source stringclasses 1
value | problem stringlengths 121 4.96k | solution stringlengths 458 6.95k | messages listlengths 2 2 |
|---|---|---|---|
aops_forum | A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figu... | 1. **Identify the Midpoints of Diagonals:**
Consider the midpoints of the diagonals of length $\sqrt{2}$ drawn as the rectangle is divided into the described figures.
2. **Connect Midpoints with Red Segments:**
Connect the midpoints of the diagonals as follows:
- The midpoints of the two equal sides of lengt... | [
{
"content": "A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles (1) with base of two units, squares (2) with unit side, and parallelograms (3) formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove t... |
aops_forum | In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$ . What is $FG+JH+CD$ ?
[asy]
import cse5;pathpen=black;pointpen=black;
size(2inch);
pair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162);
D(MP("A",A,A)--MP("B",B,B)--MP("C",C,C)--MP("D",D,D)--MP("E",E,E)--cycle,linewidth(1.5));
D(A--C--E--... | 1. **Identify the given and required quantities:**
- We are given a regular pentagon \(ABCDE\) with \(AG = 1\).
- We need to find the sum \(FG + JH + CD\).
2. **Use symmetry and properties of the regular pentagon:**
- In a regular pentagon, all sides and angles are equal.
- The diagonals intersect at point... | [
{
"content": "In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$ . What is $FG+JH+CD$ ?\n[asy]\nimport cse5;pathpen=black;pointpen=black;\nsize(2inch);\npair A=dir(90), B=dir(18), C=dir(306), D=dir(234), E=dir(162);\nD(MP(\"A\",A,A)--MP(\"B\",B,B)--MP(\"C\",C,C)--MP(\"D\",D,D)--MP(\"E\",E,E... |
aops_forum | On a $55\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out.
[asy]size(2.013cm);
draw ((0,0)--... | 1. **Define the problem and the grid:**
We have a \(55 \times 55\) square grid, which contains \(55^2 = 3025\) unit squares. We are cutting out 500 unit squares and 400 L-shaped pieces, each consisting of 3 unit squares.
2. **Calculate the total number of unit squares removed:**
Each L-shaped piece consists of 3... | [
{
"content": "On a $55\\times 55$ square grid, $500$ unit squares were cut out as well as $400$ L-shaped pieces consisting of 3 unit squares (each piece can be oriented in any way) [refer to the figure]. Prove that at least two of the cut out pieces bordered each other before they were cut out.\n[asy]size... |
aops_forum | 10) The handle of a gallon of milk is plugged by a manufacturing defect. After removing the cap and pouring out some milk, the level of milk in the main part of the jug is lower than in the handle, as shown in the figure. Which statement is true of the gauge pressure $P$ of the milk at the bottom of the jug? $\rho$ ... | To solve this problem, we need to understand the concept of gauge pressure and how it is affected by the height of the liquid column. Gauge pressure is the pressure relative to atmospheric pressure.
1. **Identify the relevant heights:**
- Let \( h \) be the height of the milk in the main part of the jug.
- Let \... | [
{
"content": "10) The handle of a gallon of milk is plugged by a manufacturing defect. After removing the cap and pouring out some milk, the level of milk in the main part of the jug is lower than in the handle, as shown in the figure. Which statement is true of the gauge pressure $P$ of the milk at the botto... |
aops_forum | In the figure, $ \angle EAB$ and $ \angle ABC$ are right angles. $ AB \equal{} 4, BC \equal{} 6, AE \equal{} 8$ , and $ \overline{AC}$ and $ \overline{BE}$ intersect at $ D$ . What is the difference between the areas of $ \triangle ADE$ and $ \triangle BDC$ ?
[asy]unitsize(4mm);
defaultpen(linewidth(.8pt)... | 1. **Calculate the area of $\triangle ABE$:**
\[
[ABE] = \frac{1}{2} \times AB \times AE = \frac{1}{2} \times 4 \times 8 = 16
\]
Let the area of $\triangle ADE$ be \( x \) and the area of $\triangle ABD$ be \( z \). Therefore,
\[
x + z = 16
\]
2. **Calculate the area of $\triangle ABC$:**
\[
... | [
{
"content": "In the figure, $ \\angle EAB$ and $ \\angle ABC$ are right angles. $ AB \\equal{} 4, BC \\equal{} 6, AE \\equal{} 8$ , and $ \\overline{AC}$ and $ \\overline{BE}$ intersect at $ D$ . What is the difference between the areas of $ \\triangle ADE$ and $ \\triangle BDC$ ?\r\n[asy]unitsize... |
aops_forum | Welcome to the **USAYNO**, where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them **all** correct, you will receive $\max(0, (n-1)(n-2))$ points. If any of them are wrong (or you leave them all blank), you will receive $0$ points... | To solve part (f) of the problem, we need to compare the surface areas of the 2018-dimensional unit sphere and the 2017-dimensional unit sphere.
1. **Volume of a High-Dimensional Sphere**:
The volume \( V_n \) of an \( n \)-dimensional unit sphere is given by:
\[
V_n = \frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2... | [
{
"content": "Welcome to the **USAYNO**, where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them **all** correct, you will receive $\\max(0, (n-1)(n-2))$ points. If any of them are wrong (or you leave them all blank), you wi... |
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 | In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$ .
[asy]
Label f;
xaxis(0,60,b... | 1. **Define Points and Areas:**
- Let the points \(A\), \(P\), and \(B\) be \((t, f(t))\), \((\sqrt{\frac{f(t)}{2}}, f(t))\), and \((\sqrt{\frac{f(t)}{2}}, \frac{f(t)}{2})\) respectively.
- Denote the areas by \([ \ ]\).
2. **Express Areas \(OAP\) and \(OBP\):**
- The area \([OAP]\) can be expressed as:
... | [
{
"content": "In the accompanying figure , $y=f(x)$ is the graph of a one-to-one continuous function $f$ . At each point $P$ on the graph of $y=2x^2$ , assume that the areas $OAP$ and $OBP$ are equal . Here $PA,PB$ are the horizontal and vertical segments . Determine the function $f$ . \n\n\n[asy... |
aops_forum | An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$ . A *move* consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells... | 1. **Initial Setup and Parity Argument:**
- Consider an equilateral triangle with side length 3, divided into 9 smaller congruent equilateral triangles.
- Initially, all cells contain the number 0.
- A move consists of selecting two adjacent cells and either increasing or decreasing the numbers in both cells b... | [
{
"content": "An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$ . A *move* consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the num... |
aops_forum | The figure $ABCDEF$ is a regular hexagon. Find all points $M$ belonging to the hexagon such that
Area of triangle $MAC =$ Area of triangle $MCD$ . | 1. **Understanding the Problem:**
We are given a regular hexagon \(ABCDEF\) and need to find all points \(M\) within the hexagon such that the area of triangle \(MAC\) is equal to the area of triangle \(MCD\).
2. **Properties of a Regular Hexagon:**
A regular hexagon can be divided into 6 equilateral triangles. ... | [
{
"content": "The figure $ABCDEF$ is a regular hexagon. Find all points $M$ belonging to the hexagon such that \n\nArea of triangle $MAC =$ Area of triangle $MCD$ .",
"role": "user"
},
{
"content": "1. **Understanding the Problem:**\n We are given a regular hexagon \\(ABCDEF\\) and need to ... |
aops_forum | An equilateral triangle of side $n$ has been divided into little equilateral triangles of side $1$ in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the $\frac{(n+1)(n+2)}{2}$ vertices. What is the minimum number of times the path can c... | To solve this problem, we need to determine the minimum number of times a path can change its direction while visiting each vertex of an equilateral triangle of side \( n \) exactly once. The total number of vertices in such a triangle is given by \(\frac{(n+1)(n+2)}{2}\).
1. **Understanding the Path and Direction Cha... | [
{
"content": "An equilateral triangle of side $n$ has been divided into little equilateral triangles of side $1$ in the usual way. We draw a path over the segments of this triangulation, in such a way that it visits exactly once each one of the $\\frac{(n+1)(n+2)}{2}$ vertices. What is the minimum number ... |
aops_forum | In the figure below, how many ways are there to select two squares which do not share an edge?
[asy]
size(3cm);
for (int t = -2; t <= 2; t=t+1) { draw( shift((t,0))*unitsquare ) ; if (t!=0) draw( shift((0,t))*unitsquare ); }
[/asy]
*Proposed by Evan Chen* | 1. **Identify the total number of squares**: The figure is a 3x3 grid, so there are \(9\) squares in total.
2. **Calculate the total number of ways to choose 2 squares**: The number of ways to choose 2 squares out of 9 is given by the combination formula \(\binom{n}{k}\), where \(n\) is the total number of items, and ... | [
{
"content": "In the figure below, how many ways are there to select two squares which do not share an edge?\n\n[asy]\nsize(3cm);\nfor (int t = -2; t <= 2; t=t+1) { draw( shift((t,0))*unitsquare ) ; if (t!=0) draw( shift((0,t))*unitsquare ); }\n[/asy]\n\n*Proposed by Evan Chen*",
"role": "user"
},
{
... |
aops_forum | In the figure, $AB \perp BC$ , $BC \perp CD$ , and $BC$ is tangent to the circle with center $O$ and diameter $AD$ . In which one of the following cases is the area of $ABCD$ an integer?
[asy]
size(170);
defaultpen(fontsize(10pt)+linewidth(.8pt));
pair O=origin, A=(-1/sqrt(2),1/sqrt(2)), B=(-1/sqrt(2),-1), C... | 1. **Identify the geometric properties and relationships:**
- Given that \(AB \perp BC\) and \(BC \perp CD\), we know that \(ABCD\) forms a rectangle.
- \(BC\) is tangent to the circle with center \(O\) and diameter \(AD\). This implies that \(BC\) is perpendicular to the radius at the point of tangency.
2. **De... | [
{
"content": "In the figure, $AB \\perp BC$ , $BC \\perp CD$ , and $BC$ is tangent to the circle with center $O$ and diameter $AD$ . In which one of the following cases is the area of $ABCD$ an integer?\n\n[asy]\nsize(170);\ndefaultpen(fontsize(10pt)+linewidth(.8pt));\npair O=origin, A=(-1/sqrt(2),1/s... |
aops_forum | In the figure below, $E$ is the midpoint of the arc $ABEC$ and the segment $ED$ is perpendicular to the chord $BC$ at $D$ . If the length of the chord $AB$ is $l_1$ , and that of the segment $BD$ is $l_2$ , determine the length of $DC$ in terms of $l_1, l_2$ .
[asy]
unitsize(1 cm);
pair A=2dir(240),B=... | 1. **Identify the key points and properties:**
- \( E \) is the midpoint of the arc \( ABEC \).
- \( ED \) is perpendicular to the chord \( BC \) at \( D \).
- \( AB = l_1 \) and \( BD = l_2 \).
2. **Introduce point \( P \):**
- Extend \( CB \) and take a point \( P \) such that \( PD = DC \).
3. **Analyz... | [
{
"content": "In the figure below, $E$ is the midpoint of the arc $ABEC$ and the segment $ED$ is perpendicular to the chord $BC$ at $D$ . If the length of the chord $AB$ is $l_1$ , and that of the segment $BD$ is $l_2$ , determine the length of $DC$ in terms of $l_1, l_2$ .\n[asy]\nunitsize(1 ... |
aops_forum | **p1.** Al usually arrives at the train station on the commuter train at $6:00$ , where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at $5:00$ . Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when he saw her. As a result... | ### Part (a)
1. Let \( m \) be the middle term of the sequence of 19 consecutive integers. The sequence can be written as:
\[
m-9, m-8, \ldots, m-1, m, m+1, \ldots, m+8, m+9
\]
2. The sum \( S \) of these 19 consecutive integers is:
\[
S = 19m
\]
3. We want this sum to be as close to 1000 as possib... | [
{
"content": "**p1.** Al usually arrives at the train station on the commuter train at $6:00$ , where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at $5:00$ . Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when h... |
aops_forum | The game of **Hive**is played on a regular hexagonal grid (as shown in the figure) by 3 players. The grid consists of $k$ layers (where $k$ is a natural number) surrounding a regular hexagon, with each layer constructed around the previous layer. The figure below shows a grid with 2 layers.
The players, *Ali*, *S... | 1. **Understanding the Grid Structure:**
- The grid is a regular hexagonal grid with \( k \) layers surrounding a central hexagon. Each layer is constructed around the previous layer.
- For \( k = 2 \), the grid consists of a central hexagon and two surrounding layers, forming a larger hexagon.
2. **Game Rules a... | [
{
"content": "The game of **Hive**is played on a regular hexagonal grid (as shown in the figure) by 3 players. The grid consists of $k$ layers (where $k$ is a natural number) surrounding a regular hexagon, with each layer constructed around the previous layer. The figure below shows a grid with 2 layers.\n\... |
aops_forum | Durer drew a regular triangle and then poked at an interior point. He made perpendiculars from it sides and connected it to the vertices. In this way, $6$ small triangles were created, of which (moving clockwise) all the second one is painted gray, as shown in figure. Show that the sum of the gray areas is just half ... | 1. **Define the vertices and point \( P \):**
Given a regular triangle \( \triangle ABC \) with vertices \( A(-a,0) \), \( B(a,0) \), and \( C(0,\sqrt{3}a) \). Let \( P(p,q) \) be an interior point.
2. **Calculate the area of the entire triangle \( \triangle ABC \):**
The area of \( \triangle ABC \) can be calcu... | [
{
"content": "Durer drew a regular triangle and then poked at an interior point. He made perpendiculars from it sides and connected it to the vertices. In this way, $6$ small triangles were created, of which (moving clockwise) all the second one is painted gray, as shown in figure. Show that the sum of the gr... |
aops_forum | The figure below was formed by taking four squares, each with side length 5, and putting one on each side of a square with side length 20. Find the perimeter of the figure below.
[center][/center] | 1. **Identify the structure of the figure:**
- The figure consists of a central square with side length 20.
- Four smaller squares, each with side length 5, are attached to each side of the central square.
2. **Calculate the perimeter of the central square:**
- The perimeter of a square is given by \(4 \times... | [
{
"content": "The figure below was formed by taking four squares, each with side length 5, and putting one on each side of a square with side length 20. Find the perimeter of the figure below.\n\n[center][/center]",
"role": "user"
},
{
"content": "1. **Identify th... |
aops_forum | For a positive integer $n$ , let denote $C_n$ the figure formed by the inside and perimeter of the circle with center the origin, radius $n$ on the $x$ - $y$ plane.
Denote by $N(n)$ the number of a unit square such that all of unit square, whose $x,\ y$ coordinates of 4 vertices are integers, and the vertic... | 1. **Restate the problem in terms of a generalized region:**
- Let \( D_n \) be the region defined by \( x^2 + y^2 \leq n \) on the \( xy \)-plane.
- Let \( M(n) \) be the number of unit squares whose vertices lie entirely within \( D_n \).
2. **Express \( M(n) \) in terms of lattice points:**
- Let \( \nu(n)... | [
{
"content": "For a positive integer $n$ , let denote $C_n$ the figure formed by the inside and perimeter of the circle with center the origin, radius $n$ on the $x$ - $y$ plane.\n\nDenote by $N(n)$ the number of a unit square such that all of unit square, whose $x,\\ y$ coordinates of 4 vertices are... |
aops_forum | Kalyn cut rectangle R from a sheet of paper and then cut figure S from R. All the cuts were made
parallel to the sides of the original rectangle. In comparing R to S
(A) the area and perimeter both decrease
(B) the area decreases and the perimeter increases
(C) the area and perimeter both increase
(D) the area increas... | 1. Let's denote the dimensions of the original rectangle \( R \) as \( l \) (length) and \( w \) (width). The area of \( R \) is given by:
\[
\text{Area of } R = l \times w
\]
The perimeter of \( R \) is given by:
\[
\text{Perimeter of } R = 2l + 2w
\]
2. Kalyn cuts figure \( S \) from \( R \), an... | [
{
"content": " Kalyn cut rectangle R from a sheet of paper and then cut figure S from R. All the cuts were made\nparallel to the sides of the original rectangle. In comparing R to S\n(A) the area and perimeter both decrease\n(B) the area decreases and the perimeter increases\n(C) the area and perimeter both inc... |
aops_forum | Two flag poles of height $11$ and $13$ are planted vertically in level ground, and an equilateral triangle is hung as shown in the figure so that [the] lowest vertex just touches the ground. What is the length of the side of the equilateral triangle?
[asy]
pair A, B, C, D, E;
A = (0,11);
B = origin;
C = (13.856406... | 1. Let \( x \) be the side length of the equilateral triangle.
2. The heights of the flag poles are given as \( 11 \) and \( 13 \).
3. The lowest vertex of the equilateral triangle touches the ground, and the other two vertices are at the tops of the flag poles.
4. The height of an equilateral triangle with side length... | [
{
"content": "Two flag poles of height $11$ and $13$ are planted vertically in level ground, and an equilateral triangle is hung as shown in the figure so that [the] lowest vertex just touches the ground. What is the length of the side of the equilateral triangle?\n\n[asy]\npair A, B, C, D, E;\nA = (0,11);\... |
aops_forum | Starting with two points A and B, some circles and points are constructed as shown in
the figure:
- the circle with centre A through B
- the circle with centre B through A
- the circle with centre C through A
- the circle with centre D through B
- the circle with centre E through A
- the circle with centre F through A... | 1. **Identify the equilateral triangles:**
- The triangle $\triangle ABC$ is equilateral, hence $\angle ABC = 60^\circ$.
- Similarly, the triangles $\triangle BCD$ and $\triangle BDE$ are equilateral, hence $\angle CBD = \angle DBE = 60^\circ$.
2. **Calculate the angle $\angle ABE$:**
- Since $\angle ABC = 60... | [
{
"content": "Starting with two points A and B, some circles and points are constructed as shown in\r\nthe figure:\n- the circle with centre A through B\n- the circle with centre B through A\n- the circle with centre C through A\n- the circle with centre D through B\n- the circle with centre E through A\n- the ... |
aops_forum | In the figure below, area of triangles $AOD, DOC,$ and $AOB$ is given. Find the area of triangle $OEF$ in terms of area of these three triangles.
[asy]
import graph; size(11.52cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.4,xmax=9.12,ymin=-6.6,ymax=5.16;
pai... | 1. **Identify the given areas:**
Let the areas of triangles \( \triangle AOD \), \( \triangle DOC \), and \( \triangle AOB \) be denoted as \( S_1 \), \( S_2 \), and \( S_3 \) respectively.
2. **Understand the configuration:**
The points \( A, B, C, D, E, F, \) and \( O \) are given in the figure. We need to fin... | [
{
"content": "In the figure below, area of triangles $AOD, DOC,$ and $AOB$ is given. Find the area of triangle $OEF$ in terms of area of these three triangles.\n\n[asy]\nimport graph; size(11.52cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-2.4,xmax=9.12,... |
aops_forum | Given a figure, containing $16$ segments. You should prove that there is no curve, that intersect each segment exactly once. The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices.
[asy]
draw((0,0)--(6,0)--(6,3)--(0,3)--(0,0));
draw((0,3/2)--(6,3... | 1. **Identify the structure of the problem:**
The given figure is a rectangle divided into smaller regions by segments. We need to prove that there is no curve that intersects each of the 16 segments exactly once.
2. **Define the regions:**
The figure can be divided into six regions (including the outside). Let'... | [
{
"content": "Given a figure, containing $16$ segments. You should prove that there is no curve, that intersect each segment exactly once. The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices.\n[asy]\ndraw((0,0)--(6,0)--(6,3)--(0,3)--(0... |
aops_forum | The sequence of functions $f_n:[0,1]\to\mathbb R$ $(n\ge2)$ is given by $f_n=1+x^{n^2-1}+x^{n^2+2n}$ . Let $S_n$ denote the area of the figure bounded by the graph of the function $f_n$ and the lines $x=0$ , $x=1$ , and $y=0$ . Compute $$ \lim_{n\to\infty}\left(\frac{\sqrt{S_1}+\sqrt{S_2}+\ldots+\sqrt{S_n}... | 1. First, we need to compute the area \( S_n \) under the curve \( f_n(x) = 1 + x^{n^2-1} + x^{n^2+2n} \) from \( x = 0 \) to \( x = 1 \). This is given by the integral:
\[
S_n = \int_{0}^{1} \left(1 + x^{n^2-1} + x^{n^2+2n}\right) \, dx
\]
2. We can split the integral into three parts:
\[
S_n = \int_{0... | [
{
"content": "The sequence of functions $f_n:[0,1]\\to\\mathbb R$ $(n\\ge2)$ is given by $f_n=1+x^{n^2-1}+x^{n^2+2n}$ . Let $S_n$ denote the area of the figure bounded by the graph of the function $f_n$ and the lines $x=0$ , $x=1$ , and $y=0$ . Compute $$ \\lim_{n\\to\\infty}\\left(\\frac{\\sqrt{S_... |
aops_forum | The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$ . Suppose that the radius of the circle is 5, that $BC = 6$ , and that $AD$ is bisected by $BC$ . Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$ . It follows that the sine of ... | 1. **Identify the center and key points:**
Let the center of the circle be \( O \). Given that \( AD \) is bisected by \( BC \), let the intersection of \( AD \) and \( BC \) be \( E \). Since \( AD \) is bisected by \( BC \), \( E \) is the midpoint of \( AD \).
2. **Calculate the length of \( AD \):**
Since \(... | [
{
"content": "The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$ . Suppose that the radius of the circle is 5, that $BC = 6$ , and that $AD$ is bisected by $BC$ . Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$ . It fol... |
aops_forum | Let $n$ be a positive integer and let $k$ be an integer between $1$ and $n$ inclusive. There is a white board of $n \times n$ . We do the following process. We draw $k$ rectangles with integer sides lenghts and sides parallel to the ones of the $n \times n$ board, and such that each rectangle covers the to... | 1. **Understanding the Problem:**
We need to determine the number of different white figures that can be created on an \( n \times n \) board by drawing \( k \) rectangles, each covering the top-right corner of the board. The white figure should be such that it cannot be created with fewer than \( k \) rectangles.
... | [
{
"content": "Let $n$ be a positive integer and let $k$ be an integer between $1$ and $n$ inclusive. There is a white board of $n \\times n$ . We do the following process. We draw $k$ rectangles with integer sides lenghts and sides parallel to the ones of the $n \\times n$ board, and such that each... |
aops_forum | Consider a fixed regular $n$ -gon of unit side. When a second regular $n$ -gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line $\kappa$ as in the figure.
[asy]
int n=9;
draw(polygon(n));
for (int i = 0; i<n;++i) {
draw(reflect(dir(360*i/n + ... | 1. **Labeling and Initial Setup**:
- Let the vertices of the fixed regular $n$-gon be labeled as $A_0, A_1, \ldots, A_{n-1}$.
- Assume $A_0$ is the initial vertex of the rolling $n$-gon.
- When the rolling $n$-gon reaches the side $\overline{A_jA_{j+1}}$, the moving vertex is placed on the $j$-th vertex of thi... | [
{
"content": "Consider a fixed regular $n$ -gon of unit side. When a second regular $n$ -gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line $\\kappa$ as in the figure.\n\n[asy]\nint n=9;\ndraw(polygon(n));\nfor (int i = 0; i<n;++i) {\... |
aops_forum | Let $n\ge 2$ be an integer. Thibaud the Tiger lays $n$ $2\times 2$ overlapping squares out on a table, such that the centers of the squares are equally spaced along the line $y=x$ from $(0,0)$ to $(1,1)$ (including the two endpoints). For example, for $n=4$ the resulting figure is shown below, and it cove... | 1. **Understanding the Problem:**
We need to find the minimum integer \( n \geq 2 \) such that the total area covered by \( n \) overlapping \( 2 \times 2 \) squares, whose centers are equally spaced along the line \( y = x \) from \((0,0)\) to \((1,1)\), is at least \(\sqrt{63}\).
2. **Analyzing the Overlapping Sq... | [
{
"content": "Let $n\\ge 2$ be an integer. Thibaud the Tiger lays $n$ $2\\times 2$ overlapping squares out on a table, such that the centers of the squares are equally spaced along the line $y=x$ from $(0,0)$ to $(1,1)$ (including the two endpoints). For example, for $n=4$ the resulting figure is ... |
aops_forum | In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$ . If angle $D$ is $60^o$ and angle $E$ is $40^o$ , how large is then angle $u$ ?
![Image](https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0... | 1. **Identify the given angles and their relationships:**
- Given that $\angle D = 60^\circ$ and $\angle E = 40^\circ$.
- Since triangle $ADE$ is produced from triangle $ABC$ by a $90^\circ$ rotation about point $A$, we know that $\angle DAE = \angle BAC$.
2. **Determine the angles in triangle $ADE$:**
- The ... | [
{
"content": "In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$ . If angle $D$ is $60^o$ and angle $E$ is $40^o$ , how large is then angle $u$ ?\n and \( DE \) have the same length, we will demonstrate that the midpoints of \( BE \) and \( CD \) coincide.
1. **Define the Geometry and Variables:**
- Let \( O \) be the center of the original semicircle.
- Let \( A \) be an arbitrary point on the diameter of the semicircle.... | [
{
"content": "Given a semicirle with center $O$ an arbitrary inner point of the diameter divides it into two segments. Let there be semicircles above the two segments as visible in the below figure. The line $\\ell$ passing through the point $A$ intersects the semicircles in $4$ points: $B, C, D$ and ... |
aops_forum | Let $ a$ be a positive real numbers. In the coordinate plane denote by $ S$ the area of the figure bounded by the curve $ y=\sin x\ (0\leq x\leq \pi)$ and the $ x$ -axis and denote $ T$ by the area of the figure bounded by the curves $ y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\ y=a\cos x\ \left(0\leq ... | 1. First, we need to find the area \( S \) bounded by the curve \( y = \sin x \) from \( x = 0 \) to \( x = \pi \) and the \( x \)-axis. This can be calculated using the definite integral:
\[
S = \int_{0}^{\pi} \sin x \, dx
\]
To solve this integral, we use the antiderivative of \( \sin x \), which is \( -\... | [
{
"content": "Let $ a$ be a positive real numbers. In the coordinate plane denote by $ S$ the area of the figure bounded by the curve $ y=\\sin x\\ (0\\leq x\\leq \\pi)$ and the $ x$ -axis and denote $ T$ by the area of the figure bounded by the curves $ y=\\sin x\\ \\left(0\\leq x\\leq \\frac{\\pi}{2... |
aops_forum | The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square.
 | 1. **Identify the given information and the goal:**
- We are given a square \(ABCD\) with points \(A\), \(B\), and \(C\) lying on three parallel lines.
- The distances between these points are indicated in the figure.
- We need to find the area of the square \(ABCD\).
2. **Understand the geometric relationshi... | [
{
"content": "The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square.\n![Image](https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMo... |
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