(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$. (b) Show that a corresponding decomposition into $30$ squares is also possible. [i](Walther Janous)[/i]

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]

Cut a rectangle into $18$ rectangles so that no two adjacent ones form a rectangle.

An equilateral triangle side $10$ is divided into $100$ equilateral triangles of side $1$ by lines parallel to its sides. There are m equilateral tiles of $4$ unit triangles and $25 - m$ straight tiles of $4$ unit triangles (as shown below). For which values of $m$ can they be used to tile the original triangle. [The straight tiles may be turned over.]

A square of side $1$ is divided into rectangles . We choose one of the two smaller sides of each rectangle (if the rectangle is a square, then we choose any of the four sides) . Prove that the sum of the lengths of all the chosen sides is at least $1$ . (Folklore)

A square is divided into $25$ small squares. We draw diagonals of some of the small squares so that no two diagonals share a common point (not even a common endpoint). What is the largest possible number of diagonals that we can draw? (I Rubanov)

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]

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.

Show that there exists an integer $N$ such that for all $n \ge N$ a square can be partitioned into $n$ smaller squares.

Determine for which natural numbers $n$ it is possible to completely cover a board of $ n \times n$, divided into $1 \times 1$ squares, with pieces like the one in the figure, without gaps or overlays and without leaving the board. Each of the pieces covers exactly six boxes. Note: Parts can be rotated. [img]https://cdn.artofproblemsolving.com/attachments/c/2/d87d234b7f9799da873bebec845c721e4567f9.png[/img]

An infinite network is partitioned with dominos. Prove there exist three other tilings with dominos, have neither common domino with the existing tiling nor with each other. Clarifications for network: It means an infinite board consisting of square cells.

Determine for which values of $n$ it is possible to tile a square of side $n$ with figures of the type shown in the picture [asy] unitsize(0.4 cm); draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((1,2)--(4,2)); draw((2,3)--(3,3)); draw((0,0)--(0,1)); draw((1,0)--(1,2)); draw((2,0)--(2,3)); draw((3,0)--(3,3)); draw((4,0)--(4,2)); draw((5,0)--(5,1)); [/asy]

Crocodile chooses $1$ x $4$ tile from $2018$ x $2018$ square.The bear has tilometer that checks $3$x$3$ square of $2018$ x $2018$ is there any of choosen cells by crocodile.Tilometer says "YES" if there is at least one choosen cell among checked $3$ x $3$ square.For what is the smallest number of such questions the Bear can certainly get an affirmative answer?

On a $ 30 \times30 $ square board or placed figures of shape 1 (of 5 squares) (in all four possible positions) and shaped figures of shape 2 (of 4 squares) . The figures do not overlap, they do not pass through the edges of the board and the squares of which they are drawn lie exactly through the squares of the board. a) Prove that the board can be fully covered using $100$ figures of both shapes. b) Prove that if there are already $50$ shaped figures on the board of shape 1, then at least one more figure can be placed on the board. c) Prove that if there are already $28$ figures of both shapes on the board then at least one more figure of both shapes can be placed on the board. [img]https://cdn.artofproblemsolving.com/attachments/3/f/f20d5a91d61557156edf203ff43acac461d9df.png[/img]

For $n$ an odd positive integer, the unit squares of an $n\times n$ chessboard are coloured alternately black and white, with the four corners coloured black. A it tromino is an $L$-shape formed by three connected unit squares. For which values of $n$ is it possible to cover all the black squares with non-overlapping trominos? When it is possible, what is the minimum number of trominos needed?

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.

Describe a method to convert any triangle into a rectangle with side 1 and area equal to the original triangle by dividing that triangle into finitely many subtriangles.

For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1 \times 2, ..., 1 \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically? Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.

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.

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".

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.

For a positive integer $n$, we consider an $n \times n$ board and tiles with dimensions $1 \times 1, 1 \times 2, ..., 1 \times n$. In how many ways exactly can $\frac12 n (n + 1)$ cells of the board are colored red, so that the red squares can all be covered by placing the $n$ tiles all horizontally, but also by placing all $n$ tiles vertically? Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.

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]

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 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.