Found problems: 187
1997 Tournament Of Towns, (550) 4
We want to draw a number of straight lines such that for each square of a chessboard, at least one of the lines passes through an interior point of the square. What is the smallest number of lines needed for a
(a) $3\times 3$;
(b) $4\times 4$
chessboard? Use a picture to show that this many lines are enough, and prove that no smaller number would do.
(M Vyalyi)
1963 All Russian Mathematical Olympiad, 033
A chess-board $6\times 6$ is tiled with the $2\times 1$ dominos. Prove that you can cut the board onto two parts by a straight line that does not cut dominos.
1983 Tournament Of Towns, (052) 5
A set $A$ of squares is given on a chessboard which is infinite in all directions. On each square of this chessboard which does not belong to $A$ there is a king. On a command all kings may be moved in such a way that each king either remains on its square or is moved to an adjacent square, which may have been occupied by another king before the command. Each square may be occupied by at most one king. Does there exist such a number $k$ and such a way of moving the kings that after $k$ moves the kings will occupy all squares of the chessboard? Consider the following cases:
(a) $A$ is the set of all squares, both of whose coordinates are multiples of $100$. (There is a horizontal line numbered by the integers from $-\infty$ to $+\infty$, and a similar vertical line. Each square of the chessboard may be denoted by two numbers, its coordinates with respect to these axes.)
(b) $A$ is the set of all squares which are covered by $100$ fixed arbitrary queens (i.e. each square covered by at least one queen).
Remark:
If $A$ consists of just one square, then $k = 1$ and the required way is the following:
all kings to the left of the square of $A$ make one move to the right.
2023 Ukraine National Mathematical Olympiad, 8.1
Oleksiy placed positive integers in the cells of the $8\times 8$ chessboard. For each pair of adjacent-by-side cells, Fedir wrote down the product of the numbers in them and added all the products. Oleksiy wrote down the sum of the numbers in each pair of adjacent-by-side cells and multiplied all the sums. It turned out that the last digits of both numbers are equal to $1$. Prove that at least one of the boys made a mistake in the calculation.
For example, for a square $3\times 3$ and the arrangement of numbers shown below, Fedir would write the following numbers: $2, 6, 8, 24, 15, 35, 2, 6, 8, 20, 18, 42$, and their sum ends with a digit $6$; Oleksiy would write the following numbers: $3, 5, 6, 10, 8, 12, 3, 5, 6, 9, 9, 13$, and their product ends with a digit $0$.
\begin{tabular}{| c| c | c |}
\hline
1 & 2 & 3 \\
\hline
2 & 4 & 6 \\
\hline
3 & 5 & 7 \\
\hline
\end{tabular}
[i]Proposed by Oleksiy Masalitin and Fedir Yudin[/i]
1930 Eotvos Mathematical Competition, 2
A straight line is drawn across an $8\times 8$ chessboard. It is said to [i]pierce [/i]a square if it passes through an interior point of the square. At most how many of the $64$ squares can this line [i]pierce[/i]?
2003 Estonia National Olympiad, 5
For which positive integers $n$ is it possible to cover a $(2n+1) \times (2n+1)$ chessboard which has one of its corner squares cut out with tiles shown in the figure (each tile covers exactly $4$ squares, tiles can be rotated and turned around)?
[img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]
2010 Contests, 3
In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.
2008 HMNT, Chess
[u]Chessboards [/u]
Joe B. is playing with some chess pieces on a $6\times 6$ chessboard. Help him find out some things.
[b]p1.[/b] Joe B. first places the black king in one corner of the board. In how many of the $35$ remaining squares can he place a white bishop so that it does not check the black king?
[b]p2.[/b] Joe B. then places a white king in the opposite corner of the board. How many total ways can he place one black bishop and one white bishop so that neither checks the king of the opposite color?
[b]p3.[/b] Joe B. now clears the board. How many ways can he place $3$ white rooks and $3$ black rooks on the board so that no two rooks of opposite color can attack each other?
[b]p4.[/b] Joe B. is frustrated with chess. He breaks the board, leaving a $4\times 4$ board, and throws $3$ black knights and $3$ white kings at the board. Miraculously, they all land in distinct squares! What is the expected number of checks in the resulting position? (Note that a knight can administer multiple checks and a king can be checked by multiple knights.)
[b]p5.[/b] Suppose that at some point Joe B. has placed $2$ black knights on the original board, but gets bored of chess. He now decides to cover the $34$ remaining squares with $17$ dominos so that no two overlap and the dominos cover the entire rest of the board. For how many initial arrangements of the two pieces is this possible?
Note: Chess is a game played with pieces of two colors, black and white, that players can move between squares on a rectangular grid. Some of the pieces
move in the following ways:
$\bullet$ Bishop: This piece can move any number of squares diagonally if there are no other pieces along its path.
$\bullet$ Rook: This piece can move any number of squares either vertically or horizontally if there are no other pieces along its path.
$\bullet$ Knight: This piece can move either two squares along a row and one square along a column or two squares along a column and one square along a row.
$\bullet$ King: This piece can move to any open adjacent square (including diagonally).
If a piece can move to a square occupied by a king of the opposite color, we say that it is checking the king.
If a piece moves to a square occupied by another piece, this is called attacking.
2009 IMO Shortlist, 6
On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over.
How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit?
[i]Proposed by Nikolay Beluhov, Bulgaria[/i]
1999 Estonia National Olympiad, 4
Let us put pieces on some squares of $2n \times 2n$ chessboard in such a way that on every horizontal and vertical line there is an odd number of pieces. Prove that the whole number of pieces on the black squares is even.
2019 USEMO, 3
Consider an infinite grid $\mathcal G$ of unit square cells. A [i]chessboard polygon[/i] is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$.
Nikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\mathcal G$ green, such that any chessboard polygon congruent to $F$ has at least $1$ green cell but at most $2020$ green cells. Can Nikolai choose $F$ to make your job impossible?
[i]Nikolai Beluhov[/i]
1999 Tournament Of Towns, 6
On a large chessboard $2n$ of its $1 \times 1$ squares have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles.
(A Shapovalov)
2015 QEDMO 14th, 11
Let $m, n$ be natural numbers and let $m\cdot n$ be a multiple of $4$. A chessboard with $m \times n$ fields are covered with $1 \times 2$ large dominoes without gaps and without overlapping. Show that the number of dominoes that are parallel to a edge of the chess board is fixed .
[hide=original wording] Seien m, n natu¨rliche Zahlen und sei m · n ein Vielfaches von 4. Ein Schachbrett mit m × n Feldern sei mit 1 × 2 großen Dominosteinen lu¨ckenlos und u¨berlappungsfrei u¨berdeckt. Zeige, dass die Anzahl der Dominosteine, die zu einer fest gew¨ahlten Kante des Schachbrettes parallel sind, gerade ist. [/hide]
2010 IMO Shortlist, 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]
1988 Tournament Of Towns, (198) 1
What is the smallest number of squares of a chess board that can be marked in such a manner that
(a) no two marked squares may have a common side or a common vertex, and
(b) any unmarked square has a common side or a common vertex with at least one marked square?
Indicate a specific configuration of marked squares satisfying (a) and (b) and show that a lesser number of marked squares will not suffice.
(A. Andjans, Riga)
2013 Balkan MO Shortlist, C4
A closed, non-self-intersecting broken line $L$ is drawn over a $(2n+1) \times (2n+1)$ chessboard in such a way that the set of L's vertices coincides with the set of the vertices of the board’s squares and every edge in $L$ is a side of some board square. All board squares lying in the interior of $L$ are coloured in red. Prove that the number of neighbouring pairs of red squares in every row of the board is even.
2022 Brazil National Olympiad, 6
Some cells of a $10 \times 10$ are colored blue. A set of six cells is called [i]gremista[/i] when the cells are the intersection of three rows and two columns, or two rows and three columns, and are painted blue. Determine the greatest value of $n$ for which it is possible to color $n$ chessboard cells blue such that there is not a [i]gremista[/i] set.
2018 Pan-African Shortlist, C3
A game is played on an $m \times n$ chessboard. At the beginning, there is a coin on one of the squares. Two players take turns to move the coin to an adjacent square (horizontally or vertically). The coin may never be moved to a square that has been occupied before. If a player cannot move any more, he loses. Prove:
[list]
[*] If the size (number of squares) of the board is even, then the player to move first has a winning strategy, regardless of the initial position.
[*] If the size of the board is odd, then the player to move first has a winning strategy if and only if the coin is initially placed on a square whose colour is not the same as the colour of the corners.
[/list]
1987 Tournament Of Towns, (149) 6
Two players play a game on an $8$ by $8$ chessboard according to the following rules. The first player places a knight on the board. Then each player in turn moves the knight , but cannot place it on a square where it has been before. The player who has no move loses. Who wins in an errorless game , the first player or the second one? (The knight moves are the normal ones. )
(V . Zudilin , year 12 student , Beltsy (Moldova))
1983 Czech and Slovak Olympiad III A, 3
An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.
2009 IMO Shortlist, 4
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]
2015 Indonesia MO Shortlist, C1
Given natural number n. Suppose that $N$ is the maximum number of elephants that can be placed on a chessboard measuring $2 \times n$ so that no two elephants are mutually under attack. Determine the number of ways to put $N$ elephants on a chessboard sized $2 \times n$ so that no two elephants attack each other.
Alternative Formulation:
Determine the number of ways to put $2015$ elephants on a chessboard measuring $2 \times 2015$ so there are no two elephants attacking each othe
PS. Elephant = Bishop
2020 HMIC, 2
Some bishops and knights are placed on an infinite chessboard, where each square has side length $1$ unit. Suppose that the following conditions hold:
[list]
[*] For each bishop, there exists a knight on the same diagonal as that bishop (there may be another piece between the bishop and the knight).
[*] For each knight, there exists a bishop that is exactly $\sqrt{5}$ units away from it.
[*] If any piece is removed from the board, then at least one of the above conditions is no longer satisfied.
[/list]
If $n$ is the total number of pieces on the board, find all possible values of $n$.
[i]Sheldon Kieren Tan[/i]
2010 Grand Duchy of Lithuania, 1
Sixteen points are placed in the centers of a $4 \times 4$ chess table in the following way:
• • • •
• • • •
• • • •
• • • •
(a) Prove that one may choose $6$ points such that no isoceles triangle can be drawn with the vertices at these points.
(b) Prove that one cannot choose $7$ points with the above property.
2021 Regional Olympiad of Mexico Center Zone, 4
Two types of pieces, bishops and rooks, are to be placed on a $10\times 10$ chessboard (without necessarily filling it) such that each piece occupies exactly one square of the board. A bishop $B$ is said to [i]attack[/i] a piece $P$ if $B$ and $P$ are on the same diagonal and there are no pieces between $B$ and $P$ on that diagonal; a rook $R$ is said to attack a piece $P$ if $R$ and $P$ are on the same row or column and there are no pieces between $R$ and $P$ on that row or column.
A piece $P$ is [i]chocolate[/i] if no other piece $Q$ attacks $P$.
What is the maximum number of chocolate pieces there may be, after placing some pieces on the chessboard?
[i]Proposed by José Alejandro Reyes González[/i]