a) Given real numbers $a_1,a_2,b_1,b_2$ and positive $p_1,p_2,q_1,q_2$. Prove that in the table $2\times 2$ $$(a_1 + b_1)/(p_1 + q_1) , (a_1 + b_2)/(p_1 + q_2) $$ $$(a_2 + b_1)/(p_2 + q_1) , (a_2 + b_2)/(p_2 + q_2)$$ there is a number in the table, that is not less than another number in the same row and is not greater than another number in the same column (a saddle point). b) Given real numbers $a_1, a_2, ... , a_n, b_1, b_2, ... , b_n$ and positive $p_1, p_2, ... , p_n, q_1, q_2, ... , q_n$. We construct the table $n\times n$, with the numbers ($0 < i,j \le n$) $$(a_i + b_j)/(p_i + q_j)$$ in the intersection of the $i$-th row and $j$-th column. Prove that there is a number in the table, that is not less than arbitrary number in the same row and is not greater than arbitrary number in the same column (a saddle point).

What is the greatest amount of rooks that can be placed on an $n\times n$ board, such that each rooks beats an even number of rooks? A rook is considered to beat another rook, if they lie on one vertical or one horizontal line and no rooks are between them. [I]Proposed by D. Karpov[/i]

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)

In table of dimensions $2n \times 2n$ there are positive integers not greater than $10$, such that numbers lying in unit squares with common vertex are coprime. Prove that there exist at least one number which occurs in table at least $\frac{2n^2}{3}$ times

A plane is cut into unit squares, each of which is colored in black or white. It is known that each rectangle 3 x 4 or 4 x 3 contains exactly 8 white squares. In how many ways can this plane be colored?

Albrecht fills in each cell of an $8 \times 8$ table with a $0$ or a $1$. Then at the end of each row and column he writes down the sum of the $8$ digits in that row or column, and then he erases the original digits in the table. Afterwards, he claims to Berthold that given only the sums, it is possible to restore the $64$ digits in the table uniquely. Show that the $8 \times 8$ table contained either a row full of $0$’s or a column full of $1$’s

In every cell of a table with $n$ rows and $m$ columns is written one of the letters $a$, $b$, $c$. Every two rows of the table have the same letter in at most $k\geq 0$ positions and every two columns coincide at most $k$ positions. Find $m$, $n$, $k$ if \[\frac{2mn+6k}{3(m+n)}\geq k+1\]

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?

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 $6$?

Each of the $16$ squares in a $4 \times 4$ table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to $1$. Determine the sum of all $16$ numbers in the table. (R Zhenodarov)

We color each unit square of a $8\times 8$ table into green or blue such that there are $a$ green unit squares in each $3 \times 3$ square and $b$ green unit squares in each $2 \times 4$ rectangle. Find all possible values of $(a, b)$. (Le Anh Vinh)

Prove that the unit square can be tiled with rectangles (not necessarily of the same size) similar to a rectangle of size $1\times(3+\sqrt[3]{3})$.

Let $n$ be a positive integer and let $t$ be an integer. $n$ distinct integers are written on a table. Bob, sitting in a room nearby, wants to know whether there exist some of these numbers such that their sum is equal to $t$. Alice is standing in front of the table and she wants to help him. At the beginning, she tells him only the initial sum of all numbers on the table. After that, in every move he says one of the $4$ sentences: $i.$ Is there a number on the table equal to $k$? $ii.$ If a number $k$ exists on the table, erase him. $iii.$ If a number $k$ does not exist on the table, add him. $iv.$ Do the numbers written on the table can be arranged in two sets with equal sum of elements? On these questions Alice answers yes or no, and the operations he says to her she does (if it is possible) and does not tell him did she do it. Prove that in less than $3n$ moves, Bob can find out whether there exist numbers initially written on the board such that their sum is equal to $t$

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

The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice can win the game no matter how Bob plays.

In the cells of a square table $n$ x $n$ the numbers $1,2,...,n^2$ are written in an arbitrary way. Prove that there exist two adjacent cells, for which the difference between the numbers written in them is no lesser than $n$.

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 $6$?

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.

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$.

Given is a table 4x4 and in every square there is 0 or 1. In a move we choose row or column and we change the numbers there. Call the square "zero" if we cannot decrease the number of zeroes in it. Call "degree of the square" the number zeroes in a "zero" square. Find all possible values of the degree.

$n$ is a natural number. Given $3n \cdot 3n$ table, the unit cells are colored white and black such that starting from the left up corner diagonals are colored in pure white or black in ratio of 2:1 respectively. ( See the picture below). In one step any chosen $2 \cdot 2$ square's white cells are colored orange, orange are colored black and black are colored white. Find all $n$ such that with finite steps, all the white cells in the table turns to black, and all black cells in the table turns to white. ( From starting point)

The square table $n\times n$ is filled by integers. If the fields have common side, the difference of numbers in them doesn't exceed $1$. Prove that some number is encountered not less than a) not less than $[n/2]$ times ($[ ...]$ mean the whole part), b) not less than $n$ times.

Is it possible to color the cells of a table 19 x 19 in yellow, blue, red, and green so that each rectangle $a$ x $b$ ($a,b\geq 2$) in the table has at least 2 cells in different color?

In a table consisting of $n$ by $n$ small squares some squares are coloured black and the other 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 $n$?

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.