problem stringlengths 21 3.37k | text stringlengths 44 5.6k | model stringclasses 2
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(a) To which class (type) of goods (services) can the described service (complete apartment renovation) be attributed? Explain why. What are the characteristics that it has (as opposed to typical consumer goods and services)? Explain.
(b) The decision between a private repair crew and a construction company was diffi... |
(a) To which class (type) of goods (services) can the described service (complete apartment renovation) be attributed? Explain why. What are the characteristics that it has (as opposed to typical consumer goods and services)? Explain.
(b) The decision between a private repair crew and a construction company was diffi... | r1 |
(a) To which class (type) of goods (services) can the described service (complete apartment renovation) be attributed? Explain why. What are the characteristics that it has (as opposed to typical consumer goods and services)? Explain.
(b) The decision between a private repair crew and a construction company was diffi... |
(a) To which class (type) of goods (services) can the described service (complete apartment renovation) be attributed? Explain why. What are the characteristics that it has (as opposed to typical consumer goods and services)? Explain.
(b) The decision between a private repair crew and a construction company was diffi... | qwen |
A $10 \times 10$ table is filled with numbers from 1 to 100: in the first row, the numbers from 1 to 10 are listed in ascending order from left to right; in the second row, the numbers from 11 to 20 are listed in the same way, and so on; in the last row, the numbers from 91 to 100 are listed from left to right. Can a ... |
A $10 \times 10$ table is filled with numbers from 1 to 100: in the first row, the numbers from 1 to 10 are listed in ascending order from left to right; in the second row, the numbers from 11 to 20 are listed in the same way, and so on; in the last row, the numbers from 91 to 100 are listed from left to right. Can a ... | r1 |
A $10 \times 10$ table is filled with numbers from 1 to 100: in the first row, the numbers from 1 to 10 are listed in ascending order from left to right; in the second row, the numbers from 11 to 20 are listed in the same way, and so on; in the last row, the numbers from 91 to 100 are listed from left to right. Can a ... |
A $10 \times 10$ table is filled with numbers from 1 to 100: in the first row, the numbers from 1 to 10 are listed in ascending order from left to right; in the second row, the numbers from 11 to 20 are listed in the same way, and so on; in the last row, the numbers from 91 to 100 are listed from left to right. Can a ... | qwen |
A grid strip of size \(1 \times 1000000\) is divided into 100 segments. An integer is written in each cell, and the numbers in cells lying within the same segment are the same. A token is placed in each cell. Then, an operation is performed where all tokens are simultaneously moved, each token moving to the right by t... |
A grid strip of size \(1 \times 1000000\) is divided into 100 segments. An integer is written in each cell, and the numbers in cells lying within the same segment are the same. A token is placed in each cell. Then, an operation is performed where all tokens are simultaneously moved, each token moving to the right by t... | r1 |
A grid strip of size \(1 \times 1000000\) is divided into 100 segments. An integer is written in each cell, and the numbers in cells lying within the same segment are the same. A token is placed in each cell. Then, an operation is performed where all tokens are simultaneously moved, each token moving to the right by t... |
A grid strip of size \(1 \times 1000000\) is divided into 100 segments. An integer is written in each cell, and the numbers in cells lying within the same segment are the same. A token is placed in each cell. Then, an operation is performed where all tokens are simultaneously moved, each token moving to the right by t... | qwen |
A high-tech Japanese company has presented a unique robot capable of producing construction blocks that can be sold for 90 monetary units each. Due to a shortage of special chips, it is impossible to replicate or even repair this robot if it goes out of order in the near future. If the robot works for $\mathrm{L}$ hou... |
A high-tech Japanese company has presented a unique robot capable of producing construction blocks that can be sold for 90 monetary units each. Due to a shortage of special chips, it is impossible to replicate or even repair this robot if it goes out of order in the near future. If the robot works for $\mathrm{L}$ hou... | r1 |
A high-tech Japanese company has presented a unique robot capable of producing construction blocks that can be sold for 90 monetary units each. Due to a shortage of special chips, it is impossible to replicate or even repair this robot if it goes out of order in the near future. If the robot works for $\mathrm{L}$ hou... |
A high-tech Japanese company has presented a unique robot capable of producing construction blocks that can be sold for 90 monetary units each. Due to a shortage of special chips, it is impossible to replicate or even repair this robot if it goes out of order in the near future. If the robot works for $\mathrm{L}$ hou... | qwen |
A natural number \( N \) is represented as \( N = a_1 - a_2 = b_1 - b_2 = c_1 - c_2 = d_1 - d_2 \), where \( a_1 \) and \( a_2 \) are squares, \( b_1 \) and \( b_2 \) are cubes, \( c_1 \) and \( c_2 \) are fifth powers, and \( d_1 \) and \( d_2 \) are seventh powers of natural numbers. Is it necessary that among the n... |
A natural number \( N \) is represented as \( N = a_1 - a_2 = b_1 - b_2 = c_1 - c_2 = d_1 - d_2 \), where \( a_1 \) and \( a_2 \) are squares, \( b_1 \) and \( b_2 \) are cubes, \( c_1 \) and \( c_2 \) are fifth powers, and \( d_1 \) and \( d_2 \) are seventh powers of natural numbers. Is it necessary that among the n... | r1 |
A natural number \( N \) is represented as \( N = a_1 - a_2 = b_1 - b_2 = c_1 - c_2 = d_1 - d_2 \), where \( a_1 \) and \( a_2 \) are squares, \( b_1 \) and \( b_2 \) are cubes, \( c_1 \) and \( c_2 \) are fifth powers, and \( d_1 \) and \( d_2 \) are seventh powers of natural numbers. Is it necessary that among the n... |
A natural number \( N \) is represented as \( N = a_1 - a_2 = b_1 - b_2 = c_1 - c_2 = d_1 - d_2 \), where \( a_1 \) and \( a_2 \) are squares, \( b_1 \) and \( b_2 \) are cubes, \( c_1 \) and \( c_2 \) are fifth powers, and \( d_1 \) and \( d_2 \) are seventh powers of natural numbers. Is it necessary that among the n... | qwen |
A point starts from the origin on a line and makes $a$ steps to the right and $b$ steps to the left in some order, where $a > b$. The range of the point's walk is defined as the difference between the maximum and minimum coordinates of the point during its entire walk.
a) Find the maximum possible range of the walk.
... |
A point starts from the origin on a line and makes $a$ steps to the right and $b$ steps to the left in some order, where $a > b$. The range of the point's walk is defined as the difference between the maximum and minimum coordinates of the point during its entire walk.
a) Find the maximum possible range of the walk.
... | r1 |
A point starts from the origin on a line and makes $a$ steps to the right and $b$ steps to the left in some order, where $a > b$. The range of the point's walk is defined as the difference between the maximum and minimum coordinates of the point during its entire walk.
a) Find the maximum possible range of the walk.
... |
A point starts from the origin on a line and makes $a$ steps to the right and $b$ steps to the left in some order, where $a > b$. The range of the point's walk is defined as the difference between the maximum and minimum coordinates of the point during its entire walk.
a) Find the maximum possible range of the walk.
... | qwen |
A random walk of a point starts from the origin on a line and makes \(a\) steps to the right, \(b\) steps to the left in some order, with \(a > b\). The range of the walk is defined as the difference between the highest and lowest coordinates of the point during the walk.
a) Find the maximum possible range of the wal... |
A random walk of a point starts from the origin on a line and makes \(a\) steps to the right, \(b\) steps to the left in some order, with \(a > b\). The range of the walk is defined as the difference between the highest and lowest coordinates of the point during the walk.
a) Find the maximum possible range of the wal... | qwen |
A random walk of a point starts from the origin on a line and makes \(a\) steps to the right, \(b\) steps to the left in some order, with \(a > b\). The range of the walk is defined as the difference between the highest and lowest coordinates of the point during the walk.
a) Find the maximum possible range of the wal... |
A random walk of a point starts from the origin on a line and makes \(a\) steps to the right, \(b\) steps to the left in some order, with \(a > b\). The range of the walk is defined as the difference between the highest and lowest coordinates of the point during the walk.
a) Find the maximum possible range of the wal... | r1 |
Angle $\widehat{A}$ is the smallest in triangle $ABC$. Points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be a point on the arc bounded by $B$ and $C$ that does not contain $A$.
The perpendicular bisectors of segments $[AB]$ and $[AC]$ intersect line $(AU)$ at points $V$ and $W$ respect... |
Angle $\widehat{A}$ is the smallest in triangle $ABC$. Points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be a point on the arc bounded by $B$ and $C$ that does not contain $A$.
The perpendicular bisectors of segments $[AB]$ and $[AC]$ intersect line $(AU)$ at points $V$ and $W$ respect... | qwen |
Angle $\widehat{A}$ is the smallest in triangle $ABC$. Points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be a point on the arc bounded by $B$ and $C$ that does not contain $A$.
The perpendicular bisectors of segments $[AB]$ and $[AC]$ intersect line $(AU)$ at points $V$ and $W$ respect... |
Angle $\widehat{A}$ is the smallest in triangle $ABC$. Points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be a point on the arc bounded by $B$ and $C$ that does not contain $A$.
The perpendicular bisectors of segments $[AB]$ and $[AC]$ intersect line $(AU)$ at points $V$ and $W$ respect... | r1 |
As depicted, chords \(AC\) and \(BD\) intersect at \(E\). Given that \(AB = BC = CD\) and \(\angle BEC = 130^\circ\), what is the measure of \(\angle ACD\)?
(A) \(15^\circ\)
(B) \(30^\circ\)
(C) \(84^\circ\)
(D) \(105^\circ\) |
As depicted, chords \(AC\) and \(BD\) intersect at \(E\). Given that \(AB = BC = CD\) and \(\angle BEC = 130^\circ\), what is the measure of \(\angle ACD\)?
(A) \(15^\circ\)
(B) \(30^\circ\)
(C) \(84^\circ\)
(D) \(105^\circ\) <think>
Alright, let's try to tackle this geometry problem step by step. I know that some... | r1 |
As depicted, chords \(AC\) and \(BD\) intersect at \(E\). Given that \(AB = BC = CD\) and \(\angle BEC = 130^\circ\), what is the measure of \(\angle ACD\)?
(A) \(15^\circ\)
(B) \(30^\circ\)
(C) \(84^\circ\)
(D) \(105^\circ\) |
As depicted, chords \(AC\) and \(BD\) intersect at \(E\). Given that \(AB = BC = CD\) and \(\angle BEC = 130^\circ\), what is the measure of \(\angle ACD\)?
(A) \(15^\circ\)
(B) \(30^\circ\)
(C) \(84^\circ\)
(D) \(105^\circ\) To solve this problem, we need to use some geometric properties and angle relationships. ... | qwen |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\arcsin 3x}{\sqrt{2+x}-\sqrt{2}}
$$ |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\arcsin 3x}{\sqrt{2+x}-\sqrt{2}}
$$ <think>
Okay, so I need to find the limit as x approaches 0 of the function arcsin(3x) divided by the square root of (2 + x) minus the square root of 2. Hmm, let me write that down to visualize it better:
$$
\li... | r1 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\arcsin 3x}{\sqrt{2+x}-\sqrt{2}}
$$ |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\arcsin 3x}{\sqrt{2+x}-\sqrt{2}}
$$
To find the limit of the function
$$
\lim _{x \rightarrow 0} \frac{\arcsin 3x}{\sqrt{2+x}-\sqrt{2}},
$$
we can use L'Hôpital's Rule since the limit evaluates to the indeterminate form $\frac{0}{0}$ when $x$ ap... | qwen |
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{n^{2}-\sqrt{3 n^{5}-7}}{\left(n^{2}-n \cos n+1\right) \sqrt{n}}
\] |
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{n^{2}-\sqrt{3 n^{5}-7}}{\left(n^{2}-n \cos n+1\right) \sqrt{n}}
\] To find the limit of the sequence
\[
\lim_{n \rightarrow \infty} \frac{n^{2}-\sqrt{3 n^{5}-7}}{\left(n^{2}-n \cos n+1\right) \sqrt{n}},
\]
we can break it down into... | qwen |
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{n^{2}-\sqrt{3 n^{5}-7}}{\left(n^{2}-n \cos n+1\right) \sqrt{n}}
\] |
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{n^{2}-\sqrt{3 n^{5}-7}}{\left(n^{2}-n \cos n+1\right) \sqrt{n}}
\] <think>
Alright, so I have this limit problem to solve:
\[
\lim _{n \rightarrow \infty} \frac{n^{2}-\sqrt{3 n^{5}-7}}{\left(n^{2}-n \cos n+1\right) \sqrt{n}}
\]
Hmm... | r1 |
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(1, 1, -1)$
$A_{2}(2, 3, 1)$
$A_{3}(3, 2, 1)$
$A_{4}(5, 9, -8)$ |
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(1, 1, -1)$
$A_{2}(2, 3, 1)$
$A_{3}(3, 2, 1)$
$A_{4}(5, 9, -8)$ <think>
Okay, so I need to calculate the volume of a tetrahedron with given ver... | r1 |
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(1, 1, -1)$
$A_{2}(2, 3, 1)$
$A_{3}(3, 2, 1)$
$A_{4}(5, 9, -8)$ |
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(1, 1, -1)$
$A_{2}(2, 3, 1)$
$A_{3}(3, 2, 1)$
$A_{4}(5, 9, -8)$ To find the volume of the tetrahedron with vertices \(A_1(1, 1, -1)\), \(A_2(2,... | qwen |
Compute an approximation using a differential.
\[ y = \sqrt[3]{3x + \cos x}, \quad x = 0.01 \] |
Compute an approximation using a differential.
\[ y = \sqrt[3]{3x + \cos x}, \quad x = 0.01 \] <think>
Okay, so I need to approximate the value of y when x is 0.01 using a differential. The function given is y = cube root of (3x + cos x). Hmm, okay, let me write that down to make sure I understand it correctly.
So, ... | r1 |
Compute an approximation using a differential.
\[ y = \sqrt[3]{3x + \cos x}, \quad x = 0.01 \] |
Compute an approximation using a differential.
\[ y = \sqrt[3]{3x + \cos x}, \quad x = 0.01 \] To approximate the value of \( y = \sqrt[3]{3x + \cos x} \) at \( x = 0.01 \) using a differential, we can use the linear approximation formula:
\[ y \approx y_0 + y'_0 (x - x_0) \]
where \( y_0 = \sqrt[3]{3x_0 + \cos x_0... | qwen |
Compute the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, dx}{(1+\cos x+\sin x)^{2}}
$$ |
Compute the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, dx}{(1+\cos x+\sin x)^{2}}
$$ <think>
Okay, so I have this integral to compute: the integral from 0 to pi/2 of sin x dx divided by (1 + cos x + sin x)^2. Hmm, that seems a bit complicated, but maybe I can simplify it somehow. Let me write it d... | r1 |
Construct the image of the square with vertices \( A(0,0), B(0,2), C(2,2), D(2,0) \) under the following transformations:
a) \( w = i z \)
b) \( w = 2 i z - 1 \)
c) \( w = z^{2} \)
d) \( w = z^{-1} \) |
Construct the image of the square with vertices \( A(0,0), B(0,2), C(2,2), D(2,0) \) under the following transformations:
a) \( w = i z \)
b) \( w = 2 i z - 1 \)
c) \( w = z^{2} \)
d) \( w = z^{-1} \) <think>
Alright, so I have this problem where I need to construct the image of a square with vertices A(0,0), B(0,2), ... | r1 |
Construct the image of the square with vertices \( A(0,0), B(0,2), C(2,2), D(2,0) \) under the following transformations:
a) \( w = i z \)
b) \( w = 2 i z - 1 \)
c) \( w = z^{2} \)
d) \( w = z^{-1} \) |
Construct the image of the square with vertices \( A(0,0), B(0,2), C(2,2), D(2,0) \) under the following transformations:
a) \( w = i z \)
b) \( w = 2 i z - 1 \)
c) \( w = z^{2} \)
d) \( w = z^{-1} \) To solve this problem, we need to apply each transformation to the vertices of the square and then determine the coord... | qwen |
Dasha cut a rectangular strip of paper into two rectangles: one white and one gray, both of which have integer side lengths. She noted the areas of these two rectangles in her notebook.
Then she overlaid the rectangles as shown in the figure. Three new rectangles were formed, and Dasha also noted the areas of these t... |
Dasha cut a rectangular strip of paper into two rectangles: one white and one gray, both of which have integer side lengths. She noted the areas of these two rectangles in her notebook.
Then she overlaid the rectangles as shown in the figure. Three new rectangles were formed, and Dasha also noted the areas of these t... | qwen |
Dasha cut a rectangular strip of paper into two rectangles: one white and one gray, both of which have integer side lengths. She noted the areas of these two rectangles in her notebook.
Then she overlaid the rectangles as shown in the figure. Three new rectangles were formed, and Dasha also noted the areas of these t... |
Dasha cut a rectangular strip of paper into two rectangles: one white and one gray, both of which have integer side lengths. She noted the areas of these two rectangles in her notebook.
Then she overlaid the rectangles as shown in the figure. Three new rectangles were formed, and Dasha also noted the areas of these t... | r1 |
Determine how much money the Romanov family will save by using a multi-tariff meter over three years.
The cost of the meter is 3500 rubles. The installation cost is 1100 rubles. On average, the family's electricity consumption is 300 kWh per month, with 230 kWh used from 23:00 to 07:00.
Electricity rates with a mult... |
Determine how much money the Romanov family will save by using a multi-tariff meter over three years.
The cost of the meter is 3500 rubles. The installation cost is 1100 rubles. On average, the family's electricity consumption is 300 kWh per month, with 230 kWh used from 23:00 to 07:00.
Electricity rates with a mult... | r1 |
Determine how much money the Romanov family will save by using a multi-tariff meter over three years.
The cost of the meter is 3500 rubles. The installation cost is 1100 rubles. On average, the family's electricity consumption is 300 kWh per month, with 230 kWh used from 23:00 to 07:00.
Electricity rates with a mult... |
Determine how much money the Romanov family will save by using a multi-tariff meter over three years.
The cost of the meter is 3500 rubles. The installation cost is 1100 rubles. On average, the family's electricity consumption is 300 kWh per month, with 230 kWh used from 23:00 to 07:00.
Electricity rates with a mult... | qwen |
Find all functions \( f: \mathbf{Z}^2 \rightarrow [0,1] \) such that for any integers \( x \) and \( y \), the following holds:
\[
f(x, y) = \frac{f(x-1, y) + f(x, y-1)}{2}.
\] |
Find all functions \( f: \mathbf{Z}^2 \rightarrow [0,1] \) such that for any integers \( x \) and \( y \), the following holds:
\[
f(x, y) = \frac{f(x-1, y) + f(x, y-1)}{2}.
\] <think>
Alright, so I need to find all functions \( f: \mathbf{Z}^2 \rightarrow [0,1] \) that satisfy the equation:
\[
f(x, y) = \frac{f(x-... | r1 |
Find all functions \( f: \mathbf{Z}^2 \rightarrow [0,1] \) such that for any integers \( x \) and \( y \), the following holds:
\[
f(x, y) = \frac{f(x-1, y) + f(x, y-1)}{2}.
\] |
Find all functions \( f: \mathbf{Z}^2 \rightarrow [0,1] \) such that for any integers \( x \) and \( y \), the following holds:
\[
f(x, y) = \frac{f(x-1, y) + f(x, y-1)}{2}.
\] To solve the problem, we need to find all functions \( f: \mathbb{Z}^2 \rightarrow [0,1] \) such that for any integers \( x \) and \( y \), t... | qwen |
Find all real numbers \( x, y, z \) that satisfy:
$$
\begin{cases}
(x+1) y z = 12 \\
(y+1) z x = 4 \\
(z+1) x y = 4
\end{cases}
$$ |
Find all real numbers \( x, y, z \) that satisfy:
$$
\begin{cases}
(x+1) y z = 12 \\
(y+1) z x = 4 \\
(z+1) x y = 4
\end{cases}
$$ <think>
Okay, so I have this system of equations to solve:
$$
\begin{cases}
(x+1) y z = 12 \\
(y+1) z x = 4 \\
(z+1) x y = 4
\end{cases}
$$
Hmm, all three equations look kind of similar... | r1 |
Find all real numbers \( x, y, z \) that satisfy:
$$
\begin{cases}
(x+1) y z = 12 \\
(y+1) z x = 4 \\
(z+1) x y = 4
\end{cases}
$$ |
Find all real numbers \( x, y, z \) that satisfy:
$$
\begin{cases}
(x+1) y z = 12 \\
(y+1) z x = 4 \\
(z+1) x y = 4
\end{cases}
$$
To solve the system of equations
\[
\begin{cases}
(x+1) y z = 12 \\
(y+1) z x = 4 \\
(z+1) x y = 4
\end{cases}
\]
we can use a systematic approach. Let's start by expressing \( yz \), \... | qwen |
Find the derivative.
$$
y=\frac{2}{3}\left(4 x^{2}-4 x+3\right) \sqrt{x^{2}-x}+(2 x-1)^{4} \cdot \arcsin \frac{1}{2 x-1}, \quad 2 x-1>0
$$ |
Find the derivative.
$$
y=\frac{2}{3}\left(4 x^{2}-4 x+3\right) \sqrt{x^{2}-x}+(2 x-1)^{4} \cdot \arcsin \frac{1}{2 x-1}, \quad 2 x-1>0
$$
To find the derivative of the given function \( y = \frac{2}{3}\left(4 x^{2}-4 x+3\right) \sqrt{x^{2}-x}+(2 x-1)^{4} \cdot \arcsin \frac{1}{2 x-1} \), we can use the sympy library... | qwen |
Find the derivative.
$$
y=\frac{2}{3}\left(4 x^{2}-4 x+3\right) \sqrt{x^{2}-x}+(2 x-1)^{4} \cdot \arcsin \frac{1}{2 x-1}, \quad 2 x-1>0
$$ |
Find the derivative.
$$
y=\frac{2}{3}\left(4 x^{2}-4 x+3\right) \sqrt{x^{2}-x}+(2 x-1)^{4} \cdot \arcsin \frac{1}{2 x-1}, \quad 2 x-1>0
$$ <think>
Alright, so I need to find the derivative of this function y with respect to x. Let me write it down again to make sure I have it correctly:
$$
y=\frac{2}{3}\left(4 x^{2}... | r1 |
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(-1, 2, -3) \)
\( M_{2}(4, -1, 0) \)
\( M_{3}(2, 1, -2) \)
\( M_{0}(1, -6, -5) \) |
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(-1, 2, -3) \)
\( M_{2}(4, -1, 0) \)
\( M_{3}(2, 1, -2) \)
\( M_{0}(1, -6, -5) \) To find the distance from a point \( M_0 \) to a plane defined by three points \( M_1, M_2, M_3 \), we can follow t... | qwen |
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(-1, 2, -3) \)
\( M_{2}(4, -1, 0) \)
\( M_{3}(2, 1, -2) \)
\( M_{0}(1, -6, -5) \) |
Find the distance from point \( M_{0} \) to the plane passing through three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(-1, 2, -3) \)
\( M_{2}(4, -1, 0) \)
\( M_{3}(2, 1, -2) \)
\( M_{0}(1, -6, -5) \) <think>
Alright, so I need to find the distance from point \( M_{0} \) to the plane defined by three other points \... | r1 |
First, determine the quantity of gas in each of the containers after equilibrium is established:
\[
\begin{aligned}
& \left\{\begin{array}{l}
v_{1}+v_{2}=4 \\
v_{2}-v_{1}=0.62
\end{array}\right. \\
& v_{2}=v_{1}+0.62 \\
& 2v_{1}+0.62=4 \\
& v_{1}=1.69 \text{ moles,} \\
& v_{2}=4-1.69=2.31 \text{ moles.}
\end{aligned}... |
First, determine the quantity of gas in each of the containers after equilibrium is established:
\[
\begin{aligned}
& \left\{\begin{array}{l}
v_{1}+v_{2}=4 \\
v_{2}-v_{1}=0.62
\end{array}\right. \\
& v_{2}=v_{1}+0.62 \\
& 2v_{1}+0.62=4 \\
& v_{1}=1.69 \text{ moles,} \\
& v_{2}=4-1.69=2.31 \text{ moles.}
\end{aligned}... | r1 |
First, determine the quantity of gas in each of the containers after equilibrium is established:
\[
\begin{aligned}
& \left\{\begin{array}{l}
v_{1}+v_{2}=4 \\
v_{2}-v_{1}=0.62
\end{array}\right. \\
& v_{2}=v_{1}+0.62 \\
& 2v_{1}+0.62=4 \\
& v_{1}=1.69 \text{ moles,} \\
& v_{2}=4-1.69=2.31 \text{ moles.}
\end{aligned}... |
First, determine the quantity of gas in each of the containers after equilibrium is established:
\[
\begin{aligned}
& \left\{\begin{array}{l}
v_{1}+v_{2}=4 \\
v_{2}-v_{1}=0.62
\end{array}\right. \\
& v_{2}=v_{1}+0.62 \\
& 2v_{1}+0.62=4 \\
& v_{1}=1.69 \text{ moles,} \\
& v_{2}=4-1.69=2.31 \text{ moles.}
\end{aligned}... | qwen |
From point \(D\), the midpoint of the base \(BC\) of an isosceles triangle \(ABC\), a perpendicular \(DE\) is dropped onto the side \(AC\). The circumcircle of triangle \(ABD\) intersects line \(BE\) at points \(B\) and \(F\). Prove that line \(AF\) passes through the midpoint of segment \(DE\). |
From point \(D\), the midpoint of the base \(BC\) of an isosceles triangle \(ABC\), a perpendicular \(DE\) is dropped onto the side \(AC\). The circumcircle of triangle \(ABD\) intersects line \(BE\) at points \(B\) and \(F\). Prove that line \(AF\) passes through the midpoint of segment \(DE\). To solve this problem,... | qwen |
Given a number $n$, let $J(n)$ be the position of the last remaining person. Prove that:
a) $J(2n) = 2J(n) - 1$
b) $J(2n + 1) = 2J(n) + 1$
c) If $n = \left(1 b_{\text{m}-1} b_{\text{m}-2} \ldots b_1 b_0\right)_2$, then $J(n) = \left(b_{\text{m}-1} b_{\text{m}-2} \ldots b_1 b_0 1\right)_2$. |
Given a number $n$, let $J(n)$ be the position of the last remaining person. Prove that:
a) $J(2n) = 2J(n) - 1$
b) $J(2n + 1) = 2J(n) + 1$
c) If $n = \left(1 b_{\text{m}-1} b_{\text{m}-2} \ldots b_1 b_0\right)_2$, then $J(n) = \left(b_{\text{m}-1} b_{\text{m}-2} \ldots b_1 b_0 1\right)_2$. <think>
Okay, so I have t... | r1 |
Given a number $n$, let $J(n)$ be the position of the last remaining person. Prove that:
a) $J(2n) = 2J(n) - 1$
b) $J(2n + 1) = 2J(n) + 1$
c) If $n = \left(1 b_{\text{m}-1} b_{\text{m}-2} \ldots b_1 b_0\right)_2$, then $J(n) = \left(b_{\text{m}-1} b_{\text{m}-2} \ldots b_1 b_0 1\right)_2$. |
Given a number $n$, let $J(n)$ be the position of the last remaining person. Prove that:
a) $J(2n) = 2J(n) - 1$
b) $J(2n + 1) = 2J(n) + 1$
c) If $n = \left(1 b_{\text{m}-1} b_{\text{m}-2} \ldots b_1 b_0\right)_2$, then $J(n) = \left(b_{\text{m}-1} b_{\text{m}-2} \ldots b_1 b_0 1\right)_2$. To solve the Josephus pro... | qwen |
Given a triangle \( ABC \). A circle with radius \( R \) touches side \( AC \) at point \( M \) and side \( BC \) at point \( P \). Side \( AB \) intersects this circle at points \( K \) and \( E \) (point \( E \) lies on segment \( BK \)). Find \( BE \), knowing that \( BC = a \), \( CM = b < a \), and \( \angle KME ... |
Given a triangle \( ABC \). A circle with radius \( R \) touches side \( AC \) at point \( M \) and side \( BC \) at point \( P \). Side \( AB \) intersects this circle at points \( K \) and \( E \) (point \( E \) lies on segment \( BK \)). Find \( BE \), knowing that \( BC = a \), \( CM = b < a \), and \( \angle KME ... | r1 |
Given a triangle \( ABC \). A circle with radius \( R \) touches side \( AC \) at point \( M \) and side \( BC \) at point \( P \). Side \( AB \) intersects this circle at points \( K \) and \( E \) (point \( E \) lies on segment \( BK \)). Find \( BE \), knowing that \( BC = a \), \( CM = b < a \), and \( \angle KME ... |
Given a triangle \( ABC \). A circle with radius \( R \) touches side \( AC \) at point \( M \) and side \( BC \) at point \( P \). Side \( AB \) intersects this circle at points \( K \) and \( E \) (point \( E \) lies on segment \( BK \)). Find \( BE \), knowing that \( BC = a \), \( CM = b < a \), and \( \angle KME ... | qwen |
Given that \( H \) is the orthocenter of \( \triangle ABC \), the circumradius of \( \triangle ABC \) is \( R \), and the circumradius of \( \triangle BHC \) is \( r \). Determine the relationship between \( R \) and \( r \):
(A) \( R = r \).
(B) \( R > r \).
(C) \( R < r \).
(D) If \( \triangle ABC \) is an acute tri... |
Given that \( H \) is the orthocenter of \( \triangle ABC \), the circumradius of \( \triangle ABC \) is \( R \), and the circumradius of \( \triangle BHC \) is \( r \). Determine the relationship between \( R \) and \( r \):
(A) \( R = r \).
(B) \( R > r \).
(C) \( R < r \).
(D) If \( \triangle ABC \) is an acute tri... | qwen |
Given that \( H \) is the orthocenter of \( \triangle ABC \), the circumradius of \( \triangle ABC \) is \( R \), and the circumradius of \( \triangle BHC \) is \( r \). Determine the relationship between \( R \) and \( r \):
(A) \( R = r \).
(B) \( R > r \).
(C) \( R < r \).
(D) If \( \triangle ABC \) is an acute tri... |
Given that \( H \) is the orthocenter of \( \triangle ABC \), the circumradius of \( \triangle ABC \) is \( R \), and the circumradius of \( \triangle BHC \) is \( r \). Determine the relationship between \( R \) and \( r \):
(A) \( R = r \).
(B) \( R > r \).
(C) \( R < r \).
(D) If \( \triangle ABC \) is an acute tri... | r1 |
Given the sequence $\left\{a_{n}\right\}$ defined by: $a_{1}=\frac{1}{4}$, $a_{2}=\frac{3}{4}$, and $a_{n+1}=2a_{n}-a_{n-1}$ for $n \geq 2$.
Additionally, the sequence $\left\{b_{n}\right\}$ is defined by: $b_{1} \neq \frac{1}{4}$ and $3b_{n}-b_{n-1}=n$ for $n \geq 2$. The sum of the first $n$ terms of the sequence ... |
Given the sequence $\left\{a_{n}\right\}$ defined by: $a_{1}=\frac{1}{4}$, $a_{2}=\frac{3}{4}$, and $a_{n+1}=2a_{n}-a_{n-1}$ for $n \geq 2$.
Additionally, the sequence $\left\{b_{n}\right\}$ is defined by: $b_{1} \neq \frac{1}{4}$ and $3b_{n}-b_{n-1}=n$ for $n \geq 2$. The sum of the first $n$ terms of the sequence ... | qwen |
How many employees of the enterprise can be insured by the given insurance company if the enterprise is willing to pay an insurance premium of 5,000,000 rubles?
The cost of service for one person in the clinic (without hospitalization), with which the insurance company has a contract, averages 18,000 rubles. The ave... |
How many employees of the enterprise can be insured by the given insurance company if the enterprise is willing to pay an insurance premium of 5,000,000 rubles?
The cost of service for one person in the clinic (without hospitalization), with which the insurance company has a contract, averages 18,000 rubles. The ave... | r1 |
If \( x_{1} \) satisfies \( 2x + 2^{x} = 5 \) and \( x_{2} \) satisfies \( 2x + 2 \log_{2}(x - 1) = 5 \), then \( x_{1} + x_{2} = \) ? |
If \( x_{1} \) satisfies \( 2x + 2^{x} = 5 \) and \( x_{2} \) satisfies \( 2x + 2 \log_{2}(x - 1) = 5 \), then \( x_{1} + x_{2} = \) ? <think>
Okay, so I have this problem where I need to find the sum of two solutions, x₁ and x₂, from two different equations. The first equation is 2x + 2^x = 5, and the second equation... | r1 |
In a city, there are 10 parallel streets running from north to south, and 10 streets running from west to east, forming a $9 \times 9$ grid. There is a square at each intersection, totaling 100 squares. The distance between any two adjacent squares along a street is 1 km. A walking distance pharmacy is defined as a ph... |
In a city, there are 10 parallel streets running from north to south, and 10 streets running from west to east, forming a $9 \times 9$ grid. There is a square at each intersection, totaling 100 squares. The distance between any two adjacent squares along a street is 1 km. A walking distance pharmacy is defined as a ph... | r1 |
In a city, there are 10 parallel streets running from north to south, and 10 streets running from west to east, forming a $9 \times 9$ grid. There is a square at each intersection, totaling 100 squares. The distance between any two adjacent squares along a street is 1 km. A walking distance pharmacy is defined as a ph... |
In a city, there are 10 parallel streets running from north to south, and 10 streets running from west to east, forming a $9 \times 9$ grid. There is a square at each intersection, totaling 100 squares. The distance between any two adjacent squares along a street is 1 km. A walking distance pharmacy is defined as a ph... | qwen |
In a mini-football tournament, bets are accepted on four teams. Bets on the first team have odds of $1: 5$ (if the first team wins, the player receives the amount they bet on this team plus five times that amount, i.e., six times the bet amount, and if they lose, the money is not returned). Bets on the second team hav... |
In a mini-football tournament, bets are accepted on four teams. Bets on the first team have odds of $1: 5$ (if the first team wins, the player receives the amount they bet on this team plus five times that amount, i.e., six times the bet amount, and if they lose, the money is not returned). Bets on the second team hav... | qwen |
In a mini-football tournament, bets are accepted on four teams. Bets on the first team have odds of $1: 5$ (if the first team wins, the player receives the amount they bet on this team plus five times that amount, i.e., six times the bet amount, and if they lose, the money is not returned). Bets on the second team hav... |
In a mini-football tournament, bets are accepted on four teams. Bets on the first team have odds of $1: 5$ (if the first team wins, the player receives the amount they bet on this team plus five times that amount, i.e., six times the bet amount, and if they lose, the money is not returned). Bets on the second team hav... | r1 |
In a reference frame associated with Gavrila, balls are thrown upwards from a single point with the same speed at intervals of $\tau$ time. Introducing the $y$-axis directed vertically upwards with its origin at the point of throw and measuring time from the moment the first ball is thrown, the motion equations for bo... |
In a reference frame associated with Gavrila, balls are thrown upwards from a single point with the same speed at intervals of $\tau$ time. Introducing the $y$-axis directed vertically upwards with its origin at the point of throw and measuring time from the moment the first ball is thrown, the motion equations for bo... | qwen |
In a reference frame associated with Gavrila, balls are thrown upwards from a single point with the same speed at intervals of $\tau$ time. Introducing the $y$-axis directed vertically upwards with its origin at the point of throw and measuring time from the moment the first ball is thrown, the motion equations for bo... |
In a reference frame associated with Gavrila, balls are thrown upwards from a single point with the same speed at intervals of $\tau$ time. Introducing the $y$-axis directed vertically upwards with its origin at the point of throw and measuring time from the moment the first ball is thrown, the motion equations for bo... | r1 |
In a right-angled triangle $ABC$ with the right angle at $C$ and angle $A$ equal to $30^{\circ}$, point $I$ is the incenter of triangle $ABC$, and point $D$ is the intersection of segment $BI$ with the incircle. Prove that the segments $AI$ and $CD$ are perpendicular. |
In a right-angled triangle $ABC$ with the right angle at $C$ and angle $A$ equal to $30^{\circ}$, point $I$ is the incenter of triangle $ABC$, and point $D$ is the intersection of segment $BI$ with the incircle. Prove that the segments $AI$ and $CD$ are perpendicular. <think>
Alright, so I have this geometry problem h... | r1 |
In a right-angled triangle $ABC$ with the right angle at $C$ and angle $A$ equal to $30^{\circ}$, point $I$ is the incenter of triangle $ABC$, and point $D$ is the intersection of segment $BI$ with the incircle. Prove that the segments $AI$ and $CD$ are perpendicular. |
In a right-angled triangle $ABC$ with the right angle at $C$ and angle $A$ equal to $30^{\circ}$, point $I$ is the incenter of triangle $ABC$, and point $D$ is the intersection of segment $BI$ with the incircle. Prove that the segments $AI$ and $CD$ are perpendicular. To prove that the segments \(AI\) and \(CD\) are p... | qwen |
In each cell of a strip of length 100 there is a chip. You can swap any two adjacent chips for 1 ruble, and you can also swap any two chips for free if there are exactly three chips between them. What is the minimum number of rubles needed to reverse the order of the chips? |
In each cell of a strip of length 100 there is a chip. You can swap any two adjacent chips for 1 ruble, and you can also swap any two chips for free if there are exactly three chips between them. What is the minimum number of rubles needed to reverse the order of the chips? <think>
Okay, so I have this problem where t... | r1 |
In each cell of a strip of length 100 there is a chip. You can swap any two adjacent chips for 1 ruble, and you can also swap any two chips for free if there are exactly three chips between them. What is the minimum number of rubles needed to reverse the order of the chips? |
In each cell of a strip of length 100 there is a chip. You can swap any two adjacent chips for 1 ruble, and you can also swap any two chips for free if there are exactly three chips between them. What is the minimum number of rubles needed to reverse the order of the chips? To solve this problem, we need to determine ... | qwen |
In the Umnovka River live Wise gudgeons, and they have a law: as soon as one's eye color changes from gray to green, they must swim away to the Mozgovka River the same night they realize this transformation. There are no mirrors underwater, and although the gudgeons understand everything, they cannot speak or gesture.... |
In the Umnovka River live Wise gudgeons, and they have a law: as soon as one's eye color changes from gray to green, they must swim away to the Mozgovka River the same night they realize this transformation. There are no mirrors underwater, and although the gudgeons understand everything, they cannot speak or gesture.... | qwen |
In the Umnovka River live Wise gudgeons, and they have a law: as soon as one's eye color changes from gray to green, they must swim away to the Mozgovka River the same night they realize this transformation. There are no mirrors underwater, and although the gudgeons understand everything, they cannot speak or gesture.... |
In the Umnovka River live Wise gudgeons, and they have a law: as soon as one's eye color changes from gray to green, they must swim away to the Mozgovka River the same night they realize this transformation. There are no mirrors underwater, and although the gudgeons understand everything, they cannot speak or gesture.... | r1 |
In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of sides \(AC\) and \(AB\) respectively. On the median \(BM\), a point \(P\) is chosen such that it does not lie on line \(CN\). It turns out that \(PC = 2PN\). Prove that \(AP = BC\). |
In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of sides \(AC\) and \(AB\) respectively. On the median \(BM\), a point \(P\) is chosen such that it does not lie on line \(CN\). It turns out that \(PC = 2PN\). Prove that \(AP = BC\). <think>
Okay, so I have this geometry problem here. Let me try to unders... | r1 |
In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of sides \(AC\) and \(AB\) respectively. On the median \(BM\), a point \(P\) is chosen such that it does not lie on line \(CN\). It turns out that \(PC = 2PN\). Prove that \(AP = BC\). |
In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of sides \(AC\) and \(AB\) respectively. On the median \(BM\), a point \(P\) is chosen such that it does not lie on line \(CN\). It turns out that \(PC = 2PN\). Prove that \(AP = BC\). To solve this problem, we will use coordinate geometry. Let's place the ... | qwen |
King Arthur wants to order a new knight's shield from the blacksmith according to his sketch. The king used a compass and drew three arcs with a radius of 1 yard, as shown in the figure. What is the area of the shield? Round your answer to two decimal places. Recall that the area of a circle with radius \( r \) is \( ... |
King Arthur wants to order a new knight's shield from the blacksmith according to his sketch. The king used a compass and drew three arcs with a radius of 1 yard, as shown in the figure. What is the area of the shield? Round your answer to two decimal places. Recall that the area of a circle with radius \( r \) is \( ... | qwen |
King Arthur wants to order a new knight's shield from the blacksmith according to his sketch. The king used a compass and drew three arcs with a radius of 1 yard, as shown in the figure. What is the area of the shield? Round your answer to two decimal places. Recall that the area of a circle with radius \( r \) is \( ... |
King Arthur wants to order a new knight's shield from the blacksmith according to his sketch. The king used a compass and drew three arcs with a radius of 1 yard, as shown in the figure. What is the area of the shield? Round your answer to two decimal places. Recall that the area of a circle with radius \( r \) is \( ... | r1 |
Kostya planted some number of tulip bulbs along a path. Then Tanya came and planted a new bulb between each pair of neighboring bulbs planted by Kostya. After that, Inna came and planted a new bulb between each pair of neighboring bulbs already planted. Finally, Dima came and did the same thing.
All planted bulbs hav... |
Kostya planted some number of tulip bulbs along a path. Then Tanya came and planted a new bulb between each pair of neighboring bulbs planted by Kostya. After that, Inna came and planted a new bulb between each pair of neighboring bulbs already planted. Finally, Dima came and did the same thing.
All planted bulbs hav... | qwen |
Kostya planted some number of tulip bulbs along a path. Then Tanya came and planted a new bulb between each pair of neighboring bulbs planted by Kostya. After that, Inna came and planted a new bulb between each pair of neighboring bulbs already planted. Finally, Dima came and did the same thing.
All planted bulbs hav... |
Kostya planted some number of tulip bulbs along a path. Then Tanya came and planted a new bulb between each pair of neighboring bulbs planted by Kostya. After that, Inna came and planted a new bulb between each pair of neighboring bulbs already planted. Finally, Dima came and did the same thing.
All planted bulbs hav... | r1 |
Let $\mathrm{ABC}$ be a triangle where the angle bisectors $\mathrm{AL}$ and $\mathrm{BN}$ intersect at the point $\mathrm{O}$. Denote by $h_{a}$ the altitude dropped onto side $a$, and $l_{a}$ the angle bisector on side $a$. (See Figure 1.) |
Let $\mathrm{ABC}$ be a triangle where the angle bisectors $\mathrm{AL}$ and $\mathrm{BN}$ intersect at the point $\mathrm{O}$. Denote by $h_{a}$ the altitude dropped onto side $a$, and $l_{a}$ the angle bisector on side $a$. (See Figure 1.) <think>
Alright, so I've got this geometry problem here. It says, let ABC be ... | r1 |
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