Let $n$ be a fixed positive integer and consider an $n\times n$ grid of real numbers. Determine the greatest possible number of cells $c$ in the grid such that the entry in $c$ is both strictly greater than the average of $c$'s column and strictly less than the average of $c$'s row. [i]Proposed by Holden Mui[/i]

Let $n$ be a posititve integer. On a $n \times n$ grid there are $n^2$ unit squares and on these we color the sides with blue such that every unit square has exactly one side with blue. [b]a)[/b] Find the maximun number of blue unit sides we can have on the $n \times n$ grid. [b]b)[/b] Find the minimun number of blue unit sides we can have on the $n \times n$ grid.

A $100 \times100$ chessboard has a non-negative real number in each of its cells. A chessboard is [b]balanced[/b] if and only if the numbers sum up to one for each column of cells as well as each row of cells. Find the largest positive real number $x$ so that, for any balanced chessboard, we can find $100$ cells of it so that these cells all have number greater or equal to $x$, and no two of these cells are on the same column or row. [i]Proposed by CSJL.[/i]

Let $n\ge 1$ be a positive integer. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of side length $1$. Find the number of parallelograms which have vertices among the vertices of the $n^2$ squares of side length $1$, with both sides smaller or equal to $2$, and which have tha area equal to $2$. (Greece)

Consider an $n\times{n}$ grid formed by $n^2$ unit squares. We define the centre of a unit square as the intersection of its diagonals. Find the smallest integer $m$ such that, choosing any $m$ unit squares in the grid, we always get four unit squares among them whose centres are vertices of a parallelogram.

The teacher reported to Peter an odd integer $m \le 2013$ and gave the guy a homework. Petrick should star the cells in the $2013 \times 2013$ table so to make the condition true: if there is an asterisk in some cell in the table, then or in row or column containing this cell should be no more than $m$ stars (including this one). Thus in each cell of the table the guy can put at most one star. The teacher promised Peter that his assessment would be just the number of stars that the guy will be able to place. What is the greatest number will the stars be able to place in the table Petrick?

In a $m\times n$ square grid, with top-left corner is $A$, there is route along the edges of the grid starting from $A$ and visits all lattice points (called "nodes") exactly once and ending also at $A$. a. Prove that this route exists if and only if at least one of $m,\ n$ is odd. b. If such a route exists, then what is the least possible of turning points? *A turning point is a node that is different from $A$ and if two edges on the route intersect at the node are perpendicular.

You are given a square $n \times n$. The centers of some of some $m$ of its $1\times 1$ cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer $n \geq 3$, find the largest value of $m$ for which it is possible. [i]Proposed by Oleksiy Masalitin, Fedir Yudin[/i]

Let $n$ be a positive integer. Every square in a $n \times n$-square grid is either white or black. How many such colourings exist, if every $2 \times 2$-square consists of exactly two white and two black squares? The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from each other by rotation are different.

The square $ABCD$ is divided into $n^2$ equal small (elementary) squares by parallel lines to its sides, (see the figure for the case $n = 4$). A spider starts from point$ A$ and moving only to the right and up tries to arrive at point $C$. Every ” movement” of the spider consists of: ”$k$ steps to the right and $m$ steps up” or ”$m$ steps to the right and $k$ steps up” (which can be performed in any way). The spider first makes $l$ ”movements” and in then, moves to the right or up without any restriction. If $n = m \cdot l$, find all possible ways the spider can approach the point $C$, where $n, m, k, l$ are positive integers with $k < m$. [img]https://cdn.artofproblemsolving.com/attachments/2/d/4fb71086beb844ca7c492a30c7d333fa08d381.png[/img]

Let $k$,$n$ be positive integers, $k,n>1$, $k<n$ and a $n \times n$ grid of unit squares is given. Ana and Maya take turns in coloring the grid in the following way: in each turn, a unit square is colored black in such a way that no two black cells have a common side or vertex. Find the smallest positive integer $n$ , such that they can obtain a configuration in which each row and column contains exactly $k$ black cells. Draw one example.

Determine all integer $n \ge 2$ such that it is possible to construct an $n * n$ array where each entry is either $-1, 0, 1$ so that the sums of elements in every row and every column are distinct

A polyomino is region with connected interior that is a union of a finite number of squares from a grid of unit squares. Do there exist a positive integer $n>4$ and a polyomino $P$ contained entirely within and $n$-by-$n$ grid such that $P$ contains exactly $3$ unit squares in every row and every column of the grid? Proposed by [i]Nikolai Beluhov[/i]

Given the sets $R_{mn} = \{ (x,y) \mid x=0,1,\dots,m; y=0,1,\dots,n \}$, consider functions $f:R_{mn}\to \{-1,0,1\}$ with the following property: for each quadruple of points $A_1,A_2,A_3,A_4\in R_{mn}$ which form a square with side length $0<s<3$, we have $$f(A_1)+f(A_2)+f(A_3)+f(A_4)=0.$$ For each pair $(m,n)$ of positive integers, determine $F(m,n)$, the number of such functions $f$ on $R_{mn}$.