source stringclasses 1
value | problem stringlengths 53 2.01k | solution stringlengths 732 6.8k | messages listlengths 2 2 |
|---|---|---|---|
olympiads | The founder of a noble family received a plot of land. Each man in the family, upon his death, divided the land equally among his sons. If a man had no sons, the land was transferred to the state. No other family members gained or lost land in any other way. There were a total of 180 people in the family. What is the s... |
Let's consider a problem where each male in a noble family divides his land equally among his sons upon death. If a person has no sons, his land reverts to the state. Given there are 180 members in the family, we need to find the smallest share of the original land that any member could inherit.
1. **Understanding th... | [
{
"content": "The founder of a noble family received a plot of land. Each man in the family, upon his death, divided the land equally among his sons. If a man had no sons, the land was transferred to the state. No other family members gained or lost land in any other way. There were a total of 180 people in the... |
olympiads | Determine the values of \(\lambda\) for which the function
$$
y=\frac{x^{2}-6 x+5}{x^{2}+4 x+\lambda}
$$
does not have any extrema. Examine the behavior of this function and plot it for \(\lambda = -12\). |
1. **Understanding Function Properties and Differentiability:**
Given the function
\[
y = \frac{x^2 - 6x + 5}{x^2 + 4x + \lambda}
\]
we need to determine the values of \(\lambda\) for which the function does not have any extremum (neither maximum nor minimum).
2. **Condition for No Extremum:**
... | [
{
"content": "Determine the values of \\(\\lambda\\) for which the function\n\n$$\ny=\\frac{x^{2}-6 x+5}{x^{2}+4 x+\\lambda}\n$$\n\ndoes not have any extrema. Examine the behavior of this function and plot it for \\(\\lambda = -12\\).",
"role": "user"
},
{
"content": "\n1. **Understanding Function P... |
olympiads | For a positive integer \( n \geq 3 \), plot \( n \) equally spaced points around a circle. Label one of them \( A \), and place a marker at \( A \). One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of \( 2n \) distinct moves available; tw... |
We will present the solution in a detailed step-by-step manner, referencing the initial problem statement and ensuring all necessary justifications are provided.
**First solution:**
1. **Problem Setup:**
We will consider $n$ equally spaced points labeled $0, 1, 2, \ldots, n-1$ placed around a circle. We are t... | [
{
"content": "For a positive integer \\( n \\geq 3 \\), plot \\( n \\) equally spaced points around a circle. Label one of them \\( A \\), and place a marker at \\( A \\). One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of \\( 2n... |
olympiads | A rectangular plot for rye cultivation was modified: one side was increased by 20%, while the other side was decreased by 20%. Will the rye yield change as a result, and if so, by how much? | 1. Let \(a\) and \(b\) be the original sides of the rectangular field.
2. After the restructuring, one side of the rectangle is increased by \(20\%\) and the other side is decreased by \(20\%\).
- Increasing \(a\) by \(20\%\) means the new length is:
\[
a_{\text{new}} = 1.2a
\]
- Decreasing \... | [
{
"content": "A rectangular plot for rye cultivation was modified: one side was increased by 20%, while the other side was decreased by 20%. Will the rye yield change as a result, and if so, by how much?",
"role": "user"
},
{
"content": "1. Let \\(a\\) and \\(b\\) be the original sides of the rectan... |
olympiads |
The cost of traveling on a toll road section depends on the vehicle class: passenger cars belong to the first class, for which the travel cost is 200 rubles, while light trucks and minibuses belong to the second class, for which the cost is 300 rubles.
At the toll payment point, an automatic system is installed which... |
1. **Problem Restatement**:
We need to find the optimal threshold value \( h \) for an automated toll system that minimizes the classification errors for vehicles passing through a toll booth. Vehicles are classified into two classes based on their height:
- Class 1: Heights less than \( h \), costing 20... | [
{
"content": "\nThe cost of traveling on a toll road section depends on the vehicle class: passenger cars belong to the first class, for which the travel cost is 200 rubles, while light trucks and minibuses belong to the second class, for which the cost is 300 rubles.\n\nAt the toll payment point, an automatic ... |
olympiads | Plot the solution set of the system of equations \(\left\{\begin{array}{l} x - 2y = 1 \\ x^{3} - 6xy - 8y^{3} = 1 \end{array}\right.\) on the coordinate plane. |
1. **Given system of equations**:
\[
\begin{cases}
x - 2y = 1 \\
x^3 - 6xy - 8y^3 = 1
\end{cases}
\]
2. **Rewriting the first equation**:
Given:
\[
x - 2y = 1
\]
We can express \(x\) in terms of \(y\):
\[
x = 2y + 1
\]
3. **Substituting \(x = 2y + 1\) into the second equati... | [
{
"content": "Plot the solution set of the system of equations \\(\\left\\{\\begin{array}{l} x - 2y = 1 \\\\ x^{3} - 6xy - 8y^{3} = 1 \\end{array}\\right.\\) on the coordinate plane.",
"role": "user"
},
{
"content": "\n1. **Given system of equations**:\n \\[\n \\begin{cases}\n x - 2y = 1 \\\\\... |
olympiads | Plot the solution set of the system of equations \(\left\{\begin{array}{l} x - 2y = 1 \\ x^{3} - 6xy - 8y^{3} = 1 \end{array}\right.\) on the coordinate plane. |
1. **Given system of equations**:
\[
\begin{cases}
x - 2y = 1 \\
x^3 - 6xy - 8y^3 = 1
\end{cases}
\]
2. **Rewriting the first equation**:
Given:
\[
x - 2y = 1
\]
We can express \(x\) in terms of \(y\):
\[
x = 2y + 1
\]
3. **Substituting \(x = 2y + 1\) into the second equati... | [
{
"content": "Plot the solution set of the system of equations \\(\\left\\{\\begin{array}{l} x - 2y = 1 \\\\ x^{3} - 6xy - 8y^{3} = 1 \\end{array}\\right.\\) on the coordinate plane.",
"role": "user"
},
{
"content": "\n1. **Given system of equations**:\n \\[\n \\begin{cases}\n x - 2y = 1 \\\\\... |
olympiads | On both sides of a 1200 m long street, there are rectangular strips of land designated for plots, one with a width of 50 m and the other with a width of 60 m. How many plots is the entire village divided into if the narrower strip contains 5 more plots than the wider strip, given that each plot on the narrower strip is... |
1. Let's denote the number of plots on the wide strip as \( x \), and the number of plots on the narrow strip as \( x + 5 \).
2. We are additionally given that each plot on the narrow strip is 1200 square meters smaller than each plot on the wide strip.
3. Let \( y \) be the area of one plot on the narrow strip, th... | [
{
"content": "On both sides of a 1200 m long street, there are rectangular strips of land designated for plots, one with a width of 50 m and the other with a width of 60 m. How many plots is the entire village divided into if the narrower strip contains 5 more plots than the wider strip, given that each plot on... |
olympiads | Let \( n \) be a positive integer. Roger has a \( (2n+1) \times (2n+1) \) square garden. He puts down fences to divide his garden into rectangular plots. He wants to end up with exactly two horizontal \( k \times 1 \) plots and exactly two vertical \( 1 \times k \) plots for each even integer \( k \) between 1 and \( 2... |
We are asked to determine the number of ways Roger can divide his \((2n+1) \times (2n+1)\) garden such that it contains exactly two horizontal \(k \times 1\) plots, two vertical \(1 \times k\) plots for each even integer \(k \in [2, 2n+1]\), and a single \(1 \times 1\) plot. Let's consider the problem step-by-step.
... | [
{
"content": "Let \\( n \\) be a positive integer. Roger has a \\( (2n+1) \\times (2n+1) \\) square garden. He puts down fences to divide his garden into rectangular plots. He wants to end up with exactly two horizontal \\( k \\times 1 \\) plots and exactly two vertical \\( 1 \\times k \\) plots for each even i... |
olympiads | A farmer fenced an outside plot of land and divided it into triangles with sides measuring 50 meters. In some of the triangles, he planted cabbage, while in others, he allowed goats to graze. Help the farmer to build additional fences along the grid lines with the shortest total length possible to protect all the cabba... | 1. **Objective Clarification**
The problem requires us to construct fences along the lines of a triangular grid to protect cabbage from goats. The goal is to use the minimum total fence length. The triangular grid consists of triangles with sides of 50 meters each.
2. **Initial Observation**
The solution illus... | [
{
"content": "A farmer fenced an outside plot of land and divided it into triangles with sides measuring 50 meters. In some of the triangles, he planted cabbage, while in others, he allowed goats to graze. Help the farmer to build additional fences along the grid lines with the shortest total length possible to... |
olympiads | On a rectangular plot, a house is being built. In a right-angled coordinate system defined by two adjacent sides of the plot, the coordinates of two adjacent corners of the house are: $(1 ; 14),(17 ; 2)$. Where should the other two corners of the house be staked out if they want the shorter walls of the house to be hal... | ### Problem:
On a rectangular lot, a house is being built. In a right-angle coordinate system defined by two adjacent sides of the lot, the coordinates of two adjacent corners of the house are given as \((1; 14)\) and \((17; 2)\). Where should the other two corners of the house be marked if it is desired that the shor... | [
{
"content": "On a rectangular plot, a house is being built. In a right-angled coordinate system defined by two adjacent sides of the plot, the coordinates of two adjacent corners of the house are: $(1 ; 14),(17 ; 2)$. Where should the other two corners of the house be staked out if they want the shorter walls ... |
olympiads | On a rectangular plot measuring $5 \times 6$ meters, there is a triangular flower bed. Find the area of this flower bed (in square meters). | 1. The rectangular plot has dimensions \(5 \, \text{m} \times 6 \, \text{m}\). First, calculate the area of the entire plot:
\[
\text{Area of the plot} = 5 \, \text{m} \times 6 \, \text{m} = 30 \, \text{m}^2
\]
2. Next, identify the areas of the two triangular parts that are not occupied by the triangular flo... | [
{
"content": "On a rectangular plot measuring $5 \\times 6$ meters, there is a triangular flower bed. Find the area of this flower bed (in square meters).",
"role": "user"
},
{
"content": "1. The rectangular plot has dimensions \\(5 \\, \\text{m} \\times 6 \\, \\text{m}\\). First, calculate the area... |
olympiads | At 2 p.m., Sanjay measures the temperature to be \(3^{\circ} \mathrm{C}\). He measures the temperature every hour after this until 10 p.m. He plots the temperatures that he measures on the graph shown. At what time after 2 p.m. does he again measure a temperature of \(3^{\circ} \mathrm{C}\)?
(A) 9 p.m.
(B) 5 p.m.
(C) ... |
1. **Identify the Relevant Temperature Line**:
- The problem requires identifying the time at which the temperature is measured again at \(3^{\circ} \, \mathrm{C}\).
- On the vertical axis, locate the horizontal line passing halfway between \(2^{\circ} \, \mathrm{C}\) and \(4^{\circ} \, \mathrm{C}\). This line r... | [
{
"content": "At 2 p.m., Sanjay measures the temperature to be \\(3^{\\circ} \\mathrm{C}\\). He measures the temperature every hour after this until 10 p.m. He plots the temperatures that he measures on the graph shown. At what time after 2 p.m. does he again measure a temperature of \\(3^{\\circ} \\mathrm{C}\\... |
olympiads | Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet. |
To solve the problem of finding the number of trees planted at 5-foot intervals on a rectangular plot of land with sides of 120 feet and 70 feet, we will follow these steps:
1. **Determine the number of intervals along the length and width of the plot:**
Since the trees are planted 5 feet apart, we can find the n... | [
{
"content": "Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet.",
"role": "user"
},
{
"content": "\nTo solve the problem of finding the number of trees planted at 5-foot intervals on a rectangular ... |
olympiads | A farmer fenced off a plot of land and divided it into triangles with sides of $50 \mathrm{~m}$. In some triangles, he planted cabbage, while in others, he let goats graze. Help the farmer build additional fences along the grid lines with the smallest possible total length to protect all the cabbage from the goats.
|
To solve this problem and construct additional fences of minimal total length to protect all the cabbage from the goats, follow these steps:
1. **Understanding the Setup:**
- The farmer has divided the land into equilateral triangles with each side measuring \(50 \mathrm{~m}\).
- Some triangles have cabbage, an... | [
{
"content": "A farmer fenced off a plot of land and divided it into triangles with sides of $50 \\mathrm{~m}$. In some triangles, he planted cabbage, while in others, he let goats graze. Help the farmer build additional fences along the grid lines with the smallest possible total length to protect all the cabb... |
olympiads | Given the probability densities of two independent uniformly distributed random variables $X$ and $Y$:
$f_{1}(x)=\frac{1}{2}$ in the interval $(0, 2)$, outside this interval $f_{1}(x)=0$
$f_{2}(y)=\frac{1}{2}$ in the interval $(0, 2)$, outside this interval $f_{2}(y)=0$
Find the cumulative distribution function (CDF... |
1. **Given Information and Definitions:**
The densities of uniformly distributed independent random variables \(X\) and \(Y\) are given as:
\[
f_{1}(x) = \begin{cases}
\frac{1}{2} & 0 < x < 2 \\
0 & \text{otherwise}
\end{cases}
\]
\[
f_{2}(y) = \begin{cases}
\frac{1... | [
{
"content": "Given the probability densities of two independent uniformly distributed random variables $X$ and $Y$:\n\n$f_{1}(x)=\\frac{1}{2}$ in the interval $(0, 2)$, outside this interval $f_{1}(x)=0$\n\n$f_{2}(y)=\\frac{1}{2}$ in the interval $(0, 2)$, outside this interval $f_{2}(y)=0$\n\nFind the cumulat... |
olympiads | A square plot measuring 14 by 14 cells needs to be paved with rectangular tiles sized 1 by 4. The tiles can only be laid along the grid (not diagonally), and tiles cannot be broken. What is the maximum number of tiles required? Will there be any unpaved area left? |
1. **Calculate the total number of cells in the grid:**
The given grid is a square of size \(14 \text{ by } 14\).
\[
14 \times 14 = 196
\]
Therefore, there are 196 cells in total.
2. **Determine the number of tiles needed:**
Each tile is of size \(1 \text{ by } 4\), so it covers 4 cells.
\[
\f... | [
{
"content": "A square plot measuring 14 by 14 cells needs to be paved with rectangular tiles sized 1 by 4. The tiles can only be laid along the grid (not diagonally), and tiles cannot be broken. What is the maximum number of tiles required? Will there be any unpaved area left?",
"role": "user"
},
{
... |
olympiads | In the 6 plots of land shown in Figure 3-11, plant either type A or type B vegetables (you can plant only one type or both). The requirement is that two adjacent plots should not both have type A vegetables. Determine the total number of possible planting schemes. |
1. **Understanding the constraints and setup**: We have 6 plots of land where we can plant two types of vegetables, labeled as type "A" and type "B". The problem states that no two adjacent plots can both have type "A" vegetables.
2. **Identify the maximum number of type "A" vegetables we can plant**:
- Given the ... | [
{
"content": "In the 6 plots of land shown in Figure 3-11, plant either type A or type B vegetables (you can plant only one type or both). The requirement is that two adjacent plots should not both have type A vegetables. Determine the total number of possible planting schemes.",
"role": "user"
},
{
... |
olympiads | When land plots were being allocated, farmer Novoselov was assigned 2 square plots of different areas with integer side lengths. Is it possible to assign farmer Malinnikov 2 square plots with integer side lengths such that the total area of Malinnikov's plots is 2 times greater than the total area of Novoselov's plots? |
1. Denote the sides of the squares allocated to farmer Novoselov as \( x \) and \( y \) such that \( x > y \).
- Hence, the areas of these squares are \( x^2 \) and \( y^2 \).
2. The total area of the squares allocated to Novoselov is:
\[
a = x^2 + y^2
\]
3. According to the problem, the total area of th... | [
{
"content": "When land plots were being allocated, farmer Novoselov was assigned 2 square plots of different areas with integer side lengths. Is it possible to assign farmer Malinnikov 2 square plots with integer side lengths such that the total area of Malinnikov's plots is 2 times greater than the total area... |
olympiads | In the school campus, there is a rectangular plot that is 18 meters long and 12 meters wide. The plan is to plant red flowers, yellow flowers, and green grass. According to one design plan, what is the area of the red flowers in square meters? |
1. Determine the total area of the rectangular plot:
\[
\text{Total area} = \text{length} \times \text{width} = 18 \, \text{meters} \times 12 \, \text{meters} = 216 \, \text{square meters}
\]
2. According to the problem, the area for red flowers and yellow flowers together forms half of the total area of ... | [
{
"content": "In the school campus, there is a rectangular plot that is 18 meters long and 12 meters wide. The plan is to plant red flowers, yellow flowers, and green grass. According to one design plan, what is the area of the red flowers in square meters?",
"role": "user"
},
{
"content": "\n1. Det... |
olympiads | In the town of Oropis, there is a lake in the shape of a triangle with each of the three sides being part of the perimeter of a square plot, with areas $370 \mathrm{~m}^{2}, 116 \mathrm{~m}^{2},$ and $74 \mathrm{~m}^{2}$, as shown in the first figure. The mayor of Oropis, Arnaldo, wants to calculate the area of the lak... |
#### (a) Determine valores inteiros de \( a, b, c \) e \( d \) que satisfaçam as condições geométricas da figura de Bernaldo.
We are given a series of geometric constraints based on the areas:
\[
\begin{aligned}
a^2 + c^2 &= 74 \\
b^2 + d^2 &= 116 \\
(a + b)^2 + (c + d)^2 &= 370
\end{aligned}
\]
To solve for the in... | [
{
"content": "In the town of Oropis, there is a lake in the shape of a triangle with each of the three sides being part of the perimeter of a square plot, with areas $370 \\mathrm{~m}^{2}, 116 \\mathrm{~m}^{2},$ and $74 \\mathrm{~m}^{2}$, as shown in the first figure. The mayor of Oropis, Arnaldo, wants to calc... |
olympiads | Find such \( n \) and such numbers \( a_{i}, b_{i}, c_{i} \) \((i=1,2, \ldots, n)\) that the condition
\[
\sum_{i=1}^{n}\left|a_{i} x+b_{i} y+c_{i}\right|=10
\]
is satisfied, such that the pairs \((x, y)\) plotted in the coordinate system form the points of a regular octagon. | 1. We need to find a value of \( n \) and numbers \( a_{i}, b_{i}, c_{i} \) for \( i=1,2, \ldots, n \) such that the condition
\[
\sum_{i=1}^{n} \left|a_{i} x + b_{i} y + c_{i}\right| = 10
\]
is satisfied by the coordinates of the vertices of a regular octagon when plotted on the coordinate system.
2. L... | [
{
"content": "Find such \\( n \\) and such numbers \\( a_{i}, b_{i}, c_{i} \\) \\((i=1,2, \\ldots, n)\\) that the condition\n\n\\[ \n\\sum_{i=1}^{n}\\left|a_{i} x+b_{i} y+c_{i}\\right|=10 \n\\]\n\nis satisfied, such that the pairs \\((x, y)\\) plotted in the coordinate system form the points of a regular octago... |
olympiads | Investigate the function \( y = \ln \left| \frac{x-1}{x+1} \right| \) and plot its graph. |
1. **Identify the Domain**:
The function \(y = \ln \left| \frac{x-1}{x+1} \right|\) is undefined when the argument of the natural logarithm is non-positive. Specifically, the function is not defined at \(x = -1\) and \(x = 1\).
Therefore, the domain of the function consists of the intervals \( (-\infty, -1) \),... | [
{
"content": "Investigate the function \\( y = \\ln \\left| \\frac{x-1}{x+1} \\right| \\) and plot its graph.",
"role": "user"
},
{
"content": "\n1. **Identify the Domain**:\n The function \\(y = \\ln \\left| \\frac{x-1}{x+1} \\right|\\) is undefined when the argument of the natural logarithm is n... |
olympiads | In triangle \( ABC \), point \( D \) is the midpoint of side \( AB \), \( E \) is the midpoint of side \( AC \), and \( F \) is the midpoint of the bisector \( AL \). Given that \( D F = 1 \) and \( E F = 2 \). Points \( D \), \( E \), and \( F \) are plotted on a plane such that line \( DF \) is horizontal, and the ot... |
#### Step 1: Establish basic conditions
In a triangle $ABC$:
- Point $D$ is the midpoint of side $AB$.
- Point $E$ is the midpoint of side $AC$.
- Point $F$ is the midpoint of the bisector $AL$ of angle $A$.
Given:
\[ DF = 1 \quad \text{and} \quad EF = 2 \]
#### Step 2: Visualize the configuration
We are given that ... | [
{
"content": "In triangle \\( ABC \\), point \\( D \\) is the midpoint of side \\( AB \\), \\( E \\) is the midpoint of side \\( AC \\), and \\( F \\) is the midpoint of the bisector \\( AL \\). Given that \\( D F = 1 \\) and \\( E F = 2 \\). Points \\( D \\), \\( E \\), and \\( F \\) are plotted on a plane suc... |
olympiads | A rectangular plot was allocated for a sports field, with a diagonal measuring 185 meters. During construction, the length of each side was reduced by 4 meters, maintaining the rectangular shape, but the area was reduced by $1012 \mathrm{~m}^{2}$. What are the actual dimensions of the sports field? | 1. Let \( x \) be the length and \( y \) the width of the original rectangular plot. According to the Pythagorean Theorem, we have:
\[
x^2 + y^2 = 185^2
\]
\[
x^2 + y^2 = 34225
\]
2. Both the length and width were reduced by 4 meters, thus the new dimensions are \( x - 4 \) and \( y - 4 \).
The ar... | [
{
"content": "A rectangular plot was allocated for a sports field, with a diagonal measuring 185 meters. During construction, the length of each side was reduced by 4 meters, maintaining the rectangular shape, but the area was reduced by $1012 \\mathrm{~m}^{2}$. What are the actual dimensions of the sports fiel... |
olympiads | Remove the highest and lowest scores out of nine scores of a certain contestant. The average of the remaining seven scores is 91. In the stem-and-leaf plot of the nine scores, one score is unclear and is represented as $x$ in Figure 1. What is the variance of the remaining seven scores?
(A) $\frac{116}{9}$
(B) $\fr... |
1. Let's identify the provided points for the analysis:
\[ 87, 87, 94, 90, 91, 90, 9x, 99, 91 \]
2. From the problem, we are given that the average score of the seven remaining scores (after removing the highest and lowest scores) is \(91\).
3. Determine the highest and lowest scores from the provided data:
\[... | [
{
"content": "Remove the highest and lowest scores out of nine scores of a certain contestant. The average of the remaining seven scores is 91. In the stem-and-leaf plot of the nine scores, one score is unclear and is represented as $x$ in Figure 1. What is the variance of the remaining seven scores? \n\n(A) $\... |
olympiads | A rectangular plot has a length of 4 meters and a width of 1 meter.
Is it possible to plant three trees on it so that the distance between any two trees is at least 2.5 meters? |
1. **Define the dimensions of the rectangular plot**:
- Length: \(4\) meters
- Width: \(1\) meter.
2. **Setting up the problem**:
- We need to determine if it is possible to plant three trees on this plot such that the distance between any two trees is at least \(2.5\) meters.
3. **Divide the rectangular p... | [
{
"content": "A rectangular plot has a length of 4 meters and a width of 1 meter.\n\nIs it possible to plant three trees on it so that the distance between any two trees is at least 2.5 meters?",
"role": "user"
},
{
"content": "\n1. **Define the dimensions of the rectangular plot**:\n - Length: \\... |
olympiads | As shown in the figure, Rourou's vegetable garden consists of 4 square plots and 1 small rectangular pool forming a large rectangle. If the area of each plot is 20 square meters and the garden's length is 10 meters, what is the perimeter of the pool (the shaded area in the figure) in meters? |
1. **Analyzing the given data:**
- There are 4 squares and 1 small rectangle forming a larger rectangle.
- Each square has an area of 20 square meters.
- The length of the garden (entire large rectangle) is given as 10 meters.
2. **Finding the dimensions of each square:**
Since each square has an area o... | [
{
"content": "As shown in the figure, Rourou's vegetable garden consists of 4 square plots and 1 small rectangular pool forming a large rectangle. If the area of each plot is 20 square meters and the garden's length is 10 meters, what is the perimeter of the pool (the shaded area in the figure) in meters?",
... |
olympiads | Plot on the $(x, y)$ plane the set of points whose coordinates satisfy the system:
\[
\left\{
\begin{array}{l}
(| x | + x)^{2} + (| y | + y)^{2} \leq 4, \\
3y + x \leq 0
\end{array}
\right.
\] |
We need to find the points \((x, y)\) on the coordinate plane that satisfy the system of inequalities:
\[
\begin{cases}
(|x| + x)^2 + (|y| + y)^2 \leq 4, \\
3y + x \leq 0.
\end{cases}
\]
Let's start by examining the first inequality, \((|x| + x)^2 + (|y| + y)^2 \leq 4\).
Notice that the expression \((|x| + x)\) and ... | [
{
"content": "Plot on the $(x, y)$ plane the set of points whose coordinates satisfy the system:\n\\[\n\\left\\{\n\\begin{array}{l}\n(| x | + x)^{2} + (| y | + y)^{2} \\leq 4, \\\\\n3y + x \\leq 0\n\\end{array}\n\\right.\n\\]",
"role": "user"
},
{
"content": "\nWe need to find the points \\((x, y)\\... |
olympiads | A plot of land is sown with seeds of smooth brome. On one section of this plot, weeds make up $0.4\%$ of the vegetation. What is the probability that among 125 plants randomly selected from this section, there are:
a) exactly 3 weeds;
b) no more than 3 weeds. |
To solve the problem, we use the Poisson distribution, which is suitable here given that the number of trials is large ($n = 125$) and the probability of each individual event occurring is very small ($p = 0.004$). Thus, we approximate the binomial distribution using the Poisson distribution with $\lambda = np$.
For ... | [
{
"content": "A plot of land is sown with seeds of smooth brome. On one section of this plot, weeds make up $0.4\\%$ of the vegetation. What is the probability that among 125 plants randomly selected from this section, there are:\n\na) exactly 3 weeds;\n\nb) no more than 3 weeds.",
"role": "user"
},
{
... |
olympiads | Given the vibrations \( y_{1} = \sin x \) and \( y_{2} = \sin^2 x \), study the properties of their combined motion \( y = y_{1} + y_{2} = \sin x + \sin^2 x \) (range of \( y \), maximum value, minimum value, zeros, variation, symmetry, periodicity, and plot the graph). |
Given the function $y = y_1 + y_2 = \sin x + \sin^2 x$, we need to analyze its properties, including its value range, maximum and minimum values, zeros, variation, symmetry, periodicity, and plot its graph.
1. **Value Range, Maximum, and Minimum Values:**
\[
y = \sin x + \sin^2 x
\]
To determine the ... | [
{
"content": "Given the vibrations \\( y_{1} = \\sin x \\) and \\( y_{2} = \\sin^2 x \\), study the properties of their combined motion \\( y = y_{1} + y_{2} = \\sin x + \\sin^2 x \\) (range of \\( y \\), maximum value, minimum value, zeros, variation, symmetry, periodicity, and plot the graph).",
"role": "... |
olympiads | A father had three sons and left them an inheritance of 900 square meters of land - a rectangle measuring 25 m × 36 m. The brothers decided to divide the land into three rectangular plots of 300 square meters each. How many ways are there to divide it (in terms of the lengths and widths of the plots), and in which opti... |
Given a rectangular plot of land with dimensions \(25 \, m \times 36 \, m\) or an area of \(900 \, m^2\), it needs to be divided into three rectangular sections such that each section has an area of \(3 \times 100 = 300 \, m^2\). We need to find how many ways this division can be made, and determine in which scenario ... | [
{
"content": "A father had three sons and left them an inheritance of 900 square meters of land - a rectangle measuring 25 m × 36 m. The brothers decided to divide the land into three rectangular plots of 300 square meters each. How many ways are there to divide it (in terms of the lengths and widths of the plo... |
olympiads | Simplify the expression \( y = \sqrt{x + 2 \sqrt{x - 1}} + \sqrt{x - 2 \sqrt{x - 1}} \), and then plot the graph of the function \( y \) for \( 1 \leq x < \infty \). |
Let's simplify the given expression \( y = \sqrt{x + 2 \sqrt{x-1}} + \sqrt{x - 2 \sqrt{x-1}} \) step-by-step.
1. **Considering the domain of the expression:**
\[
\text{For the expression to be real, the terms under the square root must be non-negative. Thus, } x - 1 \geq 0 \implies x \geq 1.
\]
Therefore,... | [
{
"content": "Simplify the expression \\( y = \\sqrt{x + 2 \\sqrt{x - 1}} + \\sqrt{x - 2 \\sqrt{x - 1}} \\), and then plot the graph of the function \\( y \\) for \\( 1 \\leq x < \\infty \\).",
"role": "user"
},
{
"content": "\nLet's simplify the given expression \\( y = \\sqrt{x + 2 \\sqrt{x-1}} + ... |
olympiads | A rectangular plot of 800 square meters is to be fenced using 100 meters of fencing. What are the dimensions of the plot? |
1. **Understanding the problem setup:**
- We have a rectangular plot of land by a river with an area of \(800 \, \text{m}^2\).
- We can only use \(100 \, \text{m}\) of fencing to enclose this plot.
- Assume the rectangular plot is positioned such that one of its sides is along the river, which means this side... | [
{
"content": "A rectangular plot of 800 square meters is to be fenced using 100 meters of fencing. What are the dimensions of the plot?",
"role": "user"
},
{
"content": "\n1. **Understanding the problem setup:**\n - We have a rectangular plot of land by a river with an area of \\(800 \\, \\text{m}... |
olympiads | Given the function \( f(x) = x^2 - 6x + 5 \), determine the region in the plane where the points \((x, y)\) satisfy both \( f(x) + f(y) \leq 0 \) and \( f(x) - f(y) \geq 0 \). Also, plot the graph of this region. |
1. **Transform the constraint \( f(x) + f(y) \leq 0 \) into a geometric condition**:
Given the function \( f(x) = x^2 - 6x + 5 \), we first compute \( f(x) \) at two different variables, \( x \) and \( y \).
Thus, we have:
\[
f(x) = x^2 - 6x + 5
\]
\[
f(y) = y^2 - 6y + 5
\]
Therefo... | [
{
"content": "Given the function \\( f(x) = x^2 - 6x + 5 \\), determine the region in the plane where the points \\((x, y)\\) satisfy both \\( f(x) + f(y) \\leq 0 \\) and \\( f(x) - f(y) \\geq 0 \\). Also, plot the graph of this region.",
"role": "user"
},
{
"content": "\n1. **Transform the constrai... |
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