The checkered plane (infinitely large house paper) is given. For which pairs (a,, b) one can color each of the squares with one of $a \cdot b$ colors, so that each rectangle of size $ a \times b$ or $b \times a$, placed appropriately in the checkered plane, always contains a unit square with each color ?

(a) A square is cut into right triangles with legs of lengths $3$ and $4$. Prove that the total number of the triangles is even. (b) A rectangle is cut into right triangles with legs of lengths $1$ and $2$. Prove that the total number of the triangles is even. (A Shapovalov)

A crymble is a solid consisting of four white and one black unit cubes as shown in the picture. Find the side length of the smallest cube that can be exactly filled up with crymbles. [img]https://cdn.artofproblemsolving.com/attachments/b/0/b1e50f7abbfb7d356913d746d653fd3875f5ae.png[/img]

What is the smallest number of $3$-cell corners that you need to paint in a $5 \times5$ square so that you cannot paint more than one corner of one it? (Shaded corners should not overlap.)

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]

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?

Prove that the unit square can be tiled with rectangles (not necessarily of the same size) similar to a rectangle of size $1\times(3+\sqrt[3]{3})$.

Fix an integer $n \geq 2$. An $n\times n$ sieve is an $n\times n$ array with $n$ cells removed so that exactly one cell is removed from every row and every column. A stick is a $1\times k$ or $k\times 1$ array for any positive integer $k$. For any sieve $A$, let $m(A)$ be the minimal number of sticks required to partition $A$. Find all possible values of $m(A)$, as $A$ varies over all possible $n\times n$ sieves. [i]Palmer Mebane[/i]

Let $n$ be a natural number. A tiling of a $2n \times 2n$ board is a placing of $2n^2$ dominos (of size $2 \times 1$ or $1 \times 2$) such that each of them covers exactly two squares of the board and they cover all the board.Consider now two [i]sepearate tilings[/i] of a $2n \times 2n$ board: one with red dominos and the other with blue dominos. We say two squares are red neighbours if they are covered by the same red domino in the red tiling; similarly define blue neighbours. Suppose we can assign a non-zero integer to each of the squares such that the number on any square equals the difference between the numbers on it's red and blue neighbours i.e the number on it's red neigbhbour minus the number on its blue neighbour. Show that $n$ is divisible by $3$ [i] Proposed by Tejaswi Navilarekallu [/i]

A rectangular building consists of $30$ square rooms situated like the cells of a $2 \times 15$ board. In each room there are three doors, each of which leads to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the building?

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.

We call a tiling of an $m\times$ n rectangle with arabos (see figure below) [i]regular[/i] if there is no sub-rectangle which is tiled with arabos. Prove that if for some $m$ and $n$ there exists a [i]regular[/i] tiling of the $m\times n$ rectangle then there exists a [i]regular[/i] tiling also for the $2m \times 2n$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/1/1/2ab41cc5107a21760392253ed52d9e4ecb22d1.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]

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 square with the dimension $ 1 \times1 $ has been removed from a square board $ 3 ^n \times 3 ^n $ ($ n \in \mathbb {N}, $ $ n> 1 $). a) Prove that any defective board with the dimension $ 3 ^ n \times3 ^ n $ can be covered with shaped figures of shape 1 (the 3 squares' one) and of shape 2 (the 5 squares' one). Figures covering the board must not overlap each other and must not cross the edge of the board. Also the squares removed from the board must not be covered. (b) How many small figures in shape 2 must be used to cover the board? [img]https://cdn.artofproblemsolving.com/attachments/4/7/e970fadd7acc7fd6f5897f1766a84787f37acc.png[/img]

You have a $7\times 7$ board divided into $49$ boxes. Mateo places a coin in a box. a) Prove that Mateo can place the coin so that it is impossible for Emi to completely cover the $48$ remaining squares, without gaps or overlaps, using $15$ $3\times1$ rectangles and a cubit of three squares, like those in the figure. [img]https://cdn.artofproblemsolving.com/attachments/6/9/a467439094376cd95c6dfe3e2ad3e16fe9f124.png[/img] b) Prove that no matter which square Mateo places the coin in, Emi will always be able to cover the 48 remaining squares using $14$ $3\times1$ rectangles and two cubits of three squares.

A cube of edge $2$ with one of the corner unit cubes removed is called a [i]piece[/i]. Prove that if a cube $T$ of edge $2^n$ is divided into $2^{3n}$ unit cubes and one of the unit cubes is removed, then the rest can be cut into [i]pieces[/i].

A $7\times 7$ square is cut into pieces following types: [img]https://cdn.artofproblemsolving.com/attachments/e/d/458b252c719946062b655340cbe8415d1bdaf9.png[/img] Show that exactly one of the pieces is of type (b). [img]https://cdn.artofproblemsolving.com/attachments/4/9/f3dd0e13fed9838969335c82f5fe866edc83e8.png[/img]

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

Let $m, n$ be positive integers with $m<n$ and consider an $n\times n$ board from which its upper left $ m\times m$ part has been removed. An example of such board for $n=5$ and $m=2$ is shown below. Determine for which pairs $(m, n)$ this board can be tiled with $3\times 1$ tiles. Each tile can be positioned either horizontally or vertically so that it covers exactly three squares of the board. The tiles should not overlap and should not cover squares outside of the board.

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]

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.

Let $n \ge 3$ be a positive integer. We call a $3 \times 3$ grid [i]beautiful[/i] if the cell located at the center is colored white and all other cells are colored black, or if it is colored black and all other cells are colored white. Determine the minimum value of $a+b$ such that there exist positive integers $a$, $b$ and a coloring of an $a \times b$ grid with black and white, so that it contains $n^2 - n$ [i]beautiful[/i] subgrids.

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.