Found problems: 187
2006 ISI B.Math Entrance Exam, 1
Bishops on a chessboard move along the diagonals ( that is , on lines parallel to the two main diagonals ) . Prove that the maximum number of non-attacking bishops on an $n*n$ chessboard is $2n-2$. (Two bishops are said to be attacking if they are on a common diagonal).
2014 Contests, 4
The numbers from $1$ to $64$ must be written on the small squares of a chessboard, with a different number in each small square. Consider the $112$ numbers you can make by adding the numbers in two small squares which have a common edge. Is it possible to write the numbers in the squares so that these $112$ sums are all different?
2000 IMO Shortlist, 4
Let $ n$ and $ k$ be positive integers such that $ \frac{1}{2} n < k \leq \frac{2}{3} n.$ Find the least number $ m$ for which it is possible to place $ m$ pawns on $ m$ squares of an $ n \times n$ chessboard so that no column or row contains a block of $ k$ adjacent unoccupied squares.
1989 Tournament Of Towns, (208) 2
On a square of a chessboard there is a pawn . Two players take turns to move it to another square, subject to the rule that , at each move the distance moved is strictly greater than that of the previous move. A player loses when unable to make a move on his turn. Who wins if the players always choose the best strategy? (The pawn is always placed in the centre of its square. )
( F . L . Nazarov)
1998 Tournament Of Towns, 3
On an $8 \times 8$ chessboard, $17$ cells are marked. Prove that one can always choose two cells among the marked ones so that a Knight will need at least three moves to go from one of the chosen cells to the other.
(R Zhenodarov)
2014 Nordic, 4
A game is played on an ${n \times n}$ chessboard. At the beginning there are ${99}$ stones on each square. Two players ${A}$ and ${B}$ take turns, where in each turn the player chooses either a row or a column and removes one stone from each square in the chosen row or column. They are only allowed to choose a row or a column, if it has least one stone on each square. The first player who cannot move, looses the game. Player ${A}$ takes the first turn. Determine all n for which player ${A}$ has a winning strategy.
2019 South Africa National Olympiad, 4
The squares of an $8 \times 8$ board are coloured alternatingly black and white. A rectangle consisting of some of the squares of the board is called [i]important[/i] if its sides are parallel to the sides of the board and all its corner squares are coloured black. The side lengths can be anything from $1$ to $8$ squares. On each of the $64$ squares of the board, we write the number of important rectangles in which it is contained. The sum of the numbers on the black squares is $B$, and the sum of the numbers on the white squares is $W$. Determine the difference $B - W$.
1947 Moscow Mathematical Olympiad, 129
How many squares different in size or location can be drawn on an $8 \times 8$ chess board? Each square drawn must consist of whole chess board’s squares.
1985 All Soviet Union Mathematical Olympiad, 397
What maximal number of the men in checkers game can be put on the chess-board $8\times 8$ so, that every man can be taken by at least one other man ?
2022 Indonesia TST, C
Distinct pebbles are placed on a $1001 \times 1001$ board consisting of $1001^2$ unit tiles, such that every unit tile consists of at most one pebble. The [i]pebble set[/i] of a unit tile is the set of all pebbles situated in the same row or column with said unit tile. Determine the minimum amount of pebbles that must be placed on the board so that no two distinct tiles have the same [i]pebble set[/i].
[hide=Where's the Algebra Problem?]It's already posted [url=https://artofproblemsolving.com/community/c6h2742895_simple_inequality]here[/url].[/hide]
1971 Poland - Second Round, 1
In how many ways can you choose $ k $ squares on a chessboard $ n \times n $ ( $ k \leq n $) so that no two of the chosen squares lie in the same row or column?
2016 Postal Coaching, 4
Consider a $2n\times 2n$ chessboard with all the $4n^2$ cells being white to start with. The following operation is allowed to be performed any number of times:
"Three consecutive cells (in a row or column) are recolored (a white cell is colored black and a black cell is colored white)."
Find all possible values of $n\ge 2$ for which using the above operation one can obtain the normal chess coloring of the given board.
2024 Indonesia Regional, 2
Given an $n \times n$ board which is divided into $n^2$ squares of size $1 \times 1$, all of which are white. Then, Aqua selects several squares from this board and colors them black. Ruby then places exactly one $1\times 2$ domino on the board, so that the domino covers exactly two squares on the board. Ruby can rotate the domino into a $2\times 1$ domino.
After Aqua colors, it turns out there are exactly $2024$ ways for Ruby to place a domino on the board so that it covers exactly $1$ black square and $1$ white square.
Determine the smallest possible value of $n$ so that Aqua and Ruby can do this.
[i]Proposed by Muhammad Afifurrahman, Indonesia [/i]
2015 IFYM, Sozopol, 4
In how many ways can $n$ rooks be placed on a $2n$ x $2n$ chessboard, so that they cover all the white fields?
1984 All Soviet Union Mathematical Olympiad, 390
The white fields of $1983\times 1984 $1983x1984 are filled with either $+1$ or $-1$. For every black field, the product of neighbouring numbers is $+1$. Prove that all the numbers are $+1$.
2022 OMpD, 1
Consider a chessboard $6 \times 6$, made up of $36$ single squares. We want to place $6$ chess rooks on this board, one rook on each square, so that there are no two rooks on the same row, nor two rooks on the same column. Note that, once the rooks have been placed in this way, we have that, for every square where a rook has not been placed, there is a rook in the same row as it and a rook in the same column as it. We will say that such rooks are in line with this square.
For each of those $30$ houses without rooks, color it green if the two rooks aligned with that same house are the same distance from it, and color it yellow otherwise. For example, when we place the $6$ rooks ($T$) as below, we have:
(a) Is it possible to place the rooks so that there are $30$ green squares?
(b) Is it possible to place the rooks so that there are $30$ yellow squares?
(c) Is it possible to place the rooks so that there are $15$ green and $15$ yellow squares?
2011 QEDMO 8th, 1
A $T$-tetromino is a non-convex as well as non-rotationally symmetrical tetromino, which has a maximum number of outside corners (popularly also "Tetris Stone "called). Find all natural numbers $n$ for which, a $n \times n$ chessboard is found that can be covered only with such $T$-tetrominos.
2006 Dutch Mathematical Olympiad, 5
Player $A$ and player $B$ play the next game on an $8$ by $8$ square chessboard.
They in turn color a field that is not yet colored. One player uses red and the other blue. Player $A$ starts. The winner is the first person to color the four squares of a square of $2$ by $2$ squares with his color somewhere on the board.
Prove that player $B$ can always prevent player $A$ from winning.
2023 Poland - Second Round, 6
Given a chessboard $n \times n$, where $n\geq 4$ and $p=n+1$ is a prime number. A set of $n$ unit squares is called [i]tactical[/i] if after putting down queens on these squares, no two queens are attacking each other. Prove that there exists a partition of the chessboard into $n-2$ tactical sets, not containing squares on the main diagonals.
Queens are allowed to move horizontally, vertically and diagonally.
2020 LIMIT Category 1, 15
In a $4\times 4$ chessboard, in how many ways can you place $3$ rooks and one bishop such that none of these pieces threaten another piece?
2012 BAMO, 1
Hugo places a chess piece on the top left square of a $20 \times 20$ chessboard and makes $10$ moves with it. On each of these $10$ moves, he moves the piece either one square horizontally (left or right) or one square vertically (up or down). After the last move, he draws an $X$ on the square that the piece occupies. When Hugo plays the game over and over again, what is the largest possible number of squares that could eventually be marked with an $X$? Prove that your answer is correct.
1989 IMO Shortlist, 19
A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.
2005 Tournament of Towns, 3
Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.)
[i](5 points)[/i]
1999 Tournament Of Towns, 4
(a) On each of the $1 \times 1$ squares of the top row of an $8 \times 8$ chessboard there is a black pawn, and on each of the $1 \times 1$ squares of the bottom row of this chessboard there is a white pawn. On each move one can shift any pawn vertically or horizontally to any adjacent empty $1 \times 1$ square. What is the smallest number of moves that are needed to move all white pawns to the top row and all black pawns to the bottom one?
(b) The same question for a $7 \times 7$ board.
(A Shapovalov_
2023 JBMO TST - Turkey, 2
A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?