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aops_forum	p1. When Charles traveled from Hawaii to Chicago, he moved his watch $5$ hours backwards instead of $5$ hours forwards. He plans to wake up at $7:00$ the next morning (Chicago time). When he wakes up during the night and sees that his watch says $6:00$ , how many more hours should he sleep? (He has a $12$ -...	1. Partition the given set into the ordered sets: \[ S_1 = \{1, 3, 5, 7, 9, 11\}, \quad S_2 = \{2, 4, 6, 8, 10\} \] Note that any subset of the given set corresponds to a unique pair of subsets, one of \( S_1 \) and \( S_2 \), and vice versa. 2. The given condition is equivalent to not having any two conse...	[ { "content": "p1. When Charles traveled from Hawaii to Chicago, he moved his watch $5$ hours backwards instead of $5$ hours forwards. He plans to wake up at $7:00$ the next morning (Chicago time). When he wakes up during the night and sees that his watch says $6:00$ , how many more hours should he sl...
aops_forum	There are $2022$ grids in a row. Two people A and B play a game with these grids. At first, they mark each odd-numbered grid from the left with A's name, and each even-numbered grid from the left with B's name. Then, starting with the player A, they take turns performing the following action: $\bullet$ One should se...	1. Generalization and Definition: Let \( f(n) \) be the largest number of grids that A can guarantee to be marked with A's name for any even number \( n \). We need to prove that \( f(4n+2) = f(4n) = 2n+1 \). 2. Base Cases: - For \( n = 1 \): \[ f(2) = 1 \] This is because there are o...	[ { "content": "There are $2022$ grids in a row. Two people A and B play a game with these grids. At first, they mark each odd-numbered grid from the left with A's name, and each even-numbered grid from the left with B's name. Then, starting with the player A, they take turns performing the following action: $\...
aops_forum	On a $55\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out. [asy]size(2.013cm); draw ((0,0)--...	1. Define the problem and the grid: We have a \(55 \times 55\) square grid, which contains \(55^2 = 3025\) unit squares. We are cutting out 500 unit squares and 400 L-shaped pieces, each consisting of 3 unit squares. 2. Calculate the total number of unit squares removed: Each L-shaped piece consists of 3...	[ { "content": "On a $55\\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out.\n[asy]size...
aops_forum	Given is an $n\times n$ board, with an integer written in each grid. For each move, I can choose any grid, and add $1$ to all $2n-1$ numbers in its row and column. Find the largest $N(n)$ , such that for any initial choice of integers, I can make a finite number of moves so that there are at least $N(n)$ even ...	1. Interpret the board modulo 2: - Consider the board where each cell contains either 0 or 1, representing the parity of the original integers modulo 2. - A move consists of toggling (flipping) the parity of all cells in a chosen row and column. 2. Define a cross at cell \( c \): - A cross at cell \( ...	[ { "content": "Given is an $n\\times n$ board, with an integer written in each grid. For each move, I can choose any grid, and add $1$ to all $2n-1$ numbers in its row and column. Find the largest $N(n)$ , such that for any initial choice of integers, I can make a finite number of moves so that there are ...
aops_forum	Find all positive integers $1 \leq k \leq 6$ such that for any prime $p$ , satisfying $p^2=a^2+kb^2$ for some positive integers $a, b$ , there exist positive integers $x, y$ , satisfying $p=x^2+ky^2$ . <details><summary>Remark on 10.4</summary>It also appears as ARO 2010 10.4 with the grid changed to $10 \tim...	To solve the problem, we need to find all positive integers \(1 \leq k \leq 6\) such that for any prime \(p\) satisfying \(p^2 = a^2 + k b^2\) for some positive integers \(a, b\), there exist positive integers \(x, y\) satisfying \(p = x^2 + k y^2\). 1. General Claim (GC): According to Thue's lemma, there exist po...	[ { "content": "Find all positive integers $1 \\leq k \\leq 6$ such that for any prime $p$ , satisfying $p^2=a^2+kb^2$ for some positive integers $a, b$ , there exist positive integers $x, y$ , satisfying $p=x^2+ky^2$ .\n\n<details><summary>Remark on 10.4</summary>It also appears as ARO 2010 10.4 with the...
aops_forum	Consider an infinite grid $\mathcal G$ of unit square cells. A chessboard polygon is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$ . Nikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\mathcal G$ green, such that any chessbo...	1. Description of $\mathcal{F}$: Denote each cell by the corresponding coordinate in $\mathbb{Z}$. Let $n=5000$ and $d>10^{10}$ be a positive integer. Let $N=(2n+1)^d$, and let $\mathcal{A}$ be the set of integers in $[0, N-1]$ not containing the digit $n$ when written in base $2n+1$. We claim that the chessboa...	[ { "content": "Consider an infinite grid $\\mathcal G$ of unit square cells. A chessboard polygon is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\\mathcal G$ . \n\nNikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\\mathcal G$ g...
aops_forum	p1A Positive reals $x$ , $y$ , and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$ . There are two possible values for $z/x + x/z;$ find the greater value.p1B Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$ . Find $x^2+y^2$ .p2 Set $A = \{5, 6, 8, 13, 20, 22, 33, 42\}...	Given the equations: \[ x + y = 2 \] \[ x^3 + y^3 = 3 \] We start by using the identity for the sum of cubes: \[ (x + y)^3 = x^3 + y^3 + 3xy(x + y) \] Substituting the given values: \[ 2^3 = 3 + 3xy \cdot 2 \] \[ 8 = 3 + 6xy \] Solving for \(xy\): \[ 8 - 3 = 6xy \] \[ 5 = 6xy \] \[ xy = \frac{5}{6} \] Next, we use ...	[ { "content": "p1A Positive reals $x$ , $y$ , and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$ . There are two possible values for $z/x + x/z;$ find the greater value.p1B Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$ . Find $x^2+y^2$ .p2 Set $A = \\{5, 6, ...
aops_forum	Let $n>1$ be an even positive integer. An $2n \times 2n$ grid of unit squares is given, and it is partitioned into $n^2$ contiguous $2 \times 2$ blocks of unit squares. A subset $S$ of the unit squares satisfies the following properties: (i) For any pair of squares $A,B$ in $S$ , there is a sequence of sq...	1. Define the problem and constraints: We are given a \(2n \times 2n\) grid of unit squares, partitioned into \(n^2\) contiguous \(2 \times 2\) blocks. We need to find the minimum number of elements in a subset \(S\) of the unit squares such that: - For any pair of squares \(A, B \in S\), there is a sequence ...	[ { "content": "Let $n>1$ be an even positive integer. An $2n \\times 2n$ grid of unit squares is given, and it is partitioned into $n^2$ contiguous $2 \\times 2$ blocks of unit squares. A subset $S$ of the unit squares satisfies the following properties:\n\n(i) For any pair of squares $A,B$ in $S$ ,...
aops_forum	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 f...	To solve this problem, we need to determine the number of valid colorings of an \( n \times n \) grid such that every \( 2 \times 2 \) sub-grid contains exactly two white and two black squares. Let's break down the solution step by step. 1. Understanding the Constraints: - Each \( 2 \times 2 \) sub-grid must ha...	[ { "content": "Let $n$ be a positive integer. Every square in a $n \\times n$ -square grid is either white or black.\nHow many such colourings exist, if every $2 \\times 2$ -square consists of exactly two white and two black squares? \nThe squares in the grid are identified as e.g. in a chessboard, so in gen...
aops_forum	Each square of an $n \times n$ grid is coloured either blue or red, where $n$ is a positive integer. There are $k$ blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form $S_B$ . He then perform...	1. Let \( b_i \) be the number of blue cells in the \( i \)-th row. Then, the total number of blue cells in the grid is given by: \[ b_1 + b_2 + \dots + b_n = k \] The number of red cells in the \( i \)-th row is \( n - b_i \). 2. The sum of the squares of the number of blue cells in each row is: \[ ...	[ { "content": "Each square of an $n \\times n$ grid is coloured either blue or red, where $n$ is a positive integer. There are $k$ blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in each column to form ...
aops_forum	Let $n \geq 4$ be an even integer. Consider an $n \times n$ grid. Two cells ( $1 \times 1$ squares) are neighbors if they share a side, are in opposite ends of a row, or are in opposite ends of a column. In this way, each cell in the grid has exactly four neighbors. An integer from 1 to 4 is written inside each ...	1. Define Variables and Constraints: Let \( n \geq 4 \) be an even integer. We are given an \( n \times n \) grid where each cell contains an integer from 1 to 4. The constraints are: - If a cell has a 2, at least two of its neighbors contain a 1. - If a cell has a 3, at least three of its neighbors contai...	[ { "content": "Let $n \\geq 4$ be an even integer. Consider an $n \\times n$ grid. Two cells ( $1 \\times 1$ squares) are neighbors if they share a side, are in opposite ends of a row, or are in opposite ends of a column. In this way, each cell in the grid has exactly four neighbors.\nAn integer from 1 to...
aops_forum	Two players, $B$ and $R$ , play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with $B$ . On $B$ 's turn, $B$ selects one white unit square and colors it blue. On $R$ 's turn, $R$ selects two white unit squares and colors them red. The play...	1. Claim: \( B \) can score at least \( 4 \). Proof: \( B \) can color \( 2 \) adjacent squares blue on his first two turns. Let's analyze the possible responses by \( R \) in his first \( 4 \) moves: - If \( R \) colors squares that do not border \( B \)'s \( 2 \times 1 \) rectangle, \( B \) can continu...	[ { "content": "Two players, $B$ and $R$ , play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with $B$ . On $B$ 's turn, $B$ selects one white unit square and colors it blue. On $R$ 's turn, $R$ selects two white unit squares and col...
aops_forum	Toph wants to tile a rectangular $m\times n$ square grid with the $6$ types of tiles in the picture (moving the tiles is allowed, but rotating and reflecting is not). For which pairs $(m,n)$ is this possible?	1. Assigning Numbers to the Grid: We assign to each cell \((j, k)\) in the \(m \times n\) grid the number \(a_{jk} = i^{j+k}\), where \(i\) is the imaginary unit such that \(i^2 = -1\). This means that the numbers assigned to the cells will cycle through \(1, i, -1, -i\). 2. Sum of Numbers in Each Tile: ...	[ { "content": "Toph wants to tile a rectangular $m\\times n$ square grid with the $6$ types of tiles in the picture (moving the tiles is allowed, but rotating and reflecting is not). For which pairs $(m,n)$ is this possible?", "role": "user" }, { "content": "1. *Assigning Numbers to the Grid:...
aops_forum	Each of the $n^2$ cells of an $n \times n$ grid is colored either black or white. Let $a_i$ denote the number of white cells in the $i$ -th row, and let $b_i$ denote the number of black cells in the $i$ -th column. Determine the maximum value of $\sum_{i=1}^n a_ib_i$ over all coloring schemes of the grid. ...	1. Understanding the Problem: We are given an \( n \times n \) grid where each cell is either black or white. We need to maximize the sum \( \sum_{i=1}^n a_i b_i \), where \( a_i \) is the number of white cells in the \( i \)-th row and \( b_i \) is the number of black cells in the \( i \)-th column. 2. **Rearr...	[ { "content": "Each of the $n^2$ cells of an $n \\times n$ grid is colored either black or white. Let $a_i$ denote the number of white cells in the $i$ -th row, and let $b_i$ denote the number of black cells in the $i$ -th column. Determine the maximum value of $\\sum_{i=1}^n a_ib_i$ over all colorin...
aops_forum	p1. Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $\|S\|$ , the number of elements ...	To solve the problem, we need to find the value of the expression: \[ \frac{bc}{a^2} + \frac{ac}{b^2} + \frac{ab}{c^2} \] where \(a\), \(b\), and \(c\) are the roots of the polynomial \(x^3 - 20x^2 + 22\). 1. Apply Vieta's Formulas: By Vieta's formulas for the polynomial \(x^3 - 20x^2 + 22\), we have: \[ ...	[ { "content": "p1. Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $\|S\|$ , t...
aops_forum	The game of Hiveis played on a regular hexagonal grid (as shown in the figure) by 3 players. The grid consists of $k$ layers (where $k$ is a natural number) surrounding a regular hexagon, with each layer constructed around the previous layer. The figure below shows a grid with 2 layers. The players, Ali, *S...	1. Understanding the Grid Structure: - The grid is a regular hexagonal grid with \( k \) layers surrounding a central hexagon. Each layer is constructed around the previous layer. - For \( k = 2 \), the grid consists of a central hexagon and two surrounding layers, forming a larger hexagon. 2. **Game Rules a...	[ { "content": "The game of Hiveis played on a regular hexagonal grid (as shown in the figure) by 3 players. The grid consists of $k$ layers (where $k$ is a natural number) surrounding a regular hexagon, with each layer constructed around the previous layer. The figure below shows a grid with 2 layers.\n\...
aops_forum	There is a $2n\times 2n$ rectangular grid and a chair in each cell of the grid. Now, there are $2n^2$ pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at $(3,3)$ and $(2,5)...	1. Define the problem and coordinates: We have a $2n \times 2n$ grid with $2n^2$ pairs of couples. Each couple's distance is defined as the sum of the absolute differences in their row and column coordinates. We aim to find the maximal total distance among all possible seating arrangements. 2. **Contribution of...	[ { "content": "There is a $2n\\times 2n$ rectangular grid and a chair in each cell of the grid. Now, there are $2n^2$ pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating a...
aops_forum	Vulcan and Neptune play a turn-based game on an infinite grid of unit squares. Before the game starts, Neptune chooses a finite number of cells to be flooded. Vulcan is building a levee, which is a subset of unit edges of the grid (called walls) forming a connected, non-self-intersecting path or loop*. The game ...	To prove that Vulcan can always build a levee that encloses all initially flooded cells, we will use a strategy that involves constructing a levee in a systematic manner. We will show that Vulcan can always build the levee into a closed loop such that all flooded cells are contained within the loop, regardless of the i...	[ { "content": "Vulcan and Neptune play a turn-based game on an infinite grid of unit squares. Before the game starts, Neptune chooses a finite number of cells to be flooded. Vulcan is building a levee, which is a subset of unit edges of the grid (called walls) forming a connected, non-self-intersecting pat...
aops_forum	In a rectangular $57\times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest positive integer $k$ such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells? [i]Proposed by James Lin	To solve this problem, we need to determine the smallest number \( k \) such that in a \( 57 \times 57 \) grid, there must exist a rectangle with its four vertices at the centers of distinct black cells. 1. Understanding the Problem: - We have a \( 57 \times 57 \) grid. - We need to find the smallest \( k \)...	[ { "content": "In a rectangular $57\\times 57$ grid of cells, $k$ of the cells are colored black. What is the smallest positive integer $k$ such that there must exist a rectangle, with sides parallel to the edges of the grid, that has its four vertices at the center of distinct black cells?\n\n[i]Proposed ...
aops_forum	A $\triangle$ or $\bigcirc$ is placed in each of the nine squares in a 3-by-3 grid. Shown below is a sample configuration with three $\triangle$ s in a line. [asy] //diagram by kante314 size(3.3cm); defaultpen(linewidth(1)); real r = 0.37; path equi = r * dir(-30) -- (r+0.03) * dir(90) -- r * dir(210) -- cycle; d...	1. Identify the possible lines for triangles and circles: - There are 6 possible lines in a 3x3 grid: 3 rows and 3 columns. - We need to place three $\triangle$s in one line and three $\bigcirc$s in another line. 2. Calculate the number of ways to choose lines for triangles and circles: - We can choos...	[ { "content": "A $\\triangle$ or $\\bigcirc$ is placed in each of the nine squares in a 3-by-3 grid. Shown below is a sample configuration with three $\\triangle$ s in a line.\n\n[asy]\n//diagram by kante314\nsize(3.3cm);\ndefaultpen(linewidth(1));\nreal r = 0.37;\npath equi = r * dir(-30) -- (r+0.03) * dir...
aops_forum	<u>Round 5</u>p13. Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n...	To solve the given problem, we start by analyzing the recurrence relation provided: \[ 2(a_{k-1} + a_k + a_{k+1}) = a_k a_{k-1} + 8 \] First, we rearrange the equation to isolate \(a_{k+1}\): \[ 2a_{k-1} + 2a_k + 2a_{k+1} = a_k a_{k-1} + 8 \] \[ 2a_{k+1} = a_k a_{k-1} - 2a_k - 2a_{k-1} + 8 \] \[ 2a_{k+1} = (a_k - 2)...	[ { "content": "<u>Round 5</u>p13. Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$...
aops_forum	Kevin colors three distinct squares in a $3\times 3$ grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.	1. Identify the problem constraints: - We have a $3 \times 3$ grid. - Kevin colors three distinct squares red. - There exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line. 2. Analyze the grid structure: - A $3 \times 3$ grid has 9 squares. - ...	[ { "content": "Kevin colors three distinct squares in a $3\\times 3$ grid red. Given that there exist two uncolored squares such that coloring one of them would create a horizontal or vertical red line, find the number of ways he could have colored the original three squares.", "role": "user" }, { ...
aops_forum	Starting with a $5 \times 5$ grid, choose a $4 \times 4$ square in it. Then, choose a $3 \times 3$ square in the $4 \times 4$ square, and a $2 \times 2$ square in the $3 \times 3$ square, and a $1 \times 1$ square in the $2 \times 2$ square. Assuming all squares chosen are made of unit squares inside th...	1. Choosing the $4 \times 4$ square: - The $5 \times 5$ grid has $2$ possible positions for the $4 \times 4$ square horizontally and $2$ possible positions vertically. Therefore, there are $2 \times 2 = 4$ ways to choose the $4 \times 4$ square. 2. **Choosing the $3 \times 3$ square within the $4 \times 4$ squa...	[ { "content": "Starting with a $5 \\times 5$ grid, choose a $4 \\times 4$ square in it. Then, choose a $3 \\times 3$ square in the $4 \\times 4$ square, and a $2 \\times 2$ square in the $3 \\times 3$ square, and a $1 \\times 1$ square in the $2 \\times 2$ square. Assuming all squares chosen are ...
aops_forum	Consider an equilateral triangular grid $G$ with $20$ points on a side, where each row consists of points spaced $1$ unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and $20$ points in the last row, for a total of $210$ points. Let $S$ be a closed n...	1. Understanding the Grid and Points: - The grid \( G \) is an equilateral triangular grid with 20 points on each side. - The total number of points in the grid is given by the sum of the first 20 natural numbers: \[ \sum_{k=1}^{20} k = \frac{20 \cdot 21}{2} = 210 \] - Each row \( k \) has \...	[ { "content": "Consider an equilateral triangular grid $G$ with $20$ points on a side, where each row consists of points spaced $1$ unit apart. More specifically, there is a single point in the first row, two points in the second row, ..., and $20$ points in the last row, for a total of $210$ points. L...
aops_forum	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 col...	1. Grid and Turtle Path Setup: - Consider a \((4n + 2) \times (4n + 2)\) square grid. - The turtle starts at a corner and must visit each square exactly once, returning to the starting square. 2. Gridline Enumeration: - Enumerate the horizontal and vertical gridlines. - Let \(r_i\) be the number of...	[ { "content": " 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 the...
aops_forum	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.	1. Initial Assumption and Counterexample for \( m = 2n - 1 \): - We need to show that \( m = 2n - 1 \) is not sufficient. Consider choosing all the unit squares in the first row and one additional square from the diagonal. This results in \( 2n - 1 \) squares. - The centers of these squares lie on two lines: ...	[ { "content": "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.\nFind 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 vertice...
aops_forum	In a $25 \times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ coloumns and $4$ rows such that the $16$ squares of the interesections are all of the same ...	To solve this problem, we need to ensure that in a \(25 \times n\) grid, we can always find 4 columns and 4 rows such that the 16 squares at their intersections are all of the same color. We will use the Pigeonhole Principle and combinatorial arguments to find the minimal \(n\). 1. Pigeonhole Principle Application...	[ { "content": "In a $25 \\times n$ grid, each square is colored with a color chosen among $8$ different colors. Let $n$ be as minimal as possible such that, independently from the coloration used, it is always possible to select $4$ coloumns and $4$ rows such that the $16$ squares of the interesectio...
aops_forum	A token starts at the point $(0,0)$ of an $xy$ -coordinate grid and them makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token ends ...	1. Define the problem and notation: - The token starts at \((0,0)\) and makes 6 moves. - Each move is 1 unit in one of the four directions: up (\(U\)), down (\(D\)), left (\(L\)), or right (\(R\)). - We need to find the probability that the token ends on the graph of \(\|y\| = \|x\|\). 2. **Calculate the tota...	[ { "content": "A token starts at the point $(0,0)$ of an $xy$ -coordinate grid and them makes a sequence of six moves. Each move is $1$ unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The proba...
aops_forum	Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ...	1. Understanding the Game Rules: - The game is played on an \( n \times n \) lattice grid. - Players alternately mark points on the grid. - A point cannot be marked if it is on or inside a non-degenerate triangle formed by three previously marked points. - Each point can be marked only once. - The ga...	[ { "content": "Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \\times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be mark...
aops_forum	Let $n\ge 2$ be a natural number. An $n \times n$ grid is drawn on a blackboard and each field with one of the numbers $-1$ or $+1$ labeled. Then the $n$ row and also the $n$ column sums calculated and the sum $S_n$ of all these $2n$ sums determined. (a) Show that for no odd number $n$ there is a lab...	### Part (a) 1. Let \( a \) be the number of fields labeled with \( +1 \) and \( b \) be the number of fields labeled with \( -1 \). Then we have: \[ a + b = n^2 \] 2. The sum \( S_n \) of all the row and column sums is calculated. Each field in the grid is counted twice in \( S_n \) (once in its row sum and o...	[ { "content": "Let $n\\ge 2$ be a natural number.\nAn $n \\times n$ grid is drawn on a blackboard and each field with one of the numbers $-1$ or $+1$ labeled. Then the $n$ row and also the $n$ column sums calculated and the sum $S_n$ of all these $2n$ sums determined.\n(a) Show that for no odd n...
aops_forum	Let $n$ be a positive integer. In how many ways can a $4 \times 4n$ grid be tiled with the following tetromino? [asy] size(4cm); draw((1,0)--(3,0)--(3,1)--(0,1)--(0,0)--(1,0)--(1,2)--(2,2)--(2,0)); [/asy]	1. Define the Problem and Initial Conditions: Let \( T(n) \) be the number of ways to tile a \( 4 \times 4n \) grid using the given tetromino. We start by noting that \( T(1) = 2 \), as there are exactly two ways to tile a \( 4 \times 4 \) grid with the given tetromino. 2. Establish the Recurrence Relation:...	[ { "content": "Let $n$ be a positive integer. In how many ways can a $4 \\times 4n$ grid be tiled with the following tetromino?\n[asy]\nsize(4cm);\ndraw((1,0)--(3,0)--(3,1)--(0,1)--(0,0)--(1,0)--(1,2)--(2,2)--(2,0));\n[/asy]", "role": "user" }, { "content": "1. **Define the Problem and Initial Co...
aops_forum	In each box of a $ 1 \times 2009$ grid, we place either a $ 0$ or a $ 1$ , such that the sum of any $ 90$ consecutive boxes is $ 65$ . Determine all possible values of the sum of the $ 2009$ boxes in the grid.	1. Understanding the Problem: We have a $1 \times 2009$ grid where each box contains either a $0$ or a $1$. The sum of any $90$ consecutive boxes is $65$. We need to determine the possible values of the sum of all $2009$ boxes. 2. Analyzing the Sum of Consecutive Boxes: Let $a_i$ be the value in the $i$-...	[ { "content": " In each box of a $ 1 \\times 2009$ grid, we place either a $ 0$ or a $ 1$ , such that the sum of any $ 90$ consecutive boxes is $ 65$ . Determine all possible values of the sum of the $ 2009$ boxes in the grid.\n", "role": "user" }, { "content": "1. **Understanding the Probl...
aops_forum	On a $49\times 69$ rectangle formed by a grid of lattice squares, all $50\cdot 70$ lattice points are colored blue. Two persons play the following game: In each step, a player colors two blue points red, and draws a segment between these two points. (Different segments can intersect in their interior.) Segments are...	1. Initial Setup and Strategy: - The game is played on a $49 \times 69$ rectangle formed by a grid of lattice squares, with $50 \cdot 70$ lattice points. - Player 1's strategy is to create $840$ pairs of horizontally adjacent lattice points and $910$ pairs of vertically adjacent lattice points. - To achie...	[ { "content": "On a $49\\times 69$ rectangle formed by a grid of lattice squares, all $50\\cdot 70$ lattice points are colored blue. Two persons play the following game: In each step, a player colors two blue points red, and draws a segment between these two points. (Different segments can intersect in their...
aops_forum	A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent ...	1. Initial Setup and Colorings: - The hunter pre-designates a square as the origin \((0,0)\). - The hunter uses four different colorings of the grid to gather information about the rabbit's movements and initial position. 2. First Coloring: - Color \((x,y)\) with \(2x + y \pmod{5}\). - This colorin...	[ { "content": "A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly...
aops_forum	Find all positive integers $n$ such that the unit segments of an $n \times n$ grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex.	1. Determine the total number of segments in an \( n \times n \) grid: - Each unit square in the grid has 4 segments (top, bottom, left, right). - However, each internal segment is shared by two squares. - Therefore, the total number of segments in the grid is: \[ 2n(n+1) \] This is bec...	[ { "content": "Find all positive integers $n$ such that the unit segments of an $n \\times n$ grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex.", "role": "user" }, { "content": "1. **Determine the total number of segments in a...
aops_forum	How many ways are there to fill the $2 \times 2$ grid below with $0$ 's and $1$ 's such that no row or column has duplicate entries? [asy] size(2cm); draw(unitsquare); draw( (0.5,0)--(0.5,1) ); draw( (0,0.5)--(1,0.5) ); [/asy]	1. Consider the $2 \times 2$ grid. We need to fill it with $0$'s and $1$'s such that no row or column has duplicate entries. This means each row and each column must contain exactly one $0$ and one $1$. 2. Let's denote the cells of the grid as follows: \[ \begin{array}{\|c\|c\|} \hline A & B \\ \hline C...	[ { "content": "How many ways are there to fill the $2 \\times 2$ grid below with $0$ 's and $1$ 's such that no row or column has duplicate entries?\n\n[asy]\n\tsize(2cm);\n\tdraw(unitsquare);\n\tdraw( (0.5,0)--(0.5,1) );\n\tdraw( (0,0.5)--(1,0.5) );\n[/asy]", "role": "user" }, { "content": "1. C...
aops_forum	An eel is a polyomino formed by a path of unit squares which makes two turns in opposite directions (note that this means the smallest eel has four cells). For example, the polyomino shown below is an eel. What is the maximum area of a $1000 \times 1000$ grid of unit squares that can be covered by eels without overla...	** - For every increase in \(k\) by 1, we can increase the number of rectangular "shells" surrounding the center by 1. - The width and height of the square increase by 6 (3 units on each side). - The middle square has dimensions \(10 \times 10\), so the configuration works for \(n \times n\) grids where \(n = ...	[ { "content": "An eel is a polyomino formed by a path of unit squares which makes two turns in opposite directions (note that this means the smallest eel has four cells). For example, the polyomino shown below is an eel. What is the maximum area of a $1000 \\times 1000$ grid of unit squares that can be covered...
aops_forum	The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of $1201$ colours so that no rectangle with perimeter $100$ contains two squares of the same colour. Show that no rectangle of size $1\times1201$ or $1201\times1$ contains two squares...	1. Define the Problem and Key Constraints: We are given an infinite grid of unit squares, each colored with one of 1201 colors. The key constraint is that no rectangle with a perimeter of 100 contains two squares of the same color. We need to show that no rectangle of size \(1 \times 1201\) or \(1201 \times 1\) ...	[ { "content": "The plane is divided into squares by two sets of parallel lines, forming an infinite grid. Each unit square is coloured with one of $1201$ colours so that no rectangle with perimeter $100$ contains two squares of the same colour. Show that no rectangle of size $1\\times1201$ or $1201\\times...
aops_forum	A $3 \times 3$ grid of unit cells is given. A snake of length $k$ is an animal which occupies an ordered $k$ -tuple of cells in this grid, say $(s_1, \dots, s_k)$ . These cells must be pairwise distinct, and $s_i$ and $s_{i+1}$ must share a side for $i = 1, \dots, k-1$ . After being placed in a finite $n \...	1. Label the Grid: We start by labeling the $3 \times 3$ grid as follows: \[ \begin{array}{ccc} A & B & C \\ D & E & F \\ G & H & I \\ \end{array} \] 2. Example of Snake Turning Around: Consider a snake of length $k = 5$ placed on $(A, B, C, F, E)$. We will show that this snake can turn ar...	[ { "content": "A $3 \\times 3$ grid of unit cells is given. A snake of length $k$ is an animal which occupies an ordered $k$ -tuple of cells in this grid, say $(s_1, \\dots, s_k)$ . These cells must be pairwise distinct, and $s_i$ and $s_{i+1}$ must share a side for $i = 1, \\dots, k-1$ . After being...
aops_forum	<u>Round 5</u>p13. Find the number of six-digit positive integers that satisfy all of the following conditions: (i) Each digit does not exceed $3$ . (ii) The number $1$ cannot appear in two consecutive digits. (iii) The number $2$ cannot appear in two consecutive digits.p14. Find the sum of all distinct pr...	1. We start with the function \( f(x) = x^2 - 2 \) and need to compute \( f(f(f(f(f(f(f(2.5))))))) \). 2. First, compute \( f(2.5) \): \[ f(2.5) = (2.5)^2 - 2 = 6.25 - 2 = 4.25 \] 3. Notice that \( 2.5 = 2 + \frac{1}{2} \). We can generalize this form for further computations: \[ 2.5 = 2^1 + \frac{1}{2^1...	[ { "content": "<u>Round 5</u>p13. Find the number of six-digit positive integers that satisfy all of the following conditions:\n(i) Each digit does not exceed $3$ .\n(ii) The number $1$ cannot appear in two consecutive digits.\n(iii) The number $2$ cannot appear in two consecutive digits.p14. Find t...