A unit square is removed from the corner of an $n \times n$ grid, where $n \geq 2$. Prove that the remainder can be covered by copies of the figures of $3$ or $5$ unit squares depicted in the drawing below. [asy] import geometry; draw((-1.5,0)--(-3.5,0)--(-3.5,2)--(-2.5,2)--(-2.5,1)--(-1.5,1)--cycle); draw((-3.5,1)--(-2.5,1)--(-2.5,0)); draw((0.5,0)--(0.5,3)--(1.5,3)--(1.5,1)--(3.5,1)--(3.5,0)--cycle); draw((1.5,0)--(1.5,1)); draw((2.5,0)--(2.5,1)); draw((0.5,1)--(1.5,1)); draw((0.5,2)--(1.5,2)); [/asy] [b]Note:[/b] Every square must be covered once and figures must not go over the bounds of the grid.

A rectangular floor is covered by a certain number of equally large quadratic tiles. The tiles along the edge are red, and the rest are white. There are equally many red and white tiles. How many tiles can there be?

Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. [asy] unitsize(0.5 cm); draw((0,0)--(1,0)); draw((0,1)--(1,1)); draw((2,1)--(3,1)); draw((0,2)--(3,2)); draw((0,3)--(3,3)); draw((0,0)--(0,3)); draw((1,0)--(1,3)); draw((2,1)--(2,3)); draw((3,1)--(3,3)); [/asy] Determine all $ m\times n$ rectangles that can be covered without gaps and without overlaps with hooks such that - the rectangle is covered without gaps and without overlaps - no part of a hook covers area outside the rectangle.

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?

The workers laid a floor of size $n \times n$ with tiles of two types: $2 \times 2$ and $3 \times 1$. It turned out that they were able to completely lay the floor in such a way that the same number of tiles of each type was used. Under what conditions could this happen? (You can’t cut tiles and also put them on top of each other.)

A square \( K = 2025 \times 2025 \) is given. We define a [i]stick[/i] as a rectangle where one of its sides is \( 1 \), and the other side is a positive integer from \( 1 \) to \( 2025 \). Find the largest positive integer \( C \) such that the following condition holds: [list] [*] If several sticks with a total area not exceeding \( C \) are taken, it is always possible to place them inside the square \( K \) so that each stick fully completely covers an integer number of \( 1 \times 1 \) squares, and no \( 1 \times 1 \) square is covered by more than one stick. [/list] [i](Basically, you can rotate sticks, but they have to be aligned by lines of the grid)[/i] [i]Proposed by Anton Trygub[/i]

Is it possible to divide a $8 \times 8$ chessboard into $32$ rectangles, each either $1 \times 2$ or $2 \times 1$, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint? (A Shapovalov)

Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that from the remaining part of the table $36$ $1\times2$ dominos can be cut [I]Proposed by S. Berlov[/i]

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 $\frac15 MN$.

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]

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]

We are given a supply of $a \times b$ tiles with $a$ and $b$ distinct positive integers. The tiles are to be used to tile a $28 \times 48$ rectangle. Find $a, b$ such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find $a, b$ such that the tile has the largest possible area and there is more than one possible tiling.

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?

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]

Given a positive integer $n$, an $n$-gun is a $2n$-mino that is formed by putting a $1 \times n$ grid and an $n \times 1$ grid side by side so that one of the corner unit squares of the first grid is next to one of the corner unit squares of the second grid. Find the minimum possible $k$ such that it is possible to color the infinite planar grid with $k$ colors such that any $n$-gun cannot cover two different squares with the same color. [i]Itf0501[/i]

A board of $6\times 6$ is totally covered by $18$ dominoes (of $2\times 1$), that is, there are no overlaps, gaps, and the tiles do not come off the board. Prove that, regardless of the arrangement of the tiles, there is always a line that divides the board into two non-empty parts, and without cutting tiles.

A rectangle is tiled with rectangles of size $6\times 1$. Prove that one of its side lengths is divisible by $6$.

Can an arbitrary polygon be cut into isosceles trapezoids?

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.

A set of identical square tiles with side length $1$ is placed on a (very large) floor. Every tile after the first shares an entire edge with at least one tile that has already been placed. - What is the largest possible perimeter for a figure made of $10$ tiles? - What is the smallest possible perimeter for a figure made of $10$ tiles? - What is the largest possible perimeter for a figure made of $2011$ tiles? - What is the smallest possible perimeter for a figure made of $2011$ tiles? Prove that your answers are correct.

Consider a $1 \times n$ rectangle and some tiles of size $1 \times 1$ of four different colours. The rectangle is tiled in such a way that no two neighboring square tiles have the same colour. a) Find the number of distinct symmetrical tilings. b) Find the number of tilings such that any consecutive square tiles have distinct colours.

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.)

How does one tile a plane, without gaps or overlappings, with the tiles equal to a given irregular quadrilateral?

The sword is a figure consisting of $6$ unit squares presented in the picture below (and any other figure obtained from it by rotation). [img]https://cdn.artofproblemsolving.com/attachments/4/3/08494627d043ea575703564e9e6b5ba63dc2ef.png[/img] Determine the largest number of swords that can be cut from a $6\times 11$ piece of paper divided into unit squares (each sword should consist of six such squares).

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]