source stringclasses 1
value | problem stringlengths 147 5.08k | solution stringlengths 113 6.3k | 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 | **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 | **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 | 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 $\\angle BAC$ 
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 | 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 = 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 | 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 | 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**p2.** If $a, b$ , and $c$ are numbers so... | 1. Given the conditions \(a + b + c = 0\) and \(a^2 + b^2 + c^2 = 1\), we need to find \(a^4 + b^4 + c^4\).
2. Consider the polynomial \(P(x) = x^3 + px^2 + qx + r\) with roots \(a, b, c\). By Vieta's formulas, we know:
\[
a + b + c = -p, \quad ab + bc + ca = q, \quad abc = -r
\]
3. Given \(a + b + c = 0\), we... | [
{
"content": "**p1.** A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the square $ABCD$ is $14$ cm.\n**p2.** If $a, b$ , a... |
aops_forum | In each square of the table below, we must write a different integer from $1$ to $17$ , such that the sum of the numbers in each of the eight columns is the same, and the sum of the numbers in the top row is twice the sum of the numbers in the bottom row. Which number from $1$ to $17$ can be omitted?
 positive integers is given by the formula:
\[
S = \frac{n(n+1)}{2}
\]
For \( n = 17 \):
\[
S = \frac{17 \times 18}{2} = 153
\]
2. **Sum of the numbers in the top and bottom rows**:
Let the sum of the numbers in the bottom row ... | [
{
"content": "In each square of the table below, we must write a different integer from $1$ to $17$ , such that the sum of the numbers in each of the eight columns is the same, and the sum of the numbers in the top row is twice the sum of the numbers in the bottom row. Which number from $1$ to $17$ can b... |
aops_forum | Points $A,B,C,D$ have been marked on checkered paper (see fig.). Find the tangent of the angle $ABD$ .
 | 1. **Identify the coordinates of the points:**
- Let's assume the coordinates of the points on the checkered paper.
- Suppose \( A = (0, 0) \), \( B = (1, 0) \), \( C = (1, 1) \), and \( D = (3, 1) \).
2. **Verify the right angle at \( B \):**
- The triangle \( ABC \) is right-angled at \( B \) because \( AB ... | [
{
"content": "Points $A,B,C,D$ have been marked on checkered paper (see fig.). Find the tangent of the angle $ABD$ . \n",
"role": "user"
},
{
"content": "1. **Identify the coordinates of the po... |
aops_forum | Three $n\times n$ squares form the figure $\Phi$ on the checkered plane as shown on the picture. (Neighboring squares are tpuching along the segment of length $n-1$ .)
Find all $n > 1$ for which the figure $\Phi$ can be covered with tiles $1\times 3$ and $3\times 1$ without overlapping. the figure \(\Phi\) can be covered with \(1 \times 3\) and \(3 \times 1\) tiles without overlapping. The figure \(\Phi\) consists of three \(n \times n\) squares arranged such that neighboring squares touch along a segment of length \(n-1\).
1. **Case \(3 \mid n\):*... | [
{
"content": "Three $n\\times n$ squares form the figure $\\Phi$ on the checkered plane as shown on the picture. (Neighboring squares are tpuching along the segment of length $n-1$ .)\nFind all $n > 1$ for which the figure $\\Phi$ can be covered with tiles $1\\times 3$ and $3\\times 1$ without ove... |
aops_forum | p1. Find all the primes $p$ such that $p^2 + 2$ is a prime number.
[p2.](https://artofproblemsolving.com/community/c4h1846777p12437991) In the drawing, the five circles are tangent to each other and tangents to the lines $L_1$ and $L_2$ as shown in the following figure. The smallest of the circles has radius ... | To solve the problem, we need to consider the worst-case scenario where Constanza takes out the maximum number of pencils without meeting the requirement of having at least 1 red, 2 blue, and 3 green pencils.
1. **Identify the worst-case scenario:**
- Constanza could take out all 13 blue pencils and all 8 green pe... | [
{
"content": "p1. Find all the primes $p$ such that $p^2 + 2$ is a prime number.\n\n\n[p2.](https://artofproblemsolving.com/community/c4h1846777p12437991) In the drawing, the five circles are tangent to each other and tangents to the lines $L_1$ and $L_2$ as shown in the following figure. The smallest o... |
aops_forum | The circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$ such that point $C$ lies on segment $\overline{AB}$ . Point $M$ is on the circle centered at $A$ and point $N$ is o... | 1. **Identify the given information and set up the problem:**
- Circle centered at \( A \) with radius \( 19 \).
- Circle centered at \( B \) with radius \( 32 \).
- Both circles are internally tangent to a larger circle centered at \( C \) with radius \( 100 \).
- Point \( C \) lies on segment \( \overline... | [
{
"content": "The circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$ such that point $C$ lies on segment $\\overline{AB}$ . Point $M$ is on the circle centered at $A... |
aops_forum | There is a board with the shape of an equilateral triangle with side $n$ divided into triangular cells with the shape of equilateral triangles with side $ 1$ (the figure below shows the board when $n = 4$ ). Each and every triangular cell is colored either red or blue. What is the least number of cells that can be... | To solve this problem, we need to determine the least number of blue cells such that no two red cells share a side. We will use a systematic approach to find the minimum number of blue cells required.
1. **Understanding the Problem:**
- We have an equilateral triangle of side length \( n \) divided into smaller equ... | [
{
"content": "There is a board with the shape of an equilateral triangle with side $n$ divided into triangular cells with the shape of equilateral triangles with side $ 1$ (the figure below shows the board when $n = 4$ ). Each and every triangular cell is colored either red or blue. What is the least numbe... |
aops_forum | p1. Bahri lives quite close to the clock gadang in the city of Bukit Tinggi West Sumatra. Bahri has an antique clock. On Monday $4$ th March $2013$ at $10.00$ am, Bahri antique clock is two minutes late in comparison with Clock Tower. A day later, the antique clock was four minutes late compared to the Clock Tower... | To solve the problem, we need to find the sum of all the digits that make up \( B \), where \( B \) is the sum of the digits of \( A \), and \( A \) is the sum of the digits of \( M = 2014^{2014} \).
1. **Estimate the number of digits in \( M \):**
\[
M = 2014^{2014}
\]
We can approximate \( 2014 \) as sli... | [
{
"content": "p1. Bahri lives quite close to the clock gadang in the city of Bukit Tinggi West Sumatra. Bahri has an antique clock. On Monday $4$ th March $2013$ at $10.00$ am, Bahri antique clock is two minutes late in comparison with Clock Tower. A day later, the antique clock was four minutes late compa... |
aops_forum | Two infinite rows of evenly-spaced dots are aligned as in the figure below. Arrows point from every dot in the top row to some dot in the lower row in such a way that:
- No two arrows point at the same dot.
- Now arrow can extend right or left by more than 1006 positions.
.
2. **Consider the points directly above these 2013 points in the top row.**
Denote these points as \( a_1, a_2, \ldots, a_{2013} \), where \( a_i \) is direc... | [
{
"content": "Two infinite rows of evenly-spaced dots are aligned as in the figure below. Arrows point from every dot in the top row to some dot in the lower row in such a way that:\n\n- No two arrows point at the same dot.\n- Now arrow can extend right or left by more than 1006 positions.\n\n. Let also triangles are painted like on the chess board. Prove that exactly $n$ coins lie on black tr... | 1. **Define the coordinates and properties of the triangles:**
- Assume the height of each small triangle is 1.
- Define the \(a\)-coordinate, \(b\)-coordinate, and \(c\)-coordinate of a triangle as the minimum distance from a point on its boundary to the top side, the left-bottom side, and the right-bottom side ... | [
{
"content": "Let regular hexagon is divided into $6n^2$ regular triangles. Let $2n$ coins are put in different triangles such, that no any two coins lie on the same layer (layer is area between two consecutive parallel lines). Let also triangles are painted like on the chess board. Prove that exactly $n$ ... |
aops_forum | In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$ , and the angular bisector from $B$ intersects the side $AC$ at the point $E$ . Furthermore $|AE| + |BD| = |AB|$ . Prove that $\angle C = 60^o$ , \( BC = a \), and \( CA = b \).
- The angle bisecto... | [
{
"content": "In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$ , and the angular bisector from $B$ intersects the side $AC$ at the point $E$ . Furthermore $|AE| + |BD| = |AB|$ . Prove that $\\angle C = 60^o$ .
- The bisector of \(\angle A\) intersects side \(BC\) at \(M\).
- The bisector of \(\angle AMC\) passes through point \(D\).
- \(\angle MDC = 45^\circ\).
2. **Define the angles:**
- Let \(\angle BAM = \thet... | [
{
"content": "The bisector of angle $A$ in parallelogram $ABCD$ intersects side $BC$ at $M$ and the bisector of $\\angle AMC$ passes through point $D$ . Find angles of the parallelogram if it is known that $\\angle MDC = 45^o$ .\n where \(a_1 < a_2 < \cdots < a_k\).
2. The sum of the first \(k\) positive integers is given by the formula:
\[
S_k = \frac{k(k+1)}{2}
\]
3. We need to find the largest ... | [
{
"content": "**p1.** The number $100$ is written as a sum of distinct positive integers. Determine, with proof, the maximum number of terms that can occur in the sum.**p2.** Inside an equilateral triangle of side length $s$ are three mutually tangent circles of radius $1$ , each one of which is also tange... |
aops_forum | Colin has a peculiar $12$ -sided dice: it is made up of two regular hexagonal pyramids. Colin wants to paint each face one of three colors so that no two adjacent faces on the same pyramid have the same color. How many ways can he do this? Two paintings are considered identical if there is a way to rotate or flip the ... | 1. **Understanding the Problem:**
Colin has a 12-sided die made up of two regular hexagonal pyramids. He wants to paint each face one of three colors such that no two adjacent faces on the same pyramid have the same color. We need to find the number of ways to do this, considering that two paintings are identical if... | [
{
"content": "Colin has a peculiar $12$ -sided dice: it is made up of two regular hexagonal pyramids. Colin wants to paint each face one of three colors so that no two adjacent faces on the same pyramid have the same color. How many ways can he do this? Two paintings are considered identical if there is a way ... |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 150