Found problems: 187
1988 Tournament Of Towns, (167) 4
The numbers from $1$ to $64$ are written on the squares of a chessboard (from $1$ to $8$ from left to right on the first row , from $9$ to $16$ from left to right on the second row , and so on). Pluses are written before some of the numbers, and minuses are written before the remaining numbers in such a way that there are $4$ pluses and $4$ minuses in each row and in each column . Prove that the sum of the written numbers is equal to zero.
2018 Peru Cono Sur TST, 6
Let $n$ be a positive integer. In an $n \times n$ board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place $n$ queens on the board such that no two threaten each other when:
$a)\:$ $n=14$.
$b)\:$ $n=15$.
2016 Saint Petersburg Mathematical Olympiad, 2
On a $300 \times 300$ board, several rooks are placed that beat the entire board. Within this case, each rook beats no more than one other rook. At what least $k$, it is possible to state that there is at least one rook in each $k\times k$ square ?
2017 Baltic Way, 8
A chess knight has injured his leg and is limping. He alternates between a normal move and a short move where he moves to any diagonally neighbouring cell.
The limping knight moves on a $5 \times 6$ cell chessboard starting with a normal move. What is the largest number of moves he can make if he is starting from a cell of his own choice and is not allowed to visit any cell (including the initial cell) more than once?
2017 Tournament Of Towns, 7
$1\times 2$ dominoes are placed on an $8 \times 8$ chessboard without overlapping. They may partially
stick out from the chessboard but the center of each domino must be strictly inside the
chessboard (not on its border). Place on the chessboard in such a way:
a) at least $40$ dominoes, (3 points)
b) at least $41$ dominoes, (3 points)
c) more than $41$ dominoes. (6 points)
[i](Mikhail Evdokimov)[/i]
2022 Taiwan TST Round 2, 2
A $100 \times100$ chessboard has a non-negative real number in each of its cells. A chessboard is [b]balanced[/b] if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number $x$ so that, for any balanced chessboard, we can find $100$ cells of it so that these cells all have number
greater or equal to $x$, and no two of these cells are on the same column or row.
[i]Proposed by CSJL.[/i]
2005 Tournament of Towns, 6
A [i]lazy[/i] rook can only move from a square to a vertical or a horizontal neighbour. It follows a path which visits each square of an $8 \times 8$ chessboard exactly once. Prove that the number of such paths starting at a corner square is greater than the number of such paths starting at a diagonal neighbour of a corner square.
[i](7 points)[/i]
2000 BAMO, 5
Alice plays the following game of solitaire on a $20 \times 20$ chessboard.
She begins by placing $100$ pennies, $100$ nickels, $100$ dimes, and $100$ quarters on the board so that each of the $400$ squares contains exactly one coin. She then chooses $59$ of these coins and removes them from the board.
After that, she removes coins, one at a time, subject to the following rules:
- A penny may be removed only if there are four squares of the board adjacent to its square (up, down, left, and right) that are vacant (do not contain coins). Squares “off the board” do not count towards this four: for example, a non-corner square bordering the edge of the board has three adjacent squares, so a penny in such a square cannot be removed under this rule, even if all three adjacent squares are vacant.
- A nickel may be removed only if there are at least three vacant squares adjacent to its square. (And again, “off the board” squares do not count.)
- A dime may be removed only if there are at least two vacant squares adjacent to its square (“off the board” squares do not count).
- A quarter may be removed only if there is at least one vacant square adjacent to its square (“off the board” squares do not count).
Alice wins if she eventually succeeds in removing all the coins. Prove that it is impossiblefor her to win.
2015 Switzerland Team Selection Test, 1
What is the maximum number of 1 × 1 boxes that can be colored black in a n × n chessboard so that any 2 × 2 square contains a maximum of 2 black boxes?
1981 All Soviet Union Mathematical Olympiad, 304
Two equal chess-boards ($8\times 8$) have the same centre, but one is rotated by $45$ degrees with respect to another. Find the total area of black fields intersection, if the fields have unit length sides.
2011 QEDMO 8th, 4
How many
a) bishops
b) horses
can be positioned on a chessboard at most, so that no one threatens another?
2017 Saudi Arabia JBMO TST, 4
Find the number of ways one can put numbers $1$ or $2$ in each cell of an $8\times 8$ chessboard in such a way that the sum of the numbers in each column and in each row is an odd number. (Two ways are considered different if the number in some cell in the first way is different from the number in the cell situated in the corresponding position in the second way)
2018 Junior Balkan Team Selection Tests - Romania, 4
What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.)
Alexandru Mihalcu
2013 Bosnia And Herzegovina - Regional Olympiad, 4
$a)$ Is it possible, on modified chessboard $20 \times 30$, to draw a line which cuts exactly $50$ cells where chessboard cells are squares $1 \times 1$
$b)$ What is the maximum number of cells which line can cut on chessboard $m \times n$, $m,n \in \mathbb{N}$
2007 JBMO Shortlist, 3
The nonnegative integer $n$ and $ (2n + 1) \times (2n + 1)$ chessboard with squares colored alternatively black and white are given. For every natural number $m$ with $1 < m < 2n+1$, an $m \times m$ square of the given chessboard that has more than half of its area colored in black, is called a $B$-square. If the given chessboard is a $B$-square, fi nd in terms of $n$ the total number of $B$-squares of this chessboard.
2005 Tournament of Towns, 4
A chess piece moves as follows: it can jump 8 or 9 squares either vertically or horizontally. It is not allowed to visit the same square twice. At most, how many squares can this piece visit on a $15 \times 15$ board (it can start from any square)?
[i](4 points)[/i]
2011 Ukraine Team Selection Test, 2
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
[i]Proposed by Sergei Berlov, Russia[/i]
2020 Romanian Master of Mathematics Shortlist, C3
Determine the smallest positive integer $k{}$ satisfying the following condition: For any configuration of chess queens on a $100 \times 100$ chequered board, the queens can be coloured one of $k$ colours so that no two queens of the same colour attack each other.
[i]Russia, Sergei Avgustinovich and Dmitry Khramtsov[/i]
2014 Gulf Math Olympiad, 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?
KoMaL A Problems 2023/2024, A. 881
We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk.
[i]Proposed by Dömötör Pálvölgyi, Budapest[/i]
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](6 points)[/i]
2010 Germany Team Selection Test, 2
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
1976 Dutch Mathematical Olympiad, 3
In how many ways can the king in the chessboard reach the eighth rank in $7$ moves from its original square on the first row?
2021 Middle European Mathematical Olympiad, 2
Let $m$ and $n$ be positive integers. Some squares of an $m \times n$ board are coloured red. A sequence $a_1, a_2, \ldots , a_{2r}$ of $2r \ge 4$ pairwise distinct red squares is called a [i]bishop circuit[/i] if for every $k \in \{1, \ldots , 2r \}$, the squares $a_k$ and $a_{k+1}$ lie on a diagonal, but the squares $a_k$ and $a_{k+2}$ do not lie on a diagonal (here $a_{2r+1}=a_1$ and $a_{2r+2}=a_2$).
In terms of $m$ and $n$, determine the maximum possible number of red squares on an $m \times n$ board without a bishop circuit.
([i]Remark.[/i] Two squares lie on a diagonal if the line passing through their centres intersects the sides of the board at an angle of $45^\circ$.)
2011 Belarus Team Selection Test, 3
2500 chess kings have to be placed on a $100 \times 100$ chessboard so that
[b](i)[/b] no king can capture any other one (i.e. no two kings are placed in two squares sharing a common vertex);
[b](ii)[/b] each row and each column contains exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)
[i]Proposed by Sergei Berlov, Russia[/i]