

Can we build parallelepiped $6 \times 7 \times 7$ from $1 \times 1 \times 2$ bricks, such that we have same amount bricks of one of $3$ directions ?

Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$

If $f(x)$ is the generating function of the sequence $a_1,a_2,\ldots$ and if $f(x)=\frac{r(x)}{s(x)}$ holds such that $r(x)$ and $s(x)$ are polynomials show that $a_n$ has a homogenous recurrence.

We choose $n > 3$ points on a circle and number them $1$ to $ n$ in some order. We say that two non-adjacent points $A$ and $B$ are related if, in one of the arcs $AB$, all the points are marked with numbers less than those at $A,B$. Show that the number of pairs of related points is exactly $n-3$.

Is it possible to arrange everything in all cells of an infinite checkered plane all natural numbers (once) so that for each $n$ in each square $n \times n$ the sum of the numbers is a multiple of $n$?

In a group of $20$ people, each person sends a letter to $10$ of the others. Prove that there are two persons who send a letter to each other.

For points $ A_1, \ldots ,A_5$ on the sphere of radius 1, what is the maximum value that $ min_{1 \leq i,j \leq 5} A_iA_j$ can take? Determine all configurations for which this maximum is attained. (Or: determine the diameter of any set $ \{A_1, \ldots ,A_5\}$ for which this maximum is attained.)

Ana, Bruno and Carolina played table tennis with each other. In each game, only two of the friends played, with the third one resting. Every time one of the friends won a game, they rested during the next game. Ana played $12$ games, Bruno played $21$ games and Carolina rested for$ 8$ games. Who rested in the last game?

Let $n\ge3$ be a natural number. For a set $S$ of $n$ real numbers, $A(S)$ denotes the set of all strictly increasing arithmetic sequences of three terms in $S$. At most, how many elements can the set $A(S)$ have?

Let $A = \{ 1,2, 3 , \ldots, 100 \}$ and $B$ be a subset of $A$ having $53$ elements. Show that $B$ has 2 distinct elements $x$ and $y$ whose sum is divisible by $11$.

Given positive integers $n$ and $k$, $n > k^2 >4.$ In a $n \times n$ grid, a $k$[i]-group[/i] is a set of $k$ unit squares lying in different rows and different columns. Determine the maximal possible $N$, such that one can choose $N$ unit squares in the grid and color them, with the following condition holds: in any $k$[i]-group[/i] from the colored $N$ unit squares, there are two squares with the same color, and there are also two squares with different colors.

A family of finite sets $\left\{ A_{1},A_{2},.......,A_{m}\right\} $is called [i]equipartitionable [/i] if there is a function $\varphi:\cup_{i=1}^{m}$$\rightarrow\left\{ -1,1\right\} $ such that $\sum_{x\in A_{i}}\varphi\left(x\right)=0$ for every $i=1,.....,m.$ Let $f\left(n\right)$ denote the smallest possible number of $n$-element sets which form a non-equipartitionable family. Prove that a) $f(4k +2) = 3$ for each nonnegative integer $k$, b) $f\left(2n\right)\leq1+m d\left(n\right)$, where $m d\left(n\right)$ denotes the least positive non-divisor of $n.$

A $T$-tetromino is a non-convex as well as non-rotationally symmetrical tetromino, which has a maximum number of outside corners (popularly also "Tetris Stone "called). Find all natural numbers $n$ for which, a $n \times n$ chessboard is found that can be covered only with such $T$-tetrominos.

Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?

A teacher is known to have $2^k$ apples for some $k \in \mathbb{N}$. He ets one of the apples and distributes the rest of the apples to his students $A$ and $B$. The students do not see how many apples the other gets, and they do not know the number $k$. However, they have pre-selected a discreet way to reveal one another something about the number of apples: each of the students scratches their head either by their right, left or both hands, depending on the number of apples they have received. To the teacher's surprise, the students will always know which one of the students got more apples, or that the teacher ate the only apple by herself. How is this possible?

Eve and Odette play a game on a $3\times 3$ checkerboard, with black checkers and white checkers. The rules are as follows: $\text{I.}$ They play alternately. $\text{II.}$ A turn consists of placing one checker on an unoccupied square of the board. $\text{III.}$ In her turn, a player may select either a white checker or a black checker and need not always use the same colour. $\text{IV.}$ When the board is full, Eve obtains one point for every row, column or diagonal that has an even number of black checkers, and Odette obtains one point for very row, column or diagonal that has an odd number of black checkers. $\text{V.}$ The player obtaining at least five of the eight points WINS. $\text{(a)}$ Is a $4-4$ tie possible? Explain. $\text{(b)}$ Describe a winning strategy for the girl who is first to play.

Prove that any group of people can be divided into two disjoint groups $A$ and $B$ such that any member from $A$ has at least half of his acquaintances in $B$ and any member from $B$ has at least half of his acquaintances in $A$ (acquaintance is reciprocal).

In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that [list] [*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and [*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color. [/list]

Alice asserts that after her recent visit to Addis-Ababa she now has spent the New Year inside every possible hemisphere of Earth except one. What is the minimal number of places where Alice has spent the New Year? Note: we consider places of spending the New Year to be points on the sphere. A point on the border of a hemisphere does not lie inside the hemisphere. Ilya Dumansky, Roman Krutovsky

There are $3$ students from Korea, China, and Japan, so total of $9$ students are present. How many ways are there to make them sit down in a circular table, with equally spaced and equal chairs, such that the students from the same country do not sit next to each other? If array $A$ can become array $B$ by rotation, these two arrays are considered equal.