Found problems: 124
2019 Tournament Of Towns, 5
One needs to ffll the cells of an $n\times n$ table ($n > 1$) with distinct integers from $1$ to $n^2$ so that every two consecutive integers are placed in cells that share a side, while every two integers with the same remainder if divided by $n$ are placed in distinct rows and distinct columns. For which $n$ is this possible?
(Alexandr Gribalko)
2012 Tournament of Towns, 4
Each entry in an $n\times n$ table is either $+$ or $-$. At each step, one can choose a row or a column and reverse all signs in it. From the initial position, it is possible to obtain the table in which all signs are $+$. Prove that this can be accomplished in at most $n$ steps.
1999 Tournament Of Towns, 5
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)
2021 Thailand TST, 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.
1998 Tournament Of Towns, 3
Nine numbers are arranged in a square table:
$a_1 \,\,\, a_2 \,\,\,a_3$
$b_1 \,\,\,b_2 \,\,\,b_3$
$c_1\,\,\, c_2 \,\,\,c_3$ .
It is known that the six numbers obtained by summing the rows and columns of the table are equal:
$a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = c_1 + c_2 + c_3 = a_1 + b_1 + c_1 = a_2 + b_2 + c_2 = a_3 + b_3 + c_3$ .
Prove that the sum of products of numbers in the rows is equal to the sum of products of numbers in the columns:
$a_1 b_1 c_1 + a_2 b_2c_2 + a_3b_3c_3 = a_1a_2a_3 + b_1 b_2 b_3 + c_1 c_2c_3$ .
(V Proizvolov)
1975 All Soviet Union Mathematical Olympiad, 208
a) Given a big square consisting of $7\times 7$ squares. You should mark the centres of $k$ points in such a way, that no quadruple of the marked points will be the vertices of a rectangle with the sides parallel to the sides of the given squares. What is the greatest $k$ such that the problem has solution?
b) The same problem for $13\times 13$ square.
1986 Austrian-Polish Competition, 8
Pairwise distinct real numbers are arranged into an $m \times n$ rectangular array. In each row the entries are arranged increasingly from left to right. Each column is then rearranged in decreasing order from top to bottom. Prove that in the reorganized array, the rows remain arranged increasingly.
2017 Estonia Team Selection Test, 7
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 $6$?
2007 Peru Iberoamerican Team Selection Test, P4
Each of the squares on a $15$×$15$ board has a zero. At every step you choose a row or a column, we delete all the numbers from it and then we write the numbers from $1$ to $15$ in the empty cells, in an arbitrary order. find the sum
possible maximum of the numbers on the board that can be achieved after a number finite number of steps.
1987 All Soviet Union Mathematical Olympiad, 453
Each field of the $1987\times 1987$ board is filled with numbers, which absolute value is not greater than one. The sum of all the numbers in every $2\times 2$ square equals $0$. Prove that the sum of all the numbers is not greater than $1987$.
2016 Iran MO (3rd Round), 3
A $30\times30$ table is given. We want to color some of it's unit squares such that any colored square has at most $k$ neighbors. ( Two squares $(i,j)$ and $(x,y)$ are called neighbors if $i-x,j-y\equiv0,-1,1 \pmod {30}$ and $(i,j)\neq(x,y)$. Therefore, each square has exactly $8$ neighbors)
What is the maximum possible number of colored squares if$:$
$a) k=6$
$b)k=1$
2007 Bulgarian Autumn Math Competition, Problem 10.4
Find all pairs of natural numbers $(m,n)$, $m\leq n$, such that there exists a table with $m$ rows and $n$ columns filled with the numbers 1 and 0, satisfying the following property: If in a cell there's a 0 (respectively a 1), then the number of zeros (respectively ones) in the row of this cell is equal to the number of zeros (respectively ones) in the column of this cell.
2013 Portugal MO, 5
Liliana wants to paint a $m\times n$ board. Liliana divides each unit square by one of its diagonals and paint one of the halves of the square with black and the other half with white in such a way that triangles that have a common side haven't the same colour. How many possibilities has Liliana to paint the board?
1986 All Soviet Union Mathematical Olympiad, 435
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?
2010 Bosnia And Herzegovina - Regional Olympiad, 4
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
2020 Tournament Of Towns, 4
For which integers $N$ it is possible to write real numbers into the cells of a square of size $N \times N$ so that among the sums of each pair of adjacent cells there are all integers from $1$ to $2(N-1)N$ (each integer once)?
Maxim Didin
2017 Istmo Centroamericano MO, 2
On a $3 \times 3$ board the numbers from $1$ to $9$ are written in some order and without repeating. We say that the arrangement obtained is [i]Isthmian [/i] if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.
Note: Two arrangements are considered equal if one can be obtained from the other by rotating the board.
2015 IFYM, Sozopol, 4
A plane is cut into unit squares, which are then colored in $n$ colors. A polygon $P$ is created from $n$ unit squares that are connected by their sides. It is known that any cell polygon created by $P$ with translation, covers $n$ unit squares in different colors. Prove that the plane can be covered with copies of $P$ so that each cell is covered exactly once.
1987 Czech and Slovak Olympiad III A, 5
Consider a table with three rows and eleven columns. There are zeroes prefilled in the cell of the first row and the first column and in the cell of the second row and the last column. Determine the least real number $\alpha$ such that the table can be filled with non-negative numbers and the following conditions hold simultaneously:
(1) the sum of numbers in every column is one,
(2) the sum of every two neighboring numbers in the first row is at most one,
(3) the sum of every two neighboring numbers in the second row is at most one,
(4) the sum of every two neighboring numbers in the third row is at most $\alpha$.
2019 Greece JBMO TST, 4
Consider a $8\times 8$ chessboard where all $64$ unit squares are at the start white. Prove that, if any $12$ of the $64$ unit square get painted black, then we can find $4$ lines and $4$ rows that have all these $12$ unit squares.
2014 IFYM, Sozopol, 1
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?
2014 Grand Duchy of Lithuania, 3
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 $n$?
2018 IFYM, Sozopol, 4
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 $n$.
2017 Thailand TSTST, 1
1.1 Let $f(A)$ denote the difference between the maximum value and the minimum value of a set $A$. Find the sum of $f(A)$ as $A$ ranges over the subsets of $\{1, 2, \dots, n\}$.
1.2 All cells of an $8 × 8$ board are initially white. A move consists of flipping the color (white to black or vice versa) of cells in a $1\times 3$ or $3\times 1$ rectangle. Determine whether there is a finite sequence of moves resulting in the state where all $64$ cells are black.
1.3 Prove that for all positive integers $m$, there exists a positive integer $n$ such that the set $\{n, n + 1, n + 2, \dots , 3n\}$ contains exactly $m$ perfect squares.
2002 Tuymaada Olympiad, 6
In the cells of the table $ 100 \times100 $ are placed in pairs different numbers. Every minute each of the numbers changes to the largest of the numbers in the adjacent cells on the side. Can after $4$ hours all the numbers in the table be the same?