A $30\times30$ table is given. We want to color some of it's unit squares such that any colored square has at most $k$ neighbors. ( Two squares $(i,j)$ and $(x,y)$ are called neighbors if $i-x,j-y\equiv0,-1,1 \pmod {30}$ and $(i,j)\neq(x,y)$. Therefore, each square has exactly $8$ neighbors) What is the maximum possible number of colored squares if$:$ $a) k=6$ $b)k=1$

A magician and his assistant perform in front of an audience of many people. On the stage there is an $8$×$8$ board, the magician blindfolds himself, and then the assistant goes inviting people from the public to write down the numbers $1, 2, 3, 4, . . . , 64$ in the boxes they want (one number per box) until completing the $64$ numbers. After the assistant covers two adyacent boxes, at her choice. Finally, the magician removes his blindfold and has to $“guess”$ what number is in each square that the assistant. Explain how they put together this trick. $Clarification:$ Two squares are adjacent if they have a common side

The square table $n\times n$ is filled by integers. If the fields have common side, the difference of numbers in them doesn't exceed $1$. Prove that some number is encountered not less than a) not less than $[n/2]$ times ($[ ...]$ mean the whole part), b) not less than $n$ times.

Liliana wants to paint a $m\times n$ board. Liliana divides each unit square by one of its diagonals and paint one of the halves of the square with black and the other half with white in such a way that triangles that have a common side haven't the same colour. How many possibilities has Liliana to paint the board?

An $n \times m$ table ( $n \le m$ ) is filled in accordance with the rules of the game "Minesweeper": mines are placed at some cells (not more than one mine at the cell) and in the remaining cells one writes the number of the mines in the neighboring (by side or by vertex) cells. Then the sum of allnumbers in the table is computed (this sum is equal to $9$ for the picture). What is the largest possible value of this sum? (V. Lebed) [img]https://cdn.artofproblemsolving.com/attachments/2/9/726ccdbc57807788a5f6e88a5acb42b10a6cc0.png[/img]

In an $8 \times 8$ grid, $n$ disks, numbered $1$ to $n$ are stacked, with random order, in a pile in the bottom left comer. The disks can be moved one at a time to a neighbouring cell either to the right or top. The aim to move all the disks to the cell at the top right comer and stack them in the order $1,2,...,n$ from the bottom. Each cell, except the bottom left and top right cell, can have at most one disk at any given time. Find the largest value of $n$ so that the aim can be achieved.

Pairwise distinct real numbers are arranged into an $m \times n$ rectangular array. In each row the entries are arranged increasingly from left to right. Each column is then rearranged in decreasing order from top to bottom. Prove that in the reorganized array, the rows remain arranged increasingly.

In the cells of the table $ 100 \times100 $ are placed in pairs different numbers. Every minute each of the numbers changes to the largest of the numbers in the adjacent cells on the side. Can after $4$ hours all the numbers in the table be the same?

For which integers $N$ it is possible to write real numbers into the cells of a square of size $N \times N$ so that among the sums of each pair of adjacent cells there are all integers from $1$ to $2(N-1)N$ (each integer once)? Maxim Didin

On a $3 \times 3$ board the numbers from $1$ to $9$ are written in some order and without repeating. We say that the arrangement obtained is [i]Isthmian [/i] if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements. Note: Two arrangements are considered equal if one can be obtained from the other by rotating the board.

a) Given real numbers $a_1,a_2,b_1,b_2$ and positive $p_1,p_2,q_1,q_2$. Prove that in the table $2\times 2$ $$(a_1 + b_1)/(p_1 + q_1) , (a_1 + b_2)/(p_1 + q_2) $$ $$(a_2 + b_1)/(p_2 + q_1) , (a_2 + b_2)/(p_2 + q_2)$$ there is a number in the table, that is not less than another number in the same row and is not greater than another number in the same column (a saddle point). b) Given real numbers $a_1, a_2, ... , a_n, b_1, b_2, ... , b_n$ and positive $p_1, p_2, ... , p_n, q_1, q_2, ... , q_n$. We construct the table $n\times n$, with the numbers ($0 < i,j \le n$) $$(a_i + b_j)/(p_i + q_j)$$ in the intersection of the $i$-th row and $j$-th column. Prove that there is a number in the table, that is not less than arbitrary number in the same row and is not greater than arbitrary number in the same column (a saddle point).

Each entry in an $n\times n$ table is either $+$ or $-$. At each step, one can choose a row or a column and reverse all signs in it. From the initial position, it is possible to obtain the table in which all signs are $+$. Prove that this can be accomplished in at most $n$ steps.

A $1\times 2019$ board is filled with numbers $1, 2, \dots, 2019$ in an increasing order. In each step, three consecutive tiles are selected, then one of the following operations is performed: $\text{(i)}$ the number in the middle is increased by $2$ and its neighbors are decreased by $1$, or $\text{(ii)}$ the number in the middle is decreased by $2$ and its neighbors are increased by $1$. After several such operations, the board again contains all the numbers $1, 2,\dots, 2019$. Prove that each number is in its original position.

We will call a rectangular table filled with natural numbers [i]“good”[/i], if for each two rows, there exist a column for which its two cells that are also in these two rows, contain numbers of different parity. Prove that for $\forall$ $n>2$ we can erase a column from a [i]good[/i] $n$ x $n$ table so that the remaining $n$ x $(n-1)$ table is also [i]good[/i].

Each of the squares on a $15$×$15$ board has a zero. At every step you choose a row or a column, we delete all the numbers from it and then we write the numbers from $1$ to $15$ in the empty cells, in an arbitrary order. find the sum possible maximum of the numbers on the board that can be achieved after a number finite number of steps.

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 $\frac{3}{2}(x+y)-6$

Consider a triangular table of positive integers \[ \begin{matrix} & & & a_{11} & a_{12} & a_{13} & & & \\ & & a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & & \\ & a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36} & a_{37} & \\ \iddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \] The first row consists of odd numbers only. For $i>1,j\ge1$ we have \[a_{ij}=a_{i-1,j-2}+a_{i-1,j-1}+a_{i-1,j}.\] If we get out of range with the second index, we consider such $a$ to be zero (eg. $a_{22}=0+a_{11}+a_{12}$ and $a_{37}=a_{25}+0+0$). Show that for every $i>1$ there is $j\in\{1,\ldots,2i+1\}$ such that $a_{ij}$ is even.