Each of the cells of an $m \times n$ table is coloured either black or white. For each cell, the total number of the cells which are in the same row or in the same column and of the same colour as this cell is strictly less than the total number of the cells which are in the same row or in the same column and of the other colour as this cell. Prove that in each row and in each column the number of white cells is the same as the number of black ones. (A Shapovalov)

Cells of a $2000\times2000$ board are colored according to the following rules: 1)At any moment a cell can be colored, if none of its neighbors are colored 2)At any moment a $1\times2$ rectangle can be colored, if exactly two of its neighbors are colored. 3)At any moment a $2\times2$ squared can be colored, if 8 of its neighbors are colored (Two cells are considered to be neighboring, if they share a common side). Can the entire $2000\times2000$ board be colored? [I]Proposed by K. Kohas[/i]

On an infinite sheet of graph paper a table is drawn so that in each square of the table stands a number equal to the arithmetic mean of the four adjacent numbers. Out of the table a piece is cut along the lines of the graph paper. Prove that the largest number on the piece always occurs at an edge, where $x = \frac14 (a + b + c + d)$.

Two people play a game on a $9 \times 9$ board. They move alternately. On each move, the first player draws a cross in an empty cell, and the second player draws a nought in an empty cell. When all $81$ cells are filled, the number $K$ of rows and columns in which there are more crosses and the number $H$ of rows and columns in which there are more noughts are counted. The score for the first player is the difference $B = K- H$. Find a value of $B$ such that the first player can guarantee a score of at least $B$, while the second player can hold the first player's score to at most B, regardless how the opponent plays. (A Kanel)

Prove that it is impossible to fill the cells of an $8 \times 8$ table with the numbers from $ 1$ to $64$ (each number must be used once) so that for each $2\times 2$ square, the difference between products of the numbers on it’s diagonals will be equal to $ 1$.

A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid? (pictured right) so that no two of them cover the same square? [img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]

In an $8 \times 8$ grid, $n$ disks, numbered $1$ to $n$ are stacked, with random order, in a pile in the bottom left comer. The disks can be moved one at a time to a neighbouring cell either to the right or top. The aim to move all the disks to the cell at the top right comer and stack them in the order $1,2,...,n$ from the bottom. Each cell, except the bottom left and top right cell, can have at most one disk at any given time. Find the largest value of $n$ so that the aim can be achieved.

In table [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC9hLzBjNjFlZWFjM2ZlOTQzMTk2YTdkMzQ2MjJiYzYyMWFlN2Y0ZGZlLnBuZw==&rn=dGFibGljYWEucG5n[/img] $10$ numbers are circled, in every row and every column exactly one. Prove that among them, there are at least two equal

The cells of a table [b]m x n[/b], $m \geq 5$, $n \geq 5$ are colored in 3 colors where: (i) Each cell has an equal number of adjacent (by side) cells from the other two colors; (ii) Each of the cells in the 4 corners of the table doesn’t have an adjacent cell in the same color. Find all possible values for $m$ and $n$.

Let $n$ be a positive integer. In how many ways can an $n \times n$ table be filled with integers from $0$ to $5$ such that a) the sum of each row is divisible by $2$ and the sum of each column is divisible by $3$ b) the sum of each row is divisible by $2$, the sum of each column is divisible by $3$ and the sum of each of the two diagonals is divisible by $6$?

Take a $n \times n$ chess page.Determine the $n$ such that we can put the numbers $1,2,3, \ldots ,n$ in the squares of the page such that we know the following two conditions are true: a) for each row we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares of that row have the same sum as the sum of the numbers in the white squares of that row. b) for each column we know all the numbers $1,2,3, \ldots ,n$ have appeared on it and the numbers that are in the black squares in that column have the same sum as the sum of the numbers in the white squares of that column.

In each cell of the table $4 \times 4$, in which the lines are labeled with numbers $1,2,3,4$, and columns with letters $a,b,c,d$, one number is written: $0$ or $1$ . Such a table is called [i]valid [/i] if there are exactly two units in each of its rows and in each column. Determine the number of [i]valid [/i] tables.

We call a table of size $n \times n$ self-describing if each cell of the table contains the total number of even numbers in its row and column other than itself. How many self-describing tables of size a) $3 \times 3$ exist? b) $4 \times 4$ exist? c) $5 \times 5$ exist? Two tables are different if they differ in at least one cell.

Is it possible to fill a $40 \times 41$ table with integers so that each integer equals the number of adjacent (by an edge) cells with the same integer? Alexandr Gribalko

Numbers $1992,1993, ... ,2000$ are written in a $3 \times 3$ table to form a magic square (i.e. the sums of numbers in rows, columns and big diagonals are all equal). Prove that the number in the center is $1996$. Which numbers are placed in the corners?

Is it possible to fill a rectangular table with black and white squares (only) so, that the number of black squares will equal to the number of white squares, and each row and each column will have more than $75\%$ squares of the same colour?

Consider a triangular table of positive integers \[ \begin{matrix} & & & a_{11} & a_{12} & a_{13} & & & \\ & & a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & & \\ & a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & \\ \iddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \] The first row consists of odd numbers only. For $i>1,j\ge1$ we have \[a_{ij}=a_{i-1,j-2}+a_{i-1,j-1}+a_{i-1,j}.\] If we get out of range with the second index, we consider such $a$ to be zero (eg. $a_{22}=0+a_{11}+a_{12}$ and $a_{37}=a_{25}+0+0$). Show that for every $i>1$ there is $j\in\{1,\ldots,2i+1\}$ such that $a_{ij}$ is even.

The natural numbers $0, 1, 2, 3, . . .$ are written on the square table $2015\times 2015$ in a circular order (anti-clockwise) such that $0$ is in the center of the table. The rows and columns are labelled from bottom to top and from left to right respectively. (see figure below) 1. The number $2015$ is in which row and which column? 2. We are allowed to perform the following operations: First, we replace the number $0$ in the center by $14$, after that, each time, we can add $1$ to each of $12$ numbers on $12$ consecutive unit squares in a row, or $12$ consecutive unit squares in a column, or $12$ unit squares in a rectangle $3\times 4$. After a finite number of steps, can we make all numbers on the table are multiples of $2016$? [img]https://cdn.artofproblemsolving.com/attachments/c/d/223b32c0e3f58f62d0d40fa78c09a2cd035ed5.png[/img]

Consider a table whose entries are integers. Adding a same integer to all entries on a same row, or on a same column, is called an [i]operation[/i]. It is given that, for infinitely many positive integers $n$, one can obtain, through a finite number of operations, a table having all entries divisible by $n$. Prove that, through a finite number of operations, one can obtain the table whose all entries are zeroes.

A magician and his assistant perform in front of an audience of many people. On the stage there is an $8$×$8$ board, the magician blindfolds himself, and then the assistant goes inviting people from the public to write down the numbers $1, 2, 3, 4, . . . , 64$ in the boxes they want (one number per box) until completing the $64$ numbers. After the assistant covers two adyacent boxes, at her choice. Finally, the magician removes his blindfold and has to $“guess”$ what number is in each square that the assistant. Explain how they put together this trick. $Clarification:$ Two squares are adjacent if they have a common side

All the fields of a square $n\times n$ (n>2) table are filled with $+1$ or $-1$ according to the rules: [i]At the beginning $-1$ are put in all the boundary fields. The number put in the field in turn (the field is chosen arbitrarily) equals to the product of the closest, from the different sides, numbers in its row or in its column. [/i] a) What is the minimal b) What is the maximal possible number of $+1$ in the obtained table?

Let $n > 4$ be an integer. A square is divided into $n^2$ smaller identical squares, in some of which were [b]1’s[/b] and in the other – [b]0's[/b]. It is not allowed in one row or column to have the following arrangements of adjacent digits in this order: $101$, $111$ or $1001$. What is the biggest number of [b]1’s[/b] in the table? (The answer depends on $n$.)

A $1\times 2019$ board is filled with numbers $1, 2, \dots, 2019$ in an increasing order. In each step, three consecutive tiles are selected, then one of the following operations is performed: $\text{(i)}$ the number in the middle is increased by $2$ and its neighbors are decreased by $1$, or $\text{(ii)}$ the number in the middle is decreased by $2$ and its neighbors are increased by $1$. After several such operations, the board again contains all the numbers $1, 2,\dots, 2019$. Prove that each number is in its original position.

Find all positive integers $n$ for which a square[b][i] n x n[/i][/b] can be covered with rectangles [b][i]k x 1[/i][/b] and one square [b][i]1 x 1[/i][/b] when: a) $k = 4$ b) $k = 8$

In each cell of the table $ 3 \times 3 $ there is one of the numbers $1, 2$ and $3$. Dima counted the sum of the numbers in each row and in each column. What is the greatest number of different sums he could get?