source stringclasses 1
value | problem stringlengths 314 1.18k | solution stringlengths 715 2.97k | messages listlengths 2 2 |
|---|---|---|---|
amc_aime | The graph of the line $y=mx+b$ is shown. Which of the following is true?
[2004 AMC 12A Problem 5.png](https://artofproblemsolving.com/wiki/index.php/File:2004_AMC_12A_Problem_5.png)
$\mathrm {(A)} mb<-1 \qquad \mathrm {(B)} -1<mb<0 \qquad \mathrm {(C)} mb=0 \qquad \mathrm {(D)}$ $0<mb<1 \qquad \mathrm {(E)} mb>1$ | 1. **Identify the slope and y-intercept from the graph**:
From the graph, we observe that the slope $m$ of the line is negative and appears to be $-\frac{1}{2}$. The y-intercept $b$ is positive and appears to be $\frac{4}{5}$.
2. **Calculate the product $mb$**:
\[
m = -\frac{1}{2}, \quad b = \frac{4}{5}
\... | [
{
"content": "The graph of the line $y=mx+b$ is shown. Which of the following is true?\n[2004 AMC 12A Problem 5.png](https://artofproblemsolving.com/wiki/index.php/File:2004_AMC_12A_Problem_5.png)\n$\\mathrm {(A)} mb<-1 \\qquad \\mathrm {(B)} -1<mb<0 \\qquad \\mathrm {(C)} mb=0 \\qquad \\mathrm {(D)}$ $0<mb<1 \... |
amc_aime | In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\overline{BC}$, and $\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\overline{AB}$ and $\overline{AC}$, respectively, so that $AE=3$ and $AF=10$. Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCF... | 1. **Calculate $BC$ using the Pythagorean Theorem**:
Since $\triangle ABC$ is a right triangle with $AB = 12$ and $AC = 35$, we apply the Pythagorean Theorem:
\[
BC = \sqrt{AB^2 + AC^2} = \sqrt{12^2 + 35^2} = \sqrt{144 + 1225} = \sqrt{1369} = 37.
\]
2. **Apply the Angle Bisector Theorem to find $BD$ and $D... | [
{
"content": "In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\\overline{BC}$, and $\\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\\overline{AB}$ and $\\overline{AC}$, respectively, so that $AE=3$ and $AF=10$. Given that $EB=9$ and $FC=27$, find the integer closest to the ... |
amc_aime | In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through the eight exterior segments of the four squares.
[2003amc8prob23a.png](https://artofproblemsolving.com/wiki/index.php/File:2003amc8prob23a.png)
If the pattern is continued, where would the cat and m... | To solve this problem, we need to determine the positions of both the cat and the mouse after the 247th move. We will analyze their movements separately and then combine the results to find the correct answer.
#### Cat's Movement:
1. **Cycle Length:** The cat moves through four squares, completing a cycle every 4 move... | [
{
"content": "In the pattern below, the cat moves clockwise through the four squares, and the mouse moves counterclockwise through the eight exterior segments of the four squares.\n\n\n[2003amc8prob23a.png](https://artofproblemsolving.com/wiki/index.php/File:2003amc8prob23a.png)\n\n\nIf the pattern is continue... |
amc_aime | Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.
[1988 AIME-12.png](https://artofproblemsolving.com/wiki/in... | 1. **Understanding the Problem**: We are given a triangle $ABC$ with an interior point $P$. Lines are extended from the vertices through $P$ to the opposite sides, creating segments of lengths $a$, $b$, $c$, and $d$. We need to find the product $abc$ given that $a + b + c = 43$ and $d = 3$.
2. **Using Ceva's Theorem**... | [
{
"content": "Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.\n[1988 AIME-12.png](https://artofpro... |
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 | [Pdfresizer.com-pdf-convert.png](https://artofproblemsolving.com/wiki/index.php/File:Pdfresizer.com-pdf-convert.png)
In the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$,
and sides $BC$ and $ED$. Each side has length $1$. Also, $\angle FAB = \angle BCD = 60^\circ$.
The area of t... | 1. **Identify the Shape and Symmetry**: The given figure can be visualized as composed of several triangles. Notably, the problem states that sides $AF$ and $CD$ are parallel, as are $AB$ and $EF$, and $BC$ and $ED$. Each side has a length of $1$, and the angles $\angle FAB$ and $\angle BCD$ are each $60^\circ$.
2. **... | [
{
"content": "[Pdfresizer.com-pdf-convert.png](https://artofproblemsolving.com/wiki/index.php/File:Pdfresizer.com-pdf-convert.png)\nIn the adjoining plane figure, sides $AF$ and $CD$ are parallel, as are sides $AB$ and $EF$, \nand sides $BC$ and $ED$. Each side has length $1$. Also, $\\angle FAB = \\angle BCD =... |
amc_aime | Two thousand points are given on a [circle](https://artofproblemsolving.com/wiki/index.php/Circle). Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. (See f... | 1. **Understanding the Problem**: We are given 2000 points on a circle and a specific labeling process. We need to find the smallest integer that labels the same point as $1993$.
2. **Labeling Process**: The labeling process involves moving around the circle in a specific pattern:
- Start at a point labeled $1$.
... | [
{
"content": "Two thousand points are given on a [circle](https://artofproblemsolving.com/wiki/index.php/Circle). Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label ... |
amc_aime | Squares $S_1$ and $S_2$ are [inscribed](https://artofproblemsolving.com/wiki/index.php/Inscribe) in [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$, as shown in the figures below. Find $AC + CB$ if area $(S_1) = 441$ and area $(S_2) = 440$.
[AIME 1987 Problem 15.png](https://artofp... | 1. **Understanding the Problem**: We are given two squares inscribed in a right triangle $ABC$ with areas $441$ and $440$. We need to find the sum of the lengths of the legs $AC$ and $CB$ of the triangle.
2. **Using Area Ratios**: Given that the areas of the squares $S_1$ and $S_2$ are $441$ and $440$ respectively, an... | [
{
"content": "Squares $S_1$ and $S_2$ are [inscribed](https://artofproblemsolving.com/wiki/index.php/Inscribe) in [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$, as shown in the figures below. Find $AC + CB$ if area $(S_1) = 441$ and area $(S_2) = 440$.\n[AIME 1987 Problem... |
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 | [Pdfresizer.com-pdf-convert-q17.png](https://artofproblemsolving.com/wiki/index.php/File:Pdfresizer.com-pdf-convert-q17.png)
The diagram above shows several numbers in the complex plane. The circle is the unit circle centered at the origin.
One of these numbers is the reciprocal of $F$. Which one?
$\textbf{(A)} \ A \... | 1. **Representing $F$ in Complex Form**:
Let $F = a + bi$, where $a$ and $b$ are real numbers. From the diagram, it is evident that both $a$ and $b$ are positive, and since $F$ is outside the unit circle, we have $a^2 + b^2 > 1$.
2. **Finding the Reciprocal of $F$**:
The reciprocal of a complex number $F = a... | [
{
"content": "[Pdfresizer.com-pdf-convert-q17.png](https://artofproblemsolving.com/wiki/index.php/File:Pdfresizer.com-pdf-convert-q17.png)\nThe diagram above shows several numbers in the complex plane. The circle is the unit circle centered at the origin. \nOne of these numbers is the reciprocal of $F$. Which o... |
amc_aime | In the figure, the length of side $AB$ of square $ABCD$ is $\sqrt{50}$ and $BE=1$. What is the area of the inner square $EFGH$?
[AMC102005Aq.png](https://artofproblemsolving.com/wiki/index.php/File:AMC102005Aq.png)
$\textbf{(A)}\ 25\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 40\qquad\textbf{(E)}\ 42... | 1. **Identify the Geometry and Given Information**: We are given a square $ABCD$ with side length $\sqrt{50}$, and a smaller square $EFGH$ inside it. The segment $BE$ is given as $1$.
2. **Analyze the Right Triangle Formed**: Since $ABCD$ is a square, each angle is $90^\circ$. The segment $BE$ forms a right triangle $... | [
{
"content": "In the figure, the length of side $AB$ of square $ABCD$ is $\\sqrt{50}$ and $BE=1$. What is the area of the inner square $EFGH$?\n[AMC102005Aq.png](https://artofproblemsolving.com/wiki/index.php/File:AMC102005Aq.png)\n$\\textbf{(A)}\\ 25\\qquad\\textbf{(B)}\\ 32\\qquad\\textbf{(C)}\\ 36\\qquad\\te... |
amc_aime | Three 12 cm $\times$12 cm [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the [midpoints](https://artofproblemsolving.com/wiki/index.php/Midpoint) of two adjacent sides. These six pieces are then attach... | 1. **Understanding the Construction**: Each 12 cm x 12 cm square is cut into two pieces by joining the midpoints of two adjacent sides. This results in two right-angled isosceles triangles per square. Since there are three squares, we obtain six such triangles.
2. **Formation of the Polyhedron**: These six triangles a... | [
{
"content": "Three 12 cm $\\times$12 cm [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the [midpoints](https://artofproblemsolving.com/wiki/index.php/Midpoint) of two adjacent sides. These six... |
amc_aime | [Circles](https://artofproblemsolving.com/wiki/index.php/Circle) with [ centers](https://artofproblemsolving.com/wiki/index.php/Center_(geometry)) $(2,4)$ and $(14,9)$ have [ radii](https://artofproblemsolving.com/wiki/index.php/Radius) $4$ and $9$, respectively. The equation of a common external [tangent](https://arto... | 1. **Identify the centers and radii of the circles**:
- Circle 1 has center $(2,4)$ and radius $4$.
- Circle 2 has center $(14,9)$ and radius $9$.
2. **Calculate the slope of the line connecting the centers**:
- The slope of the line $L_1$ connecting $(2,4)$ and $(14,9)$ is calculated as follows:
\[
... | [
{
"content": "[Circles](https://artofproblemsolving.com/wiki/index.php/Circle) with [ centers](https://artofproblemsolving.com/wiki/index.php/Center_(geometry)) $(2,4)$ and $(14,9)$ have [ radii](https://artofproblemsolving.com/wiki/index.php/Radius) $4$ and $9$, respectively. The equation of a common external ... |
amc_aime | The diagram shows twenty congruent [circles](https://artofproblemsolving.com/wiki/index.php/Circle) arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The [ratio](https://artofproblemsolving.com/wiki/index.php/Ratio) of t... | 1. **Identify the dimensions of the rectangle**:
- Let the radius of each circle be $r$.
- The longer dimension of the rectangle is determined by the arrangement of the circles. There are 7 circles along the length, each touching the next, so the total length is $2r \times 7 = 14r$.
- The shorter dimension is... | [
{
"content": "The diagram shows twenty congruent [circles](https://artofproblemsolving.com/wiki/index.php/Circle) arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The [ratio](https://artofproblemsolving.com/wiki... |
amc_aime | One commercially available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below has $\{1,2,3,6,9\}$ as its [combination](https://artofproblemsolving.com/wiki/index.php/Combination). Suppose that these locks are redesigned so that sets of as many as nine buttons o... | 1. **Current Combinations**: The lock can currently be opened by pressing exactly five buttons out of ten. The number of ways to choose 5 buttons from 10 is calculated using the binomial coefficient:
\[
{10 \choose 5}
\]
2. **Calculation of Current Combinations**:
\[
{10 \choose 5} = \frac{10!}{5!(10-5)... | [
{
"content": "One commercially available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below has $\\{1,2,3,6,9\\}$ as its [combination](https://artofproblemsolving.com/wiki/index.php/Combination). Suppose that these locks are redesigned so that sets of a... |
amc_aime | A $9 \times 9 \times 9$ [cube](https://artofproblemsolving.com/wiki/index.php/Cube) is composed of twenty-seven $3 \times 3 \times 3$ cubes. The big cube is ‘tunneled’ as follows: First, the six $3 \times 3 \times 3$ cubes which make up the center of each [face](https://artofproblemsolving.com/wiki/index.php/Face) as w... |
#### Step-by-step Analysis:
1. **Understanding the Structure**:
- The original structure is a $9 \times 9 \times 9$ cube, which is composed of twenty-seven $3 \times 3 \times 3$ cubes.
- The center cube of each face and the very center cube of the large cube are removed, leaving 20 smaller cubes.
2. **Initial ... | [
{
"content": "A $9 \\times 9 \\times 9$ [cube](https://artofproblemsolving.com/wiki/index.php/Cube) is composed of twenty-seven $3 \\times 3 \\times 3$ cubes. The big cube is ‘tunneled’ as follows: First, the six $3 \\times 3 \\times 3$ cubes which make up the center of each [face](https://artofproblemsolving.c... |
amc_aime | Square $S_{1}$ is $1\times 1.$ For $i\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular b... | 1. **Identify the sequence of areas for each square**:
Given that each square $S_{i+1}$ has side lengths half of $S_i$, the area of each square $S_i$ is $\left(\frac{1}{2^{i-1}}\right)^2 = \frac{1}{4^{i-1}}$. Thus, the areas of the squares $S_1, S_2, S_3, S_4, S_5$ are:
\[
1, \frac{1}{4}, \frac{1}{16}, \frac{... | [
{
"content": "Square $S_{1}$ is $1\\times 1.$ For $i\\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$... |
amc_aime | The faces of a cube are painted in six different colors: red $(R)$, white $(W)$, green $(G)$, brown $(B)$, aqua $(A)$, and purple $(P)$. Three views of the cube are shown below. What is the color of the face opposite the aqua face?
[2019AMC8Prob12.png](https://artofproblemsolving.com/wiki/index.php/File:2019AMC8Prob12.... | 1. **Identify visible faces from each view:**
- From the first image, we see that the top face is brown $(B)$, the front face is red $(R)$, and the right face is green $(G)$.
- From the second image, the top face is still brown $(B)$, the front face is white $(W)$, and the right face is green $(G)$.
- From the... | [
{
"content": "The faces of a cube are painted in six different colors: red $(R)$, white $(W)$, green $(G)$, brown $(B)$, aqua $(A)$, and purple $(P)$. Three views of the cube are shown below. What is the color of the face opposite the aqua face?\n[2019AMC8Prob12.png](https://artofproblemsolving.com/wiki/index.p... |
amc_aime | The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\leq x \leq 4$?
[2002AMC12A25.png](https://artofproblemsolving.com/wiki/index.php/File:2002AMC12A25.png) | 1. **Understanding the Problem**: We are given a polynomial $P(x)$ with real coefficients. A new polynomial $Q(x)$ is formed by replacing each nonzero coefficient of $P(x)$ with their mean. We need to identify which graph could represent both $P(x)$ and $Q(x)$.
2. **Key Property**: The sum of the coefficients of a pol... | [
{
"content": "The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4\\leq x \\leq 4$?\n[2002AMC12A25.png](https://artofproblemsolving.com/wiki/index.ph... |
amc_aime | [Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is divided into four parts of equal [area](https://artofproblemsolving.com/wiki/index.php/Area) by five [ segments](https://artofproblemsolving.com/wiki/index.php/Line_segment) as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + ... |
1. **Understanding the Problem:**
- We are given a rectangle $ABCD$ with $BC = 19$ cm.
- The rectangle is divided into four equal areas by segments $XY$, $PQ$, and $ZW$.
- $PQ$ is parallel to $AB$ and $PQ = 87$ cm.
- We need to find the length of $AB$.
2. **Analyzing the Given Information:**
- Since $X... | [
{
"content": "[Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is divided into four parts of equal [area](https://artofproblemsolving.com/wiki/index.php/Area) by five [ segments](https://artofproblemsolving.com/wiki/index.php/Line_segment) as shown in the figure, where $XY = YB + BC ... |
amc_aime | A ship travels from point $A$ to point $B$ along a semicircular path, centered at Island $X$. Then it travels along a straight path from $B$ to $C$. Which of these graphs best shows the ship's distance from Island $X$ as it moves along its course?
[2003amc8prob24ans.png](https://artofproblemsolving.com/wiki/index.ph... | 1. **Understanding the Path**: The ship travels from point $A$ to point $B$ along a semicircular path centered at Island $X$. This means that the distance from any point on the semicircle to Island $X$ is the radius of the semicircle, say $r$. This distance remains constant as the ship moves from $A$ to $B$.
2. **Grap... | [
{
"content": "A ship travels from point $A$ to point $B$ along a semicircular path, centered at Island $X$. Then it travels along a straight path from $B$ to $C$. Which of these graphs best shows the ship's distance from Island $X$ as it moves along its course?\n\n\n\n[2003amc8prob24ans.png](https://artofproble... |
amc_aime | The letter F shown below is rotated $90^\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?
[Foriginal.png](https://artofproblemsolving.com/wiki/index.php/File:Foriginal.png)
[Fproblem.png](https://artofproblemsolving.com/wiki/inde... | To solve this problem, we will follow the transformations step by step and determine the final position of the letter F after each transformation.
#### Step 1: Rotation by $90^\circ$ Clockwise Around the Origin
- **Original Position**: The base of the F is along the negative x-axis, and the stem is along the negative ... | [
{
"content": "The letter F shown below is rotated $90^\\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?\n[Foriginal.png](https://artofproblemsolving.com/wiki/index.php/File:Foriginal.png)\n[Fproblem.png](https://artofpro... |
amc_aime | A beam of light strikes $\overline{BC}\,$ at point $C\,$ with angle of incidence $\alpha=19.94^\circ\,$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\overline{AB}\,$ and $\overline{BC}\,$ according to the rule: angle of incidence equals angle... | 1. **Understanding the Reflections**:
The problem states that a beam of light reflects off two line segments $\overline{AB}$ and $\overline{BC}$, where $AB = BC$. The angle of incidence $\alpha = 19.94^\circ$ and the angle $\beta = \alpha/10 = 1.994^\circ$. The light beam reflects according to the law of reflection,... | [
{
"content": "A beam of light strikes $\\overline{BC}\\,$ at point $C\\,$ with angle of incidence $\\alpha=19.94^\\circ\\,$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\\overline{AB}\\,$ and $\\overline{BC}\\,$ according to the rule:... |
amc_aime | (Reid Barton) An animal with $n$ cells is a connected figure consisting of $n$ equal-sized [square](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) cells.${}^1$ The figure below shows an 8-cell animal.
[2007 USAMO-4.PNG](https://artofproblemsolving.com/wiki/index.php/File:2007_USAMO-4.PNG)
A dinosaur... |
We will prove that the maximum number of cells in a primitive dinosaur is $8025$ cells. We will use the concept of connectivity and partitioning in polyominoes (animals) to establish the maximum size of a primitive dinosaur.
#### Step 1: Definitions and Initial Observations
- **Dinosaur**: An animal with at least 200... | [
{
"content": "(Reid Barton) An animal with $n$ cells is a connected figure consisting of $n$ equal-sized [square](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) cells.${}^1$ The figure below shows an 8-cell animal.\n\n[2007 USAMO-4.PNG](https://artofproblemsolving.com/wiki/index.php/File:2007... |
amc_aime | Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$.
[AIME 1989 Problem 15.png](https://artofproblemsolving.com/wiki/index.php... |
1. **Use of Ceva's Theorem and Area Ratios**:
- We start by noting that $AP = PD = 6$, which implies that $D$ is the midpoint of $BC$. This is because the line segment $APD$ bisects $\triangle ABC$ into two smaller triangles of equal area, hence $[APB] = [APC]$.
- Since $PE = 3$ and $BP = 9$, the ratio of the ar... | [
{
"content": "Point $P$ is inside $\\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\\triangle ABC$.\n\n[AIME 1989 Problem 15.png](https://artofproblem... |
amc_aime | A small [ square](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) is constructed inside a square of [area](https://artofproblemsolving.com/wiki/index.php/Area) 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the [ vertices](https://artofproblemsolving.com/wiki/inde... | 1. **Understanding the Problem Setup:**
- We have a unit square with area 1.
- We divide each side of the square into $n$ equal parts.
- We connect vertices to the division points closest to the opposite vertices to form a smaller square inside the unit square.
- The area of the smaller square is given as $... | [
{
"content": "A small [ square](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) is constructed inside a square of [area](https://artofproblemsolving.com/wiki/index.php/Area) 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the [ vertices](https://artofproble... |
amc_aime | A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find th... | To determine which graph best represents the race between the tortoise and the hare, we need to analyze the characteristics of their movements as described in the problem:
1. **Tortoise's Movement:**
- The tortoise moves at a constant speed throughout the race. This means its graph should be a straight line with a ... | [
{
"content": "A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish... |
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