Found problems: 31
2002 Mexico National Olympiad, 4
A domino has two numbers (which may be equal) between $0$ and $6$, one at each end. The domino may be turned around. There is one domino of each type, so $28$ in all. We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends. Dominos can be added to the chain at either end. We want to form the chain so that after each domino has been added the total of all the numbers is odd. For example, we could place first the domino $(3,4)$, total $3 + 4 = 7$. Then $(1,3)$, total $1 + 3 + 3 + 4 = 11$, then $(4,4)$, total $11 + 4 + 4 = 19$. What is the largest number of dominos that can be placed in this way? How many maximum-length chains are there?
Kvant 2019, M2576
A $8\times 8$ board is divided in dominoes (rectangles with dimensions $1 \times 2$ or $2 \times 1$).
[list=a]
[*] Prove that the total length of the border between horizontal and vertical dominoes is at most $52$.
[*] Determine the maximum possible total length of the border between horizontal and vertical dominoes.
[/list]
[i]Proposed by B. Frenkin, A. Zaslavsky, E. Arzhantseva[/i]
2013 Peru MO (ONEM), 4
The next board is completely covered with dominoes in an arbitrary manner.
[img]https://cdn.artofproblemsolving.com/attachments/8/9/b4b791e55091e721c8d6040a65ae6ba788067c.png[/img]
a) Prove that we can paint $21$ dominoes in such a way that there are not two dominoes painted forming a $S$-tetramino.
b) What is the largest positive integer $k$ for which it is always possible to paint $k$ dominoes (without matter how the board is filled) in such a way that there are not two painted dominoes forming a $S$-tetramine?
Clarification: A domino is a $1 \times 2$ or $2 \times 1$ rectangle; the $S$-tetraminos are the figures of the following types:
[img]https://cdn.artofproblemsolving.com/attachments/d/f/8480306382d6b87ddb8b2a7ca96c91ee45bc6e.png[/img]
2011 Peru MO (ONEM), 4
A domino is a $1 \times 2$ (or 2 $\times 1$) rectangular piece; namely, made up of two squares. There is an $8 \times 8$ board such that each domino can be cover exactly two of its squares. John places $n$ dominoes on the board, so that each one covers exactly two squares of the board and it is no longer possible to place a piece more without overlapping with any of those already placed. Determine the smallest value of $n$ for which the described situation is possible.
2023 Brazil EGMO Team Selection Test, 4
A cricket wants to move across a $2n \times 2n$ board that is entirely covered by dominoes, with no overlap. He jumps along the vertical lines of the board, always going from the midpoint of the vertical segment of a $1 \times 1$ square to another midpoint of the vertical segment, according to the rules:
$(i)$ When the domino is horizontal, the cricket jumps to the opposite vertical segment (such as from $P_2$ to $P_3$);
$(ii)$ When the domino is vertical downwards in relation to its position, the cricket jumps diagonally downwards (such as from $P_1$ to $P_2$);
$(iii)$ When the domino is vertically upwards relative to its position, the cricket jumps diagonally upwards (such as from $P_3$ to $P_4$).
The image illustrates a possible covering and path on the $4 \times 4$ board.
Considering that the starting point is on the first vertical line and the finishing point is on the last vertical line, prove that, regardless of the covering of the board and the height at which the cricket starts its path, the path ends at the same initial height.
2009 Peru MO (ONEM), 4
Let $ n$ be a positive integer. A $4\times n$ rectangular grid is divided in$ 2\times 1$ or $1\times 2$ rectangles (as if it were completely covered with tiles of domino, no overlaps or gaps). Then all the grid points which are vertices of one of the $2\times 1$ or $1\times 2$ rectangles, are painted red. What is the least amount of red points you can get?
2018 IFYM, Sozopol, 7
Let $x$ and $y$ be odd positive integers. A table $x$ x $y$ is given in which the squares with coordinates $(2,1)$, $(x - 2, y)$, and $(x, y)$ are cut. The remaining part of the table is covered in dominoes and squares [b]2 x 2[/b]. Prove that the dominoes in a valid covering of the table are at least
$\frac{3}{2}(x+y)-6$
2022/2023 Tournament of Towns, P5
A $2N\times2N$ board is covered by non-overlapping dominos of $1\times2$ size. A lame rook (which can only move one cell at a time, horizontally or vertically) has visited each cell once on its route across the board. Call a move by the rook longitudinal if it is a move from one cell of a domino to another cell of the same domino. What is:
[list=a]
[*]the maximum possible number of longitudinal moves?
[*]the minimum possible number of longitudinal moves?
[/list]
Kvant 2023, M2774
In a $50\times 50$ checkered square, each cell is colored in one of the 100 given colors so that all colors are used and there does not exist a monochromatic domino. Galia wants to repaint all the cells of one of the colors in a different color (from the given 100 colors) so that a monochromatic domino still won't exist. Is it true that Galia will surely be able to do this
[i]Proposed by G. Sharafutdinova[/i]
2023 Silk Road, 2
Let $n$ be a positive integer. Each cell of a $2n\times 2n$ square is painted in one of the $4n^2$ colors (with some colors may be missing). We will call any two-cell rectangle a [i]domino[/I], and a domino is called [i]colorful[/I] if its cells have different colors. Let $k$ be the total number of colorful dominoes in our square; $l$ be the maximum integer such that every partition of the square into dominoes contains at least $l$ colorful dominoes. Determine the maximum possible value of $4l-k$ over all possible colourings of the square.
Russian TST 2016, P1
The infinite checkered plane is divided into dominoes. If we move any horizontal domino of the partition by 49 cells to the right or left, we will also get a domino of the partition. If we move any vertical domino of the partition up or down by 49 cells, we will also get a domino of the partition. Can this happen?
2022 Flanders Math Olympiad, 2
A domino is a rectangle whose length is twice its width.
Any square can be divided into seven dominoes, for example as shown in the figure below.
[img]https://cdn.artofproblemsolving.com/attachments/7/6/c055d8d2f6b7c24d38ded7305446721e193203.png[/img]
a) Show that you can divide a square into $n$ dominoes for all $n \ge 5$.
b) Show that you cannot divide a square into three or four dominoes.
2015 EGMO, 2
A [i]domino[/i] is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^2$ dominoes can be placed without overlapping on a $2n \times 2n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column.
2018 Regional Olympiad of Mexico Northeast, 5
A $300\times 300$ board is arbitrarily filled with $2\times 1$ dominoes with no overflow, underflow, or overlap. (Tokens can be placed vertically or horizontally.)
Decide if it is possible to paint the tiles with three different colors, so that the following conditions are met:
$\bullet$ Each token is painted in one and only one of the colors.
$\bullet$ The same number of tiles are painted in each color.
$\bullet$ No piece is a neighbor of more than two pieces of the same color.
Note: Two dominoes are [i]neighbors [/i]if they share an edge.
2017 Taiwan TST Round 3, 6
Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.
2016 Peru MO (ONEM), 2
How many dominoes can be placed on a at least $3 \times 12$ board, such so that it is impossible to place a $1\times 3$, $3 \times 1$, or $ 2 \times 2$ tile on what remains of the board?
Clarification: Each domino covers exactly two squares on the board. The chips cannot overlap.
Kvant 2023, M2760
The checkered plane is divided into dominoes. What is the maximum $k{}$ for which it is always possible to choose a $100\times 100$ checkered square containing at least $k{}$ whole dominoes?
[i]Proposed by S. Berlov[/i]
Russian TST 2017, P3
Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.
2021 Irish Math Olympiad, 4
You have a $3 \times 2021$ chessboard from which one corner square has been removed. You also have a set of $3031$ identical dominoes, each of which can cover two adjacent chessboard squares. Let $m$ be the number of ways in which the chessboard can be covered with the dominoes, without gaps or overlaps.
What is the remainder when $m$ is divided by $19$?
2022 Macedonian Mathematical Olympiad, Problem 4
Sofia and Viktor are playing the following game on a $2022 \times 2022$ board:
- Firstly, Sofia covers the table completely by dominoes, no two are overlapping and all are inside the table;
- Then Viktor without seeing the table, chooses a positive integer $n$;
- After that Viktor looks at the table covered with dominoes, chooses and fixes $n$ of them;
- Finally, Sofia removes the remaining dominoes that aren't fixed and tries to recover the table with dominoes differently from before.
If she achieves that, she wins, otherwise Viktor wins. What is the minimum number $n$ for which Viktor can always win, no matter the starting covering of dominoes.
[i]Proposed by Viktor Simjanoski[/i]
2017 Puerto Rico Team Selection Test, 4
Alberto and Bianca play a game on a square board. Alberto begins. On their turn, players place a $1 \times 2$ or $2 \times 1$ domino on two empty squares on the board. The player who cannot put a domino loses. Determine who has a winning strategy (and prove it) if the board is:
i) $3 \times 3$
ii) $3 \times 4$
Russian TST 2014, P1
On each non-boundary unit segment of an $8\times 8$ chessboard, we write the number of dissections of the board into dominoes in which this segment lies on the border of a domino. What is the last digit of the sum of all the written numbers?
2022-IMOC, C3
There are three types of piece shown as below. Today Alice wants to cover a $100 \times 101$ board with these pieces without gaps and overlaps. Determine the minimum number of $1\times 1$ pieces should be used to cover the whole board and not exceed the board. (There are an infinite number of these three types of pieces.)
[asy]
size(9cm,0);
defaultpen(fontsize(12pt));
draw((9,10) -- (59,10) -- (59,60) -- (9,60) -- cycle);
draw((59,10) -- (109,10) -- (109,60) -- (59,60) -- cycle);
draw((9,60) -- (59,60) -- (59,110) -- (9,110) -- cycle);
draw((9,110) -- (59,110) -- (59,160) -- (9,160) -- cycle);
draw((109,10) -- (159,10) -- (159,60) -- (109,60) -- cycle);
draw((180,11) -- (230,11) -- (230,61) -- (180,61) -- cycle);
draw((180,61) -- (230,61) -- (230,111) -- (180,111) -- cycle);
draw((230,11) -- (280,11) -- (280,61) -- (230,61) -- cycle);
draw((230,61) -- (280,61) -- (280,111) -- (230,111) -- cycle);
draw((280,11) -- (330,11) -- (330,61) -- (280,61) -- cycle);
draw((280,61) -- (330,61) -- (330,111) -- (280,111) -- cycle);
draw((330,11) -- (380,11) -- (380,61) -- (330,61) -- cycle);
draw((330,61) -- (380,61) -- (380,111) -- (330,111) -- cycle);
draw((401,11) -- (451,11) -- (451,61) -- (401,61) -- cycle);
[/asy]
[i]Proposed by amano_hina[/i]
2012 Peru MO (ONEM), 3
A domino is a $1\times2$ or $2\times 1$ rectangle. Diego wants to completely cover a $6\times 6$ board using $18$ dominoes. Determine the smallest positive integer $k$ for which Diego can place $k$ dominoes on the board (without overlapping) such that what remains of the board can be covered uniquely using the remaining dominoes.
2017 Brazil Team Selection Test, 4
Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.