Found problems: 169
2021-IMOC, C8
Find all positive integers $m,n$ such that the $m \times n$ grid can be tiled with figures formed by deleting one of the corners of a $2 \times 3$ grid.
[i]usjl, ST[/i]
2000 ITAMO, 5
A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?
2015 Caucasus Mathematical Olympiad, 3
The workers laid a floor of size $n\times n$ ($10 <n <20$) with two types of tiles: $2 \times 2$ and $5\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen? (You can’t cut tiles and also put them on top of each other.)
2004 VTRMC, Problem 4
A $9\times9$ chess board has two squares from opposite corners and its central square removed. Is it possible to cover the remaining squares using dominoes, where each domino covers two adjacent squares? Justify your answer.
2018 MMATHS, 1
Daniel has an unlimited supply of tiles labeled “$2$” and “$n$” where $n$ is an integer. Find (with proof) all the values of $n$ that allow Daniel to fill an $8 \times 10$ grid with these tiles such that the sum of the values of the tiles in each row or column is divisible by $11$.
2020 Dutch IMO TST, 4
Given are two positive integers $k$ and $n$ with $k \le n \le 2k - 1$. Julian has a large stack of rectangular $k \times 1$ tiles. Merlin calls a positive integer $m$ and receives $m$ tiles from Julian to place on an $n \times n$ board. Julian first writes on every tile whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number $m$ that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?
2001 Slovenia National Olympiad, Problem 4
Find the smallest number of squares on an $8\times8$ board that should be colored so that every $L$-tromino on the board contains at least one colored square.
2005 Estonia National Olympiad, 5
A $5\times 5$ board is covered by eight hooks (a three unit square figure, shown in the picture) so that one unit square remains free. Determine all squares of the board that can remain free after such covering.
[img]https://cdn.artofproblemsolving.com/attachments/6/8/a8c4e47ba137b904bd28c01c1d2cb765824e6a.png[/img]
2018 Junior Balkan Team Selection Tests - Romania, 4
Consider a $ 2018\times 2018$. board. An "LC-tile" is a tile consisting of $9$ unit squares, having the shape as in the gure below. What is the maximum number of "LC-tiles" that can be placed on the board without superposing them? (Each of the $9$ unit squares of the tile must cover one of the unit squares of the board; a tile may be rotated, turned upside down, etc.)
[img]https://cdn.artofproblemsolving.com/attachments/7/4/a2f992bc0341def1a6e5e26ba8a9eb3384698a.png
[/img]
Alexandru Girban
2014 Peru MO (ONEM), 2
The $U$-tile is made up of $1 \times 1$ squares and has the following shape:
[img]https://cdn.artofproblemsolving.com/attachments/8/7/5795ee33444055794119a99e675ef977add483.png[/img]
where there are two vertical rows of a squares, one horizontal row of $b$ squares, and also $a \ge 2$ and $b \ge 3$.
Notice that there are different types of tile $U$ .
For example, some types of $U$ tiles are as follows:
[img]https://cdn.artofproblemsolving.com/attachments/0/3/ca340686403739ffbbbb578d73af76e81a630e.png[/img]
Prove that for each integer $n \ge 6$, the board of $n\times n$ can be completely covered with $U$-tiles ,
with no gaps and no overlapping clicks.
Clarifications: The $U$-tiles can be rotated. Any amount can be used in the covering of tiles of each type.
2000 Chile National Olympiad, 6
With $76$ tiles, of which some are white, other blue and the remaining red, they form a rectangle of $4 \times 19$. Show that there is a rectangle, inside the largest, that has its vertices of the same color.
2006 Estonia Team Selection Test, 3
A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid?
(pictured right) so that no two of them cover the same square?
[img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]
2022 Abelkonkurransen Finale, 3
Nils has an $M \times N$ board where $M$ and $N$ are positive integers, and a tile shaped as shown below. What is the smallest number of squares that Nils must color, so that it is impossible to place the tile on the board without covering a colored square? The tile can be freely rotated and mirrored, but it must completely cover four squares.
[asy]
usepackage("tikz");
label("%
\begin{tikzpicture}
\draw[step=1cm,color=black] (0,0) grid (2,1);
\draw[step=1cm,color=black] (1,1) grid (3,2);
\fill [yellow] (0,0) rectangle (2,1);
\fill [yellow] (1,1) rectangle (3,2);
\draw[step=1cm,color=black] (0,0) grid (2,1);
\draw[step=1cm,color=black] (1,1) grid (3,2);
\end{tikzpicture}
");
[/asy]
2019 Tournament Of Towns, 5
Basil has an unrestricted supply of straight bricks $1 \times 1 \times 3$ and Γ-shape bricks made of three cubes $1\times 1\times 1$. Basil filled a whole box $m \times n \times k$ with these bricks, where $m, n$ and $k$ are integers greater than $1$. Prove that it was sufficient to use only Γ-shape bricks.
(Mikhail Evdokimov)
1998 Tournament Of Towns, 2
John and Mary each have a white $8 \times 8$ square divided into $1 \times 1$ cells. They have painted an equal number of cells on their respective squares in blue. Prove that one can cut up each of the two squares into $2 \times 1 $ dominoes so that it is possible to reassemble John's dominoes into a new square and Mary's dominoes into another square with the same pattern of blue cells.
(A Shapovalov)
2015 China Northern MO, 6
The figure obtained by removing one small unit square from the $2\times 2$ grid table is called an $L$ ''shape". .Put $k$ L-shapes in an $8\times 8$ grid table. Each $L$-shape can be rotated, but each $L$ shape is required to cover exactly three small unit squares in the grid table, and the common area covered by any two $L$ shapes is $0$, and except for these $k$ $L$ shapes, no other $L$ shapes can be placed. Find the minimum value of $k$.
1989 All Soviet Union Mathematical Olympiad, 498
A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares?
2018 Azerbaijan BMO TST, 4
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.
[i]Proposed by Jeck Lim, Singapore[/i]
1996 Swedish Mathematical Competition, 6
A rectangle is tiled with rectangles of size $6\times 1$. Prove that one of its side lengths is divisible by $6$.
2013 Greece Team Selection Test, 4
Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two).
Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids".
Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".
2021 Iranian Combinatorics Olympiad, P5
By a $\emph{tile}$ we mean a polyomino (i.e. a finite edge-connected set of cells in the infinite grid). There are many ways to place a tile in the infinite table (rotation is allowed but we cannot flip the tile). We call a tile $\textbf{T}$ special if we can place a permutation of the positive integers on all cells of the infinite table in such a way that each number would be maximum between all the numbers that tile covers in at most one placement of the tile.
1. Prove that each square is a special tile.
2. Prove that each non-square rectangle is not a special tile.
3. Prove that tile $\textbf{T}$ is special if and only if it looks the same after $90^\circ$ rotation.
India EGMO 2024 TST, 5
1. Can a $7 \times 7~$ square be tiled with the two types of tiles shown in the figure? (Tiles can be rotated and reflected but cannot overlap or be broken)
2. Find the least number $N$ of tiles of type $A$ that must be used in the tiling of a $1011 \times 1011$ square. Give an example of a tiling that contains exactly $N$ tiles of type $A$.
[asy]
size(4cm, 0);
pair a = (-10,0), b = (0, 0), c = (10, 0), d = (20, 0), e = (20, 10), f = (10, 10), g = (0, 10), h = (0, 20), ii = (-10, 20), j = (-10, 10);
draw(a--b--c--f--g--h--ii--cycle);
draw(g--b);
draw(j--g);
draw(f--c);
draw((30, 0)--(30, 20)--(50,20)--(50,0)--cycle);
draw((40,20)--(40,0));
draw((30,10)--(50,10));
label((0,0), "$(A)$", S);
label((40,0), "$(B)$", S);
[/asy]
[i]Proposed by Muralidharan Somasundaran[/i]
2002 BAMO, 2
In the illustration, a regular hexagon and a regular octagon have been tiled with rhombuses.
In each case, the sides of the rhombuses are the same length as the sides of the regular polygon.
(a) Tile a regular decagon ($10$-gon) into rhombuses in this manner.
(b) Tile a regular dodecagon ($12$-gon) into rhombuses in this manner.
(c) How many rhombuses are in a tiling by rhombuses of a $2002$-gon?
Justify your answer.
[img]https://cdn.artofproblemsolving.com/attachments/8/a/8413e4e2712609eba07786e34ba2ce4aa72888.png[/img]
1989 Tournament Of Towns, (231) 5
A rectangular $M \times N$ board is divided into $1 \times $ cells. There are also many domino pieces of size $1 \times 2$. These pieces are placed on a board so that each piece occupies two cells. The board is not entirely covered, but it is impossible to move the domino pieces (the board has a frame, so that the pieces cannot stick out of it). Prove that the number of uncovered cells is
(a) less than $\frac14 MN$,
(b) less than $\frac15 MN$.
1987 Spain Mathematical Olympiad, 3
A given triangle is divided into $n$ triangles in such a way that any line segment which is a side of a tiling triangle is either a side of another tiling triangle or a side of the given triangle. Let $s$ be the total number of sides and $v$ be the total number of vertices of the tiling triangles (counted without multiplicity).
(a) Show that if $n$ is odd then such divisions are possible, but each of them has the same number $v$ of vertices and the same number $s$ of sides. Express $v$ and $s$ as functions of $n$.
(b) Show that, for $n$ even, no such tiling is possible