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.

One needs to ffll the cells of an $n\times n$ table ($n > 1$) with distinct integers from $1$ to $n^2$ so that every two consecutive integers are placed in cells that share a side, while every two integers with the same remainder if divided by $n$ are placed in distinct rows and distinct columns. For which $n$ is this possible? (Alexandr Gribalko)

On a $(4n + 2)\times (4n + 2)$ square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms of $n$, what is the largest positive integer $k$ such that there must be a row or column that the turtle has entered at least $k$ distinct times?

Let there be a $n\times n$ board. Write down $0$ or $1$ in all $n^2$ squares. For $1 \le k \le n$, let $A_k$ be the product of all numbers in the $k$th row. How many ways are there to write down the numbers so that $A_1 + A_2 + ... + A_n$ is even?

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.

We number the columns of an $n\times n$-board from $1$ to $n$. In each cell, we place a number. This is done in such a way that each row precisely contains the numbers $1$ to $n$ (in some order), and also each column contains the numbers $1$ to $n$ (in some order). Next, each cell that contains a number greater than the cell's column number, is coloured grey. In the figure below you can see an example for the case $n = 3$. [asy] unitsize(0.6 cm); int i; fill((0,0)--(1,0)--(1,1)--(0,1)--cycle, gray(0.8)); fill(shift((1,0))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); fill(shift((0,2))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); for (i = 0; i <= 3; ++i) { draw((0,i)--(3,i)); draw((i,0)--(i,3)); } label("$1$", (0.5,3.5)); label("$2$", (1.5,3.5)); label("$3$", (2.5,3.5)); label("$3$", (0.5,2.5)); label("$1$", (1.5,2.5)); label("$2$", (2.5,2.5)); label("$1$", (0.5,1.5)); label("$2$", (1.5,1.5)); label("$3$", (2.5,1.5)); label("$2$", (0.5,0.5)); label("$3$", (1.5,0.5)); label("$1$", (2.5,0.5)); [/asy] (a) Suppose that $n = 5$. Can the numbers be placed in such a way that each row contains the same number of grey cells? (b) Suppose that $n = 10$. Can the numbers be placed in such a way that each row contains the same number of grey cells?

You have an $n \times n$ grid of empty squares. You place a cross in all the squares, one at a time. When you place a cross in an empty square, you receive $i+j$ points if there were $i$ crosses in the same row and $j$ crosses in the same column before you placed the new cross. Which are the possible total scores you can get?

In each cell of an \(n \times n\) grid, one of the numbers \(0\), \(1,\) or \(2\) must be written. Determine all positive integers \(n\) for which there exists a way to fill the \(n \times n\) grid such that, when calculating the sum of the numbers in each row and each column, the numbers \(1, 2, \ldots, 2n\) are obtained in some order.

Let $n$ be an odd natural number. We consider an $n\times n$ grid which is made up of $n^2$ unit squares and $2n(n+1)$ edges. We colour each of these edges either $\color{red} \textit{red}$ or $\color{blue}\textit{blue}$. If there are at most $n^2$ $\color{red} \textit{red}$ edges, then show that there exists a unit square at least three of whose edges are $\color{blue}\textit{blue}$.

Find the greatest constant $\lambda$ such that for any doubly stochastic matrix of order 100, we can pick $150$ entries such that if the other $9850$ entries were replaced by $0$, the sum of entries in each row and each column is at least $\lambda$. Note: A doubly stochastic matrix of order $n$ is a $n\times n$ matrix, all entries are nonnegative reals, and the sum of entries in each row and column is equal to 1.

A and B play a game on a square board consisting of $n \times n$ white tiles, where $n \ge 2$. A moves first, and the players alternate taking turns. A move consists of picking a square consisting of $2\times 2$ or $3\times 3$ white tiles and colouring all these tiles black. The first player who cannot find any such squares has lost. Show that A can always win the game if A plays the game right.

In a square board of size 1001 x 1001, we color some $m$ cells in such a way that: i. Of any two cells that share an edge, at least one is colored. ii. Of any 6 consecutive cells in a column or a row, at least 2 consecutive ones are colored. Determine the smallest possible value of $m$.

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]

Consider a $8\times 8$ chessboard where all $64$ unit squares are at the start white. Prove that, if any $12$ of the $64$ unit square get painted black, then we can find $4$ lines and $4$ rows that have all these $12$ unit squares.

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]

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 $R_{mn}$.