Sammy has a wooden board, shaped as a rectangle with length $2^{2014}$ and height $3^{2014}$. The board is divided into a grid of unit squares. A termite starts at either the left or bottom edge of the rectangle, and walks along the gridlines by moving either to the right or upwards, until it reaches an edge opposite the one from which the termite started. Depicted below are two possible paths of the termite. [img]https://cdn.artofproblemsolving.com/attachments/3/0/39f3b2aa9c61ff24ffc22b968790f4c61da6f9.png[/img] The termite’s path dissects the board into two parts. Sammy is surprised to find that he can still arrange the pieces to form a new rectangle not congruent to the original rectangle. This rectangle has perimeter $P$. How many possible values of $P$ are there?

Oleksii and Solomiya play the following game on a square $6n\times 6n$, where $n$ is a positive integer. Oleksii in his turn places a piece of type $F$, consisting of three cells, on the board. Solomia, in turn, after each move of Oleksii, places the numbers $0, 1, 2$ in the cells of the figure that Oleksii has just placed, using each of the numbers exactly once. If two of Oleksii's pieces intersect at any moment (have a common square), he immediately loses. Once the square is completely filled with numbers, the game stops. In this case, if the sum of the numbers in each row and each column is divisible by $3$, Solomiya wins, and otherwise Oleksii wins. Who can win this game if the figure of type $F$ is: a) a rectangle ; b) a corner of three cells? [i]Proposed by Oleksii Masalitin[/i]

Consider any rectangular table having finitely many rows and columns, with a real number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set of rows and $C$ a set of columns, is called a [i]saddle pair[/i] if the following two conditions are satisfied: [list] [*] $(i)$ For each row $r^{\prime}$, there is $r \in R$ such that $a(r, c) \geqslant a\left(r^{\prime}, c\right)$ for all $c \in C$; [*] $(ii)$ For each column $c^{\prime}$, there is $c \in C$ such that $a(r, c) \leqslant a\left(r, c^{\prime}\right)$ for all $r \in R$. [/list] A saddle pair $(R, C)$ is called a [i]minimal pair[/i] if for each saddle pair $\left(R^{\prime}, C^{\prime}\right)$ with $R^{\prime} \subseteq R$ and $C^{\prime} \subseteq C$, we have $R^{\prime}=R$ and $C^{\prime}=C$. Prove that any two minimal pairs contain the same number of rows.

$$Problem 4 $$ Straight lines $k,l,m$ intersecting each other in three different points are drawn on a classboard. Bob remembers that in some coordinate system the lines$ k,l,m$ have the equations $y = ax, y = bx$ and $y = c +2\frac{ab}{a+b}x$ (where $ab(a + b)$ is non zero). Misfortunately, both axes are erased. Also, Bob remembers that there is missing a line $n$ ($y = -ax + c$), but he has forgotten $a,b,c$. How can he reconstruct the line $n$?

Let $n$ be a positive integer and $t$ be a non-zero real number. Let $a_1, a_2, \ldots, a_{2n-1}$ be real numbers (not necessarily distinct). Prove that there exist distinct indices $i_1, i_2, \ldots, i_n$ such that, for all $1 \le k, l \le n$, we have $a_{i_k} - a_{i_l} \neq t$.

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

There are $12$ delegates in a mathematical conference. It is known that every two delegates share a common friend. Prove that there is a delegate who has at least five friends in that conference. [i]Proposed by Nairy Sedrakyan[/i]

Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there ?

Find the intersection of all sets of consecutive positive integers having at least four elements and the sum of elements equal to $2001$.

Francesca and Giorgia play the following game. On a table there are initially coins piled up in some stacks, possibly in different numbers in each stack, but with at least one coin. In turn, each player chooses exactly one move between the following: (i) she chooses a stack that has an even non-zero number of coins $ 2k$ and breaks it into two identical stacks of coins, i.e. each stack contains $ k$ coins; (ii) she removes from the table the stacks with coins in an odd number, i.e. all such in odd number, not just those with a specific odd number. At each turn, a player necessarily moves: if one choice is not available, the she must take the other. Francesca moves first. The one who removes the last coin from the table wins. 1. If initially there is only one stack of coins on the table, and this stack contains $ 2008^{2008}$ coins, which of the players has a winning strategy? 2. For which initial configurations of stacks of coins does Francesca have a winning strategy?

Baron Munchausen claims that he has drawn a polygon and chosen a point inside the polygon in such a way that any line passing through the chosen point divides the polygon into three polygons. Could the Baron’s claim be correct?

In a massive school which has $m$ students, and each student took at least one subject. Let $p$ be an odd prime. Given that: (i) each student took at most $p+1$ subjects. \\ (ii) each subject is taken by at most $p$ students. \\ (iii) any pair of students has at least $1$ subject in common. \\ Find the maximum possible value of $m$.

Some squares of a $1999\times 1999$ board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least $1999$.

Find all finite sequences $(x_0, x_1, \ldots,x_n)$ such that for every $j$, $0 \leq j \leq n$, $x_j$ equals the number of times $j$ appears in the sequence.

Two players $A$ and $B$ play the following game. An even number of cells are placed on a circle. $A$ begins and $A$ and $B$ play alternately, where each move consists of choosing a free cell and writing either $O$ or $M$ in it. The player after whose move the word $OMO$ (OMO = [i]Osterreichische Mathematik Olympiade[/i]) occurs for the first time in three successive cells wins the game. If no such word occurs, then the game is a draw. Prove that if player $B$ plays correctly, then player $A$ cannot win.

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

In the following $6\times 6$ matrix, one can choose any $k\times k$ submatrix, with $1<k\leq6 $ and add $1$ to all its entries. Is it possible to perform the operation a finite number of times so that all the entries in the $6\times 6$ matrix are multiples of $3$? $ \begin{pmatrix} 2 & 0 & 1 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 2 & 0 \\ 1 & 0 & 2 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 2 & 0 \\ 1 & 1 & 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} $ Note: A $p\times q$ submatrix of a $m\times n$ matrix (with $p\leq m$, $q\leq n$) is a $p\times q$ matrix formed by taking a block of the entries of this size from the original matrix.

Using only the digits $2,3$ and $9$, how many six-digit numbers can be formed which are divisible by $6$?

A board of $2n$ x $2n$ is colored chess style, a movement is the changing of colors of a $2$ x $2$ square. For what integers $n$ is possible to complete the board with one color using a finite number of movements?

Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the sequence $1$, $2$, $\dots$ , $n$ satisfying $$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$. Proposed by United Kingdom

The square $0\le x\le 1$, $0\le y\le 1$ is drawn in the plane $Oxy$. A grasshopper sitting at a point $M$ with noninteger coordinates outside this square jumps to a new point which is symmetrical to $M$ with respect to the leftmost (from the grasshopper’s point of view) vertex of the square. Prove that no matter how many times the grasshopper jumps, it will never reach the distance more than $10 d$ from the center $C$ of the square, where $d$ is the distance between the initial position $M$ and the center $C$. (A Kanel)

An entry in a grid is called a [i]saddle [/i] point if it is the largest number in its row and the smallest number in its column. Suppose that each cell in a $ 3 \times 3$ grid is filled with a real number, each chosen independently and uniformly at random from the interval $[0, 1]$. Compute the probability that this grid has at least one saddle point.

A word is a sequence of n letters of the alphabet {a, b, c, d}. A word is said to be complicated if it contains two consecutive groups of identic letters. The words caab, baba and cababdc, for example, are complicated words, while bacba and dcbdc are not. A word that is not complicated is a simple word. Prove that the numbers of simple words with n letters is greater than $2^n$, if n is a positive integer.

A closed broken self-intersecting line is drawn in the plane. Each of the links of this line is intersected exactly once and no three links intersect at the same point. Further, there are no self-intersections at the vertices and no two links have a common segment. Can it happen that every point of self-intersection divides both links in halves?

In a village (where only dwarfs live) there are $k$ streets, and there are $k(n-1)+1$ clubs each containing $n$ dwarfs. A dwarf can be in more than one clubs, and two dwarfs know each other if they live in the same street or they are in the same club (there is a club they are both in). Prove that is it possible to choose $n$ different dwarfs from $n$ different clubs (one dwarf from each club), such that the $n$ dwarfs know each other!