Olympiad problems

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?

2024 Czech-Polish-Slovak Junior Match, 6

Tags: rectangular grid , table , numbers in a table , combinatorics

We are given a rectangular table with a positive integer written in each of its cells. For each cell of the table, the number in it is equal to the total number of different values in the cells that are in the same row or column (including itself). Find all tables with this property.

1989 Tournament Of Towns, (242) 6

Tags: combinatorics , table

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)

2006 Belarusian National Olympiad, 6

Tags: combinatorics , table , sum , max

An $n \times m$ table ( $n \le m$ ) is filled in accordance with the rules of the game "Minesweeper": mines are placed at some cells (not more than one mine at the cell) and in the remaining cells one writes the number of the mines in the neighboring (by side or by vertex) cells. Then the sum of allnumbers in the table is computed (this sum is equal to $9$ for the picture). What is the largest possible value of this sum? (V. Lebed) [img]https://cdn.artofproblemsolving.com/attachments/2/9/726ccdbc57807788a5f6e88a5acb42b10a6cc0.png[/img]

2012 Tournament of Towns, 2

Tags: prime , combinatorics , table , rectangle table

The cells of a $1\times 2n$ board are labelled $1,2,...,, n, -n,..., -2, -1$ from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then $2n + 1$ is prime.

2016 Saudi Arabia BMO TST, 4

Tags: combinatorics , combinatorial geometry , table

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.)

2016 Iran MO (3rd Round), 3

Tags: modular arithmetic , combinatorics , table , coloring

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$:$ $a) k=6$ $b)k=1$

2002 Tuymaada Olympiad, 6

Tags: table , combinatorics

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?

2012 Tournament of Towns, 4

Tags: combinatorics , square table , table , signs

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.

1996 Estonia National Olympiad, 3

Tags: algebra , magic square , table

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?

2007 Peru Iberoamerican Team Selection Test, P4

Tags: combinatorics , board , table

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.

2018 IFYM, Sozopol, 2

Tags: table , combinatorics

A square is divided into 169 identical small squares and in every small square is written 0 or 1. It isn’t allowed in one row or column to have the following arrangements of adjacent digits in this order: 101, 111 or 1001. What is the the biggest possible number of 1’s in the table?

2017 Thailand TSTST, 4

Tags: combinatorics , coloring , table , cell , construction

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?

2016 Iran MO (3rd Round), 2

Tags: combinatorics , table

Is it possible to divide a $7\times7$ table into a few $\text{connected}$ parts of cells with the same perimeter? ( A group of cells is called $\text{connected}$ if any cell in the group, can reach other cells by passing through the sides of cells.)

1996 Estonia National Olympiad, 5

Tags: table , combinatorics , board

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?

2017 Istmo Centroamericano MO, 2

Tags: table , combinatorics

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.

2020 Dürer Math Competition (First Round), P5

Tags: counting , table , combinatorics

We call a table of size $n \times n$ self-describing if each cell of the table contains the total number of even numbers in its row and column other than itself. How many self-describing tables of size a) $3 \times 3$ exist? b) $4 \times 4$ exist? c) $5 \times 5$ exist? Two tables are different if they differ in at least one cell.

2019 Tournament Of Towns, 5

Tags: square grid , numbers in a table , table , combinatorics

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)

2018 IFYM, Sozopol, 3

Tags: table , combinatorics

We will call one of the cells of a rectangle 11 x 13 “[i]peculiar[/i]” , if after removing it the remaining figure can be cut into squares 2 x 2 and 3 x 3. How many of the 143 cells are “[i]peculiar[/i]”?

2014 IFYM, Sozopol, 6

Tags: combinatorics , covering , table

We have 19 triminos (2 x 2 squares without one unit square) and infinite amount of 2 x 2 squares. Find the greatest odd number $n$ for which a square $n$ x $n$ can be covered with the given figures.

1992 All Soviet Union Mathematical Olympiad, 565

Tags: table , combinatorics , combinatorial geometry

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?

2007 Bulgarian Autumn Math Competition, Problem 10.4

Tags: combinatorics , table

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.

1991 All Soviet Union Mathematical Olympiad, 549

Tags: combinatorics , square table , table

An $h \times k$ minor of an $n \times n$ table is the $hk$ cells which lie in $h$ rows and $k$ columns. The semiperimeter of the minor is $h + k$. A number of minors each with semiperimeter at least $n$ together include all the cells on the main diagonal. Show that they include at least half the cells in the table.

2018 IFYM, Sozopol, 4

Tags: table , coloring , combinatorics

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 $n$.