Found problems: 107
2020 Bosnia and Herzegovina Junior BMO TST, 2
A board $n \times n$ is divided into $n^2$ unit squares and a number is written in each unit square.
Such a board is called [i] interesting[/i] if the following conditions hold:
$\circ$ In all unit squares below the main diagonal, the number $0$ is written;
$\circ$ Positive integers are written in all other unit squares.
$\circ$ When we look at the sums in all $n$ rows, and the sums in all $n$ columns, those $2n$ numbers
are actually the numbers $1,2,...,2n$ (not necessarily in that order).
$a)$ Determine the largest number that can appear in a $6 \times 6$ [i]interesting[/i] board.
$b)$ Prove that there is no [i]interesting[/i] board of dimensions $7\times 7$.
Mathematical Minds 2024, P7
In every cell of an $n\times n$ board is written $1$ or $-1$. At each step we may choose any of the $4n-2$ diagonals of the board and change the signs of all the numbers on that diagonal. Determine the number of initial configurations from which, after a finite number of steps, we may arrive at a configuration where all products of numbers on rows and columns equal to $1$.
[i]Proposed by Pavel Ciurea[/i]
2024 Tuymaada Olympiad, 5
Given a board with size $25\times 25$. Some $1\times 1$ squares are marked, so that for each $13\times 13$ and $4\times 4$ sub-boards, there are atleast $\frac{1}{2}$ marked parts of the sub-board. Find the least possible amount of marked squares in the entire board.
2009 Puerto Rico Team Selection Test, 6
The entries on an $ n$ × $ n$ board are colored black and white like it is usually done in a chessboard, and the upper left hand corner is black.
We color the entries on the chess board black according to the following rule:
In each step we choose an arbitrary $ 2$×$ 3$ or $ 3$× $ 2$ rectangle that still contains $ 3$ white entries, and we color these three entries black.
For which values of $ n$ can the whole board be colored black in a finite number of steps
2002 Mexico National Olympiad, 1
The numbers $1$ to $1024$ are written one per square on a $32 \times 32$ board, so that the first row is $1, 2, ... , 32$, the second row is $33, 34, ... , 64$ and so on. Then the board is divided into four $16 \times 16$ boards and the position of these boards is moved round clockwise, so that
$AB$ goes to $DA$
$DC \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \, CB$
then each of the $16 \times 16 $ boards is divided into four equal $8 \times 8$ parts and each of these is moved around in the same way (within the $ 16 \times 16$ board). Then each of the $8 \times 8$ boards is divided into four $4 \times 4$ parts and these are moved around, then each $4 \times 4$ board is divided into $2 \times 2$ parts which are moved around, and finally the squares of each $2 \times 2$ part are moved around. What numbers end up on the main diagonal (from the top left to bottom right)?
2020 China Northern MO, P4
Two students $A$ and $B$ play a game on a $20 \text{ x } 20$ chessboard. It is known that two squares are said to be [i]adjacent[/i] if the two squares have a common side. At the beginning, there is a chess piece in a certain square of the chessboard. Given that $A$ will be the first one to move the chess piece, $A$ and $B$ will alternately move this chess piece to an adjacent square. Also, the common side of any pair of adjacent squares can only be passed once. If the opponent cannot move anymore, then he will be declared the winner (to clarify since the wording wasn’t that good, you lose if you can’t move). Who among $A$ and $B$ has a winning strategy? Justify your claim.
2024 Brazil National Olympiad, 5
Esmeralda chooses two distinct positive integers \(a\) and \(b\), with \(b > a\), and writes the equation
\[
x^2 - ax + b = 0
\]
on the board. If the equation has distinct positive integer roots \(c\) and \(d\), with \(d > c\), she writes the equation
\[
x^2 - cx + d = 0
\]
on the board. She repeats the procedure as long as she obtains distinct positive integer roots. If she writes an equation for which this does not occur, she stops.
a) Show that Esmeralda can choose \(a\) and \(b\) such that she will write exactly 2024 equations on the board.
b) What is the maximum number of equations she can write knowing that one of the initially chosen numbers is 2024?