A matrix $A=(a_{ij})$ is called [i]nice[/i], if it has the following properties: (i) the set of all entries of $A$ is $\{1,2,\dots,2t\}$ for some integer $t$; (ii) the entries are non-decreasing in every row and in every column: $a_{i,j} \le a_{i,j+1}$ and $a_{i,j} \le a_{i+1,j}$; (iii) equal entries can appear only in the same row or the same column: if $a_{i,j}=a_{k,\ell}$, then either $i=k$ or $j=\ell$; (iv) for each $s=1,2,\dots,2t-1$, there exist $i \ne k$ and $j \ne \ell$ such that $a_{i,j}=s$ and $a_{k,\ell}=s+1$. Prove that for any positive integers $m$ and $n$, the number of nice $m \times n$ matrixes is even. For example, the only two nice $2 \times 3$ matrices are $\begin{pmatrix} 1 & 1 & 1\\2 & 2 & 2 \end{pmatrix}$ and $\begin{pmatrix} 1 & 1 & 3\\2 & 4 & 4 \end{pmatrix}$.

$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $handshakes, what is $N$?

A rectangular array of positive integers has $4$ rows. The sum of the entries in each column is $20$. Within each row, all entries are distinct. What is the maximum possible number of columns?

Positive integers are put into the following table. \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline 2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline 4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline 7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline 11 & 17 & 24 & 32 & 41 & & & & & \\ \hline 16 & 23 & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline ... & & & & & & & & & \\ \hline \end{tabular} Find the number of the line and column where the number $2015$ stays.

Consider a rectangular array of single digits $d_{i,j}$ with 10 rows and 7 columns, such that $d_{i+1,j}-d_{i,j}$ is always 1 or -9 for all $1 \leq i \leq 9$ and all $1 \leq j \leq 7$, as in the example below. For $1 \leq i \leq 10$, let $m_i$ be the median of $d_{i,1}$, ..., $d_{i,7}$. Determine the least and greatest possible values of the mean of $m_1$, $m_2$, ..., $m_{10}$. Example: [img]https://cdn.artofproblemsolving.com/attachments/8/a/b77c0c3aeef14f0f48d02dde830f979eca1afb.png[/img]

Fix integers $m \geq 3$ and $n \geq 3$. Each cell of an array with $m$ rows and $n$ columns is coloured one of two colours such that: [b](1)[/b] Both colours occur on every column; and [b](2)[/b] On every two rows the cells on the same column share colour on exactly $k$ columns. Show that, if $m$ is odd, then \[\frac{n(m-1)}{2m}\leq k\leq \frac{n(m-2)}{m}\] [i]The Problem Selection Committee[/i]

In an $n\times n$ array all the fields are colored with a different color. In one move one can choose a row, move all the fields one place to the right, and move the last field (from the right) to the leftmost field of the row; or one can choose a column, move all the fields one place downwards, and move the field at the bottom of the column to the top field of the same column. For what values of $n$ is it possible to reach any arrangement of the $n^2$ fields using these kinds of steps? [i]Proposed by Ádám Schweitzer[/i]

A triangle is composed of circular cells arranged in $5784$ rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called [b]diagonal[/b] if the two cells in it [i]aren't[/i] in the same row. What is the minimum possible amount of diagonal pairs in the division? An example division into pairs is depicted in the image.