Found problems: 187
2015 IFYM, Sozopol, 7
A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that 2 rectangles $1$ x $(n-1)$ are connected to two adjacent sides of a square $1$ x $1$, so that their unit sides coincide.
The squares or a chessboard $100$ x $100$ are colored in 15 colors. We say that a corner with arm 8 is [i]“multicolored”[/i], if it contains each of the colors on the board. What’s the greatest number of corners with arm 8 which could be [i]“mutlticolored”[/i]?
1970 Bulgaria National Olympiad, Problem 3
On a chessboard (with $64$ squares) there are situated $32$ white and $32$ black pools. We say that two pools form a mixed pair when they are with different colors and they lie on the same row or column. Find the maximum and the minimum of the mixed pairs for all possible situations of the pools.
[i]K. Dochev[/i]
2017 Saudi Arabia IMO TST, 3
The $64$ cells of an $8 \times 8$ chessboard have $64$ different colours. A Knight stays in one cell. In each move, the Knight jumps from one cell to another cell (the $2$ cells on the diagonal of an $2 \times 3$ board) also the colours of the $2$ cells interchange. In the end, the Knight goes to a cell having common side with the cell it stays at first. Can it happen that: there are exactly $3$ cells having the colours different from the original colours?
2013 Balkan MO Shortlist, C5
The cells of an $n \times n$ chessboard are coloured in several colours so that no $2\times 2$ square contains four cells of the same colour. A [i]proper path [/i] of length $m$ is a sequence $a_1,a_2,..., a_m$ of distinct cells in which the cells $a_i$ and $a_{i+1}$ have a common side and are coloured in different colours for all $1 \le i < m$. Show that there exists a proper path of length $n$.
2013 Balkan MO Shortlist, C2
Some squares of an $n \times n$ chessboard have been marked ($n \in N^*$). Prove that if the number of marked squares is at least $n\left(\sqrt{n} + \frac12\right)$, then there exists a rectangle whose vertices are centers of marked squares.
1979 All Soviet Union Mathematical Olympiad, 275
What is the least possible number of the checkers being required
a) for the $8\times 8$ chess-board,
b) for the $n\times n$ chess-board,
to provide the property:
[i]Every line (of the chess-board fields) parallel to the side or diagonal is occupied by at least one checker[/i] ?
1974 IMO, 4
Consider decompositions of an $8\times 8$ chessboard into $p$ non-overlapping rectangles subject to the following conditions:
(i) Each rectangle has as many white squares as black squares.
(ii) If $a_i$ is the number of white squares in the $i$-th rectangle, then $a_1<a_2<\ldots <a_p$.
Find the maximum value of $p$ for which such a decomposition is possible. For this value of $p$, determine all possible sequences $a_1,a_2,\ldots ,a_p$.
2019 Tuymaada Olympiad, 3
The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to rooms adjacent by side. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess rook (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 1$)?
1983 Tournament Of Towns, (048) 5
$N^2$ pieces are placed on an $N \times N$ chessboard. Is it possible to rearrange them in such a way that any two pieces which can capture each other (when considered to be knights) after the rearrangement are on adjacent squares (i.e. squares having at least one common boundary point)? Consider two cases:
(a) $N = 3$.
(b) $N = 8$
(S Stefanov)
1996 Dutch Mathematical Olympiad, 3
What is the largest number of horses that you can put on a chessboard without there being two horses that can beat each other?
a. Describe an arrangement with that maximum number.
b. Prove that a larger number is not possible.
(A chessboard consists of $8 \times 8$ spaces and a horse jumps from one field to another field according to the line "two squares vertically and one squared horizontally" or "one square vertically and two squares horizontally")
[asy]
unitsize (0.5 cm);
int i, j;
for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
if ((i + j) % 2 == 0) {
if ((i - 2)^2 + (j - 3)^2 == 5) {
fill(shift((i,j))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red);
}
else {
fill(shift((i,j))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8));
}
}
}}
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
label("$a$", (0.5,-0.5), fontsize(10));
label("$b$", (1.5,-0.5), fontsize(10));
label("$c$", (2.5,-0.5), fontsize(10));
label("$d$", (3.5,-0.5), fontsize(10));
label("$e$", (4.5,-0.5), fontsize(10));
label("$f$", (5.5,-0.5), fontsize(10));
label("$g$", (6.5,-0.5), fontsize(10));
label("$h$", (7.5,-0.5), fontsize(10));
label("$1$", (-0.5,0.5), fontsize(10));
label("$2$", (-0.5,1.5), fontsize(10));
label("$3$", (-0.5,2.5), fontsize(10));
label("$4$", (-0.5,3.5), fontsize(10));
label("$5$", (-0.5,4.5), fontsize(10));
label("$6$", (-0.5,5.5), fontsize(10));
label("$7$", (-0.5,6.5), fontsize(10));
label("$8$", (-0.5,7.5), fontsize(10));
label("$P$", (2.5,3.5), fontsize(10));
[/asy]
1998 Tournament Of Towns, 3
What is the maximum number of colours that can be used to paint an $8 \times 8$ chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour?
(A Shapovalov)
Novosibirsk Oral Geo Oly VIII, 2020.2
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?
2018 Caucasus Mathematical Olympiad, 2
On a chessboard $8\times 8$, $n>6$ Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of $n$.
2020 Novosibirsk Oral Olympiad in Geometry, 2
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?
1953 Moscow Mathematical Olympiad, 258
A knight stands on an infinite chess board. Find all places it can reach in exactly $2n$ moves.
2011 Armenian Republican Olympiads, Problem 6
Find the smallest $n$ such that in an $8\times 8$ chessboard any $n$ cells contain two cells which are at least $3$ knight moves apart from each other.
1978 Bundeswettbewerb Mathematik, 1
A knight is modified so that it moves $p$ fields horizontally or vertically and $q$ fields in the perpendicular direction. It is placed on an infinite chessboard. If the knight returns to the initial field after $n$ moves, show that $n$ must be even.
2014 Indonesia MO Shortlist, C3
Let $n$ be a natural number. Given a chessboard sized $m \times n$. The sides of the small squares of chessboard are not on the perimeter of the chessboard will be colored so that each small square has exactly two sides colored. Prove that a coloring like that is possible if and only if $m \cdot n$ is even.
2003 Chile National Olympiad, 1
Investigate whether a chess knight can traverse a $4 \times 4$ mini-chessboard so that it reaches each of the $16$ squares only once.
Note: the drawing below shows the endpoints of the eight possible moves of the knight $(C)$ on a chessboard of size $8 \times 8$.
[asy]
unitsize(0.4 cm);
int i;
fill(shift((2,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7));
fill(shift((4,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7));
fill(shift((1,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7));
fill(shift((5,3))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7));
fill(shift((1,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7));
fill(shift((5,5))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7));
fill(shift((2,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7));
fill(shift((4,6))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.7));
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
label("C", (3.5,4.5), fontsize(8));
[/asy]
2001 Czech-Polish-Slovak Match, 3
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.
2021 Kyiv City MO Round 1, 11.2
Chess piece called [i]skew knight[/i], if placed on the black square, attacks all the gray squares.
[img]https://i.ibb.co/HdTDNjN/Kyiv-MO-2021-Round-1-11-2.png[/img]
What is the largest number of such knights that can be placed on the $8\times 8$ chessboard without them attacking each other?
[i]Proposed by Arsenii Nikolaiev[/i]
2013 Tournament of Towns, 4
There is a $8\times 8$ table, drawn in a plane and painted in a chess board fashion. Peter mentally chooses a square and an interior point in it. Basil can draws any polygon (without self-intersections) in the plane and ask Peter whether the chosen point is inside or outside this polygon. What is the minimal number of questions suffcient to determine whether the chosen point is black or white?
1974 IMO Shortlist, 11
We consider the division of a chess board $8 \times 8$ in p disjoint rectangles which satisfy the conditions:
[b]a)[/b] every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares.
[b]b)[/b] the numbers $\ a_{1}, \ldots, a_{p}$ of white squares from $p$ rectangles satisfy $a_1, , \ldots, a_p.$ Find the greatest value of $p$ for which there exists such a division and then for that value of $p,$ all the sequences $a_{1}, \ldots, a_{p}$ for which we can have such a division.
[color=#008000]Moderator says: see [url]https://artofproblemsolving.com/community/c6h58591[/url][/color]
2013 Tournament of Towns, 4
Eight rooks are placed on a $8\times 8$ chessboard, so that no two rooks attack one another.
All squares of the board are divided between the rooks as follows. A square where a rook is placed belongs to it. If a square is attacked by two rooks then it belongs to the nearest rook; in case these two rooks are equidistant from this square each of them possesses a half of the square. Prove that every rook possesses the equal area.
2010 BAMO, 4
Place eight rooks on a standard $8 \times 8$ chessboard so that no two are in the same row or column. With the standard rules of chess, this means that no two rooks are attacking each other. Now paint $27$ of the remaining squares (not currently occupied by rooks) red. Prove that no matter how the rooks are arranged and which set of $27$ squares are painted, it is always possible to move some or all of the rooks so that:
• All the rooks are still on unpainted squares.
• The rooks are still not attacking each other (no two are in the same row or same column).
• At least one formerly empty square now has a rook on it; that is, the rooks are not on the same $8$ squares as before.