source stringclasses 1
value | problem stringlengths 164 1.61k | solution stringlengths 966 3.37k | messages listlengths 2 2 |
|---|---|---|---|
amc_aime | In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, then chord $EF$ has le... | 1. **Identify the Configuration**: Given that $AB$, $BC$, and $CD$ are diameters of circles $O$, $N$, and $P$ respectively, and all circles have a radius of $15$. Since $B$ and $C$ lie on $AD$, and each segment is a diameter, $AB = BC = CD = 30$. Therefore, $AD = AB + BC + CD = 30 + 30 + 30 = 90$.
2. **Position of $N$... | [
{
"content": "In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, ... |
amc_aime | A figure is an equiangular parallelogram if and only if it is a
$\textbf{(A)}\ \text{rectangle}\qquad \textbf{(B)}\ \text{regular polygon}\qquad \textbf{(C)}\ \text{rhombus}\qquad \textbf{(D)}\ \text{square}\qquad \textbf{(E)}\ \text{trapezoid}$ | 1. **Definition and Analysis**: An equiangular parallelogram is defined as a parallelogram where all interior angles are equal. By the properties of a parallelogram, opposite angles are equal, and adjacent angles are supplementary. If all angles are equal in a parallelogram, each angle must be $90^\circ$.
2. **Examine... | [
{
"content": "A figure is an equiangular parallelogram if and only if it is a\n$\\textbf{(A)}\\ \\text{rectangle}\\qquad \\textbf{(B)}\\ \\text{regular polygon}\\qquad \\textbf{(C)}\\ \\text{rhombus}\\qquad \\textbf{(D)}\\ \\text{square}\\qquad \\textbf{(E)}\\ \\text{trapezoid}$",
"role": "user"
},
{
... |
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 | A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the side... | 1. **Understanding the Problem Setup**: We have a square sheet of wrapping paper and a box with a square base of side $w$ and height $h$. The box is placed such that its base vertices lie on the midlines of the wrapping paper. The wrapping paper is folded up to meet at a point $A$ at the center of the top of the box.
... | [
{
"content": "A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be f... |
amc_aime | The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.
[asy] dr... | We are given a ring divided into six sections, and we need to paint each section such that no two adjacent sections have the same color. We have four colors available.
1. **Label the sections** from $1$ to $6$ in a circular order.
2. **Coloring Section 1:** There are no restrictions for the first section, so it can b... | [
{
"content": "The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with th... |
amc_aime | The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner?
$\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 15$ | To solve this problem, we need to understand how a cube is formed and how the numbers on the faces are arranged. In a standard die, opposite faces sum up to 7. This means:
- If one face shows 1, the opposite face shows 6.
- If one face shows 2, the opposite face shows 5.
- If one face shows 3, the opposite face shows 4... | [
{
"content": "The figure may be folded along the lines shown to form a number cube. Three number faces come together at each corner of the cube. What is the largest sum of three numbers whose faces come together at a corner?\n\n$\\text{(A)}\\ 11 \\qquad \\text{(B)}\\ 12 \\qquad \\text{(C)}\\ 13 \\qquad \\text... |
amc_aime | Let $ABCD$ be an isosceles trapezoid with $\overline{BC}\parallel \overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\overline{AC}$ with $X$ between $A$ and $Y$, as shown in the figure. Suppose $\angle AXD = \angle BYC = 90^\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the area of $ABCD?$
$\textbf{(A)... | 1. **Setting up the coordinate system**: Place $X$ at the origin $(0,0)$, align $AC$ along the $x$-axis, and $DX$ along the $y$-axis. This gives us:
- $X = (0,0)$
- $A = (3,0)$ (since $AX = 3$)
- $Y = (-1,0)$ (since $XY = 1$)
- $C = (-3,0)$ (since $YC = 2$)
2. **Locating points $B$ and $D$**: Let $BY = u$ ... | [
{
"content": "Let $ABCD$ be an isosceles trapezoid with $\\overline{BC}\\parallel \\overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\\overline{AC}$ with $X$ between $A$ and $Y$, as shown in the figure. Suppose $\\angle AXD = \\angle BYC = 90^\\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the ... |
amc_aime | In the adjoining figure of a rectangular solid, $\angle DHG=45^\circ$ and $\angle FHB=60^\circ$. Find the cosine of $\angle BHD$.
$\text {(A)} \frac{\sqrt{3}}{6} \qquad \text {(B)} \frac{\sqrt{2}}{6} \qquad \text {(C)} \frac{\sqrt{6}}{3} \qquad \text{(D)}\frac{\sqrt{6}}{4}\qquad \text{(E)}\frac{\sqrt{6}-\sqrt{2}}{4... | 1. **Assume dimensions of the rectangular solid**: Without loss of generality, let the side length $CD = 1$.
2. **Analyze square $GHDC$**: Given $\angle DHG = 45^\circ$, we know that $\triangle DHG$ is a 45-45-90 triangle. Therefore, $DG = HG = 1$ and $DH = \sqrt{2}$.
3. **Analyze square $DHFB$ and triangle $CHB$**:... | [
{
"content": "In the adjoining figure of a rectangular solid, $\\angle DHG=45^\\circ$ and $\\angle FHB=60^\\circ$. Find the cosine of $\\angle BHD$.\n\n$\\text {(A)} \\frac{\\sqrt{3}}{6} \\qquad \\text {(B)} \\frac{\\sqrt{2}}{6} \\qquad \\text {(C)} \\frac{\\sqrt{6}}{3} \\qquad \\text{(D)}\\frac{\\sqrt{6}}{4... |
amc_aime | In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids?
[asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,6.5)); draw((9.4,9.7)--... | 1. **Identify the total area of the large triangle**: Given in the problem, the area of the outer equilateral triangle is $16$ square units.
2. **Identify the area of the inner triangle**: The area of the inner equilateral triangle is given as $1$ square unit.
3. **Calculate the area between the inner and outer trian... | [
{
"content": "In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids?\n[asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,... |
amc_aime | The [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) enclosed by the solid lines in the figure consists of 4 [congruent](https://artofproblemsolving.com/wiki/index.php/Congruent) [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) joined [edge](https://artofproblemsolving.com/w... |
To solve this problem, we need to determine how many of the nine positions for the additional square allow the resulting figure to be folded into a cube with one face missing. We start by understanding the structure of the given figure and the implications of adding a square at each position.
#### Step 1: Understand ... | [
{
"content": "The [polygon](https://artofproblemsolving.com/wiki/index.php/Polygon) enclosed by the solid lines in the figure consists of 4 [congruent](https://artofproblemsolving.com/wiki/index.php/Congruent) [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) joined [edge](https://art... |
amc_aime | Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$
inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square
until $P$ returns to its original position. The length o... | To solve this problem, we need to understand the path traced by vertex $P$ of the equilateral triangle $ABP$ as it rotates around the square $AXYZ$.
1. **Understanding the Rotation**:
- The triangle $ABP$ is equilateral with side length $2$ inches.
- The square $AXYZ$ has a side length of $4$ inches.
- Verte... | [
{
"content": "Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ \ninches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square \nuntil $P$ returns to its origina... |
amc_aime | In the figure, it is given that angle $C = 90^{\circ}$, $\overline{AD} = \overline{DB}$, $DE \perp AB$, $\overline{AB} = 20$, and $\overline{AC} = 12$. The area of quadrilateral $ADEC$ is:
$\textbf{(A)}\ 75\qquad\textbf{(B)}\ 58\frac{1}{2}\qquad\textbf{(C)}\ 48\qquad\textbf{(D)}\ 37\frac{1}{2}\qquad\textbf{(E)}\ \tex... | 1. **Identify the given information and the goal:**
- $\angle C = 90^\circ$ indicates $\triangle ABC$ is a right triangle.
- $\overline{AD} = \overline{DB}$ implies $D$ is the midpoint of $\overline{AB}$.
- $DE \perp AB$ means $\triangle BDE$ is a right triangle.
- $\overline{AB} = 20$ and $\overline{AC} = ... | [
{
"content": "In the figure, it is given that angle $C = 90^{\\circ}$, $\\overline{AD} = \\overline{DB}$, $DE \\perp AB$, $\\overline{AB} = 20$, and $\\overline{AC} = 12$. The area of quadrilateral $ADEC$ is: \n\n$\\textbf{(A)}\\ 75\\qquad\\textbf{(B)}\\ 58\\frac{1}{2}\\qquad\\textbf{(C)}\\ 48\\qquad\\textbf{(D... |
amc_aime | The following figures are composed of squares and circles. Which figure has a shaded region with largest area?
$\textbf{(A)}\ \text{A only}\qquad\textbf{(B)}\ \text{B only}\qquad\textbf{(C)}\ \text{C only}\qquad\textbf{(D)}\ \text{both A and B}\qquad\textbf{(E)}\ \text{all are equal}$ | To determine which figure has the largest shaded area, we need to calculate the shaded area for each figure.
1. **Figure A:**
- The square has a side length of 2, so its area is $2^2 = 4$.
- The circle inside the square has a radius of 1, so its area is $\pi \times 1^2 = \pi$.
- The shaded area in figure A is... | [
{
"content": "The following figures are composed of squares and circles. Which figure has a shaded region with largest area?\n\n$\\textbf{(A)}\\ \\text{A only}\\qquad\\textbf{(B)}\\ \\text{B only}\\qquad\\textbf{(C)}\\ \\text{C only}\\qquad\\textbf{(D)}\\ \\text{both A and B}\\qquad\\textbf{(E)}\\ \\text{al... |
amc_aime | In this figure $\angle RFS=\angle FDR$, $FD=4$ inches, $DR=6$ inches, $FR=5$ inches, $FS=7\dfrac{1}{2}$ inches. The length of $RS$, in inches, is:
$\textbf{(A) }\text{undetermined}\qquad\textbf{(B) }4\qquad\textbf{(C) }5\dfrac{1}{2}\qquad\textbf{(D) }6\qquad\textbf{(E) }6\dfrac{1}{2}\qquad$ | 1. **Identify Given Angles and Sides**: We are given that $\angle RFS = \angle FDR = \theta$, and the side lengths $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, and $FS = 7\frac{1}{2}$ inches.
2. **Apply the Law of Cosines in $\triangle FDR$**:
\[
FR^2 = FD^2 + DR^2 - 2 \cdot FD \cdot DR \cdot \cos(\theta)
... | [
{
"content": "In this figure $\\angle RFS=\\angle FDR$, $FD=4$ inches, $DR=6$ inches, $FR=5$ inches, $FS=7\\dfrac{1}{2}$ inches. The length of $RS$, in inches, is:\n$\\textbf{(A) }\\text{undetermined}\\qquad\\textbf{(B) }4\\qquad\\textbf{(C) }5\\dfrac{1}{2}\\qquad\\textbf{(D) }6\\qquad\\textbf{(E) }6\\dfrac{1}{... |
amc_aime | $ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Eucli... | To prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country, we follow these steps:
1. **Uniqueness of Point $O$:**
- Assume there are two points, $P$ and $Q$, on the small map such that both $P$ a... | [
{
"content": "$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the count... |
amc_aime | In this figure the center of the circle is $O$. $AB \perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then:
$\textbf{(A)} AP^2 = PB \times AB\qquad \\ \textbf{(B)}\ AP \times DO = PB \times AD\qquad \\ \textbf{(C)}\ AB^2 = AD \times DE\qquad \\ \textbf{(D)}\ AB \times AD = OB \ti... | 1. **Identify given information and relationships:**
- The center of the circle is $O$.
- $AB \perp BC$.
- $ADOE$ is a straight line.
- $AP = AD$.
- $AB = 2r$, where $r$ is the radius of the circle.
2. **Calculate $AO$:**
Since $AB \perp BC$ and $AB = 2r$, triangle $ABO$ is a right triangle with $AB$... | [
{
"content": "In this figure the center of the circle is $O$. $AB \\perp BC$, $ADOE$ is a straight line, $AP = AD$, and $AB$ has a length twice the radius. Then:\n\n$\\textbf{(A)} AP^2 = PB \\times AB\\qquad \\\\ \\textbf{(B)}\\ AP \\times DO = PB \\times AD\\qquad \\\\ \\textbf{(C)}\\ AB^2 = AD \\times DE\\qqu... |
amc_aime | The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
size(300);
defaultpen(linewidth(0.8));
real r = 0.35;
path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r);
path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r);
for(int i=0;i <= 4;i=i+1)
{
dr... | To determine which transformations will map the figure onto itself, we analyze each type of transformation given in the problem:
1. **Some rotation around a point on line $\ell$:**
- Consider a rotation of $180^\circ$ around a point exactly halfway between an up-facing square and a down-facing square on line $\ell$... | [
{
"content": "The figure below shows line $\\ell$ with a regular, infinite, recurring pattern of squares and line segments.\n\nsize(300);\ndefaultpen(linewidth(0.8));\nreal r = 0.35;\npath P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r);\npath Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r);\n... |
amc_aime | Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is
$\mathrm{(A) \ 10 } \qquad \mathrm{(B) \ 12 } \qquad \mathrm{(C) \ 14 } \qquad \mathrm{(D) \ 16 } \qquad \mathrm{(E) \ 18 }$ | To solve this problem, we need to count all the triangles formed by the lines in the given figure. The figure consists of a square, its diagonals, and segments joining the midpoints of opposite sides. Let's analyze the figure and count the triangles of different sizes.
1. **Identify the vertices and lines:**
- The ... | [
{
"content": "Consider the figure consisting of a square, its diagonals, and the segments joining the midpoints of opposite sides. The total number of triangles of any size in the figure is \n$\\mathrm{(A) \\ 10 } \\qquad \\mathrm{(B) \\ 12 } \\qquad \\mathrm{(C) \\ 14 } \\qquad \\mathrm{(D) \\ 16 } \\qquad \\m... |
amc_aime | If $a,b$, and $d$ are the lengths of a side, a shortest diagonal and a longest diagonal, respectively,
of a regular nonagon (see adjoining figure), then
$\textbf{(A) }d=a+b\qquad \textbf{(B) }d^2=a^2+b^2\qquad \textbf{(C) }d^2=a^2+ab+b^2\qquad\\ \textbf{(D) }b=\frac{a+d}{2}\qquad \textbf{(E) }b^2=ad$ | 1. **Identify the angles in a regular nonagon**: In a regular nonagon, each interior angle is given by $\frac{(9-2)\times 180^\circ}{9} = 140^\circ$. The central angle subtended by each side is $\frac{360^\circ}{9} = 40^\circ$.
2. **Use the Law of Cosines**:
- For the longest diagonal $d$, which spans 4 sides of t... | [
{
"content": "If $a,b$, and $d$ are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, \nof a regular nonagon (see adjoining figure), then\n$\\textbf{(A) }d=a+b\\qquad \\textbf{(B) }d^2=a^2+b^2\\qquad \\textbf{(C) }d^2=a^2+ab+b^2\\qquad\\\\ \\textbf{(D) }b=\\frac{a+d}{2}\\qquad \\... |
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 | Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
Suppose $PQ=6$ and... | 1. **Identify the given information and setup:**
- $ABCD$ is a parallelogram with area $15$.
- $PQ = 6$ and $RS = 8$ are the lengths of the projections of $A$ and $C$ onto $BD$, and $B$ and $D$ onto $AC$, respectively.
- We need to find $d^2$, where $d$ is the length of diagonal $BD$.
2. **Use the properties ... | [
{
"content": "Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these point... |
amc_aime | Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids:
$\circ$ Roger's cookies are rectangles:
$\circ$ Paul's... | To solve this problem, we need to determine the number of cookies Trisha can make from the same amount of dough as Art, given that all cookies have the same thickness and thus the same volume per unit area.
1. **Calculate the area of one of Art's cookies**:
Art's cookies are trapezoids. The formula for the area of ... | [
{
"content": "Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures\nFour friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.\n$\\circ$ Art's cookies are trapezoids: \n\n$\\circ$ Roger's cookies are ... |
amc_aime | An "$n$-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\cdots ,k,\cdots,n,\text{ }n\ge 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong the $n$ pairs of sides numbere... | To solve this problem, we need to determine the sum of the interior angles at the vertices of the $n$-pointed star formed by extending every second side of an $n$-sided convex polygon.
1. **Understanding the Star Formation**:
- Each vertex of the star is formed by the intersection of two lines, which are extension... | [
{
"content": "An \"$n$-pointed star\" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\\cdots ,k,\\cdots,n,\\text{ }n\\ge 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong th... |
amc_aime | As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE$?
$\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C... | 1. **Identify the given information and setup**: We know that $ABCD$ is a square, and $E$ lies on the opposite half-plane determined by line $CD$ from point $A$. The angle $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ such that $DE = DF$.
2. **Extend $\overline{AD}$ to point $G$**: By extending $\overlin... | [
{
"content": "As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\\angle CDE = 110^\\circ$. Point $F$ lies on $\\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\\angle AFE$?\n\n$\\textbf{(A) }160\\qquad\\... |
amc_aime | Let $D$ represent a repeating decimal. If $P$ denotes the $r$ figures of $D$ which do not repeat themselves, and $Q$ denotes the $s$ figures of $D$ which do repeat themselves, then the incorrect expression is:
$\text{(A) } D = .PQQQ\ldots \qquad\\ \text{(B) } 10^rD = P.QQQ\ldots \\ \text{(C) } 10^{r + s}D = PQ.QQQ\ldo... | To solve this problem, we need to analyze each expression given and determine which one incorrectly represents the repeating decimal $D$.
Let's start by defining $D$ more formally:
- $P$ is the non-repeating part of $D$ with $r$ digits.
- $Q$ is the repeating part of $D$ with $s$ digits.
#### Expression (A):
$D = .PQ... | [
{
"content": "Let $D$ represent a repeating decimal. If $P$ denotes the $r$ figures of $D$ which do not repeat themselves, and $Q$ denotes the $s$ figures of $D$ which do repeat themselves, then the incorrect expression is: \n$\\text{(A) } D = .PQQQ\\ldots \\qquad\\\\ \\text{(B) } 10^rD = P.QQQ\\ldots \\\\ \\te... |
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 | The closed curve in the figure is made up of 9 congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve?
$\textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt{3}\qquad\text... | 1. **Understanding the Geometry**: The problem describes a closed curve made up of 9 congruent circular arcs, each with a length of $\frac{2\pi}{3}$. These arcs are centered at the vertices of a regular hexagon with side length 2.
2. **Calculating the Radius of the Arcs**: The length of each arc is given by the formul... | [
{
"content": "The closed curve in the figure is made up of 9 congruent circular arcs each of length $\\frac{2\\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve?\n\n$\\textbf{(A)}\\ 2\\pi+6\\qquad\\textbf{... |
amc_aime | On the dart board shown in the figure below, the outer circle has radius $6$ and the inner circle has radius $3$. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this b... | 1. **Identify the regions and their point values**: The dartboard is divided into six regions by three radii and two concentric circles. The inner circle is divided into three regions each with a radius of $3$, and the outer ring (between the inner circle and the outer circle with radius $6$) is also divided into three... | [
{
"content": "On the dart board shown in the figure below, the outer circle has radius $6$ and the inner circle has radius $3$. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. Whe... |
amc_aime | Which of the following methods of proving a geometric figure a locus is not correct?
$\textbf{(A)}\ \text{Every point of the locus satisfies the conditions and every point not on the locus does}\\ \text{not satisfy the conditions.}$
$\textbf{(B)}\ \text{Every point not satisfying the conditions is not on the locus and... | To determine which method of proving a geometric figure as a locus is incorrect, we need to analyze each statement to see if it logically ensures that the set of all points satisfying the conditions is exactly the locus.
1. **Statement (A):** "Every point of the locus satisfies the conditions and every point not on th... | [
{
"content": "Which of the following methods of proving a geometric figure a locus is not correct? \n$\\textbf{(A)}\\ \\text{Every point of the locus satisfies the conditions and every point not on the locus does}\\\\ \\text{not satisfy the conditions.}$\n$\\textbf{(B)}\\ \\text{Every point not satisfying the c... |
amc_aime | Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?
[asy] draw((0,0)--(4,4*sqrt(3))); draw(... | 1. **Identify the total number of triangles in each half**: Each half of the figure contains 3 red triangles, 5 blue triangles, and 8 white triangles.
2. **Analyze the red triangles**: When the upper half is folded down, 2 pairs of red triangles coincide. This means 4 red triangles (2 from each half) are involved in c... | [
{
"content": "Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?\n[asy] draw((0,0)... |
amc_aime | If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that
$I.Q. (A_1BC) > I.Q.(A_2BC)$,
where the isoperimetric quotient of a figure $F$ is defined by
$I.Q.(F) = \frac{\text{Area (F)}}{\text{[Perimeter (F)]}^2}$ | 1. **Understanding the Isoperimetric Quotient (I.Q.)**:
The isoperimetric quotient for a triangle $ABC$ is defined as:
\[
I.Q.(ABC) = \frac{\text{Area}(ABC)}{(\text{Perimeter}(ABC))^2}
\]
Using the notation $[ABC]$ for the area of triangle $ABC$ and $s = \frac{a+b+c}{2}$ (where $a$, $b$, and $c$ are the ... | [
{
"content": "If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\\triangle{A_1BC}$, prove that\n$I.Q. (A_1BC) > I.Q.(A_2BC)$,\nwhere the isoperimetric quotient of a figure $F$ is defined by\n$I.Q.(F) = \\frac{\\text{Area (F)}}{\\text{[Perimeter (F)]}^2}$... |
amc_aime | A large [equilateral triangle](https://artofproblemsolving.com/wiki/index.php/Equilateral_triangle) is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many... | To solve this problem, we need to determine the number of toothpicks required to construct a large equilateral triangle made up of smaller equilateral triangles, with the base row consisting of 2003 small triangles.
#### Step 1: Understanding the structure
Each small equilateral triangle shares sides with adjacent tri... | [
{
"content": "A large [equilateral triangle](https://artofproblemsolving.com/wiki/index.php/Equilateral_triangle) is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in t... |
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 | Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$
$\textbf{(A) }240 \qquad \textbf{(B) }248 \qquad \textbf{(C) }256 \qquad \textbf{(D) }264 \qquad \textbf{(E) }272$ | 1. **Identify the Geometry and Given Values**: We are given a rectangle $ABCD$ inscribed in a semicircle with diameter $\overline{FE}$. The lengths $DA = 16$ and $FD = AE = 9$ are provided.
2. **Complete the Circle**: To utilize symmetry and the properties of a circle, consider the full circle with diameter $\overline... | [
{
"content": "Rectangle $ABCD$ is inscribed in a semicircle with diameter $\\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD?$\n\n$\\textbf{(A) }240 \\qquad \\textbf{(B) }248 \\qquad \\textbf{(C) }256 \\qquad \\textbf{(D) }264 \\qquad \\textbf{(E) }272$",
"ro... |
amc_aime | The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$
$\textbf{(A)} ~20 \qquad\textbf{(B)} ~21 \qquad\textbf{(C)} ~22 \qquad\textbf{(D)} ~23 \qquad\textbf{(E)} ~24... | 1. **Identify Key Triangles and Midpoint**: Let $M$ be the midpoint of $CD$. Since each side of the pentagon and the internal segments are of length $2$, and given the symmetry and equal lengths, we can infer that triangles $AED$ and $ABC$ are isosceles with a vertex angle of $120^\circ$ formed by extending sides of eq... | [
{
"content": "The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\\sqrt{m} + \\sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$\n\n$\\textbf{(A)} ~20 \\qquad\\textbf{(B)} ~21 \\qquad\\textbf{(C)} ~22 \\qquad\\textb... |
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... |
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