source stringclasses 1
value | problem stringlengths 185 1.31k | solution stringlengths 611 4.12k | messages listlengths 2 2 |
|---|---|---|---|
olympiads | Consider the following figure:
We seek to label the nine fields in the figure with the numbers 1, 2, .., 9, using each number exactly once. Furthermore, the ... |
1. Let's denote the numbers in the fields of the figure as follows:
2. If we denote the common value of the six sums by \( s \), we obtain the following equations from the problem ... | [
{
"content": "Consider the following figure:\n\n\n\nWe seek to label the nine fields in the figure with the numbers 1, 2, .., 9, using each number exact... |
olympiads | Some Christmas lights are arranged using magical wires. Each light can be either green or yellow. Each wire connects two lights and has a magical property: when someone touches a wire connecting two lights, each light changes color from green to yellow or from yellow to green.
a) In the following arrangement, each poi... |
**(a) Problem:**
Each edge we touch changes exactly the color of two lamps. Since there are 16 yellow lamps, we must touch at least 8 edges.
**1. Understand the mechanism:**
- Touching a wire connecting two lamps changes both lamps' colors.
- Goal is to turn all the lamps from yellow to green.
**2. Fundamental... | [
{
"content": "Some Christmas lights are arranged using magical wires. Each light can be either green or yellow. Each wire connects two lights and has a magical property: when someone touches a wire connecting two lights, each light changes color from green to yellow or from yellow to green.\n\na) In the followi... |
olympiads | Gabrielle has a $3 \times 3$ grid and L-shaped pieces that cover three squares each, as shown below. Can she cover her grid using these pieces?
 |
1. **Understand the $3 \times 3$ grid and the L-shaped piece:**
- The L-shaped piece covers exactly 3 squares on the grid.
- The grid itself is a $3 \times 3$ matrix, meaning it has a total of $3 \times 3 = 9$ squares.
2. **Consider the full coverage by multiple L-shaped pieces:**
- To cover the entire $3 \t... | [
{
"content": "Gabrielle has a $3 \\times 3$ grid and L-shaped pieces that cover three squares each, as shown below. Can she cover her grid using these pieces?\n\n",
... |
olympiads | In the diagram, the \(5 \times 6\) grid is made out of thirty \(1 \times 1\) squares. What is the total length of the six solid line segments shown?
(A) 6
(B) 12
(C) 16
(D) 18
(E) 20
 | :
To determine the total length of the six solid line segments shown in the diagram, we follow these steps:
1. **Identify each individual segment length**:
- The lengths of the solid line segments from top to bottom are:
\[
5, 1, 4, 2, 3, \text{and} 3
\]
2. **Sum these lengths**:
\[
5 + 1 + 4... | [
{
"content": "In the diagram, the \\(5 \\times 6\\) grid is made out of thirty \\(1 \\times 1\\) squares. What is the total length of the six solid line segments shown?\n(A) 6\n(B) 12\n(C) 16\n(D) 18\n(E) 20\n\n![Image](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-044.jpg?height=331&width=37... |
olympiads | The word "stop" starts in the position shown in the diagram to the right. It is then rotated $180^{\circ}$ clockwise about the origin, $O$, and this result is then reflected in the $x$-axis. Which of the following represents the final image?
(A)
![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-089.... | :
1. **Initial Position**: Let's first understand the initial position of the word "stop".
2. **Rotation by $180^\circ$**: Rotating the word $180^\circ$ clockwise abou... | [
{
"content": "The word \"stop\" starts in the position shown in the diagram to the right. It is then rotated $180^{\\circ}$ clockwise about the origin, $O$, and this result is then reflected in the $x$-axis. Which of the following represents the final image?\n(A)\n\n
(The internal intersection point is the center of the square. All segments of the squares are vis... |
We need to find two non-congruent polyhedra that both have a front view and top view matching the given diagram:
In both these views, the internal intersection point is the center of the squ... | [
{
"content": "Show two non-congruent polyhedra whose front view and top view are as follows:\n\n\n\n(The internal intersection point is the center of the square. All segme... |
olympiads | In the figure below, \(ABCD\) is a square and points \(K\), \(L\), and \(M\) are on the sides \(AB\), \(BC\), and \(CD\) respectively, such that \(\triangle KLM\) is a right isosceles triangle at \(L\). Prove that \(AL\) and \(DK\) are perpendicular.
 is a square.
- Points \(K\), \(L\), and \(M\) lie on sides \(AB\), \(BC\), and \(CD\) respectively.
- \(\triangle KLM\) is an isosceles right triangle at \(L\).
2. **Identify Key Angles:**
- Since \(\triangle KLM\) is an isosceles right triangle at \(L\), we know:
\[
... | [
{
"content": "In the figure below, \\(ABCD\\) is a square and points \\(K\\), \\(L\\), and \\(M\\) are on the sides \\(AB\\), \\(BC\\), and \\(CD\\) respectively, such that \\(\\triangle KLM\\) is a right isosceles triangle at \\(L\\). Prove that \\(AL\\) and \\(DK\\) are perpendicular.\n\n\) illuminates a right-angle region consisting of points \((x, y)\) where \(x \geq m\) and \(y \geq n\). How many lamps are on the plane if the following cells ... | 1. **Define the problem setting**:
- We have a grid with a finite number of lamps, each located at lattice points \((m, n)\) with integer coordinates.
- A lamp at point \((m, n)\) illuminates points \((x, y)\) where \(x \geq m\) and \(y \geq n\).
- We need to find the number of lamps on the grid when exactl... | [
{
"content": "\nThere is a grid plane. At the grid nodes, that is, at points with integer coordinates, there are a finite number of lamps. A lamp located at the point \\((m, n)\\) illuminates a right-angle region consisting of points \\((x, y)\\) where \\(x \\geq m\\) and \\(y \\geq n\\). How many lamps are on ... |
olympiads | Point \(O\) is the center of the circle. What is the value of angle \(x\) in degrees?
 |
1. **Understanding the problem**:
We are given a circle with center at point \( O \) and are asked to find the value of angle \( x \).
2. **Identifying given values**:
We are given two angles in the diagram: \( \angle BCO = 32^\circ \) and \( \angle ACO = x \). Since \( OB = OC \) and \( OA = OC \), triangle \(... | [
{
"content": "Point \\(O\\) is the center of the circle. What is the value of angle \\(x\\) in degrees?\n\n",
"role": "user"
},
{
"content": "\n1. **Understa... |
olympiads | In a tetrahedron, all the plane angles at one vertex are right angles. A certain point in space is at a distance of 3 from the mentioned vertex of the tetrahedron and at distances of \(\sqrt{5}\), \(\sqrt{6}\), and \(\sqrt{7}\) from the other vertices of the tetrahedron. Find the distances from the center of the circum... |
Let the tetrahedron be \(ABCD\) with all planar angles at vertex \(D\) being right angles. Let the lengths of the segments \(DA\), \(DB\), and \(DC\) be \(a\), \(b\), and \(c\) respectively. We shall connect these segments with the rectangular coordinate system \(Dxyz\) such that \(D\) has coordinates \((0,0,0)\) and ... | [
{
"content": "In a tetrahedron, all the plane angles at one vertex are right angles. A certain point in space is at a distance of 3 from the mentioned vertex of the tetrahedron and at distances of \\(\\sqrt{5}\\), \\(\\sqrt{6}\\), and \\(\\sqrt{7}\\) from the other vertices of the tetrahedron. Find the distance... |
olympiads | In the figure below, \( \triangle ABC \) is a triangle with side lengths \( AB = 4 \, \text{cm}, AC = 6 \, \text{cm} \). Additionally, \( \angle DAH = \angle HAB \), \( \angle AHB = 90^\circ \), and \( M \) is the midpoint of \( BC \). Find the length of the segment \( MH \).
 with side lengths \(AB = 4 \text{ cm}\) and \(AC = 6 \text{ cm}\).
- \(\angle DAH = \angle HAB\) and \( \angle AHB = 90^\circ \). This indicates \(AH\) is both an altitude and an angle bisector.
2. **Conclusion about triangle \(ABD... | [
{
"content": "In the figure below, \\( \\triangle ABC \\) is a triangle with side lengths \\( AB = 4 \\, \\text{cm}, AC = 6 \\, \\text{cm} \\). Additionally, \\( \\angle DAH = \\angle HAB \\), \\( \\angle AHB = 90^\\circ \\), and \\( M \\) is the midpoint of \\( BC \\). Find the length of the segment \\( MH \\)... |
olympiads | There are 7 absolutely identical dice, each of which has 1 dot on one face, 2 dots on another face, ..., and 6 dots on the sixth face. Furthermore, the sum of the dots on two opposite faces is 7.
These 7 dice are combined to form a figure, as shown in the figure, in such a way that the number of dots on each pair of g... |
1. Let's begin by understanding the distribution of dots on the cubes. Each cube has a total of 7 dot markings split across pairs of opposite faces such that the dots on two opposite faces sum to 7.
2. When arranging these 7 cubes to form the given figure, we observe that there are nine pairs of glued faces, and glu... | [
{
"content": "There are 7 absolutely identical dice, each of which has 1 dot on one face, 2 dots on another face, ..., and 6 dots on the sixth face. Furthermore, the sum of the dots on two opposite faces is 7.\n\nThese 7 dice are combined to form a figure, as shown in the figure, in such a way that the number o... |
olympiads |
On the faces of a cube are drawn six letters: А, Б, В, Г, Д, Е. The picture shows three images of the cube from different points of view. Which letter is drawn opposite the face with the letter $\mathrm{B}$?
![Image of the cube from different angles](https://cdn.mathpix.com/cropped/2024_05_06_4cfa7cdc7a8a51b9e752g-24... |
1. **Identifying the visible faces:**
- From the given images, we observe the faces of the cube labeled $\mathrm{A}$, $\mathrm{B}$, $\mathrm{D}$, and $\mathrm{E}$.
2. **Establishing neighboring relationships:**
- The face labeled $\mathrm{B}$ is adjacent to faces labeled $\mathrm{A}$, $\mathrm{D}$, and $\mathrm... | [
{
"content": "\nOn the faces of a cube are drawn six letters: А, Б, В, Г, Д, Е. The picture shows three images of the cube from different points of view. Which letter is drawn opposite the face with the letter $\\mathrm{B}$?\n\n
(d)
![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-004.jpg?height=208&... | ### Solução:
1. **Primeira Observação**:
Para formar um quadrado utilizando as peças fornecidas, o total de quadrados pequenos deve ser um quadrado perfeito. Isto é, um número inteiro que é o quadrado de algum outro número inteiro.
2. **Identificação dos Quadrados Perfeitos**:
Os quadrados perfeitos menores ... | [
{
"content": "Pedro assembled a square with four out of the five pieces below. Which piece did he not use?\n\n(a)\n\n\n(d)\n |
1. **Identify the Dimensions of the Rectangle:**
- The problem states a rectangle composed of $5 \times m$ unit squares. This means the rectangle has 5 rows and $m$ columns.
2. **Understanding the Pentomino $\mathrm{P}$:**
- A pentomino is a shape made out of 5 connected unit squares. Various shapes can be form... | [
{
"content": "We consider a rectangle composed of $5 \\times m$ unit squares. Can it be tiled by the pentomino $\\mathrm{P}$ shown below? Rotations and reflections of the pentomino $P$ are allowed.\n\n grid.
1. **Identify the sums on the diagonals:**
According to the problem statement, the sums of the numbers on one diagonal are:
\[
1 + 2 + 4 = 7
\]
and on the other diagonal:
... | [
{
"content": "The numbers from 1 to 9 are arranged in the cells of a $3 \\times 3$ table such that the sum of the numbers on one diagonal is 7, and on the other diagonal, it is 21. What is the sum of the numbers in the five shaded cells?\n\n and \( F \) are placed on a square grid, as shown. How many triangles that are not right-angled can be drawn by using 3 of these 6 points as vertices?
(A) 2
(B) 1
(C) 6
(D) 0
(E) 4
.
2. **Classify Triangles with Edges on $DEF$**:
List all possible triangles formed that have at least one edge from $D E, D F,$ or $E F$:
\[
\... | [
{
"content": "Six points \\( A, B, C, D, E, \\) and \\( F \\) are placed on a square grid, as shown. How many triangles that are not right-angled can be drawn by using 3 of these 6 points as vertices?\n(A) 2 \n(B) 1 \n(C) 6 \n(D) 0 \n(E) 4\n\n | 1. **Identify the structure of the problem**: Consider the configuration in the problem where \(\overline{OC} = 12\) and \(\overline{DM} = 10\). We aim to find the length of \(\overline{BO}\).
2. **Analyze triangle congruency**: If we extend segments \(AB\) and \(CD\), we form triangle \(APD\), which is congruent to t... | [
{
"content": "\nIn the figure below, $\\overline{OC}=12$ and $\\overline{DM}=10$.\n\nCalculate $\\overline{BO}$.\n\n",
"role": "user"
},
{
"content": "1. **... |
olympiads | In the figure, the two triangles \( \triangle C E F \) and \( \triangle D I H \) are equilateral, and the angles are given in degrees. Determine the value of \( x \).
 | ### Problem:
In the figure, the two triangles $C E F$ and $D I H$ are equilateral and the angles are given in degrees. Determine the value of $x$.
1. **Identify the given information**:
- $\triangle C E F$ and $\triangle D I H$ are equilateral triangles.
- Given angles: $\angle FCD = 75^\circ$ and $\angle MDC = ... | [
{
"content": "In the figure, the two triangles \\( \\triangle C E F \\) and \\( \\triangle D I H \\) are equilateral, and the angles are given in degrees. Determine the value of \\( x \\).\n\n?
(A) 65
(B) 75
(C) 85
(D) 95
(E) 105
 | 1. **Identify the sum of the angles in a triangle**:
The sum of the interior angles in any triangle is always \(180^\circ\).
2. **Set up the equation for the given triangle**:
We are given two angles in the triangle, \(50^\circ\) and \(45^\circ\), along with \(x^\circ\) which we need to find.
\[
50^\... | [
{
"content": "In the diagram, what is the value of \\( x \\)?\n(A) 65\n(B) 75\n(C) 85\n(D) 95\n(E) 105\n\n",
"role": "user"
},
{
"content": "1. **Identify the sum of the a... |
olympiads | We obtain the square of a three-digit number through the sketchy squaring method shown here, where $A$ denotes the same digit. What is that number?
 |
We need to find a three-digit number whose square follows a specific pattern. Here is a step-by-step detailed solution.
1. **Understanding the Given Information:
Discover a three-digit number designated by \(AAA\) such that its square follows a particular arrangement. From the given problem, we have some scattered... | [
{
"content": "We obtain the square of a three-digit number through the sketchy squaring method shown here, where $A$ denotes the same digit. What is that number?\n\n",
"role": "user"... |
olympiads | Five cards are on a table, each with a number on one side and a letter on the other. Simone must determine whether the following statement is true: "If a card has a vowel on one side, then it has an even number on the other." What is the minimum number of cards she needs to turn over to make a correct decision?
![](ht... |
We are given five cards, each with a number on one face and a letter on the other. Simone needs to determine the truth of the statement: "If a card has a vowel on one face, then it has an even number on the other."
To solve this, we need to examine which cards must be turned over to check the veracity of the statemen... | [
{
"content": "Five cards are on a table, each with a number on one side and a letter on the other. Simone must determine whether the following statement is true: \"If a card has a vowel on one side, then it has an even number on the other.\" What is the minimum number of cards she needs to turn over to make a c... |
olympiads | In the figure below, we have a circle intersected by 4 segments. Write the numbers from 1 to 9 in the circles so that the sum of the numbers written in each segment is always the same.
}{2}
\]
where \( n \) is the highest number in the series. For numbers 1 to 9:
\[
\text{Sum} = \f... | [
{
"content": "In the figure below, we have a circle intersected by 4 segments. Write the numbers from 1 to 9 in the circles so that the sum of the numbers written in each segment is always the same.\n\n could be
(A) \( (1,3) \)
(B) \( (1,-3) \)
(C) \( (-3,1) \)
(D) \( (3,-1) \)
(E) \( (-1,3) \)
 |
1. **Identify the range for the coordinates:**
- From the problem statement and the diagram, we know:
- The $x$-coordinate of point $P$ lies between $-2$ and $0$:
\[
-2 < x < 0
\]
- The $y$-coordinate of point $P$ lies between $2$ and $4$:
\[
2 < y... | [
{
"content": "In the diagram, the coordinates of point \\( P \\) could be\n(A) \\( (1,3) \\)\n(B) \\( (1,-3) \\)\n(C) \\( (-3,1) \\)\n(D) \\( (3,-1) \\)\n(E) \\( (-1,3) \\)\n\n is a parallelogram, which means it is a quadrilateral with opposite sides having the same length (and are parallel). Additionally, segment \(BE\) has a length of 6, and segments \(CD\) and \(DE\) have the same length. What is the length of segment \(FG\)?
 is a parallelogram, meaning opposite sides are equal and parallel. The length of segment \(BE\) is given as 6, and segments \(CD\) and \(DE\) have equal lengths.
2. **Creating a Congruent Parallelogram:**
To solve the problem, we draw another parallelogram below \(... | [
{
"content": "In the figure below, \\(ABCD\\) is a parallelogram, which means it is a quadrilateral with opposite sides having the same length (and are parallel). Additionally, segment \\(BE\\) has a length of 6, and segments \\(CD\\) and \\(DE\\) have the same length. What is the length of segment \\(FG\\)?\n\... |
olympiads | In the diagram below, triangle \( \triangle ABC \) is an equilateral triangle and \( CD \) is both an altitude of the triangle and a diameter of the circle. If \( AB = 10 \) cm, determine the shaded area.
 is equilateral with side length \( AB = 10 \) cm, the altitude can be calculated as:
\[
\text{Altitude} = \frac{\sqrt{3}}{2} \times \text{Side length} = \frac{\sqrt{3}}{2} \times 10 = 5\sqrt{3} \, \text{cm}
\]
... | [
{
"content": "In the diagram below, triangle \\( \\triangle ABC \\) is an equilateral triangle and \\( CD \\) is both an altitude of the triangle and a diameter of the circle. If \\( AB = 10 \\) cm, determine the shaded area.\n\n that must be used to leave the board "full," meaning no more pieces can be placed.
1. **Verification Example**:
- An example of a filled board with 9 pieces is provided. The figure below illustrates... | [
{
"content": "\nSome gray pieces in the shape of a $2 \\times 1$ domino can be used to cover the squares of a $5 \\times 5$ board. We say that the board is crowded when there is no space to place new pieces, as in the example below. What is the minimum number of pieces required to leave a board crowded?\n\n | 1. **Identify the location of points $P$ and $Q$:**
\( P \) is located 3 units above the $x$-axis.
\( Q \) is located 6 units above the $x$-axis.
2. **Reflect point $P$ across the $x$-axis:**
- The reflection of a point across the $x$-axis changes the $y$-coordinate to its opposite value.
- Since ... | [
{
"content": "In the graph shown, which of the following represents the image of the line segment $PQ$ after a reflection across the $x$-axis?\n(A) $PS$\n(B) $TU$\n(C) $MN$\n(D) $WV$\n(E) $FG$\n\n?
(A) 1
(B) 3
(C) 5
(D) 0
(E) 2
 must take on the values 2, 3, 4,... | [
{
"content": "The integers 1 to 6 are to be inserted into the grid shown. No two integers that differ by 1 may be in squares that share an edge. If the 1 is inserted as shown, how many different integers can be placed in the box labelled \\( x \\)?\n\n(A) 1 \n(B) 3 \n(C) 5 \n(D) 0 \n(E) 2 \n\n![Grid Image]... |
olympiads | Which of the points positioned on the number line best represents the value of \( S \div T \)?
(A) \( P \)
(B) \( Q \)
(C) \( R \)
(D) \( T \)
(E) \( U \)
 |
1. First, identify the approximate values of \( S \) and \( T \) on the number line:
- \( S \) is approximately 1.5
- \( T \) is approximately 1.6
2. Next, compute \( \frac{S}{T} \):
\[
\frac{S}{T} \approx \frac{1.5}{1.6}
\]
To simplify \( \frac{1.5}{1.6} \), perform the division:
\[
\frac{1.5... | [
{
"content": "Which of the points positioned on the number line best represents the value of \\( S \\div T \\)?\n(A) \\( P \\) \n(B) \\( Q \\) \n(C) \\( R \\) \n(D) \\( T \\) \n(E) \\( U \\)\n\n with orthocenter \(H\) has \(AB = 48\) and \(HC = 14\). The midpoint of side \(AB\) is \(M\) and the midpoint of segment \(HC\) is \(N\).
(a) Show that the angle \(\angle MEN\) is a right angle.
(b) Determine the length of segment \(MN\).
:
1. **Understanding the Construction**:
The triangle \(ABC\) is acute-angled, and \(H\) is its orthocenter. The points \(M\) and \(N\) are the midpoints of the segments \(AB\) and \(HC\), respectively.
2. **Using Median Properties**:
Since \(M\) is the midpoint of \(AB\):
\[
ME \text{ is ... | [
{
"content": "\nThe acute triangle \\(ABC\\) with orthocenter \\(H\\) has \\(AB = 48\\) and \\(HC = 14\\). The midpoint of side \\(AB\\) is \\(M\\) and the midpoint of segment \\(HC\\) is \\(N\\).\n\n(a) Show that the angle \\(\\angle MEN\\) is a right angle.\n\n(b) Determine the length of segment \\(MN\\).\n\n... |
olympiads | In the figure below, we have a $5 \times 5$ square that contains a central black square. There is a collection of squares with sides parallel to the sides of the board, with dimensions ranging from $1 \times 1$ to $5 \times 5$, formed by the small squares in the figure. How many elements of this collection contain the ... |
To solve this problem, we need to count the number of squares with sizes varying from $1 \times 1$ to $5 \times 5$ that contain the central $1 \times 1$ black square in the given $5 \times 5$ grid. Let's approach this by dealing with each possible size of the square separately.
1. **Squares of size $1 \times 1$**:
... | [
{
"content": "In the figure below, we have a $5 \\times 5$ square that contains a central black square. There is a collection of squares with sides parallel to the sides of the board, with dimensions ranging from $1 \\times 1$ to $5 \\times 5$, formed by the small squares in the figure. How many elements of thi... |
olympiads | Diogo and Helen play the Game of Removal, which consists of the following rules. Given a grid of \(1 \times 1\) squares, each player, on their turn, can choose a square and then remove from the grid all squares below it, all squares to the left of it, and all others that are both below and to the left of it. For exampl... |
**Part a)**
Helen starts by removing the square shown below:
After this move, Diogo has two options:
1. **Option 1:** If Diogo removes the square highlighted below,
![](https://cdn.m... | [
{
"content": "Diogo and Helen play the Game of Removal, which consists of the following rules. Given a grid of \\(1 \\times 1\\) squares, each player, on their turn, can choose a square and then remove from the grid all squares below it, all squares to the left of it, and all others that are both below and to t... |
olympiads | In the following figure, the square \(ABCD\) is divided into four rectangles, all having the same area.
Given that \(MN = 3\), calculate the area of the square ... | 1. Let’s start by analyzing the given figure where the square \(ABCD\) is divided into four rectangles with equal area. We know that \(MN = 3\).
2. We take into account the rectangles \(MNPC\) and \(NQFP\). For the areas of these rectangles to be equal, the dimensions must conform to specific constraints. Since the ent... | [
{
"content": "In the following figure, the square \\(ABCD\\) is divided into four rectangles, all having the same area.\n\n\n\nGiven that \\(MN = 3\\), cal... |
olympiads | Fix a positive integer \( n \). Two players, Phil and Ellie, play the following game. First, Phil fills the squares of an \( n \times n \) chessboard with nonnegative integers less than \( n \). Then, Ellie chooses three squares making an L, as in any of the following pictures:
 Phil wins in this case.**
1. **Sum Analysis:**
- Consider the sum $S$ of all the numbers on the chessboard.
- Each move by Ellie involves adding 1 to three chosen squares in an L-pattern.
2. **Sum Change:**
- A single move adds 3 to the sum since three squares each increase by 1.
- If any of the... | [
{
"content": "Fix a positive integer \\( n \\). Two players, Phil and Ellie, play the following game. First, Phil fills the squares of an \\( n \\times n \\) chessboard with nonnegative integers less than \\( n \\). Then, Ellie chooses three squares making an L, as in any of the following pictures:\n\n![](https... |
olympiads | Leonid has a white checkered rectangle. First, he painted every other column gray, starting with the leftmost one, and then every other row, starting with the topmost one. All cells adjacent to the border of the rectangle ended up being painted.
How many cells could be painted in the rectangle if 74 white cells remain... |
1. **Understanding the Problem:**
Leonid has a rectangular grid where he paints every other column starting from the leftmost one, and then he paints every other row starting from the top row. The boundary cells of the rectangle are thus all painted. The problem requires us to determine how many total painted cells... | [
{
"content": "Leonid has a white checkered rectangle. First, he painted every other column gray, starting with the leftmost one, and then every other row, starting with the topmost one. All cells adjacent to the border of the rectangle ended up being painted.\n\nHow many cells could be painted in the rectangle ... |
olympiads | The word "stop" starts in the position shown in the diagram. It is then rotated \(180^{\circ}\) clockwise about the origin, \(O\), and this result is then reflected in the \(x\)-axis. Which of the following represents the final image?
(A), (B), (C), (D), or (E)
![diagram](https://cdn.mathpix.com/cropped/2024_04_20_46e... |
1. **Initial Position of the Word "STOP"**:
The word "stop" starts in its normal position where 's', 't', 'o', 'p' are arranged left to right.
2. **Rotation by $180^{\circ}$ Clockwise around the Origin**:
When we rotate an object by $180^{\circ}$ clockwise, each point $(x, y)$ of the object moves to $(-x, -y)$.... | [
{
"content": "The word \"stop\" starts in the position shown in the diagram. It is then rotated \\(180^{\\circ}\\) clockwise about the origin, \\(O\\), and this result is then reflected in the \\(x\\)-axis. Which of the following represents the final image?\n(A), (B), (C), (D), or (E)\n\n
She did so in a way that all the numbers were used. Her older brother, P... |
#### (a) Show that the number in the top circle is 4 and the constant sum is 12.
1. **Define the Problem:**
- Let \( s \) be the constant sum of the numbers in each collinear trio of circles.
- Let \( x \) be the number that Luana placed in the top circle.
2. **Sum of all numbers:**
- Sum of all the ... | [
{
"content": "Inside each of the circles shown in the figure below, little Luana placed one of the numbers 1, 2, 3, 4, 5, 6, and 7.\n\n\n\nShe did so in a way that all the numbers were ... |
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