source stringclasses 1
value | problem stringlengths 239 1.18k | solution stringlengths 1.19k 3.17k | messages listlengths 2 2 |
|---|---|---|---|
amc_aime | Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place a counter in both of ... |
1. **Understanding the Game Setup:**
- Player $A$ places two counters on adjacent hexagons.
- Player $B$ can remove any one counter from the board.
- $A$ wins if he creates a line of $k$ consecutive counters.
2. **Strategy for $A$ to Win with $k=5$:**
- **Initial Placement:** $A$ starts by placing two cou... | [
{
"content": "Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place... |
amc_aime | In a game of Chomp, two players alternately take bites from a 5-by-7 grid of [unit squares](https://artofproblemsolving.com/wiki/index.php/Unit_square). To take a bite, a player chooses one of the remaining [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)), then removes ("eats") all squares i... | 1. **Understanding the Game Dynamics**: In Chomp, when a player chooses a square, all squares to the right and all squares above that square are also removed. This means that if a square at position $(i, j)$ in a $5 \times 7$ grid is chosen, all squares $(i, k)$ for $k \geq j$ and all squares $(l, j)$ for $l \leq i$ ar... | [
{
"content": "In a game of Chomp, two players alternately take bites from a 5-by-7 grid of [unit squares](https://artofproblemsolving.com/wiki/index.php/Unit_square). To take a bite, a player chooses one of the remaining [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)), then removes ... |
amc_aime | Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge... | To solve this problem, we will calculate the probability that Frieda reaches a corner square within four hops, starting from the center of a $3 \times 3$ grid. We will use a state-based approach to model Frieda's possible positions and transitions.
#### Definitions:
- **State**: Represents Frieda's position on the gri... | [
{
"content": "Frieda the frog begins a sequence of hops on a $3 \\times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she \"wraps around\" and j... |
amc_aime | Placing no more than one $\text{X}$ in each small [square](https://artofproblemsolving.com/wiki/index.php/Square), what is the greatest number of $\text{X}$'s that can be put on the grid shown without getting three $\text{X}$'s in a row vertically, horizontally, or diagonally?
$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad... | 1. **Understanding the Grid**: The grid is a $4 \times 4$ square grid, which means there are 16 small squares in total.
2. **Objective**: We need to place the maximum number of $\text{X}$'s such that no three $\text{X}$'s are aligned vertically, horizontally, or diagonally.
3. **Using the Pigeonhole Principle**: If w... | [
{
"content": "Placing no more than one $\\text{X}$ in each small [square](https://artofproblemsolving.com/wiki/index.php/Square), what is the greatest number of $\\text{X}$'s that can be put on the grid shown without getting three $\\text{X}$'s in a row vertically, horizontally, or diagonally?\n$\\text{(A)}\\ 2... |
amc_aime | The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
$\textbf{(A) }\frac{1}{21}\qquad\textbf{(B) }\frac{1}{14}\qquad\textbf{(C)... |
To solve this problem, we need to ensure that the sum of the numbers in each row and each column is odd. We can achieve this by having either three odd numbers or one odd and two even numbers in each row and column. Let's analyze the possible configurations and calculate the probability.
#### Step 1: Understanding th... | [
{
"content": "The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?\n$\\textbf{(A) }\\frac{1}{21}\\qquad\\textbf{(B) }... |
amc_aime | (Melanie Wood)
Alice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing on the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each row, the square with th... |
We will analyze and expand on Solution 1, which provides a strategic approach for Bob to ensure his victory.
#### Step 1: Strategy Setup
Bob's strategy revolves around controlling the maximum numbers in each row by designating certain squares as "useless" where Alice's entries will not affect the outcome. Specificall... | [
{
"content": "(Melanie Wood)\nAlice and Bob play a game on a 6 by 6 grid. On his or her turn, a player chooses a rational number not yet appearing on the grid and writes it in an empty square of the grid. Alice goes first and then the players alternate. When all squares have numbers written in them, in each ... |
amc_aime | In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $N$ be the number of shadings with this property. Find the remainder when $N$ is divided by 1000.
[AIME I 2007-10.png](https://artofproblemsolving.co... |
We will analyze the problem using the first solution provided and add any necessary details and calculations.
1. **Choosing Rows for the First Column:**
We start by selecting 3 rows out of 6 to shade in the first column. The number of ways to do this is given by the binomial coefficient:
\[
\binom{6}{3} = 20... | [
{
"content": "In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $N$ be the number of shadings with this property. Find the remainder when $N$ is divided by 1000.\n[AIME I 2007-10.png](https:/... |
amc_aime | Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place a counter in both of ... |
1. **Understanding the Game Setup**:
- Player $A$ places two counters on adjacent hexagons.
- Player $B$ removes one counter from the board.
- $A$ wins if he creates a line of $k$ consecutive counters.
2. **Strategy for $A$ to Win with $k = 5$**:
- **Initial Placement**: $A$ starts by placing two counter... | [
{
"content": "Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place... |
amc_aime | Two unit squares are selected at random without replacement from an $n \times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\frac{1}{2015}$. | To solve this problem, we need to calculate the probability that two randomly selected unit squares from an $n \times n$ grid are adjacent, either horizontally or vertically, and find the smallest $n$ for which this probability is less than $\frac{1}{2015}$.
#### Step 1: Count the number of adjacent pairs in an $n \ti... | [
{
"content": "Two unit squares are selected at random without replacement from an $n \\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\\frac{1}{2015}$.",
"role": "user"
},... |
amc_aime | An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:
- The two $1 \times 1$ faces of each beam co... | 1. **Setup and Definitions**:
- Consider a $2020 \times 2020 \times 2020$ cube with a coordinate system where $(x, y, z)$ denotes the center of a unit cube.
- Define the cube's corners at $(1, 1, 1)$ and $(2020, 2020, 2020)$.
- A beam is defined as a $1 \times 1 \times 2020$ rectangular prism, oriented along o... | [
{
"content": "An empty $2020 \\times 2020 \\times 2020$ cube is given, and a $2020 \\times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \\times 1 \\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:\n- The two $1 \\t... |
amc_aime | The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$?$
[asy] import olympiad; unitsize(25); for (int i = 0; i < 3; ++i) { for (int j = 0; j < 9; ++j) { pair A = (j,i); } } for (int i = 0; i... | To solve this problem, we need to calculate the total length of the line segments that form the acronym "AMC" on the given grid. We will count the number of straight line segments and slanted line segments separately, and then compute their total lengths.
1. **Identify the straight and slanted line segments:**
- **... | [
{
"content": "The acronym AMC is shown in the rectangular grid below with grid lines spaced $1$ unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC$?$\n[asy] import olympiad; unitsize(25); for (int i = 0; i < 3; ++i) { for (int j = 0; j < 9; ++j) { pair A = (j,i);... |
amc_aime | A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?
$\textbf{(A) }\frac{1}{6} \qquad \textbf{(B) }\frac{1}{5} \qquad \textbf{(C) }\frac{1}{4} \qquad \textbf{(D) }\frac{1}{3} \qquad \textbf{(E) }\frac{1}{2}$ |
To find the fraction of the grid covered by the triangle with vertices $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$, we first need to calculate the area of the triangle and then compare it to the area of the grid.
#### Step 1: Calculate the area of the triangle using the Shoelace Theorem
The Shoelace Theorem provides a formul... | [
{
"content": "A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\\times5$ grid. What fraction of the grid is covered by the triangle?\n$\\textbf{(A) }\\frac{1}{6} \\qquad \\textbf{(B) }\\frac{1}{5} \\qquad \\textbf{(C) }\\frac{1}{4} \\qquad \\textbf{(D) }\\frac{1}{3} \\qquad \\te... |
amc_aime | How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?
$\textbf{(A)} ~12\qquad\textbf{(B... | We are tasked with placing $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in a $3 \times 3$ grid such that no two chips of the same color are directly adjacent either vertically or horizontally.
#### Step 1: Fixing the position of one chip
To simplify the probl... | [
{
"content": "How many ways are there to place $3$ indistinguishable red chips, $3$ indistinguishable blue chips, and $3$ indistinguishable green chips in the squares of a $3 \\times 3$ grid so that no two chips of the same color are directly adjacent to each other, either vertically or horizontally?\n$\\textbf... |
amc_aime | The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
$\textbf{(A) }\frac{1}{21}\qquad\textbf{(B) }\frac{1}{14}\qquad\textbf{(C)... |
To solve this problem, we need to ensure that the sum of the numbers in each row and each column is odd. We know that the sum of three numbers is odd if and only if either all three numbers are odd or exactly one of them is odd (and the other two are even).
#### Step 1: Counting Odd and Even Numbers
From the numbers ... | [
{
"content": "The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?\n$\\textbf{(A) }\\frac{1}{21}\\qquad\\textbf{(B) }... |
amc_aime | The Y2K Game is played on a $1 \times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning stra... | 1. **Initial Setup**: Label the squares from $1$ to $2000$. The first player (Player 1) writes a letter in any square $k$.
2. **Second Player's Strategy**: The second player (Player 2) should choose a square $l$ such that $\min(|l-1|, |l-2000|, |l-k|) > 3$ and writes an 'S'. This ensures that no immediate 'SOS' can b... | [
{
"content": "The Y2K Game is played on a $1 \\times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second pl... |
amc_aime | Problem
Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a recta... |
1. **Understanding the Problem:**
Steve is placing $m$ indistinguishable stones on an $n \times n$ grid. The placement is such that each square can hold any number of stones. The goal is to determine the number of distinct ways to pile the stones, considering that some configurations can be transformed into others ... | [
{
"content": "Problem\nSteve is piling $m\\geq 1$ indistinguishable stones on the squares of an $n\\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, whi... |
amc_aime | A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the farmer does not want t... | We will analyze the problem by considering different cases based on the distribution of crops in the sections. The key restrictions are that corn cannot be adjacent to wheat, and soybeans cannot be adjacent to potatoes.
#### Case 1: All sections have the same crop.
- There are 4 types of crops (corn, wheat, soybeans, ... | [
{
"content": "A farmer's rectangular field is partitioned into $2$ by $2$ grid of $4$ rectangular sections as shown in the figure. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the ... |
amc_aime | Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row f... | To solve this problem, we need to understand the pattern in which numbers are placed in the grid. The numbers are placed starting from $1$ in the center of the grid and spiraling outwards in a clockwise manner. The grid is $15 \times 15$, so it contains numbers from $1$ to $225$.
1. **Identify the position of the numb... | [
{
"content": "Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appe... |
amc_aime | Define a T-grid to be a $3\times3$ matrix which satisfies the following two properties:
Exactly five of the entries are $1$'s, and the remaining four entries are $0$'s.
Among the eight rows, columns, and long diagonals (the long diagonals are $\{a_{13},a_{22},a_{31}\}$ and $\{a_{11},a_{22},a_{33}\}$, no more than one... |
#### Step 1: Understanding the Problem
We need to find the number of distinct $3 \times 3$ matrices (T-grids) where:
- Five entries are $1$s and four entries are $0$s.
- No more than one of the eight (three rows, three columns, and two long diagonals) has all three entries equal.
#### Step 2: Total Combinations
First... | [
{
"content": "Define a T-grid to be a $3\\times3$ matrix which satisfies the following two properties:\n\n\nExactly five of the entries are $1$'s, and the remaining four entries are $0$'s.\nAmong the eight rows, columns, and long diagonals (the long diagonals are $\\{a_{13},a_{22},a_{31}\\}$ and $\\{a_{11},a_{2... |
amc_aime | (Gregory Galperin) A [square](https://artofproblemsolving.com/wiki/index.php/Square) grid on the [Euclidean plane](https://artofproblemsolving.com/wiki/index.php/Cartesian_plane) consists of all [points](https://artofproblemsolving.com/wiki/index.php/Point) $(m,n)$, where $m$ and $n$ are [integers](https://artofproblem... |
#### Lemma:
Among 3 tangent circles with radius greater than or equal to 5, one can always fit a circle with radius greater than $\frac{1}{\sqrt{2}}$ between those 3 circles.
#### Proof of Lemma:
Using Descartes' Circle Theorem, which states that if $a$, $b$, $c$, and $d$ are the curvatures of four mutually tangent c... | [
{
"content": "(Gregory Galperin) A [square](https://artofproblemsolving.com/wiki/index.php/Square) grid on the [Euclidean plane](https://artofproblemsolving.com/wiki/index.php/Cartesian_plane) consists of all [points](https://artofproblemsolving.com/wiki/index.php/Point) $(m,n)$, where $m$ and $n$ are [integers... |
amc_aime | To complete the grid below, each of the digits 1 through 4 must occur once
in each row and once in each column. What number will occupy the lower
right-hand square?
\[\begin{tabular}{|c|c|c|c|}\hline 1 & & 2 &\\ \hline 2 & 3 & &\\ \hline & &&4\\ \hline & &&\\ \hline\end{tabular}\]
$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ ... | We are given a 4x4 grid where each digit from 1 through 4 must appear exactly once in each row and once in each column. The initial grid is:
\[
\begin{array}{|c|c|c|c|}
\hline
1 & & 2 & \\
\hline
2 & 3 & & \\
\hline
& & & 4 \\
\hline
& & & \\
\hline
\end{array}
\]
We need to determine the number... | [
{
"content": "To complete the grid below, each of the digits 1 through 4 must occur once\nin each row and once in each column. What number will occupy the lower\nright-hand square?\n\\[\\begin{tabular}{|c|c|c|c|}\\hline 1 & & 2 &\\\\ \\hline 2 & 3 & &\\\\ \\hline & &&4\\\\ \\hline & &&\\\\ \\hline\\end{tabular}... |
amc_aime | A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black ... | 1. **Identify the invariant**: The center square remains unchanged after a $90^\circ$ rotation. Therefore, for the entire grid to be black after the process, the center square must initially be black. The probability of the center square being black is $\frac{1}{2}$.
2. **Consider the rotation effect on other squares*... | [
{
"content": "A $3 \\times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\\,^{\\circ}$ clockwise about its center, and every white square in a position form... |
amc_aime | A $3 \times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\,^{\circ}$ clockwise about its center, and every white square in a position formerly occupied by a black ... | 1. **Identify the invariant**: The center square remains unchanged after a $90^\circ$ rotation. Since the entire grid must end up black, the center square must initially be black. The probability of this happening is $\frac{1}{2}$.
2. **Consider the effect of rotation on the other squares**: The rotation affects the p... | [
{
"content": "A $3 \\times 3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated $90\\,^{\\circ}$ clockwise about its center, and every white square in a position form... |
amc_aime | A $10\times10\times10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points. | To solve this problem, we need to find lines that contain exactly 8 points from a $10 \times 10 \times 10$ grid. Each point on the grid is represented by coordinates $(i, j, k)$ where $i, j, k \in \{1, 2, \ldots, 10\}$.
#### Step 1: Define the line
A line in three-dimensional space can be defined by a starting point $... | [
{
"content": "A $10\\times10\\times10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points.",
"role": "user"
},
{
"content": "To solve t... |
amc_aime | There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ is a line of symmetry for the square?
$\te... | 1. **Identify the total number of points and the position of $P$:**
The square contains $81$ grid points, arranged in a $9 \times 9$ grid. Point $P$, being at the center, is located at the center grid point.
2. **Determine the number of points excluding $P$:**
Excluding point $P$, there are $81 - 1 = 80$ points.... | [
{
"content": "There are $81$ grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is in the center of the square. Given that point $Q$ is randomly chosen among the other $80$ points, what is the probability that the line $PQ$ is a line of symmetry... |
amc_aime | Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is
$\textrm{(A)}\ 30\qquad\textrm{(B)}\ 200\qquad\textrm{(C)}\ 410\qquad\textrm{(D)}\ 420\qquad\textrm{(E)}\ 430$ | To determine the number of toothpicks used to build a rectangular grid that is 20 toothpicks high and 10 toothpicks wide, we need to consider both the horizontal and vertical toothpicks.
1. **Counting Horizontal Toothpicks:**
- The grid is 20 toothpicks high, which means there are 21 horizontal lines (since there i... | [
{
"content": "Toothpicks of equal length are used to build a rectangular grid as shown. If the grid is 20 toothpicks high and 10 toothpicks wide, then the number of toothpicks used is\n\n$\\textrm{(A)}\\ 30\\qquad\\textrm{(B)}\\ 200\\qquad\\textrm{(C)}\\ 410\\qquad\\textrm{(D)}\\ 420\\qquad\\textrm{(E)}\\ 430$"... |
amc_aime | An [isosceles triangle](https://artofproblemsolving.com/wiki/index.php/Isosceles_triangle) is a [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) with two [sides](https://artofproblemsolving.com/wiki/index.php/Edge) of equal [length](https://artofproblemsolving.com/wiki/index.php/Length). How many of... | To determine how many of the triangles are isosceles, we need to calculate the lengths of the sides of each triangle and check if at least two sides are equal in length.
#### Triangle 1:
Vertices: $(0,6), (2,6), (1,4)$
- Side 1: Distance between $(0,6)$ and $(2,6)$ = $2 - 0 = 2$
- Side 2: Distance between $(0,6)$ and ... | [
{
"content": "An [isosceles triangle](https://artofproblemsolving.com/wiki/index.php/Isosceles_triangle) is a [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) with two [sides](https://artofproblemsolving.com/wiki/index.php/Edge) of equal [length](https://artofproblemsolving.com/wiki/index.php... |
amc_aime | Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether?
[asy] picture corner; draw(corner,(5,0)--(35,0)); draw(corner,(0,-5)--(0,-35)); for (int i=0; i<3; ++i){for (int j=0; j>-2; --j){if ((i-j)<3){add(corner,(50i,50j));}}} draw((5,-100)--(45,-1... | To determine the total number of toothpicks used in constructing a grid that is $60$ toothpicks long and $32$ toothpicks wide, we need to consider both the vertical and horizontal toothpicks.
1. **Vertical Toothpicks:**
- Each vertical line extends from the top to the bottom of the grid.
- Since the grid is $60$... | [
{
"content": "Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether?\n[asy] picture corner; draw(corner,(5,0)--(35,0)); draw(corner,(0,-5)--(0,-35)); for (int i=0; i<3; ++i){for (int j=0; j>-2; --j){if ((i-j)<3){add(corner,(50i,50j));}}... |
amc_aime | Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge... | To solve this problem, we will calculate the probability that Frieda reaches a corner square within four hops, starting from the center of a $3 \times 3$ grid. We will use a state-based approach to model Frieda's possible positions and transitions.
#### Definitions:
- **State**: A position on the grid.
- **Transition*... | [
{
"content": "Frieda the frog begins a sequence of hops on a $3 \\times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she \"wraps around\" and j... |
amc_aime | What is the area of the shaded pinwheel shown in the $5 \times 5$ grid?
[asy] filldraw((2.5,2.5)--(0,1)--(1,1)--(1,0)--(2.5,2.5)--(4,0)--(4,1)--(5,1)--(2.5,2.5)--(5,4)--(4,4)--(4,5)--(2.5,2.5)--(1,5)--(1,4)--(0,4)--cycle, gray, black); int i; for(i=0; i<6; i=i+1) { draw((i,0)--(i,5)); draw((0,i)--(5,i)); } [/asy]
$\tex... | To find the area of the shaded pinwheel in the $5 \times 5$ grid, we can use Pick's Theorem. However, the theorem requires all vertices of the polygons to be lattice points (points with integer coordinates), and the center of the pinwheel is not a lattice point. To address this, we scale the figure by a factor of 2, ma... | [
{
"content": "What is the area of the shaded pinwheel shown in the $5 \\times 5$ grid?\n[asy] filldraw((2.5,2.5)--(0,1)--(1,1)--(1,0)--(2.5,2.5)--(4,0)--(4,1)--(5,1)--(2.5,2.5)--(5,4)--(4,4)--(4,5)--(2.5,2.5)--(1,5)--(1,4)--(0,4)--cycle, gray, black); int i; for(i=0; i<6; i=i+1) { draw((i,0)--(i,5)); draw((0,i)... |
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