A [i]stick[/i] is defined as a $1 \times k$ or $k\times 1$ rectangle for any integer $k \ge 1$. We wish to partition the cells of a $2022 \times 2022$ chessboard into $m$ non-overlapping sticks, such that any two of these $m$ sticks share at most one unit of perimeter. Determine the smallest $m$ for which this is possible. [i]Holden Mui[/i]

Positive integers are put into the following table. \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline 4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline 7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline 11 & 17 & 24 & 32 & 41 & & & & & \\ \hline 16 & 23 & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline \end{tabular} Find the number of the line and column where the number $2015$ stays.

On a table with $25$ columns and $300$ rows, Kostya painted all its cells in three colors. Then, Lesha, looking at the table, for each row names one of the three colors and marks in that row all cells of that color (if there are no cells of that color in that row, he does nothing). After that, all columns that have at least a marked square will be deleted. Kostya wants to be left as few as possible columns in the table, and Lesha wants there to be as many as possible columns in the table. What is the largest number of columns Lesha can guarantee to leave?

A positive integer $n\geq 2$ and a set $S$ consisting of $2n$ disting positive integers smaller or equal to $n^2$ are given. Prove that there exists a positive integer $r\in \{1, 2, ..., n\}$ that can be written in the form $r=a-b$, for $a, b\in \mathbb{S}$ in at least $3$ different ways.

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.

Determine the largest natural number $ N $ having the following property: every $ 5\times 5 $ array consisting of pairwise distinct natural numbers from $ 1 $ to $ 25 $ contains a $ 2\times 2 $ subarray of numbers whose sum is, at least, $ N. $ [i]Demetres Christofides[/i] and [i]Silouan Brazitikos[/i]

15 boxes are given. They all are initially empty. By one move it is allowed to choose some boxes and to put in them numbers of apricots which are pairwise distinct powers of 2. Find the least positive integer $k$ such that it is possible to have equal numbers of apricots in all the boxes after $k$ moves.

Let $A_1A_2\cdots A_{2002}$ be a regular 2002 sided polygon. Each vertex $A_i$ is associated with a positive integer $a_i$ such that the following condition is satisfied: If $j_1,j_2,\cdots, j_k$ are positive integers such that $k<500$ and $A_{j_1}, A_{j_2}, \cdots A_{j_k}$ is a regular $k$ sided polygon, then the values of $a_{j_1},A_{j_2}, \cdots A_{j_k}$ are all different. Find the smallest possible value of $a_1+a_2+\cdots a_{2002}$

Let $m,n\in \mathbb{Z}_{\ge 1}$ and a rectangular board $m\times n$ sliced by parallel lines to the rectangle's sides into $mn$ unit squares. At moment $t=0$, there is an ant inside every square, positioned exactly in its centre, such that it is oriented towards one of the rectangle's sides. Every second, all the ants move exactly a unit following their current orientation; however, if two ants meet at the centre of a unit square, both of them turn $180^o$ around (the turn happens instantly, without any loss of time) and the next second they continue their motion following their new orientation. If two ants meet at the midpoint of a side of a unit square, they just continue moving, without changing their orientation.\\ \\ Prove that, after finitely many seconds, some ant must fall off the table.\\ \\ [i](Oliver Hayman)[/i]

How many ways are there to arrange the numbers $1, 2, 3, .. , 15$ in some order such that for any two numbers which are $2$ or $3$ positions apart, the one on the left is greater?

Let $n$ be an given positive integer, the set $S=\{1,2,\cdots,n\}$.For any nonempty set $A$ and $B$, find the minimum of $|A\Delta S|+|B\Delta S|+|C\Delta S|,$ where $C=\{a+b|a\in A,b\in B\}, X\Delta Y=X\cup Y-X\cap Y.$

Find all family $\mathcal{F}$ of subsets of $[n]$ such that for any nonempty subset $X\subseteq [n]$, exactly half of the elements $A\in \mathcal{F}$ satisfies that $|A\cap X|$ is even.

Each of $2n + 1$ students chooses a finite, nonempty set of consecutive integers . Two students are friends if they have chosen a common number. Everyone student is friends with at least $n$ other students. Show that there is a student who is friends with everyone else.

There are $2017$ points on the plane, no three of them are collinear. Some pairs of the points are connected by $n$ segments. Find the smallest value of $n$ so that there always exists two disjoint segments in any case.

Let $S$ be the set of natural numbers whose digits are different and belong to the set $\{1, 3, 5, 7\}$. Calculate the sum of the elements of $S$.

Let $n$ be a positive integer. Compute the number of words $w$ that satisfy the following three properties. 1. $w$ consists of $n$ letters from the alphabet $\{a,b,c,d\}.$ 2. $w$ contains an even number of $a$'s 3. $w$ contains an even number of $b$'s. For example, for $n=2$ there are $6$ such words: $aa, bb, cc, dd, cd, dc.$

A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$

Let $P$ be a regular $2006$-gon. A diagonal is called [i]good[/i] if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called [i]good[/i]. Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.

Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le  i\le  m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$, so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j  \ge i$. Similarly, we define, for $1\le   j \le  n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$. E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$. (a) Prove that $a_j = c_j $ for $1  \le j  \le n$. (b) Prove that for $1\le  k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.

In a school there are $ 2008$ students. Students are members of certain committees. A committee has at most $ 1004$ members and every two students join a common committee. (i) Determine the smallest possible number of committees in the school. (ii) If it is further required that the union of any two committees consists of at most $ 1800$ students, will your answer in (i) still hold?

show there exist positive constants $c_1$ and $c_2$ such that for any $n\geq 3$, whenever $T_1$ and $T_2$ are two trees on the set of vertices $X = \{1, 2, ..., n\}$, there exists a function $f : X \to \{-1, +1\}$ for which $$\bigg | \sum_ {x \in P} f (x) \bigg | <c_1 \log n$$ for any path P that is a subgraph of $T_1$ or $T_2$ , but with an upper bound $c_2 \log n / \log \log n$ the statement is no longer true.

On a chessboard (with $64$ squares) there are situated $32$ white and $32$ black pools. We say that two pools form a mixed pair when they are with different colors and they lie on the same row or column. Find the maximum and the minimum of the mixed pairs for all possible situations of the pools. [i]K. Dochev[/i]

A $20 \times 20 \times 20$ cube is composed of $2000$ bricks of size $2 \times 2 \times 1$ . Prove that it is possible to pierce the cube with a needle so that the needle passes through the cube without passing through a brick . (A . Andjans , Riga)

Let $P$ be a polynomial with integer coefficients of degree $d$. For the set $A = \{ a_1, a_2, ..., a_k\}$ of positive integers we denote $S (A) = P (a_1) + P (a_2) + ... + P (a_k )$. The natural numbers $m, n$ are such that $m ^{d+ 1} | n$. Prove that the set $\{1, 2, ..., n\}$ can be subdivided into $m$ disjoint subsets $A_1, A_2, ..., A_m$ with the same number of elements such that $S (A_1) = S(A_2) = ... = S (A_m )$.

There are $n$ lamps $L_1, L_2, \dots, L_n$ arranged in a circle in that order. At any given time, each lamp is either [i]on[/i] or [i]off[/i]. Every second, each lamp undergoes a change according to the following rule: (a) For each lamp $L_i$, if $L_{i-1}, L_i, L_{i+1}$ have the same state in the previous second, then $L_i$ is [i]off[/i] right now. (Indices taken mod $n$.) (b) Otherwise, $L_i$ is [i]on[/i] right now. Initially, all the lamps are [i]off[/i], except for $L_1$ which is [i]on[/i]. Prove that for infinitely many integers $n$ all the lamps will be [i]off[/i] eventually, after a finite amount of time.