Found problems: 109
Hernan wants to paint a $8\times 8$ board such that every square is painted with blue or red. Also wants to every $3\times 3$ subsquare have exactly $a$ blue squares and every $2\times 4$ or $4\times 2$ rectangle have exactly $b$ blue squares. Find all couples $(a,b)$ such that Hernan can do the required.
Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards.
Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells.
[*]Prove that every minimal blocking set containing at most $3m^2$ cells.
A $2N\times2N$ board is covered by non-overlapping dominos of $1\times2$ size. A lame rook (which can only move one cell at a time, horizontally or vertically) has visited each cell once on its route across the board. Call a move by the rook longitudinal if it is a move from one cell of a domino to another cell of the same domino. What is:
[list=a]
[*]the maximum possible number of longitudinal moves?
[*]the minimum possible number of longitudinal moves?
[/list]
The unit cells of a 5 x 5 board are painted with 5 colors in a way that every cell is painted by exactly one color and each color is used in 5 cells. Show that exists at least one line or one column of the board in which at least 3 colors were used.
In a $44\times 44$ board, some of the cells are blue, and the rest are red. No blue cells borders another blue cell on the side. The red cells, on the other hand, form a connected component (one may get from any red cell to any other red cell only by traversing edge-adjacent red cells). Prove that less than one third of the cells on the board are blue.
[i]Proposed by B. Frenkin[/i]
Alice and Bob play on an infinite board formed by equilateral triangles. In each turn, Alice first places a white token on an unoccupied cell, and then Bob places a black token on an unoccupied cell. Alice's goal is to eventually have $k$ white tokens on a line. Determine the maximum value of $k$ for which Alice can achieve this no matter how Bob plays.
[i]Proposed by Oriol Solé[/i]
Let $n>1$ be an integer. A rook stands in one of the cells of an infinite chessboard that is initially all white. Each move of the rook is exactly $n{}$ cells in a single direction, either vertically or horizontally, and causes the $n{}$ cells passed over by the rook to be painted black. After several such moves, without visiting any cell twice, the rook returns to its starting cell, with the resulting black cells forming a closed path. Prove that the number of white cells inside the black path gives a remainder of $1{}$ when divided by $n{}$.
Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black)
The figure can be rotated $90°, 180°$ or $270°$.
Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.
You have a $9 \times 9$ board with white squares. A tile can be moved from one square to another neighbor (tiles that share one side). If we paint some squares of black, we say that such coloration is [i]good [/i] if there is a white square where we can place a chip that moving through white squares can return to the initial square having passed through at least $3$ boxes, without passing the same square twice.
Find the highest possible value of $k$ such that any form of painting $k$ squares of black are a [i]good [/i] coloring.
In each cell of a $2025 \times 2025$ board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to $1$, and the sum of the numbers in each column is equal to $1$. Define $r_i$ to be the largest value in row $i$, and let $R = r_1 + r_2 + ... + r_{2025}$. Similarly, define $c_i$ to be the largest value in column $i$, and let $C = c_1 + c_2 + ... + c_{2025}$.
What is the largest possible value of $\frac{R}{C}$?
[i]Proposed by Paulius Aleknavičius, Lithuania, and Anghel David Andrei, Romania[/i]
Determine the greatest positive integer $A{}$ with the following property: however we place the numbers $1,2,\ldots, 100$ on a $10\times 10$ board, each number appearing exactly once, we can find two numbers on the same row or column which differ by at least $A{}$.
Let $n \ge 2$ be a positive integer. Each square of an $n\times n$ board is coloured red or blue. We put dominoes on the board, each covering two squares of the board. A domino is called [i]even [/i] if it lies on two red or two blue squares and [i]colourful [/i] if it lies on a red and a blue square. Find the largest positive integer $k$ having the following property: regardless of how the red/blue-colouring of the board is done, it is always possible to put $k$ non-overlapping dominoes on the board that are either all [i]even [/i] or all [i]colourful[/i].
Ana and Beta play a turn-based game on a $m \times n$ board. Ana begins. At the beginning, there is a stone in the lower left square and the objective is to move it to the upper right corner. A move consists of the player moving the stone to the right or up as many squares as the player wants. Find all the values of $(m, n)$ for which Ana can guarantee victory.
Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black)
The figure can be rotated $90°, 180°$ or $270°$.
Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.
A knight is placed on each square of the first column of a $2017 \times 2017$ board. A [i]move[/i] consists in choosing two different knights and moving each of them to a square which is one knight-step away. Find all integers $k$ with $1 \leq k \leq 2017$ such that it is possible for each square in the $k$-th column to contain one knight after a finite number of moves.
Note: Two squares are a knight-step away if they are opposite corners of a $2 \times 3$ or $3 \times 2$ board.
An [i]$n$-type triangle[/i] where $n\geqslant 2$ is formed by the cells of a $(2n+1)\times(2n+1)$ board, situated under both main diagonals. For instance, a $3$-type triangle looks like this:[img]https://i.ibb.co/k4fmwWY/Screenshot-2024-07-31-153932.png[/img]Determine the maximal length of a sequence with pairwise distinct cells in an $n$-type triangle, such that, beggining with the second one, any cell of the sequence has a common side with the previous one.
[i]Cristi Săvescu[/i]
Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards.
Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells.
[*]Prove that every minimal blocking set containing at most $3m^2$ cells.
Cards numbered with numbers $1$ to $1000$ are to be placed on the cells of a $1\times 1994$ rectangular board one by one, according to the following rule: If the cell next to the cell containing the card $n$ is free, then the card $n+1$ must be put on it. Prove that the number of possible arrangements is not more than half a mllion.
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]
Given an $n \times n$ board which is divided into $n^2$ squares of size $1 \times 1$, all of which are white. Then, Aqua selects several squares from this board and colors them black. Ruby then places exactly one $1\times 2$ domino on the board, so that the domino covers exactly two squares on the board. Ruby can rotate the domino into a $2\times 1$ domino.
After Aqua colors, it turns out there are exactly $2024$ ways for Ruby to place a domino on the board so that it covers exactly $1$ black square and $1$ white square.
Determine the smallest possible value of $n$ so that Aqua and Ruby can do this.
[i]Proposed by Muhammad Afifurrahman, Indonesia [/i]
The numbers from $1$ to $100$ are placed without repetition in each cell of a \(10 \times 10\) board. An increasing path of length \(k\) on this board is a sequence of cells \(c_1, c_2, \ldots, c_k\) such that, for each \(i = 2, 3, \ldots, k\), the following properties are satisfied:
• The cells \(c_i\) and \(c_{i-1}\) share a side or a vertex;
• The number in \(c_i\) is greater than the number in \(c_{i-1}\).
What is the largest positive integer \(k\) for which we can always find an increasing path of length \(k\), regardless of how the numbers from 1 to 100 are arranged on the board?
A $8 \times 8$ board is given, with sides directed north-south and east-west.
It is divided into $1 \times 1$ cells in the usual manner. In each cell, there is most one [i]house[/i]. A house occupies only one cell.
A house is [i] in the shade[/i] if there is a house in each of the cells in the south, east and west sides of its cell. In particular, no house placed on the south, east or west side of the board is in the shade.
Find the maximal number of houses that can be placed on the board such that no house is in the shade.
Consider a $N\times N$ square board where $N\geq 3$ is an odd integer. The caterpillar Carl sits at the center of the square; all other cells contain distinct positive integers. An integer $n$ weights $1\slash n$ kilograms. Carl wants to leave the board but can eat at most $2$ kilograms. Determine whether Carl can always find a way out when
a) $N=2003.$
b) $N$ is an arbitrary odd integer.
We are given an $n \times n$ board. Rows are labeled with numbers $1$ to $n$ downwards and columns are labeled with numbers $1$ to $n$ from left to right. On each field we write the number $x^2 + y^2$ where $(x, y)$ are its coordinates. We are given a figure and can initially place it on any field. In every step we can move the figure from one field to another if the other field has not already been visited and if at least one of the following
conditions is satisfied:[list]
[*] the numbers in those $2$ fields give the same remainders when divided by $n$,
[*] those fields are point reflected with respect to the center of the board.[/list]Can all the fields be visited in case:
[list=a][*] $n = 4$,
[*] $n = 5$?[/list]
[i]Josip Pupić[/i]
Every square of $1000 \times 1000$ board is colored black or white. It is known that exists one square $10 \times 10$ such that all squares inside it are black and one square $10 \times 10$ such that all squares inside are white. For every square $K$ $10 \times 10$ we define its power $m(K)$ as an absolute value of difference between number of white and black squares $1 \times 1$ in square $K$. Let $T$ be a square $10 \times 10$ which has minimum power among all squares $10 \times 10$ in this board. Determine maximal possible value of $m(T)$