A rectangular array has $m$ rows and $n$ columns, where $m < n$. Some cells of the array contain stars, in such a way that there is at least one star in each column. Prove that there is at least one such star such that the row containing it has more stars than the column containing it. (A. Razborov, Moscow)

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?

An $m \times n$ rectangle is divided into mn unit squares by lines parallel to its sides. A gnomon is the figure of three unit squares formed by deleting one unit square from a $2 \times 2$ square. For what $m, n$ can we divide the rectangle into gnomons so that no two gnomons form a rectangle and no vertex is in four gnomons?

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.

Let $d\geq 1$ and $n\geq 0$ be integers. Find the number of ways to write down a nonnegative integer in each square of a $d\times d$ grid such that the numbers in any set of $d$ squares, no two in the same row or column, sum to $n$.

We will call a rectangular table filled with natural numbers [i]“good”[/i], if for each two rows, there exist a column for which its two cells that are also in these two rows, contain numbers of different parity. Prove that for $\forall$ $n>2$ we can erase a column from a [i]good[/i] $n$ x $n$ table so that the remaining $n$ x $(n-1)$ table is also [i]good[/i].

A cross with length $p$ (or [i]p-cross[/i] for short) will be called the figure formed by a unit square and 4 rectangles $p-1$ x $1$ on its sides. What’s the least amount of colors one has to use to color the cells of an infinite table, so that each [i]p-cross[/i] on it covers cells, no two of which are in the same color?

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 \[\frac{2mn+6k}{3(m+n)}\geq k+1\]

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)

Two intelligent people playing a game on the $1403 \times 1403$ table with $1403^2$ cells. The first one in each turn chooses a cell that didn't select before and draws a vertical line segment from the top to the bottom of the cell. The second person in each turn chooses a cell that didn't select before and draws a horizontal line segment from the left to the right of the cell. After $1403^2$ steps the game will be over. The first person gets points equal to the longest verticals line segment and analogously the second person gets point equal to the longest horizonal line segment. At the end the person who gets the more point will win the game. What will be the result of the game?

Consider an infinite white plane divided into square cells. For which $k$ it is possible to paint a positive finite number of cells black so that on each horizontal, vertical and diagonal line of cells there is either exactly $k$ black cells or none at all? A. Dinev, K. Garov, N Belukhov

The cells of a $8 \times 8$ table are colored either black or white so that each row has a different number of black squares, and each column has a different number of black squares. What is the maximum number of pairs of adjacent cells of different colors?

Given positive numbers $a_1, a_2, ..., a_m, b_1, b_2, ..., b_n$. Is known that $$a_1+a_2+...+a_m=b_1+b_2+...+b_n.$$ Prove that you can fill an empty table with $m$ rows and $n$ columns with no more than $(m+n-1)$ positive number in such a way, that for all $i,j$ the sum of the numbers in the $i$-th row will equal to $a_i$, and the sum of the numbers in the $j$-th column -- to $b_j$.

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]

Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.

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]

The entries of a $7 \times 5$ table are filled with numbers so that in each $2 \times 3$ rectangle (vertical or horizontal) the sum of numbers is $0$. For $100$ dollars Peter may choose any single entry and learn the number in it. What is the least amount of dollars he should spend in order to learn the total sum of numbers in the table for sure?

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.

The integers from 1 to 49 are written in a $7\times 7$ table, such that for any $k\in\{1,2,\ldots,7\}$ the product of the numbers in the $k$-th row equals the product of the numbers in the $(8-k)$-th row. [list=a] [*]Prove that there exists a row such that the sum of the numbers written on it is a prime number. [*]Give an example of such a table. [/list] [i]Cristi Săvescu[/i]

Consider a table of cyclic permutations ($n\ge2$) \[ \begin{matrix} 1, & 2, & \ldots, & n-1, & n \\ 2, & 3, & \ldots, & n, & 1, \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n, & 1, & \ldots, & n-2, & n-1. \end{matrix} \] Then multiply each number of the first row by that number of the $k$-th row that is in the same column. Sum all these products and denote $s_k$ the result (e.g. $s_2=1\cdot2+2\cdot3+\cdots+(n-1)\cdot n+n\cdot1$). a) Find a recursive relation for $s_k$ in terms of $s_{k-1}$ and determine the explicit formula for $s_k$. b) Determine both an index $k$ and the value of $s_k$ such that the sum $s_k$ is minimal.

Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that from the remaining part of the table $36$ $1\times2$ dominos can be cut [I]Proposed by S. Berlov[/i]

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 every unit square of a$ n \times n$ table ($n \ge 11$) a real number is written, such that the sum of the numbers in any $10 \times 10$ square is positive and the sum of the numbers in any $11\times 11$ square is negative. Determine all possible values for $n$

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).