For any column and any row in a rectangular numerical table, the product of the sum of the numbers in a column by the sum of the numbers in a row is equal to the number at the intersection of the column and the row. Prove that either the sum of all the numbers in the table is equal to $1$ or all the numbers are equal to $0$.

In a table $n\times n$ some unit squares are coloured black and the other unit squares are coloured white. For each pair of columns and each pair of rows the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $n$?

A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that 2 rectangles $1$ x $(n-1)$ are connected to two adjacent sides of a square $1$ x $1$, so that their unit sides coincide. The squares or a chessboard $100$ x $100$ are colored in 15 colors. We say that a corner with arm 8 is [i]“multicolored”[/i], if it contains each of the colors on the board. What’s the greatest number of corners with arm 8 which could be [i]“mutlticolored”[/i]?

In the cell $(i,j)$ of a table $n\times n$ is written the number $(i-1)n + j$. Determine all positive integers $n$ such that there are exactly $2025$ rows not containing a perfect square.

In how many ways can $31$ squares be marked on an $8 \times 8$ chessboard so that no two of the marked squares have a common side? (R Zhenodarov)

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.

A $3n$-table is a table with three rows and $n$ columns containing all the numbers $1, 2, …, 3n$. Such a table is called [i]tidy [/i] if the $n$ numbers in the first row appear in ascending order from left to right, and the three numbers in each column appear in ascending order from top to bottom. How many tidy $3n$-tables exist?

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.

Each of the cells of a table 2014 x 2014 is colored in white or black. It is known that each square 2 x 2 contains an even number of black cells and each cross (3 x 3 square without its corner cells) contains an odd number of black cells. Prove that the 4 corner cells of the table are in the same color.

On a checkered square $10 \times 10$ the cells of the upper left $5 \times 5$ square are black and all the other cells are white. What is the maximal $n$ such that the original square can be dissected (along the borders of the cells) into $n$ polygons such that in each of them the number of black cells is three times less than the number of white cells? (The polygons need not be congruent or even equal in area.)

A checkered table consists of $2012$ rows and $k > 2$ columns. A marker is placed in a cell of the left-most column. Two players move the marker in turns. During each move, the player moves the marker by $1$ cell to the right, up or down to a cell that had never been occupied by the marker before. The game is over when any of the players moves the marker to the right-most column. However, whether this player is to win or to lose, the players are advised only when the marker reaches the second column from the right. Can any player secure his win?

A table consisting of $9$ rows and $2001$ columns is filfed with integers $1,2,..., 2001$ in such a way that each of these integers occurs in the table exactly $9$ times and the integers in any column differ by no more than $3$. Find the maximum possible value of the minimal column sum (sum of the numbers in one column).

Sofia and Viktor are playing the following game on a $2022 \times 2022$ board: - Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table; - Then Viktor without seeing the table, chooses a positive integer $n$; - After that Viktor looks at the table covered with dominoes, chooses and fixes $n$ of them; - Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before. If she achieves that, she wins, otherwise Viktor wins. What is the minimum number $n$ for which Viktor can always win, no matter the starting covering of dominoes. [i]Proposed by Viktor Simjanoski[/i]

We may permute the rows and the columns of the table below. How may different tables can we generate? 1 2 3 4 5 6 7 7 1 2 3 4 5 6 6 7 1 2 3 4 5 5 6 7 1 2 3 4 4 5 6 7 1 2 3 3 4 5 6 7 1 2 2 3 4 5 6 7 1

For a positive integer $n$, consider a square cake which is divided into $n \times n$ pieces with at most one strawberry on each piece. We say that such a cake is [i]delicious[/i] if both diagonals are fully occupied, and each row and each column has an odd number of strawberries. Find all positive integers $n$ such that there is an $n \times n$ delicious cake with exactly $\left\lceil\frac{n^2}{2}\right\rceil$ strawberries on it.

Three children wanted to make a table-game. For that purpose they wished to enumerate the $mn$ squares of an $m \times n$ game-board by the numbers $1, ... ,mn$ in such way that the numbers $1$ and $mn$ lie in the corners of the board and the squares with successive numbers have a common edge. The children agreed to place the initial square (with number $1$) in one of the corners but each child wanted to have the final square (with number $mn$ ) in different corner. For which numbers $m$ and $n$ is it possible to satisfy the wish of any of the children?