a) The fields of the square table $4\times 4$ are filled with the "+" or "-" signs. You are allowed to change the signs simultaneously in the whole row, column, or diagonal to the opposite sign. That means, for example, that You can change the sign in the corner square, because it makes a diagonal itself. Prove that you will never manage to obtain a table containing pluses only. b) The fields of the square table $8\times 8$ are filled with the "+" or signs except one non-corner field with "-". You are allowed to change the signs simultaneously in the whole row, column, or diagonal to the opposite sign. That means, for example, that You can change the sign in the corner field, because it makes a diagonal itself. Prove that you will never manage to obtain a table containing pluses only.

Twenty-five people of different heights stand in a $5\times 5$ grid of squares, with one person in each square. We know that each row has a shortest person, suppose Ana is the tallest of these five people. Similarly, we know that each column has a tallest person, suppose Bev is the shortest of these five people. Assuming Ana and Bev are not the same person, who is taller: Ana or Bev? Prove that your answer is always correct.

Find the greatest number of depicted pieces composed of $4$ unit squares that can be placed without overlapping on an $n \times n$ grid (where n is a positive integer) in such a way that it is possible to move from some corner to the opposite corner via uncovered squares (moving between squares requires a common edge). The shapes can be rotated and reflected. [img]https://cdn.artofproblemsolving.com/attachments/b/d/f2978a24fdd737edfafa5927a8d2129eb586ee.png[/img]

Given a $n\times n$ table, where $n$ is odd. There is either $1$ or $-1$ in its every field. A product of the numbers in the column is written under every column. A product of the numbers in the row is written to the right of every row. Prove that the sum of $2n$ products doesn't equal to $0$.

a) The numbers $1, 2, . . . , 49$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/5/0/c2e350a6ad0ebb8c728affe0ebb70783baf913.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $36$ numbers, etc., $7$ times. Find the sum of the numbers selected. b) The numbers $1, 2, . . . , k^2$ are arranged in a square table as follows: [img]https://cdn.artofproblemsolving.com/attachments/2/d/28d60518952c3acddc303e427483211c42cd4a.png[/img] Among these numbers we select an arbitrary number and delete from the table the row and the column which contain this number. We do the same with the remaining table of $(k - 1)^2$ numbers, etc., $k$ times. Find the sum of the numbers selected.

There is a $8\times 8$ table, drawn in a plane and painted in a chess board fashion. Peter mentally chooses a square and an interior point in it. Basil can draws any polygon (without self-intersections) in the plane and ask Peter whether the chosen point is inside or outside this polygon. What is the minimal number of questions suffcient to determine whether the chosen point is black or white?

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$.

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 $6$?

Every cell of a $4\times 4$ square grid net, has $1\times 1$ size. Is it possible to represent this net as a union of the following sets: a) Eight broken lines of length five each? b) Five broken lines of length eight each?

Every unit square of a $n \times n$ board is colored either red or blue so that among all 2 $\times 2$ squares on this board all possible colorings of $2 \times 2$ squares with these two colors are represented (colorings obtained from each other by rotation and reflection are considered different). a) Find the least possible value of $n$. b) For the least possible value of $n$ find the least possible number of red unit squares

(a) On each square of a squared sheet of paper of size $20 \times 20$ there is a soldier. Vanya chooses a number $d$ and Petya moves the soldiers to new squares in such a way that each soldier is moved through a distance of at least $d$ (the distance being measured between the centres of the initial and the new squares) and each square is occupied by exactly one soldier. For which $d$ is this possible? (Give the maximum possible $d$, prove that it is possible to move the soldiers through distances not less than $d$ and prove that there is no greater $d$ for which this procedure may be carried out.) (b) Answer the same question as (a), but with a sheet of size $21 \times 21$. (SS Krotov, Moscow)

Every cell of a $20 \times 20$ table has to be coloured black or white (there are $2^{400}$ such colourings in total). Given any colouring $P$, we consider division of the table into rectangles with sides in the grid lines where no rectangle contains more than two black cells and where the number of rectangles containing at most one black cell is the least possible. We denote this smallest possible number of rectangles containing at most one black cell by $f(P)$. Determine the maximum value of $f(P)$ as $P$ ranges over all colourings.

$n^2$ real numbers are written in a square $n \times n$ table so that the sum of the numbers in each row and column equals zero. A move is to add a row to one column and subtract it from another (so if the entries are $a_{ij}$ and we select row $i$, column $h$ and column $k$, then column h becomes $a_{1h} + a_{i1}, a_{2h} + a_{i2}, ... , a_{nh} + a_{in}$, column $k$ becomes $a_{1k} - a_{i1}, a_{2k} - a_{i2}, ... , a_{nk} - a_{in}$, and the other entries are unchanged). Show that we can make all the entries zero by a series of moves.

Find the smallest positive integer $n$ with the following property: After painting black exactly $n$ cells of a $7\times 7$ board there always exists a $2\times 2$ square with at least three black cells.

Given a square $100\times 100$ on the sheet of cross-lined paper. There are several broken lines drawn inside the square. Their links consist of the small squares sides. They are neither pairwise- nor self-intersecting (have no common points). Their ends are on the big square boarder, and all the other vertices are in the big square interior. Prove that there exists (in addition to four big square angles) a node (corresponding to the cross-lining family, inside the big square or on its side) that does not belong to any broken line.

a) A square table with $49$ small squares is filled with numbers $1$ to $7$ so that in each row and in each column all numbers from $1$ to $7$ are present. Let the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , 7$ are present. b) A square table with $n^2$ small squares is filled with numbers $1$ to $n$ so that in each row and in each column all numbers from $1$ to $n$ are present. Let $n$ be odd and the table be symmetric through the main diagonal. Prove that on this diagonal all the numbers $1, 2, 3, . . . , n$ are present.

There is a $19\times19$ board. Is it possible to mark some $1\times 1$ squares so that each of $10\times 10$ squares contain different number of marked squares?

In a grid of dimensions $n \times n$, a part of the squares is marked with crosses such that in each at least half of the $4 \times 4$ squares are marked. Find the least possible the total number of marked squares in the grid.

Each $1 \times 1$ square of an $n \times n$ table contains a different number. The smallest number in each row is marked, and these marked numbers are in different columns. Then the smallest number in each column is marked, and these marked numbers are in different rows. Prove that the two sets of marked numbers are identical. (V Klepcyn)

In a $N \times N$ array of numbers, all rows are different (two rows are said to be different even if they differ only in one entry). Prove that there is a column which can be deleted in such a way that the resulting rows will still be different. (A Andjans, Riga)

Is it possible to put the numbers $0,1$ or $2$ in the unit squares of the cross-lined paper $100\times 100$ in such a way, that every rectangle $3\times 4$ (and $4\times 3$) would contain three zeros, four ones and five twos?

A sequence $(a_1, a_2,...,a_k)$ consisting of pairwise different cells of an $n\times n$ board is called a cycle if $k \ge 4$ and cell ai shares a side with cell $a_{i+1}$ for every $i = 1,2,..., k$, where $a_{k+1} = a_1$. We will say that a subset $X$ of the set of cells of a board is [i]malicious [/i] if every cycle on the board contains at least one cell belonging to $X$. Determine all real numbers $C$ with the following property: for every integer $n \ge 2$ on an $n\times n$ board there exists a malicious set containing at most $Cn^2$ cells.

The figure shows a figure of $5$ unit squares, a Greek cross. What is the largest number of Greek crosses that can be placed on a grid of dimensions $8 \times 8$ without any overlaps, with each unit square covering just one square in a grid?

Given the square table $n\times n$ with $(n-1)$ marked fields. Prove that it is possible to move all the marked fields below the diagonal by moving rows and columns.

Can the squares of a $1990 \times 1990$ chessboard be colored black or white so that half the squares in each row and column are black and cells symmetric with respect to the center are of opposite color?