source stringclasses 1
value | problem stringlengths 173 4.91k | solution stringlengths 113 6.3k | messages listlengths 2 2 |
|---|---|---|---|
aops_forum | The diagram below shows two parallel rows with seven points in the upper row and nine points in the lower row. The points in each row are spaced one unit apart, and the two rows are two units apart. How many trapezoids which are not parallelograms have vertices in this set of $16$ points and have area of at least six... | 1. **Calculate the total number of ways to choose two points from the top row and two points from the bottom row:**
The number of ways to choose 2 points from 7 points in the top row is given by:
\[
\binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \cdot 6}{2 \cdot 1} = 21
\]
The number of ways to choose 2 p... | [
{
"content": "The diagram below shows two parallel rows with seven points in the upper row and nine points in the lower row. The points in each row are spaced one unit apart, and the two rows are two units apart. How many trapezoids which are not parallelograms have vertices in this set of $16$ points and hav... |
aops_forum | **p1.** What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square?**p2.** Two angles of an isosceles triangle measure $80^o$ and $x^o$ . What is the sum of all the possible values of $x$ ?**p3.** Let $p$ and $... | 1. **p1.** What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square?
To find the maximum number of points of intersection between a square and a triangle, we need to consider the following:
- A square has 4 sides.
... | [
{
"content": "**p1.** What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square?**p2.** Two angles of an isosceles triangle measure $80^o$ and $x^o$ . What is the sum of all the possible values of $x$ ?**... |
aops_forum | The diagram below shows some small squares each with area $3$ enclosed inside a larger square. Squares that touch each other do so with the corner of one square coinciding with the midpoint of a side of the other square. Find integer $n$ such that the area of the shaded region inside the larger square but outside t... | 1. **Determine the side length of the smaller squares:**
Each smaller square has an area of \(3\). Therefore, the side length \(s\) of each smaller square is:
\[
s = \sqrt{3}
\]
2. **Calculate the diagonal of the smaller squares:**
The diagonal \(d\) of a square with side length \(s\) is given by \(d = ... | [
{
"content": "The diagram below shows some small squares each with area $3$ enclosed inside a larger square. Squares that touch each other do so with the corner of one square coinciding with the midpoint of a side of the other square. Find integer $n$ such that the area of the shaded region inside the large... |
aops_forum | Let $m, n, k$ be positive integers such that $k\le mn$ . Let $S$ be the set consisting of the $(m + 1)$ -by- $(n + 1)$ rectangular array of points on the Cartesian plane with coordinates $(i, j)$ where $i, j$ are integers satisfying $0\le i\le m$ and $0\le j\le n$ . The diagram below shows the
example whe... | 1. **Define the Problem and Setup:**
We are given an \((m+1)\)-by-\((n+1)\) rectangular array of points on the Cartesian plane with coordinates \((i, j)\) where \(0 \leq i \leq m\) and \(0 \leq j \leq n\). We need to show that for any positive integer \(k \leq mn\), there exists a triangle with vertices at these poi... | [
{
"content": "Let $m, n, k$ be positive integers such that $k\\le mn$ . Let $S$ be the set consisting of the $(m + 1)$ -by- $(n + 1)$ rectangular array of points on the Cartesian plane with coordinates $(i, j)$ where $i, j$ are integers satisfying $0\\le i\\le m$ and $0\\le j\\le n$ . The diagram ... |
aops_forum | The diagram below shows four regular hexagons each with side length $1$ meter attached to the sides of a square. This figure is drawn onto a thin sheet of metal and cut out. The hexagons are then bent upward along the sides of the square so that $A_1$ meets $A_2$ , $B_1$ meets $B_2$ , $C_1$ meets $C_2$ , and... | 1. **Identify the shape and dimensions**: The problem describes a dish formed by bending four regular hexagons attached to the sides of a square. Each hexagon has a side length of 1 meter. When bent, the hexagons form a pyramidal frustum with the square as the top base and a larger square as the bottom base.
2. **Calc... | [
{
"content": "The diagram below shows four regular hexagons each with side length $1$ meter attached to the sides of a square. This figure is drawn onto a thin sheet of metal and cut out. The hexagons are then bent upward along the sides of the square so that $A_1$ meets $A_2$ , $B_1$ meets $B_2$ , $C_... |
aops_forum | Let $ABC$ be an equilateral triangle with the side of $20$ units. Amir divides this triangle into $400$ smaller equilateral triangles with the sides of $1$ unit. Reza then picks $4$ of the vertices of these smaller triangles. The vertices lie inside the triangle $ABC$ and form a parallelogram with sides par... | 1. **Define the problem and variables:**
Let \( n \) and \( m \) be the lengths of the sides of the parallelogram in terms of the number of unit triangles. We need to find the possible values for the area of this parallelogram given that there are exactly 46 smaller triangles that have at least one point in common w... | [
{
"content": "Let $ABC$ be an equilateral triangle with the side of $20$ units. Amir divides this triangle into $400$ smaller equilateral triangles with the sides of $1$ unit. Reza then picks $4$ of the vertices of these smaller triangles. The vertices lie inside the triangle $ABC$ and form a parall... |
aops_forum | Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$ . Let $I$ be the center of the inscribed circle of $\triangle{ABC}$ . If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$ , then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$ , $b$ , and $c$ are posit... | 1. **Given Information and Setup:**
- We are given a triangle \( \triangle ABC \) with \( AB = 10 \) and \( AC = 11 \).
- \( I \) is the incenter of \( \triangle ABC \).
- \( M \) is the midpoint of \( AI \).
- We are given that \( BM = BC \) and \( CM = 7 \).
2. **Using the Given Conditions:**
- We nee... | [
{
"content": "Let $\\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$ . Let $I$ be the center of the inscribed circle of $\\triangle{ABC}$ . If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$ , then $BC$ can be expressed in the form $\\frac{\\sqrt{a}-b}{c}$ where $a$ , ... |
aops_forum | Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$ , and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to righ... | To solve this problem, we need to determine the probability that exactly 3 of the 5 trapezoidal pieces contain both dark and white chocolate. We will use combinatorial methods and probability calculations to achieve this.
1. **Understanding the Problem:**
- We have a rectangular chocolate bar divided into two halve... | [
{
"content": "Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$ , and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P... |
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 1</u>**p1.** If $2m = 200 cm$ and $m \ne 0$ , find $c$ .**p2.** A right triangle has two sides of lengths $3$ and $4$ . Find the smallest possible length of the third side.**p3.** Given that $20(x + 17) = 17(x + 20)$ , determine the value of $x$ .
<u>Round 2</u>**p4.** According to the Egyptian Metr... | 1. **Identify the number of $1 \times 1$ squares:**
The diagram is a $7 \times 7$ grid of points, which means there are $6 \times 6 = 36$ $1 \times 1$ squares. However, the diagram also includes additional points around the eyes and other areas, which contribute to more $1 \times 1$ squares. By carefully counting a... | [
{
"content": "<u>Round 1</u>**p1.** If $2m = 200 cm$ and $m \\ne 0$ , find $c$ .**p2.** A right triangle has two sides of lengths $3$ and $4$ . Find the smallest possible length of the third side.**p3.** Given that $20(x + 17) = 17(x + 20)$ , determine the value of $x$ .\n\n\n<u>Round 2</u>**p4.** Acco... |
aops_forum | To *dissect* a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:
... | 1. **Identify the problem and the goal:**
We need to determine all integers \( n \) for which it is possible to completely dissect a regular \( n \)-sided polygon into integer-ratio right triangles.
2. **Example for \( n = 4 \):**
- Consider a square (a regular 4-sided polygon).
- We can use two right triangl... | [
{
"content": "To *dissect* a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:\n![Image](https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc9041... |
aops_forum | <u>Round 1</u>**p1.** The temperature inside is $28^o$ F. After the temperature is increased by $5^o$ C, what will the new temperature in Fahrenheit be?**p2.** Find the least positive integer value of $n$ such that $\sqrt{2021+n}$ is a perfect square.**p3.** A heart consists of a square with two semicircles att... | To solve the problem, we need to determine the number of ways to fill a $12 \times 2$ rectangle using $L$-shaped trominos. Let's denote $a_n$ as the number of ways to tile a $2 \times n$ rectangle using $L$-shaped trominos.
1. **Define the recurrence relation:**
We start by considering the possible placements of th... | [
{
"content": "<u>Round 1</u>**p1.** The temperature inside is $28^o$ F. After the temperature is increased by $5^o$ C, what will the new temperature in Fahrenheit be?**p2.** Find the least positive integer value of $n$ such that $\\sqrt{2021+n}$ is a perfect square.**p3.** A heart consists of a square w... |
aops_forum | The diagram below shows nine points on a circle where $AB=BC=CD=DE=EF=FG=GH$ . Given that $\angle GHJ=117^\circ$ and $\overline{BH}$ is perpendicular to $\overline{EJ}$ , there are relatively prime positive integers $m$ and $n$ so that the degree measure of $\angle AJB$ is $\textstyle\frac mn$ . Find $m... | 1. **Determine the angle measure of each segment:**
Since \(AB = BC = CD = DE = EF = FG = GH\), the circle is divided into 7 equal arcs. Let the measure of each arc be \(x\) degrees.
2. **Calculate the total angle around the circle:**
The total angle around the circle is \(360^\circ\). Therefore, we have:
\[
... | [
{
"content": "The diagram below shows nine points on a circle where $AB=BC=CD=DE=EF=FG=GH$ . Given that $\\angle GHJ=117^\\circ$ and $\\overline{BH}$ is perpendicular to $\\overline{EJ}$ , there are relatively prime positive integers $m$ and $n$ so that the degree measure of $\\angle AJB$ is $\\te... |
aops_forum | The diagram below shows a $12$ by $20$ rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region.
 | 1. **Understanding the Problem:**
The problem involves a $12$ by $20$ rectangle split into four strips of equal widths, surrounding an isosceles triangle. We need to find the area of the shaded region.
2. **Identifying the Shapes:**
The rectangle is divided into:
- Two right triangles
- One isosceles trian... | [
{
"content": "The diagram below shows a $12$ by $20$ rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region.\n",
"role": "user"
},... |
aops_forum | The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
 | 1. **Calculate the perimeter of the original $8 \times 12$ rectangle:**
The perimeter \( P \) of a rectangle is given by the formula:
\[
P = 2 \times (\text{length} + \text{width})
\]
Here, the length is 12 units and the width is 8 units. Therefore,
\[
P = 2 \times (12 + 8) = 2 \times 20 = 40 \text... | [
{
"content": "The diagram shows a polygon made by removing six $2\\times 2$ squares from the sides of an $8\\times 12$ rectangle. Find the perimeter of this polygon.\n",
"role": "user"
},
{
"c... |
aops_forum | The diagram below is made up of a rectangle AGHB, an equilateral triangle AFG, a rectangle ADEF, and a parallelogram ABCD. Find the degree measure of ∠ABC. For diagram go to http://www.purplecomet.org/welcome/practice, the 2015 middle school contest, and go to #2 | 1. **Identify the angles in the given shapes:**
- The rectangle \(ABHG\) has all angles equal to \(90^\circ\).
- The rectangle \(AFED\) also has all angles equal to \(90^\circ\).
- The equilateral triangle \(AFG\) has all angles equal to \(60^\circ\).
2. **Calculate the sum of angles around point \(A\):**
... | [
{
"content": "The diagram below is made up of a rectangle AGHB, an equilateral triangle AFG, a rectangle ADEF, and a parallelogram ABCD. Find the degree measure of ∠ABC. For diagram go to http://www.purplecomet.org/welcome/practice, the 2015 middle school contest, and go to #2",
"role": "user"
},
{
... |
aops_forum | Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below.
\[\begin{tabular}{| c | c | c | c | c | c | c | c |}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8
\hline
9 & 10 & 11 & 12 & 13 & 14 & 15 & 16
\hline
17 & 18 & 19 & 20 & 21 & 22 & 23 & 24
\hline
25 & 26 & 27 & 28 & 2... | 1. **Understanding the Knight's Movement**:
- A knight in chess moves in an "L" shape: two squares in one direction and then one square perpendicular, or one square in one direction and then two squares perpendicular.
- This means a knight placed on a square can attack up to 8 different squares, depending on its ... | [
{
"content": "Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below.\n\n\\[\\begin{tabular}{| c | c | c | c | c | c | c | c |}\n\\hline\n1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \n\\hline\n9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \n\\hline\n17 & 18 & 19 & 20 & 21 & 22 & 23... |
aops_forum | The diagram show $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$ . Find the length of segment $AE$ .
[asy]
path seg1, seg2;
seg1=(6,0)--(0,3);
seg2=(2,0)--(4,2);
dot((0,0));
dot((1,0));
fill(circle((2,0),0.1),black);
dot((3,0));
dot((4,0));
dot((5,0));
fill... | 1. **Identify Coordinates:**
- The coordinates of the points are:
- \( A = (0, 3) \)
- \( B = (6, 0) \)
- \( C = (4, 2) \)
- \( D = (2, 0) \)
2. **Find Equations of Lines:**
- The equation of line \( AB \):
\[
\text{slope of } AB = \frac{0 - 3}{6 - 0} = -\frac{1}{2}
\]
Usi... | [
{
"content": "The diagram show $28$ lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$ . Find the length of segment $AE$ .\n\n[asy]\npath seg1, seg2;\nseg1=(6,0)--(0,3);\nseg2=(2,0)--(4,2);\ndot((0,0));\ndot((1,0));\nfill(circle((2,0),0.1),black);\ndot((3,0)... |
aops_forum | [asy]
draw((0,0)--(0,1)--(2,1)--(2,0)--cycle^^(.5,1)--(.5,2)--(1.5,2)--(1.5,1)--(.5,2)^^(.5,1)--(1.5,2)^^(1,2)--(1,0));
//Credit to Zimbalono for the diagram[/asy]
The radius of the smallest circle containing the symmetric figure composed of the $3$ unit squares shown above is $\textbf{(A) }\sqrt{2}\qquad\textbf{(B)... | 1. **Identify the vertices of the figure:**
- The figure is composed of three unit squares. Let's place the bottom-left corner of the bottom-left square at the origin \((0,0)\).
- The coordinates of the vertices of the squares are:
- Bottom-left square: \((0,0)\), \((0,1)\), \((1,0)\), \((1,1)\)
- Top-l... | [
{
"content": "[asy]\ndraw((0,0)--(0,1)--(2,1)--(2,0)--cycle^^(.5,1)--(.5,2)--(1.5,2)--(1.5,1)--(.5,2)^^(.5,1)--(1.5,2)^^(1,2)--(1,0));\n//Credit to Zimbalono for the diagram[/asy]\n\nThe radius of the smallest circle containing the symmetric figure composed of the $3$ unit squares shown above is $\\textbf{(A)... |
aops_forum | **p1.** Nicky is studying biology and has a tank of $17$ lizards. In one day, he can either remove $5$ lizards or add $2$ lizards to his tank. What is the minimum number of days necessary for Nicky to get rid of all of the lizards from his tank?**p2.** What is the maximum number of spheres with radius $1$ that ... | To solve the problem, we need to calculate the probability that a randomly drawn hand of four cards from a deck of 16 cards forms a tractor. A tractor is defined as a hand of the form $(c, c, c+1, c+1)$.
1. **Total number of ways to choose 4 cards from 16 cards:**
The total number of ways to choose 4 cards from 16 ... | [
{
"content": "**p1.** Nicky is studying biology and has a tank of $17$ lizards. In one day, he can either remove $5$ lizards or add $2$ lizards to his tank. What is the minimum number of days necessary for Nicky to get rid of all of the lizards from his tank?**p2.** What is the maximum number of spheres w... |
aops_forum | In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right?
[asy]
size(340);
int i, j;
for(i = 0; i<10; i = i+1) {
for(j = 0; j<5; j = j+1) {
if(10*j + i == 11 || 10*j + i == 12 || 10*j + i =... | To solve this problem, we need to count the number of distinct paths from the top-left corner (January 1) to the bottom-right corner (December 31) on a grid, where movement is allowed to the right, down, or diagonally down to the right. We will use dynamic programming to solve this problem.
1. **Initialize the Grid:**... | [
{
"content": "In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right?\n\n[asy]\nsize(340);\nint i, j;\nfor(i = 0; i<10; i = i+1) {\nfor(j = 0; j<5; j = j+1) {\nif(10*j + i == 11 ||... |
aops_forum | The following diagram shows equilateral triangle $\vartriangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\vartriangle ABC$ a distance $\frac18 AB$ along a side of $\vartriangle ABC$ in the directions from $A$ to $B$ , from $B$ to $C$ , an... | 1. **Understanding the Problem:**
We are given an equilateral triangle $\triangle ABC$ and three other triangles congruent to it. These triangles are obtained by sliding copies of $\triangle ABC$ a distance $\frac{1}{8} AB$ along a side of $\triangle ABC$ in the directions from $A$ to $B$, from $B$ to $C$, and from ... | [
{
"content": "The following diagram shows equilateral triangle $\\vartriangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\\vartriangle ABC$ a distance $\\frac18 AB$ along a side of $\\vartriangle ABC$ in the directions from $A$ to $B$... |
aops_forum | In the diagram, $\angle AOB = \angle BOC$ and $\angle COD = \angle DOE = \angle EOF$ . Given that $\angle AOD = 82^o$ and $\angle BOE = 68^o$ . Find $\angle AOF$ .
 | 1. Let $\angle AOB = x$ and $\angle COD = y$. Given that $\angle AOB = \angle BOC$ and $\angle COD = \angle DOE = \angle EOF$, we can set up the following equations based on the given angles:
\[
2x + y = 82^\circ \quad \text{(since } \angle AOD = \angle AOB + \angle BOC + \angle COD = 2x + y)
\]
\[
x + 2... | [
{
"content": "In the diagram, $\\angle AOB = \\angle BOC$ and $\\angle COD = \\angle DOE = \\angle EOF$ . Given that $\\angle AOD = 82^o$ and $\\angle BOE = 68^o$ . Find $\\angle AOF$ .\n",
"role":... |
aops_forum | [asy]
size(200);
pair A=(-2,0),B,C=(-1,0),D=(1,0),EE,O=(0,0);
draw(arc(O,1, 0, 180));
EE=midpoint(arc(O,1, 0, 90));
draw(A--EE);
draw(A--D);
B=intersectionpoint(arc(O,1, 180, 0),EE--A);
draw(O--EE);
label(" $A$ ",A,W);
label(" $B$ ",B,NW);
label(" $C$ ",C,S);label(" $D$ ",D,E);label(" $E$ ",EE,NE);label(" $O$ ",O,S);la... | 1. **Draw the necessary lines and identify the given angles:**
- Draw radius \( OB \).
- We have two isosceles triangles, \( \triangle ABO \) and \( \triangle OBE \).
- Given \( \angle EOD = 45^\circ \), then \( \angle EOA = 180^\circ - 45^\circ = 135^\circ \).
2. **Define the unknown angle:**
- Let \( \an... | [
{
"content": "[asy]\nsize(200);\npair A=(-2,0),B,C=(-1,0),D=(1,0),EE,O=(0,0);\ndraw(arc(O,1, 0, 180));\nEE=midpoint(arc(O,1, 0, 90));\ndraw(A--EE);\ndraw(A--D);\nB=intersectionpoint(arc(O,1, 180, 0),EE--A);\ndraw(O--EE);\nlabel(\" $A$ \",A,W);\nlabel(\" $B$ \",B,NW);\nlabel(\" $C$ \",C,S);label(\" $D$ \",D,E);l... |
aops_forum | In the following diagram, $O$ is the center of the circle. If three angles $\alpha, \beta$ and $\gamma$ be equal, find $\alpha.$ [asy]
unitsize(40);
import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffttww = rgb(1,0.2... | 1. **Identify the given information and the goal:**
- We are given a circle with center \( O \).
- Three angles \(\alpha\), \(\beta\), and \(\gamma\) are equal.
- We need to find the value of \(\alpha\).
2. **Understand the geometric configuration:**
- Since \(O\) is the center of the circle, \(OA\), \(OB\... | [
{
"content": "In the following diagram, $O$ is the center of the circle. If three angles $\\alpha, \\beta$ and $\\gamma$ be equal, find $\\alpha.$ [asy]\nunitsize(40);\nimport graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.... |
aops_forum | In the diagram, congruent rectangles $ABCD$ and $DEFG$ have a common vertex $D$ . Sides $BC$ and $EF$ meet at $H$ . Given that $DA = DE = 8$ , $AB = EF = 12$ , and $BH = 7$ . Find the area of $ABHED$ .
 | 1. **Identify the given information and the goal:**
- Congruent rectangles \(ABCD\) and \(DEFG\) share a common vertex \(D\).
- \(DA = DE = 8\), \(AB = EF = 12\), and \(BH = 7\).
- We need to find the area of the pentagon \(ABHED\).
2. **Determine the coordinates of the vertices:**
- Place \(D\) at the ori... | [
{
"content": "In the diagram, congruent rectangles $ABCD$ and $DEFG$ have a common vertex $D$ . Sides $BC$ and $EF$ meet at $H$ . Given that $DA = DE = 8$ , $AB = EF = 12$ , and $BH = 7$ . Find the area of $ABHED$ .\n![Image](https://cdn.artofproblemsolving.com/attachments/f/b/7225fa89097e7b20ea24... |
aops_forum | Given is a circle $\omega$ and a line $\ell$ tangent to $\omega$ at $Y$ . Point $X$ lies on $\ell$ to the left of $Y$ . The tangent to $\omega$ , perpendicular to $\ell$ meets $\ell$ at $A$ and touches $\omega$ at $D$ . Let $B$ a point on $\ell$ , to the right of $Y$ , such that $AX=BY$ . The... | 1. **Identify the given elements and their relationships:**
- Circle $\omega$ with center $O$.
- Line $\ell$ tangent to $\omega$ at point $Y$.
- Point $X$ on $\ell$ to the left of $Y$.
- Tangent to $\omega$ perpendicular to $\ell$ meets $\ell$ at $A$ and touches $\omega$ at $D$.
- Point $B$ on $\ell$ to ... | [
{
"content": "Given is a circle $\\omega$ and a line $\\ell$ tangent to $\\omega$ at $Y$ . Point $X$ lies on $\\ell$ to the left of $Y$ . The tangent to $\\omega$ , perpendicular to $\\ell$ meets $\\ell$ at $A$ and touches $\\omega$ at $D$ . Let $B$ a point on $\\ell$ , to the right of ... |
aops_forum | In the diagram, $OB_i$ is parallel and equal in length to $A_i A_{i + 1}$ for $i = 1$ , 2, 3, and 4 ( $A_5 = A_1$ ). Show that the area of $B_1 B_2 B_3 B_4$ is twice that of $A_1 A_2 A_3 A_4$ .
[asy]
unitsize(1 cm);
pair O;
pair[] A, B;
O = (0,0);
A[1] = (0.5,-3);
A[2] = (2,0);
A[3] = (-0.2,0.5);
A[4] = (-1... | 1. **Identify the given conditions and setup:**
- We are given a quadrilateral \(A_1A_2A_3A_4\) and another quadrilateral \(B_1B_2B_3B_4\) such that \(OB_i\) is parallel and equal in length to \(A_iA_{i+1}\) for \(i = 1, 2, 3, 4\) (with \(A_5 = A_1\)).
- We need to show that the area of \(B_1B_2B_3B_4\) is twice ... | [
{
"content": "In the diagram, $OB_i$ is parallel and equal in length to $A_i A_{i + 1}$ for $i = 1$ , 2, 3, and 4 ( $A_5 = A_1$ ). Show that the area of $B_1 B_2 B_3 B_4$ is twice that of $A_1 A_2 A_3 A_4$ .\n\n[asy]\nunitsize(1 cm);\n\npair O;\npair[] A, B;\n\nO = (0,0);\nA[1] = (0.5,-3);\nA[2] = (2,0... |
aops_forum | The diagram below shows $\vartriangle ABC$ with area $64$ , where $D, E$ , and $F$ are the midpoints of $BC, CA$ , and $AB$ , respectively. Point $G$ is the intersection of $DF$ and $BE$ . Find the area of quadrilateral $AFGE$ .
 and \(F\) are the midpoints of \(BC, CA,\) and \(AB\) respectively, and \(G\) is the intersection of \(DF\) and \(BE\), we know that \(G\) is the centroid of \(\triangle ABC\). The centroid divides each median into a ratio of \(2:1\).
2. **Area of \(\tri... | [
{
"content": "The diagram below shows $\\vartriangle ABC$ with area $64$ , where $D, E$ , and $F$ are the midpoints of $BC, CA$ , and $AB$ , respectively. Point $G$ is the intersection of $DF$ and $BE$ . Find the area of quadrilateral $AFGE$ .\n;
defaultpen(linewidth(0.7));
int i, j;
for(i = 0; i < 4; ++i)
for(j = 0; j < 4; ++j)
dot(((real)i, (real)j));[/asy]Define a *growing path* to be a sequence of d... | 1. **Define the problem and initial setup:**
We are given a \(4 \times 4\) rectangular array of points, each 1 unit away from its nearest neighbors. We need to find the maximum number of points in a growing path (where the distance between consecutive points is strictly increasing) and the number of such paths. Let ... | [
{
"content": "The diagram below shows a $ 4\\times4$ rectangular array of points, each of which is $ 1$ unit away from its nearest neighbors.\r\n[asy]unitsize(0.25inch);\ndefaultpen(linewidth(0.7));\n\nint i, j;\nfor(i = 0; i < 4; ++i)\n\tfor(j = 0; j < 4; ++j)\n\t\tdot(((real)i, (real)j));[/asy]Define a *g... |
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