This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 187

2004 South East Mathematical Olympiad, 4
Tags: combinatorics unsolved , combinatorics , Chessboard
Given a positive integer $n (n>2004)$, we put 1, 2, 3, …,$n^2$ into squares of an $n\times n$ chessboard with one number in a square. A square is called a “good square” if the square satisfies following conditions: 1) There are at least 2004 squares that are in the same row with the square such that any number within these 2004 squares is less than the number within the square. 2) There are at least 2004 squares that are in the same column with the square such that any number within these 2004 squares is less than the number within the square. Find the maximum value of the number of the “good square”.

2019 Brazil Team Selection Test, 4
Tags: Chessboard , paths , combinatorics
Consider a checkered board $2m \times 2n$, $m, n \in \mathbb{Z}_{>0}$. A stone is placed on one of the unit squares on the board, this square is different from the upper right square and from the lower left square. A snail goes from the bottom left square and wants to get to the top right square, walking from one square to other adjacent, one square at a time (two squares are adjacent if they share an edge). Determine all the squares the stone can be in so that the snail can complete its path by visiting each square exactly one time, except the square with the stone, which the snail does not visit.

2021 Durer Math Competition Finals, 9
Tags: combinatorics , Chessboard
On an $8 \times 8$ chessboard, a rook stands on the bottom left corner square. We want to move it to the upper right corner, subject to the following rules: we have to move the rook exactly $9$ times, such that the length of each move is either $3$ or $4$. (It is allowed to mix the two lengths throughout the "journey".) How many ways are there to do this? In each move, the rook moves horizontally or vertically.

2013 QEDMO 13th or 12th, 1
Tags: combinatorics , Chessboard
A lightly damaged rook moves around on a $m \times n$ chessboard by taking turns moves to a horizontal or vertical field. For which $m$ and $n$, is it possible for him to have visited each field exactly once? The starting field counts as visited, squares skipped during a move, however, are not.

2013 Romania National Olympiad, 2
Tags: combinatorics , combinatorial geometry , Chessboard
A rook starts moving on an infinite chessboard, alternating horizontal and vertical moves. The length of the first move is one square, of the second – two squares, of the third – three squares and so on. a) Is it possible for the rook to arrive at its starting point after exactly $2013$ moves? b) Find all $n$ for which it possible for the rook to come back to its starting point after exactly $n$ moves.

2008 Bulgaria Team Selection Test, 1
Tags: induction , combinatorics proposed , combinatorics , graph theory , Hamiltonian path , Chessboard
Let $n$ be a positive integer. There is a pawn in one of the cells of an $n\times n$ table. The pawn moves from an arbitrary cell of the $k$th column, $k \in \{1,2, \cdots, n \}$, to an arbitrary cell in the $k$th row. Prove that there exists a sequence of $n^{2}$ moves such that the pawn goes through every cell of the table and finishes in the starting cell.

1987 Tournament Of Towns, (138) 3
Tags: combinatorics , Chessboard
Nine pawns forming a $3$ by $3$ square are placed in the lower left hand corner of an $8$ by $8$ chessboard. Any pawn may jump over another one standing next to it into a free square, i .e. may be reflected symmetrically with respect to a neighb our's centre (jumps may be horizontal , vertical or diagonal) . It is required to rearrange the nine pawns in another corner of the chessboard (in another $3$ by $3$ square) by means of such jumps. Can the pawns be thus re-arranged in the (a) upper left hand corner? (b) upper right hand corner? (J . E . Briskin)

2019 Polish Junior MO First Round, 6
Tags: combinatorics , Chessboard , Coloring
The $14 \times 14$ chessboard squares are colored in pattern, as shown in the picture. Can you choose seven fields blacks and seven white squares of this chessboard in such a way, that there is exactly one selected field in each row and column? Justify your answer. [img]https://cdn.artofproblemsolving.com/attachments/e/4/e8ba46030cd0f0e0511f1f9e723e5bd29e9975.png[/img]

1989 All Soviet Union Mathematical Olympiad, 491
Tags: combinatorics , Chessboard
Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.

2010 Lithuania National Olympiad, 3
Tags: combinatorics , Chessboard
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.

1989 IMO Longlists, 64
Tags: combinatorics , invariant , Chessboard , algorithm , IMO Shortlist
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.

1950 Moscow Mathematical Olympiad, 173
Tags: Chessboard , circle , geometry , combinatorial geometry
On a chess board, the boundaries of the squares are assumed to be black. Draw a circle of the greatest possible radius lying entirely on the black squares.

2012 Tournament of Towns, 5
Tags: Chessboard , Rook , game , combinatorics , game strategy
In an $8\times 8$ chessboard, the rows are numbers from $1$ to $8$ and the columns are labelled from $a$ to $h$. In a two-player game on this chessboard, the fi rst player has a White Rook which starts on the square $b2$, and the second player has a Black Rook which starts on the square $c4$. The two players take turns moving their rooks. In each move, a rook lands on another square in the same row or the same column as its starting square. However, that square cannot be under attack by the other rook, and cannot have been landed on before by either rook. The player without a move loses the game. Which player has a winning strategy?

1998 Tournament Of Towns, 5
Tags: Chessboard , combinatorics
A "labyrinth" is an $8 \times 8$ chessboard with walls between some neighboring squares. If a rook can traverse the entire board without jumping over the walls, the labyrinth is "good" ; otherwise it is "bad" . Are there more good labyrinths or bad labyrinths? (A Shapovalov)

2021 Science ON all problems, 4
Tags: combinatorics , induction , Chessboard
An $n\times n$ chessboard is given, where $n$ is an even positive integer. On every line, the unit squares are to be permuted, subject to the condition that the resulting table has to be symmetric with respect to its main diagonal (the diagonal from the top-left corner to the bottom-right corner). We say that a board is [i]alternative[/i] if it has at least one pair of complementary lines (two lines are complementary if the unit squares on them which lie on the same column have distinct colours). Otherwise, we call the board [i]nonalternative[/i]. For what values of $n$ do we always get from the $n\times n$ chessboard an alternative board?\\ \\ [i](Alexandru Petrescu and Andra Elena Mircea)[/i]

1992 India National Olympiad, 7
Tags: combinatorics , counting , Chessboard , Arithmetic Progression
Let $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.

2021 Harvard-MIT Mathematics Tournament., 10
Tags: Chessboard , combinatorics
Let $n>1$ be a positive integer. Each unit square in an $n\times n$ grid of squares is colored either black or white, such that the following conditions hold: $\bullet$ Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge; $\bullet$ Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge; $\bullet$ Any $2\times 2$ subgrid contains at least one square of each color. Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of $n$).

2010 Germany Team Selection Test, 3
Tags: modular arithmetic , combinatorics , IMO Shortlist , Chessboard , Chess rook
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]

2016 Czech And Slovak Olympiad III A, 6
Tags: combinatorics , Chessboard
We put a figure of a king on some $6 \times 6$ chessboard. It can in one thrust jump either vertically or horizontally. The length of this jump is alternately one and two squares, whereby a jump of one (i.e. to the adjacent square) of the piece begins. Decide whether you can choose the starting position of the pieces so that after a suitable sequence $35$ jumps visited each box of the chessboard just once.

2003 Estonia National Olympiad, 5
Tags: Chessboard , Tiling , combinatorics
Is it possible to cover an $n \times n$ chessboard which has its center square cut out with tiles shown in the picture (each tile covers exactly $4$ squares, tiles can be rotated and turned around) if a) $n = 5$, b) $n = 2003$? [img]https://cdn.artofproblemsolving.com/attachments/6/5/8fddeefc226ee0c02353a1fc11e48ce42d8436.png[/img]

2013 Tournament of Towns, 5
Tags: Chessboard , combinatorics
Eight rooks are placed on a chessboard so that no two rooks attack each other. Prove that one can always move all rooks, each by a move of a knight so that in the final position no two rooks attack each other as well. (In intermediate positions several rooks can share the same square).

1976 All Soviet Union Mathematical Olympiad, 229
Tags: Chessboard , combinatorics , combinatorial geometry
Given a chess-board $99\times 99$ with a set $F$ of fields marked on it (the set is different in three tasks). There is a beetle sitting on every field of the set $F$. Suddenly all the beetles have raised into the air and flied to another fields of the same set. The beetles from the neighbouring fields have landed either on the same field or on the neighbouring ones (may be far from their starting point). (We consider the fields to be neighbouring if they have at least one common vertex.) Consider a statement: [i]"There is a beetle, that either stayed on the same field or moved to the neighbouring one".[/i] Is it always valid if the figure $F$ is: a) A central cross, i.e. the union of the $50$-th row and the $50$-th column? b) A window frame, i.e. the union of the $1$-st, $50$-th and $99$-th rows and the $1$-st, $50$-th and $99$-th columns? c) All the chess-board?

2012 Chile National Olympiad, 1
Tags: combinatorics , Chessboard
What is the minimum number of movements that a horse must carry out on chess, on an $8\times 8$ board, to reach the upper right square starting at the lower left? Remember that the horse moves in the usual $L$-shaped manner.

2018 India PRMO, 26
Tags: combinatorics , Chessboard
What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard such that no two chosen squares have a side in common?

1986 Brazil National Olympiad, 5
Tags: combinatorics , number theory , maximum , Chessboard
A number is written in each square of a chessboard, so that each number not on the border is the mean of the $4$ neighboring numbers. Show that if the largest number is $N$, then there is a number equal to $N$ in the border squares.