source stringclasses 1
value | problem stringlengths 81 1.23k | solution stringlengths 562 3.28k | messages listlengths 2 2 |
|---|---|---|---|
amc_aime | Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.
[AIME 2002 II Problem 4.gif](https://artofproblemsolving.com/wiki/index.php/File:AIME_2002_II_Problem_4.gif)
If ... | 1. **Identify the number of blocks in the path**: For $n > 1$, the path of blocks has $6(n-1)$ blocks in total. This is because there are six sides to the hexagon, and each side (except the first in each direction) contributes an additional block to the length of the previous side.
2. **Calculate the total number of b... | [
{
"content": "Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.\n[AIME 2002 II Problem 4.gif](https://artofproblemsolving.com/wiki/index.php/File:AIME_200... |
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 this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\sqrt{3}$ and $8\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\tfrac{1}{2}$ % of itself, while the lengths o... | To solve this problem, we first need to determine the side lengths of the equilateral triangles and the square based on the given areas.
1. **Calculate the side length of the equilateral triangles:**
- For an equilateral triangle, the area $A$ is given by $A = \frac{\sqrt{3}}{4} s^2$, where $s$ is the side length.
... | [
{
"content": "In this diagram, not drawn to scale, Figures $I$ and $III$ are equilateral triangular regions with respective areas of $32\\sqrt{3}$ and $8\\sqrt{3}$ square inches. Figure $II$ is a square region with area $32$ square inches. Let the length of segment $AD$ be decreased by $12\\tfrac{1}{2}$ % of it... |
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 | The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o' clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what poi... | 1. **Understanding the Setup**: We have a large circular clock face with a radius of $20$ cm and a smaller circular disk with a radius of $10$ cm. The smaller disk rolls around the larger clock face without slipping.
2. **Initial Position**: The smaller disk is initially tangent to the clock face at the $12$ o'clock p... | [
{
"content": "The diagram below shows the circular face of a clock with radius $20$ cm and a circular disk with radius $10$ cm externally tangent to the clock face at $12$ o' clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the c... |
amc_aime | As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X. What does the paper look like when unfolded?
[asy] draw((2,0)--(2,1)--(4,1)--(4,0)--cycle); draw(circle((2.25,.75),.225)); draw((2.05,.95)--(2.45,.55)); draw((2.45,.95)--(2.... |
To solve this problem, we need to carefully track the position of the hole punched in the paper through each stage of unfolding. We start by understanding the folding and punching process:
1. **First Fold (bottom to top)**: The paper is folded in half by bringing the bottom edge up to meet the top edge. This means th... | [
{
"content": "As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X. What does the paper look like when unfolded?\n[asy] draw((2,0)--(2,1)--(4,1)--(4,0)--cycle); draw(circle((2.25,.75),.225)); draw((2.05,.95)--(2.45,.55)... |
amc_aime | [asy] import cse5; pathpen=black; pointpen=black; dotfactor=3; pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP("$A$",A)); D(MP("$B$",B)); D(MP("$C$",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP("$B'$",BB[0],NW);MP("$C'$",CC[1],NE); //Credi... | 1. **Setting up the problem**: We are given three circles with centers at $A$, $B$, and $C$, each having radius $r$ where $1 < r < 2$. The distance between each pair of centers is $2$. We need to find the length of the line segment $B'C'$, where $B'$ and $C'$ are points of intersection of the circles centered at $A$ wi... | [
{
"content": "[asy] import cse5; pathpen=black; pointpen=black; dotfactor=3; pair A=(1,2),B=(2,0),C=(0,0); D(CR(A,1.5)); D(CR(B,1.5)); D(CR(C,1.5)); D(MP(\"$A$\",A)); D(MP(\"$B$\",B)); D(MP(\"$C$\",C)); pair[] BB,CC; CC=IPs(CR(A,1.5),CR(B,1.5)); BB=IPs(CR(A,1.5),CR(C,1.5)); D(BB[0]--CC[1]); MP(\"$B'$\",BB[0],NW... |
amc_aime | $\begin{tabular}{ccccccccccccc}& & & & & & C & & & & & &\\ & & & & & C & O & C & & & & &\\ & & & & C & O & N & O & C & & & &\\ & & & C & O & N & T & N & O & C & & &\\ & & C & O & N & T & E & T & N & O & C & &\\ & C & O & N & T & E & S & E & T & N & O & C &\\ C & O & N & T & E & S & T & S & E & T & N & O & C \end{... | To solve this problem, we need to count the number of paths that spell out the word "CONTEST" by moving horizontally or vertically between adjacent letters in the given diagram. We will consider three cases based on the starting position of the path: starting from the left side, starting from the right side, and going ... | [
{
"content": "$\\begin{tabular}{ccccccccccccc}& & & & & & C & & & & & &\\\\ & & & & & C & O & C & & & & &\\\\ & & & & C & O & N & O & C & & & &\\\\ & & & C & O & N & T & N & O & C & & &\\\\ & & C & O & N & T & E & T & N & O & C & &\\\\ & C & O & N & T & E & S & E & T & N & O & C &\\\\ C & O & N & T & E & ... |
amc_aime | A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\frac{1}{2}$ foot from the top face. The second cut is $\frac{1}{3}$ foot below the first cut, and the third cut is $\frac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labe... | 1. **Understanding the Problem**: We have a cube with a volume of 1 cubic foot, and it is cut into four pieces by three cuts parallel to the top face. The cuts are made at $\frac{1}{2}$ foot, $\frac{1}{3}$ foot, and $\frac{1}{17}$ foot below each preceding cut. The pieces are then rearranged end to end.
2. **Calculati... | [
{
"content": "A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\\frac{1}{2}$ foot from the top face. The second cut is $\\frac{1}{3}$ foot below the first cut, and the third cut is $\\frac{1}{17}$ foot below the second cut. From the top to the b... |
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 | If the pattern in the diagram continues, what fraction of eighth triangle would be shaded?
[asy] unitsize(10); draw((0,0)--(12,0)--(6,6sqrt(3))--cycle); draw((15,0)--(27,0)--(21,6sqrt(3))--cycle); fill((21,0)--(18,3sqrt(3))--(24,3sqrt(3))--cycle,black); draw((30,0)--(42,0)--(36,6sqrt(3))--cycle); fill((34,0)--(32,2sq... | 1. **Identify the pattern of shaded triangles**: Observing the given sequence of diagrams, we notice that the number of shaded triangles in each subsequent diagram follows the sequence of triangular numbers. The sequence of triangular numbers is defined by the formula $T_n = \frac{n(n+1)}{2}$, where $n$ is the term num... | [
{
"content": "If the pattern in the diagram continues, what fraction of eighth triangle would be shaded?\n[asy] unitsize(10); draw((0,0)--(12,0)--(6,6sqrt(3))--cycle); draw((15,0)--(27,0)--(21,6sqrt(3))--cycle); fill((21,0)--(18,3sqrt(3))--(24,3sqrt(3))--cycle,black); draw((30,0)--(42,0)--(36,6sqrt(3))--cycle... |
amc_aime | Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.)
[asy] size(85); ... | 1. **Identify the areas of the circles**:
- The largest circle has radius 3, so its area is $9\pi$.
- The middle circle has radius 2, so its area is $4\pi$.
- The smallest circle has radius 1, so its area is $\pi$.
2. **Set up the equations for shaded and unshaded regions**:
- Let $S$ be the area of the s... | [
{
"content": "Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\\pi$ radians is $180$ deg... |
amc_aime | The $16$ squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:
(1) fold the top half over the bottom half
(2) fold the bottom half over the top half
(3) fold the right half over the left half
(4) fold the left half ov... | To solve this problem, we need to track the position of the top square through each fold. We start by visualizing the initial configuration of the squares and then follow each fold step-by-step.
#### Initial Configuration:
The squares are arranged in a $4 \times 4$ grid, numbered from 1 to 16. The numbering is assumed... | [
{
"content": "The $16$ squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:\n(1) fold the top half over the bottom half\n(2) fold the bottom half over the top half\n(3) fold the right half over the left half\... |
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 | All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB=AC$. Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$. What is the area of trapezoid $DBCE$?
$\textbf{(A) } 16 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 20 \qquad ... | 1. **Identify the relationship between the triangles**: All of the triangles in the diagram are similar to isosceles triangle $ABC$, where $AB = AC$. Each of the 7 smallest triangles has an area of 1, and $\triangle ABC$ has an area of 40.
2. **Express the area of trapezoid $DBCE$**: We know that the area of trapezoid... | [
{
"content": "All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB=AC$. Each of the $7$ smallest triangles has area $1,$ and $\\triangle ABC$ has area $40$. What is the area of trapezoid $DBCE$?\n\n$\\textbf{(A) } 16 \\qquad \\textbf{(B) } 18 \\qquad ... |
amc_aime | In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the ... | 1. **Understanding the Pyramid Rule**: In the given sign pyramid, a cell at a higher level gets a "+" if the two cells directly below it have the same sign, and it gets a "-" if they have different signs. We need to determine how many ways we can fill the bottom row (four cells) to ensure the top of the pyramid has a "... | [
{
"content": "In a sign pyramid a cell gets a \"+\" if the two cells below it have the same sign, and it gets a \"-\" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produc... |
amc_aime | The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn?
$\text{(A)}\ 15 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 25 \qquad \text{(D)}\ 30 \qquad \text{(E)}\ 35$ | 1. **Identify the distance traveled by each biker after 4 hours**: According to the problem, Bjorn biked 45 miles and Alberto biked 60 miles in the same time period.
2. **Calculate the difference in miles traveled**: To find out how many more miles Alberto biked than Bjorn, subtract the distance biked by Bjorn from th... | [
{
"content": "The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn?\n\n$\\text{(A)}\\ 15 \\qquad \\text{(B)}\\ 20 \\qquad \\text{(C)}\\ 25 \\qquad \\text{(D)}\\ 30 \\qquad \\text{(E)}\\ 35$",
"role": "user"
},
{
"c... |
amc_aime | All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB=AC$. Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40$. What is the area of trapezoid $DBCE$?
$\textbf{(A) } 16 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 20 \qquad ... | 1. **Identify the relationship between the triangles**: All the triangles in the diagram are similar to isosceles triangle $ABC$, where $AB = AC$. Each of the 7 smallest triangles has an area of 1, and $\triangle ABC$ has an area of 40.
2. **Express the area of trapezoid $DBCE$**: We can write the area of trapezoid $D... | [
{
"content": "All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB=AC$. Each of the $7$ smallest triangles has area $1,$ and $\\triangle ABC$ has area $40$. What is the area of trapezoid $DBCE$?\n\n$\\textbf{(A) } 16 \\qquad \\textbf{(B) } 18 \\qquad ... |
amc_aime | Let $ABCD$ be an isoceles trapezoid having parallel bases $\overline{AB}$ and $\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\overline{CD}$ and moving clockwise as sh... | 1. **Identify the trapezoid and its properties**: Let $ABCD$ be an isosceles trapezoid with parallel bases $\overline{AB}$ and $\overline{CD}$, where $AB > CD$. Denote the lengths of these bases as $AB = r$ and $CD = s$.
2. **Point inside the trapezoid and triangle areas**: Let $P$ be a point inside the trapezoid such... | [
{
"content": "Let $ABCD$ be an isoceles trapezoid having parallel bases $\\overline{AB}$ and $\\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\\overline{CD}$ a... |
amc_aime | In this diagram $AB$ and $AC$ are the equal sides of an isosceles $\triangle ABC$, in which is inscribed equilateral $\triangle DEF$.
Designate $\angle BFD$ by $a$, $\angle ADE$ by $b$, and $\angle FEC$ by $c$. Then:
$\textbf{(A)}\ b=\frac{a+c}{2}\qquad \textbf{(B)}\ b=\frac{a-c}{2}\qquad \textbf{(C)}\ a=\frac{b-c}{2... | 1. **Identify the properties of the triangles**: Given that $\triangle DEF$ is equilateral, all its angles are $60^\circ$. Since $\triangle ABC$ is isosceles with $AB = AC$, the base angles $\angle ABC$ and $\angle ACB$ are equal.
2. **Use the property of a straight line**: The sum of angles on a straight line is $180... | [
{
"content": "In this diagram $AB$ and $AC$ are the equal sides of an isosceles $\\triangle ABC$, in which is inscribed equilateral $\\triangle DEF$. \nDesignate $\\angle BFD$ by $a$, $\\angle ADE$ by $b$, and $\\angle FEC$ by $c$. Then:\n\n$\\textbf{(A)}\\ b=\\frac{a+c}{2}\\qquad \\textbf{(B)}\\ b=\\frac{a-c}{... |
amc_aime | In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ is $k$, then the area ... | 1. **Identify Key Points and Relationships**: In parallelogram $ABCD$, $DP$ bisects $BC$ at $N$ and extends to meet $AB$ at $P$. Similarly, $CQ$ bisects $AD$ at $M$ and extends to meet $AB$ at $Q$. Lines $DP$ and $CQ$ intersect at point $O$.
2. **Area Decomposition**: We can decompose the area of triangle $QPO$ into s... | [
{
"content": "In parallelogram $ABCD$ of the accompanying diagram, line $DP$ is drawn bisecting $BC$ at $N$ and meeting $AB$ (extended) at $P$. From vertex $C$, line $CQ$ is drawn bisecting side $AD$ at $M$ and meeting $AB$ (extended) at $Q$. Lines $DP$ and $CQ$ meet at $O$. If the area of parallelogram $ABCD$ ... |
amc_aime | A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $\text{X}$ is
$\text{(A)}\ \text{Z} \qquad \text{(B)}\ \text{U} \qquad \text{(C)}\ \text{V} \qquad \text{(D)}\ \ \text{W} \qquad \tex... | To determine the face opposite to the face labeled $\text{X}$ on a cube formed by folding a flat arrangement of six squares, we need to identify which faces are adjacent to $\text{X}$ and which are not. The face opposite to $\text{X}$ will be the one that does not share any edge with $\text{X}$.
1. **Identify adjacent... | [
{
"content": "A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $\\text{X}$ is \n\n$\\text{(A)}\\ \\text{Z} \\qquad \\text{(B)}\\ \\text{U} \\qquad \\text{(C)}\\ \\text{V} \\qqu... |
amc_aime | Tamara has three rows of two $6$-feet by $2$-feet flower beds in her garden. The beds are separated and also surrounded by $1$-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?
$\textbf{(A)}\ 72\qquad\textbf{(B)}\ 78\qquad\textbf{(C)}\ 90\qquad\textbf{(D)}\ 120\qquad\... | 1. **Calculate the dimensions of the garden including walkways:**
- Each flower bed measures $6$ feet by $2$ feet.
- There are $1$-foot-wide walkways around and between the beds.
For the width:
- There are two beds in each row, so the total width of the beds is $2 \times 6 = 12$ feet.
- There are $3$ wa... | [
{
"content": "Tamara has three rows of two $6$-feet by $2$-feet flower beds in her garden. The beds are separated and also surrounded by $1$-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?\n\n$\\textbf{(A)}\\ 72\\qquad\\textbf{(B)}\\ 78\\qquad\\textbf{(C)}\\ ... |
amc_aime | $ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\overline{AB}$. $\overline{PS} \perp \overline{BD}$ and $\overline{PR} \perp \overline{AC}$. $\overline{AF} \perp \overline{BD}$ and $\overline{PQ} \perp \overline{AF}$. Then $PR + PS$ is equal to:
$\textbf{(A)}\ PQ\qquad \textbf{(B)}\ AE\qqu... | 1. **Identify Parallel Lines**: Since $\overline{PQ}$ and $\overline{BD}$ are both perpendicular to $\overline{AF}$, it follows that $\overline{PQ} \parallel \overline{BD}$.
2. **Angle Relationships**: Because $\overline{PQ} \parallel \overline{BD}$ and $\overline{AB}$ is a transversal, $\angle APQ = \angle ABD$. Sinc... | [
{
"content": "$ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\\overline{AB}$. $\\overline{PS} \\perp \\overline{BD}$ and $\\overline{PR} \\perp \\overline{AC}$. $\\overline{AF} \\perp \\overline{BD}$ and $\\overline{PQ} \\perp \\overline{AF}$. Then $PR + PS$ is equal to:\n\n$\\textb... |
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 | In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is , $X$ in centimeters?
$\textbf{(A)}\hspace{.05in}1\qquad\textbf{(B)}\hspace{.05in}2\qquad\textbf{(C)}\hspace{.05in}3\qquad\textbf{(D)}\hspace{.05in}4\qquad\textbf{(E)}\hspa... | To solve for $X$, we need to understand the relationship between the lengths of the sides in the diagram. The problem states that all angles are right angles, which implies that the figure is composed of rectangles or squares.
From the diagram, we can see that the total horizontal length on the top side of the figure ... | [
{
"content": "In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is , $X$ in centimeters?\n\n$\\textbf{(A)}\\hspace{.05in}1\\qquad\\textbf{(B)}\\hspace{.05in}2\\qquad\\textbf{(C)}\\hspace{.05in}3\\qquad\\textbf{(D)}\\hs... |
amc_aime | Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?
[asy] // diagram by SirCalcsALot, edited by MRENTHUSIASM size(250); path p = scale(0.8)*unitcircle; pair[] A; pen grey1 = rgb(100/25... | To solve this problem, we need to understand the pattern of how the dots are arranged in each hexagon and how the number of dots increases as the hexagons grow in size.
1. **Observing the pattern:**
- The first hexagon has 1 dot.
- The second hexagon has 7 dots: 1 central dot and 6 surrounding dots.
- The thi... | [
{
"content": "Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?\n[asy] // diagram by SirCalcsALot, edited by MRENTHUSIASM size(250); path p = scale(0.8)*unitcircle; pair[] A;... |
amc_aime | In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD$. $O$,$G$,$H$,$J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF.$ Then, as $CD$ and $EF$ a... | 1. **Identify the given information and setup the problem:**
- The center of the circle is $O$, and the radius is $a$ inches.
- Chords $CD$ and $EF$ are parallel, and $O$, $G$, $H$, $J$ are collinear with $G$ as the midpoint of $CD$.
- $OG = a - 2h$, where $h = JH = HG$.
- We need to find the ratio $K:R$ as... | [
{
"content": "In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD$. $O$,$G$,$H$,$J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF.$ Th... |
amc_aime | In the diagram, if points $A, B$ and $C$ are points of tangency, then $x$ equals: | 1. **Extend the lines**: Extend the lines containing points $A$ and $B$ so that they meet the line containing point $C$. Let's denote the vertices of the triangle formed by these extended lines as $D$, $E$, and $F$.
2. **Identify the triangle type**: Given that the angle shown in the diagram is $60^\circ$, and assumin... | [
{
"content": "In the diagram, if points $A, B$ and $C$ are points of tangency, then $x$ equals:",
"role": "user"
},
{
"content": "1. **Extend the lines**: Extend the lines containing points $A$ and $B$ so that they meet the line containing point $C$. Let's denote the vertices of the triangle formed ... |
amc_aime | The [square](https://artofproblemsolving.com/wiki/index.php/Square) in the first diagram "rolls" clockwise around the fixed regular [hexagon](https://artofproblemsolving.com/wiki/index.php/Hexagon) until it reaches the bottom. In which position will the solid [triangle](https://artofproblemsolving.com/wiki/index.php/T... |
1. **Understanding the Problem:**
The problem involves a square rolling around a fixed regular hexagon. We need to determine the position of a solid triangle marked on the square after it has rolled around the hexagon to a specific position.
2. **Analyzing the Geometry:**
- The inner angle of a regular hexagon ... | [
{
"content": "The [square](https://artofproblemsolving.com/wiki/index.php/Square) in the first diagram \"rolls\" clockwise around the fixed regular [hexagon](https://artofproblemsolving.com/wiki/index.php/Hexagon) until it reaches the bottom. In which position will the solid [triangle](https://artofproblemsolv... |
amc_aime | In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. **Identify Coordinates of Rectangle $ABCD$**:
Given $B=(0,0)$, we can determine the coordinates of the other vertices based on the side lengths:
- $A = (0, 3)$ (since $AB = 3$ and is vertical),
- $C = (11, 0)$ (since $BC = 11$ and is horizontal),
- $D = (11, 3)$ (since $CD = 3$ and is vertical).
2. **Id... | [
{
"content": "In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\\frac mn$, where $m$ and $n$ are relatively prime positive intege... |
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 | Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square?
$\textbf{(A)}\ 29\sqrt{3} \qquad\textbf{(B)}\ \frac{21}{2}\sqrt{2}+\frac{41}{2}\sqrt{3} \qquad... | 1. **Identify the relationships and congruences in the figure:**
- Let $EZ = x$ and $\angle XAB = \alpha$.
- Since $\angle BAX = \angle EYZ$ and $\angle ZEY = \angle XBA$, and $AX = YZ$, by the Angle-Angle-Side (AAS) congruence criterion, $\triangle BAX \cong \triangle EYZ$.
- Therefore, $BX = EZ = x$.
2. **U... | [
{
"content": "Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\\overline{BC}$, $Y$ on $\\overline{DE}$, and $Z$ on $\\overline{EF}$. Suppose that $AB=40$, and $EF=41(\\sqrt{3}-1)$. What is the side-length of the square?\n$\\textbf{(A)}\\ 29\\sqrt{3} \\qquad\\textbf{(B)}\\ \\frac{21}{2}\\... |
amc_aime | A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square ... | To solve this problem, we need to calculate the probability that a randomly placed coin will cover part of the black region on the square. We start by determining the total possible region where the center of the coin can land and then calculate the area of the regions where the coin would overlap with the black region... | [
{
"content": "A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is d... |
amc_aime | The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?
$\textbf{(A... | 1. **Identify the larger triangle and its dimensions**: The problem describes a larger triangle with vertices at points A, B, and D. The base of this triangle (AD) is along Main Street and measures 6 miles. The height of this triangle (from point B to line AD) is along Brown Road and measures 3 miles.
2. **Calculate t... | [
{
"content": "The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of ... |
amc_aime | [asy] real h = 7; real t = asin(6/h)/2; real x = 6-h*tan(t); real y = x*tan(2*t); draw((0,0)--(0,h)--(6,h)--(x,0)--cycle); draw((x,0)--(0,y)--(6,h)); draw((6,h)--(6,0)--(x,0),dotted); label("L",(3.75,h/2),W); label("$\theta$",(6,h-1.5),W);draw(arc((6,h),2,270,270-degrees(t)),Arrow(2mm)); label("6''",(3,0),S); draw((2.5... | 1. **Identify the Geometry and Variables**:
- Let the rectangle be $ABCD$ with $A$ at the top left, $B$ at the top right, $C$ at the bottom right, and $D$ at the bottom left.
- The crease $BE$ is formed such that $E$ lies on side $CD$.
- Define $F$ on $AD$ such that $F$ is the reflection of $C$ over line $BE$... | [
{
"content": "[asy] real h = 7; real t = asin(6/h)/2; real x = 6-h*tan(t); real y = x*tan(2*t); draw((0,0)--(0,h)--(6,h)--(x,0)--cycle); draw((x,0)--(0,y)--(6,h)); draw((6,h)--(6,0)--(x,0),dotted); label(\"L\",(3.75,h/2),W); label(\"$\\theta$\",(6,h-1.5),W);draw(arc((6,h),2,270,270-degrees(t)),Arrow(2mm)); labe... |
amc_aime | The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, what is the ratio $\dfrac{XQ}{QY}$?
$\textbf{(A)}\ \frac{2}{5}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{3}{5}... | 1. **Calculate the total area of the octagon**: The octagon is composed of $10$ unit squares. Therefore, the total area of the octagon is $10$ square units.
2. **Area bisected by $\overline{PQ}$**: Since $\overline{PQ}$ bisects the area of the octagon, the area above and below $\overline{PQ}$ must each be $5$ square u... | [
{
"content": "The diagram shows an octagon consisting of $10$ unit squares. The portion below $\\overline{PQ}$ is a unit square and a triangle with base $5$. If $\\overline{PQ}$ bisects the area of the octagon, what is the ratio $\\dfrac{XQ}{QY}$?\n\n$\\textbf{(A)}\\ \\frac{2}{5}\\qquad\\textbf{(B)}\\ \\frac{1}... |
amc_aime | The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of
$\text{(A)}\ 10.05 \qquad \text{(B)}\ 10.15 \qquad \text{(C)}\ 10.25 \qquad \text{(D)}\ 10.3 \qquad \text{(E)}\ 10.6$ | 1. **Identify the Range**: The arrow on the scale is positioned between the marks labeled $10.25$ and $10.5$. This observation immediately eliminates options $\text{(A)}\ 10.05$, $\text{(B)}\ 10.15$, and $\text{(E)}\ 10.6$ as they are outside this range.
2. **Determine Closer Mark**: We need to determine whether the a... | [
{
"content": "The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of\n\n$\\text{(A)}\\ 10.05 \\qquad \\text{(B)}\\ 10.15 \\qquad \\text{(C)}\\ 10.25 \\qquad \\text{(D)}\\ 10.3 \\qquad \\text{(E)}\\ 10.6$",
"role": "user"
},
{
"content": "1. **Identify... |
amc_aime | The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?
$\... | 1. **Original Data Collection**:
- Monday: $20$
- Tuesday: $26$
- Wednesday: $16$
- Thursday: $22$
- Friday: $16$
2. **Original Total and Mean Calculation**:
- Total number of participants = $20 + 26 + 16 + 22 + 16 = 100$
- Mean (average) number of participants = $\frac{100}{5} = 20$
3. **Origina... | [
{
"content": "The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the co... |
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