source stringclasses 1
value | problem stringlengths 147 5.08k | solution stringlengths 113 6.64k | messages listlengths 2 2 |
|---|---|---|---|
aops_forum | **p1.** When Charles traveled from Hawaii to Chicago, he moved his watch $5$ hours backwards instead of $5$ hours forwards. He plans to wake up at $7:00$ the next morning (Chicago time). When he wakes up during the night and sees that his watch says $6:00$ , how many more hours should he sleep? (He has a $12$ -... | 1. Partition the given set into the ordered sets:
\[
S_1 = \{1, 3, 5, 7, 9, 11\}, \quad S_2 = \{2, 4, 6, 8, 10\}
\]
Note that any subset of the given set corresponds to a unique pair of subsets, one of \( S_1 \) and \( S_2 \), and vice versa.
2. The given condition is equivalent to not having any two conse... | [
{
"content": "**p1.** When Charles traveled from Hawaii to Chicago, he moved his watch $5$ hours backwards instead of $5$ hours forwards. He plans to wake up at $7:00$ the next morning (Chicago time). When he wakes up during the night and sees that his watch says $6:00$ , how many more hours should he sl... |
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 | **p1.** What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square?**p2.** Two angles of an isosceles triangle measure $80^o$ and $x^o$ . What is the sum of all the possible values of $x$ ?**p3.** Let $p$ and $... | 1. **p1.** What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square?
To find the maximum number of points of intersection between a square and a triangle, we need to consider the following:
- A square has 4 sides.
... | [
{
"content": "**p1.** What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square?**p2.** Two angles of an isosceles triangle measure $80^o$ and $x^o$ . What is the sum of all the possible values of $x$ ?**... |
aops_forum | **p1A** Positive reals $x$ , $y$ , and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$ . There are two possible values for $z/x + x/z;$ find the greater value.**p1B** Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$ . Find $x^2+y^2$ .**p2** Set $A = \{5, 6, 8, 13, 20, 22, 33, 42\}... | Given the equations:
\[ x + y = 2 \]
\[ x^3 + y^3 = 3 \]
We start by using the identity for the sum of cubes:
\[ (x + y)^3 = x^3 + y^3 + 3xy(x + y) \]
Substituting the given values:
\[ 2^3 = 3 + 3xy \cdot 2 \]
\[ 8 = 3 + 6xy \]
Solving for \(xy\):
\[ 8 - 3 = 6xy \]
\[ 5 = 6xy \]
\[ xy = \frac{5}{6} \]
Next, we use ... | [
{
"content": "**p1A** Positive reals $x$ , $y$ , and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$ . There are two possible values for $z/x + x/z;$ find the greater value.**p1B** Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$ . Find $x^2+y^2$ .**p2** Set $A = \\{5, 6, ... |
aops_forum | **p1.**Ed, Fred and George are playing on a see-saw that is slightly off center. When Ed sits on the left side and George, who weighs $100$ pounds, on the right side, they are perfectly balanced. Similarly, if Fred, who weighs $400$ pounds, sits on the left and Ed sits on the right, they are also perfectly balanced... | To find the possible lengths of the other base of the trapezoid, we need to consider the two different orientations of the legs. We will use the Pythagorean theorem to determine the horizontal components of the legs.
1. **Case 1: Leg of length 15 slopes to the right**
- The horizontal component of the leg of lengt... | [
{
"content": "**p1.**Ed, Fred and George are playing on a see-saw that is slightly off center. When Ed sits on the left side and George, who weighs $100$ pounds, on the right side, they are perfectly balanced. Similarly, if Fred, who weighs $400$ pounds, sits on the left and Ed sits on the right, they are a... |
aops_forum | **p1.** A function $f$ defined on the set of positive numbers satisfies the equality $$ f(xy) = f(x) + f(y), x, y > 0. $$ Find $f(2007)$ if $f\left( \frac{1}{2007} \right) = 1$ .**p2.** The plane is painted in two colors. Show that there is an isosceles right triangle with all vertices of the same color.**p3.*... | 1. Suppose that there are no monochromatic isosceles right triangles in the plane. We will show that this assumption leads to a contradiction.
2. Let \( P \) be an arbitrary point in the plane. Define \( C_P \) as the set of points that have the same color as \( P \), and \( C_P' \) as the set of points that have a di... | [
{
"content": "**p1.** A function $f$ defined on the set of positive numbers satisfies the equality $$ f(xy) = f(x) + f(y), x, y > 0. $$ Find $f(2007)$ if $f\\left( \\frac{1}{2007} \\right) = 1$ .**p2.** The plane is painted in two colors. Show that there is an isosceles right triangle with all vertices... |
aops_forum | We call the mapping $ \Delta:\mathbb Z\backslash\{0\}\longrightarrow\mathbb N$ , a degree mapping if and only if for each $ a,b\in\mathbb Z$ such that $ b\neq0$ and $ b\not|a$ there exist integers $ r,s$ such that $ a \equal{} br\plus{}s$ , and $ \Delta(s) <\Delta(b)$ .
a) Prove that the following mapping i... | ### Part (a)
We need to prove that the mapping \(\delta(n) = \text{Number of digits in the binary representation of } n\) is a degree mapping.
1. Let \(a, b \in \mathbb{Z}\) such that \(b \neq 0\) and \(b \nmid a\).
2. Perform the division of \(a\) by \(b\) to get \(a = br' + s'\) where \(0 < s' < |b|\).
3. If \(s' < ... | [
{
"content": "We call the mapping $ \\Delta:\\mathbb Z\\backslash\\{0\\}\\longrightarrow\\mathbb N$ , a degree mapping if and only if for each $ a,b\\in\\mathbb Z$ such that $ b\\neq0$ and $ b\\not|a$ there exist integers $ r,s$ such that $ a \\equal{} br\\plus{}s$ , and $ \\Delta(s) <\\Delta(b)$ .\r... |
aops_forum | In $\triangle{ADE}$ points $B$ and $C$ are on side $AD$ and points $F$ and $G$ are on side $AE$ so that $BG \parallel CF \parallel DE$ , as shown. The area of $\triangle{ABG}$ is $36$ , the area of trapezoid $CFED$ is $144$ , and $AB = CD$ . Find the area of trapezoid $BGFC$ .
[center] and \( BC = y \).
2. Given that the area of \(\triangle ABG\) is 36, we can express this area in terms of the area of \(\triangle ADE\). Since \(BG \parallel CF \parallel DE\), the triangles \(\triangle ABG\) and \(\triangle ADE\) are similar. The ratio of their areas is the square of the rati... | [
{
"content": "In $\\triangle{ADE}$ points $B$ and $C$ are on side $AD$ and points $F$ and $G$ are on side $AE$ so that $BG \\parallel CF \\parallel DE$ , as shown. The area of $\\triangle{ABG}$ is $36$ , the area of trapezoid $CFED$ is $144$ , and $AB = CD$ . Find the area of trapezoid $B... |
aops_forum | Given $\vartriangle ABC$ . Choose two points $P, Q$ on $AB, AC$ such that $BP = CQ$ . Let $M, T$ be the midpoints of $BC, PQ$ . Show that $MT$ is parallel to the angle bisevtor of $\angle BAC$ ![Image](http://4.bp.blogspot.com/-MgMtdnPtq1c/XnSHHFl1LDI/AAAAAAAALdY/8g8541DnyGo_Gqd19-7bMBpVRFhbXeYPACK4BGAYYCw/... | 1. **Define Points and Parallelogram:**
Let $D$ be the point such that $ABDC$ forms a parallelogram. This implies $AB = DC$ and $AD = BC$.
2. **Introduce Points $R$ and $S$:**
Let $R$ and $S$ be points on $BD$ and $CD$ respectively such that $BR = CS = BP = CQ$.
3. **Show $PQSR$ is a Parallelogram:**
- Sin... | [
{
"content": "Given $\\vartriangle ABC$ . Choose two points $P, Q$ on $AB, AC$ such that $BP = CQ$ . Let $M, T$ be the midpoints of $BC, PQ$ . Show that $MT$ is parallel to the angle bisevtor of $\\angle BAC$ ![Image](http://4.bp.blogspot.com/-MgMtdnPtq1c/XnSHHFl1LDI/AAAAAAAALdY/8g8541DnyGo_Gqd19-7b... |
aops_forum | Let $ABCD$ be a cyclic quadrilateral such that $AB \cdot BC=2 \cdot AD \cdot DC$ . Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \le 9AC^2$ .
<span style="color:#FF0000">
Moderator says: Do not double post [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=590175](http://www.a... | 1. **Given Condition and Setup:**
We are given a cyclic quadrilateral \(ABCD\) such that \(AB \cdot BC = 2 \cdot AD \cdot DC\). We need to prove that the diagonals \(AC\) and \(BD\) satisfy the inequality \(8BD^2 \le 9AC^2\).
2. **Area Relationship:**
The given condition can be interpreted in terms of areas. Spe... | [
{
"content": "Let $ABCD$ be a cyclic quadrilateral such that $AB \\cdot BC=2 \\cdot AD \\cdot DC$ . Prove that its diagonals $AC$ and $BD$ satisfy the inequality $8BD^2 \\le 9AC^2$ .\n<span style=\"color:#FF0000\">\nModerator says: Do not double post [http://www.artofproblemsolving.com/Forum/viewtopic.p... |
aops_forum | **p1.** Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $|S|$ , the number of elements ... | To solve the problem, we need to find the value of the expression:
\[
\frac{bc}{a^2} + \frac{ac}{b^2} + \frac{ab}{c^2}
\]
where \(a\), \(b\), and \(c\) are the roots of the polynomial \(x^3 - 20x^2 + 22\).
1. **Apply Vieta's Formulas:**
By Vieta's formulas for the polynomial \(x^3 - 20x^2 + 22\), we have:
\[
... | [
{
"content": "**p1.** Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $|S|$ , t... |
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 | 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 | **p1.** A right triangle has a hypotenuse of length $25$ and a leg of length $16$ . Compute the length of the other leg of this triangle.**p2.** Tanya has a circular necklace with $5$ evenly-spaced beads, each colored red or blue. Find the number of distinct necklaces in which no two red beads are adjacent. If a n... | To find the minimum value of the function \( f(x) = \sqrt{x^2 - 4x + 5} + \sqrt{x^2 + 4x + 8} \), we can interpret the terms geometrically.
1. Rewrite the terms inside the square roots:
\[
\sqrt{x^2 - 4x + 5} = \sqrt{(x-2)^2 + 1^2}
\]
\[
\sqrt{x^2 + 4x + 8} = \sqrt{(x+2)^2 + 2^2}
\]
This allows us... | [
{
"content": "**p1.** A right triangle has a hypotenuse of length $25$ and a leg of length $16$ . Compute the length of the other leg of this triangle.**p2.** Tanya has a circular necklace with $5$ evenly-spaced beads, each colored red or blue. Find the number of distinct necklaces in which no two red bead... |
aops_forum | <u>Round 5</u>**p13.** Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n... | To solve the given problem, we start by analyzing the recurrence relation provided:
\[ 2(a_{k-1} + a_k + a_{k+1}) = a_k a_{k-1} + 8 \]
First, we rearrange the equation to isolate \(a_{k+1}\):
\[ 2a_{k-1} + 2a_k + 2a_{k+1} = a_k a_{k-1} + 8 \]
\[ 2a_{k+1} = a_k a_{k-1} - 2a_k - 2a_{k-1} + 8 \]
\[ 2a_{k+1} = (a_k - 2)... | [
{
"content": "<u>Round 5</u>**p13.** Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$... |
aops_forum | **p1.** Consider this statement: An equilateral polygon circumscribed about a circle is also equiangular.
Decide whether this statement is a true or false proposition in euclidean geometry.
If it is true, prove it; if false, produce a counterexample.**p2.** Show that the fraction $\frac{x^2-3x+1}{x-3}$ has no value ... | ### Problem 1
1. Consider an equilateral polygon circumscribed about a circle. By definition, an equilateral polygon has all sides of equal length.
2. Since the polygon is circumscribed about a circle, each side of the polygon is tangent to the circle.
3. In an equilateral polygon, the tangents from a point to a circle... | [
{
"content": "**p1.** Consider this statement: An equilateral polygon circumscribed about a circle is also equiangular.\nDecide whether this statement is a true or false proposition in euclidean geometry. \nIf it is true, prove it; if false, produce a counterexample.**p2.** Show that the fraction $\\frac{x^2-3... |
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 | Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$ , the point $E$ lies on the extension of $CB$ beyond $B$ , and $|CD| = |DE|$ . Prove that $|AD| = |BE|$ .
![Image](https://1.bp.blogspot.com/-QnAXFw3ijn0/XzR0YjqBQ3I/AAAAAAAAMU0/0TvhMQtBNjolYHtgXsQo2OPGJzEYSfCwACLcBGAsYH... | 1. Let $ABC$ be an equilateral triangle with side length $a$. Since $ABC$ is equilateral, we have $AB = BC = CA = a$.
2. Let $D$ be a point on the extension of $AB$ beyond $B$ such that $|CD| = |DE|$. Let $E$ be a point on the extension of $CB$ beyond $B$.
3. We need to prove that $|AD| = |BE|$.
To proceed, we will us... | [
{
"content": "Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$ , the point $E$ lies on the extension of $CB$ beyond $B$ , and $|CD| = |DE|$ . Prove that $|AD| = |BE|$ .\n(a + 2)(a + 3)$ is divisible by $24$ .**p3.** There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that
(a) for each... | 1. Let \(a, b, c\) be positive integers such that \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1\). Without loss of generality, assume \(a \leq b \leq c\).
2. Since \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1\), we have:
\[
\frac{3}{a} \geq \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1 \implies a \leq 3
\]
3. A... | [
{
"content": "**p1.** Find all triples of positive integers such that the sum of their reciprocals is equal to one.**p2.** Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$ .**p3.** There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle ... |
aops_forum | <u>Round 1</u>**p1.** If $2m = 200 cm$ and $m \ne 0$ , find $c$ .**p2.** A right triangle has two sides of lengths $3$ and $4$ . Find the smallest possible length of the third side.**p3.** Given that $20(x + 17) = 17(x + 20)$ , determine the value of $x$ .
<u>Round 2</u>**p4.** According to the Egyptian Metr... | 1. **Identify the number of $1 \times 1$ squares:**
The diagram is a $7 \times 7$ grid of points, which means there are $6 \times 6 = 36$ $1 \times 1$ squares. However, the diagram also includes additional points around the eyes and other areas, which contribute to more $1 \times 1$ squares. By carefully counting a... | [
{
"content": "<u>Round 1</u>**p1.** If $2m = 200 cm$ and $m \\ne 0$ , find $c$ .**p2.** A right triangle has two sides of lengths $3$ and $4$ . Find the smallest possible length of the third side.**p3.** Given that $20(x + 17) = 17(x + 20)$ , determine the value of $x$ .\n\n\n<u>Round 2</u>**p4.** Acco... |
aops_forum | To *dissect* a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:
... | 1. **Identify the problem and the goal:**
We need to determine all integers \( n \) for which it is possible to completely dissect a regular \( n \)-sided polygon into integer-ratio right triangles.
2. **Example for \( n = 4 \):**
- Consider a square (a regular 4-sided polygon).
- We can use two right triangl... | [
{
"content": "To *dissect* a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:\n is equilateral, so \(\angle BAC = \angle ABC = \angle BCA = 60^\circ\).
- Ray \(BE\) intersects segment \(AC\) at \(D\) such that \(\angle CBE = 20^\circ\).
- \(DE = AB\).
2. **Determine the angles in the triangle:**
- Since \(\... | [
{
"content": "The triangle $ABC$ is equilateral. The ray $[BE$ intersects the segment $[AC]$ at $D$ , such that $\\angle CBE= 20^o$ and $|DE|=|AB|$ . What is the measure of the angle $\\angle BEC$ ?\n![Image](https://1.bp.blogspot.com/-Jo_RPrXDzt0/XzAYNoxvhRI/AAAAAAAAMTo/QHWoXLrDZKUFHEkaq1DsDP71I1Ox... |
aops_forum | <u>Round 1</u>**p1.** The temperature inside is $28^o$ F. After the temperature is increased by $5^o$ C, what will the new temperature in Fahrenheit be?**p2.** Find the least positive integer value of $n$ such that $\sqrt{2021+n}$ is a perfect square.**p3.** A heart consists of a square with two semicircles att... | To solve the problem, we need to determine the number of ways to fill a $12 \times 2$ rectangle using $L$-shaped trominos. Let's denote $a_n$ as the number of ways to tile a $2 \times n$ rectangle using $L$-shaped trominos.
1. **Define the recurrence relation:**
We start by considering the possible placements of th... | [
{
"content": "<u>Round 1</u>**p1.** The temperature inside is $28^o$ F. After the temperature is increased by $5^o$ C, what will the new temperature in Fahrenheit be?**p2.** Find the least positive integer value of $n$ such that $\\sqrt{2021+n}$ is a perfect square.**p3.** A heart consists of a square w... |
aops_forum | In the octagon below all sides have the length $1$ and all angles are equal.
Determine the distance between the corners $A$ and $B$ .
 | 1. **Identify the properties of the regular octagon:**
- All sides are of equal length, \( s = 1 \).
- All internal angles are equal, each being \( 135^\circ \).
2. **Extend the non-parallel sides containing \( A \) and \( B \) to intersect at point \( C \):**
- Since the octagon is regular, the angle between... | [
{
"content": "In the octagon below all sides have the length $1$ and all angles are equal.\nDetermine the distance between the corners $A$ and $B$ .\n",
"role": "u... |
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 | The diagram below shows a $12$ by $20$ rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region.
 | 1. **Understanding the Problem:**
The problem involves a $12$ by $20$ rectangle split into four strips of equal widths, surrounding an isosceles triangle. We need to find the area of the shaded region.
2. **Identifying the Shapes:**
The rectangle is divided into:
- Two right triangles
- One isosceles trian... | [
{
"content": "The diagram below shows a $12$ by $20$ rectangle split into four strips of equal widths all surrounding an isosceles triangle. Find the area of the shaded region.\n",
"role": "user"
},... |
aops_forum | <u>Round 1</u>**p1.** A circle is inscribed inside a square such that the cube of the radius of the circle is numerically equal to the perimeter of the square. What is the area of the circle?**p2.** If the coefficient of $z^ky^k$ is $252$ in the expression $(z + y)^{2k}$ , find $k$ .**p3.** Let $f(x) = \frac{4x^... | To find the number of 5-digit numbers with strictly increasing digits, we need to choose 5 distinct digits from the set \(\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\). Note that 0 cannot be included because it cannot be a leading digit in a 5-digit number.
1. **Choose 5 digits from 9 possible digits:**
We need to select 5 digit... | [
{
"content": "<u>Round 1</u>**p1.** A circle is inscribed inside a square such that the cube of the radius of the circle is numerically equal to the perimeter of the square. What is the area of the circle?**p2.** If the coefficient of $z^ky^k$ is $252$ in the expression $(z + y)^{2k}$ , find $k$ .**p3.** ... |
aops_forum | The diagram shows a polygon made by removing six $2\times 2$ squares from the sides of an $8\times 12$ rectangle. Find the perimeter of this polygon.
 | 1. **Calculate the perimeter of the original $8 \times 12$ rectangle:**
The perimeter \( P \) of a rectangle is given by the formula:
\[
P = 2 \times (\text{length} + \text{width})
\]
Here, the length is 12 units and the width is 8 units. Therefore,
\[
P = 2 \times (12 + 8) = 2 \times 20 = 40 \text... | [
{
"content": "The diagram shows a polygon made by removing six $2\\times 2$ squares from the sides of an $8\\times 12$ rectangle. Find the perimeter of this polygon.\n",
"role": "user"
},
{
"c... |
aops_forum | Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$ . Suppose that $AH,AO$ and $AM$ are the altitude, the bisector and the median derived from $A$ , respectively. If $HO = 3 MO$ , then the length of $BC$ is
, \( AB = c = 3 \), and \( AC = b = 4 \). We are given that \( HO = 3MO \). Let \( OM = x \), then \( HO = 3x \) and \( MO = x \).
2. **Use the Angle Bisector Theorem:**
The angle bisector theorem states that the ratio of the two segments created by the ... | [
{
"content": "Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$ . Suppose that $AH,AO$ and $AM$ are the altitude, the bisector and the median derived from $A$ , respectively. If $HO = 3 \u0002MO$ , then the length of $BC$ is\n![Image](https://cdn.artofproblemsolving.com/attachments/e/c/26cc... |
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