instruction_seed stringlengths 21 1.44k | response_seed stringlengths 0 1.44k ⌀ | reasoning stringlengths 1.35k 80.8k | deepseek_solution stringlengths 237 3.8k | __original_row_idx int64 36 40k ⌀ | source stringclasses 4
values | final_reasoning_trace stringlengths 1.74k 82.6k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|
$(*)$ Let $ABC$ be a triangle with $\angle ABC$ obtuse. The $A$-excircle is a circle in the exterior of $\triangle ABC$ that is tangent to side $\overline{BC}$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respect... | null | Okay, so I need to figure out if the line EF can be tangent to the A-excircle in an obtuse triangle ABC where angle ABC is obtuse. Let me start by recalling some definitions and properties to make sure I have the right setup.
First, the A-excircle. Ex circles are also called escribed circles. The A-excircle is tangent... | Given an obtuse triangle \(ABC\) with \(\angle ABC\) obtuse, we need to determine if the line \(EF\) formed by the feet of the altitudes from \(B\) and \(C\) to \(AC\) and \(AB\) respectively can be tangent to the \(A\)-excircle.
1. **Understanding the A-excircle**: The \(A\)-excircle is tangent to \(BC\) and the exte... | null | OpenMathInstruct-AIME | <think>
Okay, so I need to figure out if the line EF can be tangent to the A-excircle in an obtuse triangle ABC where angle ABC is obtuse. Let me start by recalling some definitions and properties to make sure I have the right setup.
First, the A-excircle. Ex circles are also called escribed circles. The A-excircle is... | [
{
"from": "human",
"value": "$(*)$ Let $ABC$ be a triangle with $\\angle ABC$ obtuse. The $A$-excircle is a circle in the exterior of $\\triangle ABC$ that is tangent to side $\\overline{BC}$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from... |
15. Let the complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3,\left|z_{1}-z_{2}\right|=3 \sqrt{3}$, find the value of $\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000}+\left(\overline{z_{1}} z_{2}\right)^{2000}\right|$. | null | Okay, so I need to solve this problem involving complex numbers. Let me read it again carefully.
We have two complex numbers, z₁ and z₂. They satisfy |z₁| = |z₁ + z₂| = 3, and |z₁ - z₂| = 3√3. We need to find the value of log base 3 of the absolute value of (z₁ times the conjugate of z₂) raised to the 2000th power plu... | Given the complex numbers \( z_1 \) and \( z_2 \) satisfying \( |z_1| = |z_1 + z_2| = 3 \) and \( |z_1 - z_2| = 3\sqrt{3} \), we need to find the value of \( \log_3 \left| \left( z_1 \overline{z_2} \right)^{2000} + \left( \overline{z_1} z_2 \right)^{2000} \right| \).
First, we use the given magnitudes to set up equati... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to solve this problem involving complex numbers. Let me read it again carefully.
We have two complex numbers, z₁ and z₂. They satisfy |z₁| = |z₁ + z₂| = 3, and |z₁ - z₂| = 3√3. We need to find the value of log base 3 of the absolute value of (z₁ times the conjugate of z₂) raised to the 2000th p... | [
{
"from": "human",
"value": "15. Let the complex numbers $z_{1}, z_{2}$ satisfy $\\left|z_{1}\\right|=\\left|z_{1}+z_{2}\\right|=3,\\left|z_{1}-z_{2}\\right|=3 \\sqrt{3}$, find the value of $\\log _{3}\\left|\\left(z_{1} \\overline{z_{2}}\\right)^{2000}+\\left(\\overline{z_{1}} z_{2}\\right)^{2000}\\right|$... |
Each corner cube is removed from this $3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}$ cube. The surface area of the remaining figure is
[asy] draw((2.7,3.99)--(0,3)--(0,0)); draw((3.7,3.99)--(1,3)--(1,0)); draw((4.7,3.99)--(2,3)--(2,0)); draw((5.7,3.99)--(3,3)--(3,0)); draw((0,0)--(3,0)--(5.7,0.99)); draw((0,1)--(3... | null | Okay, let me try to figure out this problem step by step. So, we have a 3 cm × 3 cm × 3 cm cube, and each corner cube is removed. Then we need to find the surface area of the remaining figure. The answer choices are from 19 up to 72 sq.cm, so there's a big range here. Let me visualize the cube first. A 3x3x3 cube is ma... | The original 3 cm × 3 cm × 3 cm cube has a surface area of 6 faces × 9 sq.cm each = 54 sq.cm. When each corner cube is removed, the surface area is affected as follows:
1. **Removal of Corner Cubes**: Each corner cube is part of three faces of the original cube. Removing a corner cube subtracts 1 sq.cm from each of th... | null | OpenMathInstruct-AIME | <think>
Okay, let me try to figure out this problem step by step. So, we have a 3 cm × 3 cm × 3 cm cube, and each corner cube is removed. Then we need to find the surface area of the remaining figure. The answer choices are from 19 up to 72 sq.cm, so there's a big range here. Let me visualize the cube first. A 3x3x3 cu... | [
{
"from": "human",
"value": "Each corner cube is removed from this $3\\text{ cm}\\times 3\\text{ cm}\\times 3\\text{ cm}$ cube. The surface area of the remaining figure is\n[asy] draw((2.7,3.99)--(0,3)--(0,0)); draw((3.7,3.99)--(1,3)--(1,0)); draw((4.7,3.99)--(2,3)--(2,0)); draw((5.7,3.99)--(3,3)--(3,0)); ... |
Simplify
\[\frac{\sin{10^\circ}+\sin{20^\circ}}{\cos{10^\circ}+\cos{20^\circ}}.\]Enter your answer is a trigonometric function evaluated at an integer, such as "sin 7". (The angle should be positive and as small as possible.) | Simplify
\[\frac{\sin{10^\circ}+\sin{20^\circ}}{\cos{10^\circ}+\cos{20^\circ}}.\]Enter your answer is a trigonometric function evaluated at an integer, such as "sin 7". (The angle should be positive and as small as possible.) | Okay, let's see. I need to simplify this trigonometric expression: (sin10° + sin20°)/(cos10° + cos20°). Hmm, the problem says to express it as a trigonometric function evaluated at an integer angle, like "sin 7", and the angle should be as small and positive as possible. Alright, so maybe I can use some trigonometric i... | To simplify the expression \(\frac{\sin{10^\circ}+\sin{20^\circ}}{\cos{10^\circ}+\cos{20^\circ}}\), we use the sum-to-product identities for sine and cosine.
For the numerator \(\sin{10^\circ} + \sin{20^\circ}\):
\[
\sin{10^\circ} + \sin{20^\circ} = 2 \sin{\left(\frac{10^\circ + 20^\circ}{2}\right)} \cos{\left(\frac{1... | 33,695 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to simplify this trigonometric expression: (sin10° + sin20°)/(cos10° + cos20°). Hmm, the problem says to express it as a trigonometric function evaluated at an integer angle, like "sin 7", and the angle should be as small and positive as possible. Alright, so maybe I can use some trigono... | [
{
"from": "human",
"value": "Simplify\n\\[\\frac{\\sin{10^\\circ}+\\sin{20^\\circ}}{\\cos{10^\\circ}+\\cos{20^\\circ}}.\\]Enter your answer is a trigonometric function evaluated at an integer, such as \"sin 7\". (The angle should be positive and as small as possible.)"
},
{
"from": "gpt",
"valu... |
Prove that any knot or link in a $3$-highly twisted $2m$-plat diagram with $m \geq 2$ and at least three twist regions is hyperbolic. What fundamental properties of the link complement lead to this conclusion? | Okay, so I need to prove that any knot or link in a 3-highly twisted 2m-plat diagram with m ≥ 2 and at least three twist regions is hyperbolic. Hmm, let me start by breaking down the terms here to make sure I understand them correctly.
First, a 2m-plat diagram. I remember that a plat diagram is a way of representing k... | To prove that any knot or link in a 3-highly twisted 2m-plat diagram with \( m \geq 2 \) and at least three twist regions is hyperbolic, we need to show that its complement in \( S^3 \) is irreducible, atoroidal, and anannular. These properties ensure the complement admits a complete hyperbolic metric by Thurston's hyp... | 18,039 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to prove that any knot or link in a 3-highly twisted 2m-plat diagram with m ≥ 2 and at least three twist regions is hyperbolic. Hmm, let me start by breaking down the terms here to make sure I understand them correctly.
First, a 2m-plat diagram. I remember that a plat diagram is a way of repres... | [
{
"from": "human",
"value": "Prove that any knot or link in a $3$-highly twisted $2m$-plat diagram with $m \\geq 2$ and at least three twist regions is hyperbolic. What fundamental properties of the link complement lead to this conclusion?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need ... | |
A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the same size. If the bookshelf is currently empty, what is the least number of shelves that need to be used to store 24 books of different sizes, assuming that books of the same size must be placed on the same shelf? | A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the same size. If the bookshelf is currently empty, what is the least number of shelves that need to be used to store 24 books of different sizes, assuming that books of the same size must be placed on the same shelf? | Alright, so I need to figure out the least number of shelves required to store 24 books of different sizes on a bookshelf that has 5 shelves. Each shelf can hold at most 8 books of the same size. The catch here is that books of the same size must be on the same shelf. Since all 24 books are different sizes, that means ... | The problem requires determining the least number of shelves needed to store 24 books of different sizes on a bookshelf with 5 shelves, where each shelf can hold at most 8 books of the same size.
Key points:
1. Each shelf can hold at most 8 books of the same size.
2. All 24 books are of different sizes, meaning each ... | 3,003 | ai2-adapt-dev/openmath-2-math | <think>
Alright, so I need to figure out the least number of shelves required to store 24 books of different sizes on a bookshelf that has 5 shelves. Each shelf can hold at most 8 books of the same size. The catch here is that books of the same size must be on the same shelf. Since all 24 books are different sizes, tha... | [
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the same size. If the bookshelf is currently empty, what is the least number of shelves that need to be used to store 24 books of different sizes, assuming that books of the same size must be placed on the sam... |
On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process conti... | null | Okay, let me try to figure out this problem step by step. So, the question is about finding the smallest number of students such that from June 1 to June 12, they can arrange themselves in different row configurations each day, but on June 13, they can't find a new way. The answer choices are given, and we need to pick... | To determine the smallest number of students \( N \) such that the group can be arranged in different row configurations each day from June 1 to June 12, and cannot find a new configuration on June 13, we need \( N \) to have exactly 12 divisors. Additionally, \( N \) must be a multiple of 15 and 6 (as given by the pro... | null | OpenMathInstruct-AIME | <think>
Okay, let me try to figure out this problem step by step. So, the question is about finding the smallest number of students such that from June 1 to June 12, they can arrange themselves in different row configurations each day, but on June 13, they can't find a new way. The answer choices are given, and we need... | [
{
"from": "human",
"value": "On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6... |
Given the curve $\frac{x^2}{12-m}+\frac{y^2}{m-4}=1$ represents an ellipse, which of the following statements is correct?
A: The range of $m$ is $\left(4,12\right)$.
B: If the foci of the ellipse are on the $y$-axis, then $m\in \left(8,12\right)$.
C: If $m=6$, then the focal length of the ellipse is $4$.
D: If the ... | null | Okay, let's see. I have this problem about an ellipse given by the equation (x²)/(12 - m) + (y²)/(m - 4) = 1. I need to check which of the statements A, B, C, D are correct. Let's start by recalling some basics about ellipses.
First, the standard form of an ellipse is (x²/a²) + (y²/b²) = 1, where a and b are positive ... | Given the curve \(\frac{x^2}{12-m} + \frac{y^2}{m-4} = 1\) represents an ellipse, we need to determine which of the statements A, B, C, and D are correct.
**Statement A: The range of \(m\) is \((4, 12)\).**
For the equation to represent an ellipse, both denominators must be positive:
- \(12 - m > 0 \implies m < 12\)
... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I have this problem about an ellipse given by the equation (x²)/(12 - m) + (y²)/(m - 4) = 1. I need to check which of the statements A, B, C, D are correct. Let's start by recalling some basics about ellipses.
First, the standard form of an ellipse is (x²/a²) + (y²/b²) = 1, where a and b are p... | [
{
"from": "human",
"value": "Given the curve $\\frac{x^2}{12-m}+\\frac{y^2}{m-4}=1$ represents an ellipse, which of the following statements is correct?\n\nA: The range of $m$ is $\\left(4,12\\right)$.\n\nB: If the foci of the ellipse are on the $y$-axis, then $m\\in \\left(8,12\\right)$.\n\nC: If $m=6$, th... |
Consider the damped hyperbolic Allen-Cahn equation given by
\[
\tau u_{tt} + g(u) u_t = \Delta u - \varepsilon^{-2} F'(u),
\]
where \(F(u) = \frac{1}{4}(u^2 - 1)^2\) is a double well potential with wells at \(u = -1\) and \(u = 1\).
Assuming \(u(\cdot,0) = u_0\) takes values \(-1\) in one half of the domain and \(+... | Okay, so I need to figure out the motion of the interface in the damped hyperbolic Allen-Cahn equation as ε approaches zero. The equation given is:
τ u_{tt} + g(u) u_t = Δu - ε^{-2} F'(u),
where F(u) is the double well potential (1/4)(u² - 1)². The initial condition u₀ is -1 in one half of the domain and +1 in the ot... | To analyze the motion of the interface in the damped hyperbolic Allen-Cahn equation as \(\varepsilon \to 0\), we use matched asymptotic expansions. The equation is given by:
\[
\tau u_{tt} + g(u) u_t = \Delta u - \varepsilon^{-2} F'(u),
\]
where \(F(u) = \frac{1}{4}(u^2 - 1)^2\). The interface between the regions whe... | 4,303 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the motion of the interface in the damped hyperbolic Allen-Cahn equation as ε approaches zero. The equation given is:
τ u_{tt} + g(u) u_t = Δu - ε^{-2} F'(u),
where F(u) is the double well potential (1/4)(u² - 1)². The initial condition u₀ is -1 in one half of the domain and +1 i... | [
{
"from": "human",
"value": "Consider the damped hyperbolic Allen-Cahn equation given by\n\n\\[\n\\tau u_{tt} + g(u) u_t = \\Delta u - \\varepsilon^{-2} F'(u),\n\\]\n\nwhere \\(F(u) = \\frac{1}{4}(u^2 - 1)^2\\) is a double well potential with wells at \\(u = -1\\) and \\(u = 1\\). \n\nAssuming \\(u(\\cdot,0... | |
From time $t=0$ to time $t=1$ a population increased by $i\%$, and from time $t=1$ to time $t=2$ the population increased by $j\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by
$\text{(A) (i+j)\%} \quad \text{(B) } ij\%\quad \text{(C) } (i+ij)\%\quad \text{(D) } \left(i+j+\frac{ij}{100}\right)\%... | null | Okay, so I need to figure out the total percentage increase in population from time t=0 to t=2, given that it increased by i% from t=0 to t=1 and then by j% from t=1 to t=2. The answer choices are A through E, and I have to pick the right one. Let me start by recalling how percentage increases work when applied success... | Given a population that increases by \(i\%\) from time \(t=0\) to \(t=1\) and then by \(j\%\) from time \(t=1\) to \(t=2\), we need to determine the total percentage increase from \(t=0\) to \(t=2\).
1. **Initial Population**: Let the initial population at \(t=0\) be \(P_0\).
2. **Population at \(t=1\)**: After an inc... | null | OpenMathInstruct-AIME | <think>
Okay, so I need to figure out the total percentage increase in population from time t=0 to t=2, given that it increased by i% from t=0 to t=1 and then by j% from t=1 to t=2. The answer choices are A through E, and I have to pick the right one. Let me start by recalling how percentage increases work when applied... | [
{
"from": "human",
"value": "From time $t=0$ to time $t=1$ a population increased by $i\\%$, and from time $t=1$ to time $t=2$ the population increased by $j\\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by\n$\\text{(A) (i+j)\\%} \\quad \\text{(B) } ij\\%\\quad \\text{(C) } (i+ij)\\... |
Let \( w: \mathbb{R}^2 \to \mathbb{R} \) be the function defined by
\[
w(x,y) := \begin{cases}
x & \text{if } y > 0, \\
-x & \text{if } y < 0.
\end{cases}
\]
For every point \( (x_0, y_0) \neq (0,0) \), what is the area of an open neighborhood \( U \) around \( (x_0, y_0) \) such that \( w \) is a Dirichlet minimi... | Okay, let's try to figure out this problem about the Mumford-Shah functional and Dirichlet minimizers. The function given is w(x,y) which is x when y > 0 and -x when y < 0. So, essentially, it's a piecewise function that changes sign depending on whether we're in the upper half-plane or the lower half-plane. The questi... | The function \( w(x, y) \) is defined as \( x \) for \( y > 0 \) and \( -x \) for \( y < 0 \). We need to determine the area of an open neighborhood \( U \) around a point \( (x_0, y_0) \neq (0,0) \) such that \( w \) is a Dirichlet minimizer of the Mumford-Shah functional in \( U \).
1. **Case \( y_0 \neq 0 \)**:
... | 10,314 | mlfoundations-dev/gair_mathpile | <think>
Okay, let's try to figure out this problem about the Mumford-Shah functional and Dirichlet minimizers. The function given is w(x,y) which is x when y > 0 and -x when y < 0. So, essentially, it's a piecewise function that changes sign depending on whether we're in the upper half-plane or the lower half-plane. Th... | [
{
"from": "human",
"value": "Let \\( w: \\mathbb{R}^2 \\to \\mathbb{R} \\) be the function defined by\n\n\\[\nw(x,y) := \\begin{cases} \nx & \\text{if } y > 0, \\\\ \n-x & \\text{if } y < 0. \n\\end{cases}\n\\]\n\nFor every point \\( (x_0, y_0) \\neq (0,0) \\), what is the area of an open neighborhood \\( U... | |
Given two analytic self-maps \(\varphi\) and \(\psi\) on the open unit disk \(\mathbb{D}\), and two functions \(g, h \in H(\mathbb{D})\), define the generalized composition operators \(C_\varphi^g\) and \(C_\psi^h\) acting from the \(\alpha\)-Bloch space \(\mathcal{B}^\alpha\) to the \(\beta\)-Bloch space \(\mathcal{B}... | Okay, so I need to figure out the conditions under which the essential norm of the difference between two generalized composition operators \( C^g_\varphi - C^h_\psi \) from the α-Bloch space to the β-Bloch space is zero. Also, how does this relate to those three limit expressions involving the supremums?
First, let m... | To determine the conditions under which the essential norm \(\|C^g_\varphi - C^h_\psi\|_{e, \mathcal{B}^\alpha \rightarrow \mathcal{B}^\beta}\) is zero, we need to ensure that the operator \(C^g_\varphi - C^h_\psi\) is compact. Compactness of this operator implies that it maps bounded sets in the α-Bloch space \(\mathc... | 33,676 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the conditions under which the essential norm of the difference between two generalized composition operators \( C^g_\varphi - C^h_\psi \) from the α-Bloch space to the β-Bloch space is zero. Also, how does this relate to those three limit expressions involving the supremums?
Firs... | [
{
"from": "human",
"value": "Given two analytic self-maps \\(\\varphi\\) and \\(\\psi\\) on the open unit disk \\(\\mathbb{D}\\), and two functions \\(g, h \\in H(\\mathbb{D})\\), define the generalized composition operators \\(C_\\varphi^g\\) and \\(C_\\psi^h\\) acting from the \\(\\alpha\\)-Bloch space \\... | |
16. Figure (1) is a small cubic wooden block placed horizontally, Figures (2) and (3) are formed by stacking such small cubic wooden blocks according to a certain pattern. Continuing to stack them in this pattern, in the 10th stacked figure, the total number of small cubic wooden blocks in the bottom layer is $\qquad$ | null | Okay, let me try to figure out this problem. So, we have these figures made by stacking small cubic wooden blocks. Figure 1 is just a single cube placed horizontally. Then Figures 2 and 3 are formed by stacking more cubes following a certain pattern. The question is asking about the 10th figure in this sequence: how ma... | To determine the number of small cubic wooden blocks in the bottom layer of the 10th figure, we need to identify the pattern in which the blocks are stacked.
1. **Figure 1**: A single cubic block, which is a 1x1x1 cube. The bottom layer has 1 block.
2. **Figure 2**: Stacked according to a certain pattern. If each sub... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this problem. So, we have these figures made by stacking small cubic wooden blocks. Figure 1 is just a single cube placed horizontally. Then Figures 2 and 3 are formed by stacking more cubes following a certain pattern. The question is asking about the 10th figure in this sequence... | [
{
"from": "human",
"value": "16. Figure (1) is a small cubic wooden block placed horizontally, Figures (2) and (3) are formed by stacking such small cubic wooden blocks according to a certain pattern. Continuing to stack them in this pattern, in the 10th stacked figure, the total number of small cubic woode... |
The sum of two natural numbers is $17{,}402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?
$\textbf{(A)} ~10{,}272\qquad\textbf{(B)} ~11{,}700\qquad\textbf{(C)} ~13{,}362\qquad\textbf{(D)} ~14{,}238\q... | null | Okay, let's see. The problem says that the sum of two natural numbers is 17,402. One of them is divisible by 10, which means it ends with a 0 in the units digit. If we erase that 0, we get the other number. We need to find the difference between these two numbers.
Let me break it down. Let the two numbers be A and B. ... | Let the two numbers be \( A \) and \( B \), where \( A \) is divisible by 10. When the units digit of \( A \) is erased, we obtain \( B \). This means \( B = \frac{A}{10} \). Given that the sum of the two numbers is 17,402, we have:
\[
A + B = 17,402
\]
Substituting \( B = \frac{A}{10} \) into the equation:
\[
A + \... | null | OpenMathInstruct-AIME | <think>
Okay, let's see. The problem says that the sum of two natural numbers is 17,402. One of them is divisible by 10, which means it ends with a 0 in the units digit. If we erase that 0, we get the other number. We need to find the difference between these two numbers.
Let me break it down. Let the two numbers be A... | [
{
"from": "human",
"value": "The sum of two natural numbers is $17{,}402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?\n$\\textbf{(A)} ~10{,}272\\qquad\\textbf{(B)} ~11{,}700\\qquad\\text... |
Given the symplectic blob algebra $\sba$ with generators \( \mathcal{G}_n = \{ e, U_1, \ldots, U_{n-1}, f \} \) and the relations that include \( e^2 = X_2 e \) and \( f^2 = X_3 f \), can you derive the specific expressions for \( \theta(D) \), \( \theta(D_L) \), \( \theta(D_R) \), \( \theta(K_L) \), \( \theta(K_R) \),... | Okay, so I need to figure out the expressions for these theta functions θ(D), θ(D_L), θ(D_R), θ(K_L), θ(K_R), and θ(K) in terms of the parameters a, b, c, d, x, y, z, w. And also determine the conditions these parameters must satisfy to preserve the representation of the symplectic blob algebra. Hmm, let's start by rec... | To derive the specific expressions for \( \theta(D) \), \( \theta(D_L) \), \( \theta(D_R) \), \( \theta(K_L) \), \( \theta(K_R) \), and \( \theta(K) \) in terms of the parameters \( a, b, c, d, x, y, z, w \), we need to consider the structure of the symplectic blob algebra and the relations it satisfies. The key relati... | 3,632 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the expressions for these theta functions θ(D), θ(D_L), θ(D_R), θ(K_L), θ(K_R), and θ(K) in terms of the parameters a, b, c, d, x, y, z, w. And also determine the conditions these parameters must satisfy to preserve the representation of the symplectic blob algebra. Hmm, let's star... | [
{
"from": "human",
"value": "Given the symplectic blob algebra $\\sba$ with generators \\( \\mathcal{G}_n = \\{ e, U_1, \\ldots, U_{n-1}, f \\} \\) and the relations that include \\( e^2 = X_2 e \\) and \\( f^2 = X_3 f \\), can you derive the specific expressions for \\( \\theta(D) \\), \\( \\theta(D_L) \\)... | |
How many three-digit numbers satisfy the property that the middle digit is the [average](https://artofproblemsolving.com/wiki/index.php/Mean) of the first and the last digits?
$(\mathrm {A}) \ 41 \qquad (\mathrm {B}) \ 42 \qquad (\mathrm {C})\ 43 \qquad (\mathrm {D}) \ 44 \qquad (\mathrm {E})\ 45$ | null | Okay, let's see. The problem is asking for the number of three-digit numbers where the middle digit is the average of the first and the last digits. The options are from 41 to 45, so the answer is probably one of those. Let me think about how to approach this.
First, a three-digit number can be represented as ABC, whe... | To determine the number of three-digit numbers where the middle digit is the average of the first and the last digits, we represent the three-digit number as \(ABC\) where \(A\) is the hundreds digit, \(B\) is the tens digit, and \(C\) is the ones digit. The middle digit \(B\) must be the average of \(A\) and \(C\), me... | null | OpenMathInstruct-AIME | <think>
Okay, let's see. The problem is asking for the number of three-digit numbers where the middle digit is the average of the first and the last digits. The options are from 41 to 45, so the answer is probably one of those. Let me think about how to approach this.
First, a three-digit number can be represented as ... | [
{
"from": "human",
"value": "How many three-digit numbers satisfy the property that the middle digit is the [average](https://artofproblemsolving.com/wiki/index.php/Mean) of the first and the last digits?\n$(\\mathrm {A}) \\ 41 \\qquad (\\mathrm {B}) \\ 42 \\qquad (\\mathrm {C})\\ 43 \\qquad (\\mathrm {D}) ... |
In a triangular array of numbers, each number is the sum of the two numbers directly above it. The first row consists of the numbers 1, 2, and 3. The second row consists of the numbers 3, 5, and 7. The third row consists of the numbers 6, 10, and 14. And so on. Which row has the greatest number of entries that are divi... | In a triangular array of numbers, each number is the sum of the two numbers directly above it. The first row consists of the numbers 1, 2, and 3. The second row consists of the numbers 3, 5, and 7. The third row consists of the numbers 6, 10, and 14. And so on. Which row has the greatest number of entries that are divi... | Okay, let's see. I need to figure out which row in this triangular array has the most entries divisible by 5. The first row is 1, 2, 3. The second row is 3, 5, 7. The third row is 6, 10, 14. And it continues on like that. Each number is the sum of the two numbers above it. So, similar to Pascal's triangle, but starting... | The problem involves a triangular array where each number is the sum of the two numbers directly above it. The first few rows are given, and we need to determine which row has the greatest number of entries divisible by 5.
1. **Understanding the Construction**:
- The first row is \(1, 2, 3\).
- The second row is... | 7,791 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to figure out which row in this triangular array has the most entries divisible by 5. The first row is 1, 2, 3. The second row is 3, 5, 7. The third row is 6, 10, 14. And it continues on like that. Each number is the sum of the two numbers above it. So, similar to Pascal's triangle, but ... | [
{
"from": "human",
"value": "In a triangular array of numbers, each number is the sum of the two numbers directly above it. The first row consists of the numbers 1, 2, and 3. The second row consists of the numbers 3, 5, and 7. The third row consists of the numbers 6, 10, and 14. And so on. Which row has the... |
A fair six-sided die is rolled twice. What is the expected value of the product of the numbers on the two rolls? | A fair six-sided die is rolled twice. What is the expected value of the product of the numbers on the two rolls? | Okay, so I need to find the expected value of the product of two rolls of a fair six-sided die. Hmm, let me start by recalling what expected value means. The expected value is like the average outcome we would expect if we did an experiment many times. So in this case, if I roll a die twice, multiply the two numbers, a... | To find the expected value of the product of the numbers on two rolls of a fair six-sided die, we start by noting that each roll is an independent random variable. Let \(X\) and \(Y\) represent the outcomes of the first and second rolls, respectively. Both \(X\) and \(Y\) can take values from 1 to 6 with equal probabil... | 7,087 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the expected value of the product of two rolls of a fair six-sided die. Hmm, let me start by recalling what expected value means. The expected value is like the average outcome we would expect if we did an experiment many times. So in this case, if I roll a die twice, multiply the two nu... | [
{
"from": "human",
"value": "A fair six-sided die is rolled twice. What is the expected value of the product of the numbers on the two rolls?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the expected value of the product of two rolls of a fair six-sided die. Hmm, let me start ... |
A set of 8 coins consists of 2 pennies, 3 nickels, and 3 dimes. Two of the coins are selected at random and put into a box. If the identity of each coin (penny, nickel, or dime) and the total value of the coins are known, how many distinct possible combinations of coins could be in the box? | A set of 8 coins consists of 2 pennies, 3 nickels, and 3 dimes. Two of the coins are selected at random and put into a box. If the identity of each coin (penny, nickel, or dime) and the total value of the coins are known, how many distinct possible combinations of coins could be in the box? | Okay, let's see. So, we have a set of 8 coins: 2 pennies, 3 nickels, and 3 dimes. We need to pick two coins at random and put them into a box. Then, if we know the identity (type) of each coin and the total value, how many distinct possible combinations could be in the box? Hmm, the question is asking for the number of... | We have a set of 8 coins consisting of 2 pennies (P), 3 nickels (N), and 3 dimes (D). We need to determine the number of distinct possible combinations of coins that could be in the box when two coins are selected at random, considering both the identity of each coin and the total value.
First, we list all possible co... | 35,716 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. So, we have a set of 8 coins: 2 pennies, 3 nickels, and 3 dimes. We need to pick two coins at random and put them into a box. Then, if we know the identity (type) of each coin and the total value, how many distinct possible combinations could be in the box? Hmm, the question is asking for the n... | [
{
"from": "human",
"value": "A set of 8 coins consists of 2 pennies, 3 nickels, and 3 dimes. Two of the coins are selected at random and put into a box. If the identity of each coin (penny, nickel, or dime) and the total value of the coins are known, how many distinct possible combinations of coins could be... |
The coefficient of $x^7$ in the expansion of $\left(\frac{x^2}{2}-\frac{2}{x}\right)^8$ is:
$\textbf{(A)}\ 56\qquad \textbf{(B)}\ -56\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ -14\qquad \textbf{(E)}\ 0$ | null | Okay, so I need to find the coefficient of \( x^7 \) in the expansion of \( \left( \frac{x^2}{2} - \frac{2}{x} \right)^8 \). Hmm, let me think.
First, I remember that when expanding expressions like \( (a + b)^n \), we can use the binomial theorem, which states that:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n - ... | To find the coefficient of \( x^7 \) in the expansion of \( \left( \frac{x^2}{2} - \frac{2}{x} \right)^8 \), we use the binomial theorem. The general term in the expansion is given by:
\[
\binom{8}{k} \left( \frac{x^2}{2} \right)^{8 - k} \left( -\frac{2}{x} \right)^k
\]
Simplifying each part of the term, we have:
1.... | null | OpenMathInstruct-AIME | <think>
Okay, so I need to find the coefficient of \( x^7 \) in the expansion of \( \left( \frac{x^2}{2} - \frac{2}{x} \right)^8 \). Hmm, let me think.
First, I remember that when expanding expressions like \( (a + b)^n \), we can use the binomial theorem, which states that:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}... | [
{
"from": "human",
"value": "The coefficient of $x^7$ in the expansion of $\\left(\\frac{x^2}{2}-\\frac{2}{x}\\right)^8$ is:\n$\\textbf{(A)}\\ 56\\qquad \\textbf{(B)}\\ -56\\qquad \\textbf{(C)}\\ 14\\qquad \\textbf{(D)}\\ -14\\qquad \\textbf{(E)}\\ 0$"
},
{
"from": "gpt",
"value": "<think>\nOkay... |
5. Let the mass of the new alloy be 1, and the relative masses of the 1st, 2nd, and 3rd alloys in it be \( x, y, z \) respectively; the content of chromium in the new alloy is \( k \). We obtain the system of equations:
$$
\left\{\begin{array}{c}
x+y+z=1 \\
0.9 x+0.3 z=0.45 \\
0.4 x+0.1 y+0.5 z=k
\end{array}\right.
$$... | null | Okay, so I need to figure out the minimum and maximum possible percentages of chromium in the new alloy based on the given system of equations. Let me start by understanding the problem step by step.
First, the problem mentions three original alloys that are mixed to form a new alloy. The masses of these three origina... | Given the system of equations for the new alloy:
$$
\left\{\begin{array}{c}
x + y + z = 1 \\
0.9x + 0.3z = 0.45 \\
0.4x + 0.1y + 0.5z = k
\end{array}\right.
$$
1. From the second equation, solve for \( x \):
$$
0.9x + 0.3z = 0.45 \implies x = \frac{0.45 - 0.3z}{0.9} = 0.5 - \frac{1}{3}z
$$
2. Substitute \( ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out the minimum and maximum possible percentages of chromium in the new alloy based on the given system of equations. Let me start by understanding the problem step by step.
First, the problem mentions three original alloys that are mixed to form a new alloy. The masses of these three... | [
{
"from": "human",
"value": "5. Let the mass of the new alloy be 1, and the relative masses of the 1st, 2nd, and 3rd alloys in it be \\( x, y, z \\) respectively; the content of chromium in the new alloy is \\( k \\). We obtain the system of equations:\n\n$$\n\\left\\{\\begin{array}{c}\nx+y+z=1 \\\\\n0.9 x+... |
What is $100(100-3)-(100\cdot100-3)$?
$\textbf{(A)}\ -20,000 \qquad \textbf{(B)}\ -10,000 \qquad \textbf{(C)}\ -297 \qquad \textbf{(D)}\ -6 \qquad \textbf{(E)}\ 0$ | null | Okay, so I need to solve this math problem: $100(100 - 3) - (100 \cdot 100 - 3)$. Let me break it down step by step.
First, let's look at the expression. It's two parts: the first part is 100 multiplied by (100 minus 3), and then we subtract the second part, which is (100 times 100 minus 3). So, written out, it's lik... | To solve the expression \(100(100 - 3) - (100 \cdot 100 - 3)\), we break it down step by step:
1. Compute the first part: \(100(100 - 3)\)
- Calculate inside the parentheses: \(100 - 3 = 97\)
- Multiply by 100: \(100 \times 97 = 9700\)
2. Compute the second part: \((100 \cdot 100 - 3)\)
- Calculate inside th... | null | OpenMathInstruct-AIME | <think>
Okay, so I need to solve this math problem: $100(100 - 3) - (100 \cdot 100 - 3)$. Let me break it down step by step.
First, let's look at the expression. It's two parts: the first part is 100 multiplied by (100 minus 3), and then we subtract the second part, which is (100 times 100 minus 3). So, written out, ... | [
{
"from": "human",
"value": "What is $100(100-3)-(100\\cdot100-3)$?\n$\\textbf{(A)}\\ -20,000 \\qquad \\textbf{(B)}\\ -10,000 \\qquad \\textbf{(C)}\\ -297 \\qquad \\textbf{(D)}\\ -6 \\qquad \\textbf{(E)}\\ 0$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to solve this math problem: $10... |
If an item is sold for $x$ dollars, there is a loss of $15\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\%$ based on the cost. The ratio of $y:x$ is:
$\text{(A) } 23:17\quad \text{(B) } 17y:23\quad \text{(C) } 23x:17\quad \\ \text{(D) dependent upon the cost} \quad \... | null | Okay, so let me try to figure out this problem. It says that if an item is sold for $x$ dollars, there's a 15% loss based on the cost. And if it's sold for $y$ dollars, there's a 15% profit instead. We need to find the ratio of y to x, right? The options are given, but I'll try to solve it step by step.
First, let me ... | Let the cost price be denoted as \( C \).
When the item is sold for \( x \) dollars, there is a 15% loss:
\[ x = 0.85C \]
When the item is sold for \( y \) dollars, there is a 15% profit:
\[ y = 1.15C \]
The ratio \( y : x \) is:
\[ y : x = 1.15C : 0.85C \]
Dividing both terms by \( C \):
\[ y : x = 1.15 : 0.85 \]
... | null | OpenMathInstruct-AIME | <think>
Okay, so let me try to figure out this problem. It says that if an item is sold for $x$ dollars, there's a 15% loss based on the cost. And if it's sold for $y$ dollars, there's a 15% profit instead. We need to find the ratio of y to x, right? The options are given, but I'll try to solve it step by step.
First,... | [
{
"from": "human",
"value": "If an item is sold for $x$ dollars, there is a loss of $15\\%$ based on the cost. If, however, the same item is sold for $y$ dollars, there is a profit of $15\\%$ based on the cost. The ratio of $y:x$ is:\n$\\text{(A) } 23:17\\quad \\text{(B) } 17y:23\\quad \\text{(C) } 23x:17\\... |
Given the Leslie-Gower predator-prey model described in the text, under what conditions can we ascertain whether the two species will eventually spread successfully or vanish? Provide specific criteria based on the initial habitat size \( h_0 \), the parameters \(\mu\), and \(D\) as outlined in the results. | Okay, so I need to figure out under what conditions the two species in the Leslie-Gower predator-prey model will either spread successfully or vanish, based on the initial habitat size \( h_0 \), and the parameters \(\mu\) and \(D\). Let me start by recalling what the Leslie-Gower model is.
From what I remember, the ... | To determine whether the two species in the Leslie-Gower predator-prey model will spread successfully or vanish, the critical factor is the **initial habitat size \( h_0 \)** relative to a threshold determined by the parameters \(\mu\) (prey's growth rate) and \(D\) (diffusion coefficient). The specific criteria are as... | 2,058 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out under what conditions the two species in the Leslie-Gower predator-prey model will either spread successfully or vanish, based on the initial habitat size \( h_0 \), and the parameters \(\mu\) and \(D\). Let me start by recalling what the Leslie-Gower model is.
From what I rememb... | [
{
"from": "human",
"value": "Given the Leslie-Gower predator-prey model described in the text, under what conditions can we ascertain whether the two species will eventually spread successfully or vanish? Provide specific criteria based on the initial habitat size \\( h_0 \\), the parameters \\(\\mu\\), and... | |
For how many integers $n$ between $1$ and $50$, inclusive, is \[\frac{(n^2-1)!}{(n!)^n}\] an integer? (Recall that $0! = 1$.)
$\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$ | null | Okay, so I need to figure out for how many integers n between 1 and 50 inclusive, the expression (n² - 1)! divided by (n!)^n is an integer. Hmm, okay. Let's start by understanding what this expression actually represents.
First, the numerator is (n² - 1)! which is the factorial of n squared minus one. The denominator... | To determine for how many integers \( n \) between 1 and 50, inclusive, the expression \(\frac{(n^2 - 1)!}{(n!)^n}\) is an integer, we need to check if the prime factorization exponents in the numerator \((n^2 - 1)!\) are at least as large as those in the denominator \((n!)^n\).
Key steps:
1. **Prime Exponents Check*... | null | OpenMathInstruct-AIME | <think>
Okay, so I need to figure out for how many integers n between 1 and 50 inclusive, the expression (n² - 1)! divided by (n!)^n is an integer. Hmm, okay. Let's start by understanding what this expression actually represents.
First, the numerator is (n² - 1)! which is the factorial of n squared minus one. The den... | [
{
"from": "human",
"value": "For how many integers $n$ between $1$ and $50$, inclusive, is \\[\\frac{(n^2-1)!}{(n!)^n}\\] an integer? (Recall that $0! = 1$.)\n$\\textbf{(A) } 31 \\qquad \\textbf{(B) } 32 \\qquad \\textbf{(C) } 33 \\qquad \\textbf{(D) } 34 \\qquad \\textbf{(E) } 35$"
},
{
"from": "gp... |
Given a diagonal matrix sequence \( \{D_n\} \) with real entries such that \( D_n \sim_\lambda f(x) \) for some measurable function \( f:[0,1] \to \mathbb{R} \), prove that there exist permutation matrices \( P_n \) such that \( \{P_nD_nP_n^T\} \sim_{GLT} f(x) \otimes 1 \). What are the implications of this result in t... | Alright, so I need to prove that for a diagonal matrix sequence {D_n} with real entries, where D_n is distributed like f(x) in the eigenvalue sense, there exist permutation matrices P_n such that {P_n D_n P_n^T} is a GLT sequence with symbol f(x) ⊗ 1. Then, I have to discuss the implications for the spectral symbol of ... | Given a diagonal matrix sequence \( \{D_n\} \) with real entries such that \( D_n \sim_\lambda f(x) \) for some measurable function \( f:[0,1] \to \mathbb{R} \), we need to prove that there exist permutation matrices \( P_n \) such that \( \{P_nD_nP_n^T\} \sim_{GLT} f(x) \otimes 1 \). Additionally, we discuss the impli... | 26,069 | mlfoundations-dev/gair_mathpile | <think>
Alright, so I need to prove that for a diagonal matrix sequence {D_n} with real entries, where D_n is distributed like f(x) in the eigenvalue sense, there exist permutation matrices P_n such that {P_n D_n P_n^T} is a GLT sequence with symbol f(x) ⊗ 1. Then, I have to discuss the implications for the spectral sy... | [
{
"from": "human",
"value": "Given a diagonal matrix sequence \\( \\{D_n\\} \\) with real entries such that \\( D_n \\sim_\\lambda f(x) \\) for some measurable function \\( f:[0,1] \\to \\mathbb{R} \\), prove that there exist permutation matrices \\( P_n \\) such that \\( \\{P_nD_nP_n^T\\} \\sim_{GLT} f(x) ... | |
Consider the Bianchi-Cartan-Vranceanu spaces, denoted as \( M^3_{m,\ell} \), and a proper CMC Hopf cylinder \( \Sigma \) within such a space. If the geodesic curvature \( \kappa_g \) of its base curve is a constant, under what conditions on the parameters \( m \), \( \ell \), and \( r \) does \( \Sigma \) exist as a pr... | Alright, let's tackle this problem step by step. The question is about Bianchi-Cartan-Vranceanu (BCV) spaces, denoted as \( M^3_{m,\ell} \), and proper CMC Hopf cylinders within them. We need to determine under what conditions on the parameters \( m \), \( \ell \), and \( r \) such a Hopf cylinder \( \Sigma \) exists a... | To determine under what conditions a proper CMC Hopf cylinder \(\Sigma\) within a Bianchi-Cartan-Vranceanu space \(M^3_{m,\ell}\) exists as a proper \(r\)-harmonic surface, we start by considering the geometry of the Hopf cylinder and its base curve.
1. **Mean Curvature Relation**: The mean curvature \(H\) of the Hop... | 7,335 | mlfoundations-dev/gair_mathpile | <think>
Alright, let's tackle this problem step by step. The question is about Bianchi-Cartan-Vranceanu (BCV) spaces, denoted as \( M^3_{m,\ell} \), and proper CMC Hopf cylinders within them. We need to determine under what conditions on the parameters \( m \), \( \ell \), and \( r \) such a Hopf cylinder \( \Sigma \) ... | [
{
"from": "human",
"value": "Consider the Bianchi-Cartan-Vranceanu spaces, denoted as \\( M^3_{m,\\ell} \\), and a proper CMC Hopf cylinder \\( \\Sigma \\) within such a space. If the geodesic curvature \\( \\kappa_g \\) of its base curve is a constant, under what conditions on the parameters \\( m \\), \\(... | |
11. (GDR 3) The lock on a safe consists of three wheels, each of which may be set in eight different positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations that must be tried if one is to guarantee being... | null | Alright, so I have this problem about a safe with three wheels, each having eight different positions. The lock is defective, so the door will open if any two of the three wheels are in the correct position. The question is asking for the smallest number of combinations needed to guarantee opening the safe, without kno... | The problem involves a safe with three wheels, each having eight different positions. The safe will open if any two of the three wheels are in the correct position. The goal is to determine the smallest number of combinations needed to guarantee opening the safe.
1. **Understanding the Problem**:
- The safe opens i... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I have this problem about a safe with three wheels, each having eight different positions. The lock is defective, so the door will open if any two of the three wheels are in the correct position. The question is asking for the smallest number of combinations needed to guarantee opening the safe, wit... | [
{
"from": "human",
"value": "11. (GDR 3) The lock on a safe consists of three wheels, each of which may be set in eight different positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations that m... |
8.1. Prove that for any natural numbers $a$ and $b$ the number $4 a^{2}+4 a b+4 a+2 b+1$ is composite
| null | Okay, so I need to prove that for any natural numbers \( a \) and \( b \), the expression \( 4a^{2} + 4ab + 4a + 2b + 1 \) is composite. Hmm, let me start by understanding what the problem is asking. A composite number is a positive integer that has at least one positive divisor other than 1 and itself. So, I need to s... | To prove that for any natural numbers \(a\) and \(b\), the number \(4a^2 + 4ab + 4a + 2b + 1\) is composite, we start by examining the given expression:
\[
4a^2 + 4ab + 4a + 2b + 1
\]
We consider this expression as a quadratic in \(a\):
\[
4a^2 + (4b + 4)a + (2b + 1)
\]
We attempt to factor this quadratic expressio... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that for any natural numbers \( a \) and \( b \), the expression \( 4a^{2} + 4ab + 4a + 2b + 1 \) is composite. Hmm, let me start by understanding what the problem is asking. A composite number is a positive integer that has at least one positive divisor other than 1 and itself. So, I n... | [
{
"from": "human",
"value": "8.1. Prove that for any natural numbers $a$ and $b$ the number $4 a^{2}+4 a b+4 a+2 b+1$ is composite\n\n"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to prove that for any natural numbers \\( a \\) and \\( b \\), the expression \\( 4a^{2} + 4ab + 4a + 2b ... |
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8$ | null | Okay, so I need to find the middle number when three whole numbers are taken in pairs to get the sums 12, 17, and 19. Hmm. Let me think. Let's call the three numbers a, b, and c. Then, the sums taken in pairs would be a+b, a+c, and b+c. The problem states that these sums are 12, 17, and 19. But I don't know which sum c... | Let the three whole numbers be \(x\), \(y\), and \(z\). The sums of these numbers taken in pairs are given as 12, 17, and 19. We need to find the middle number.
First, we add all three pairwise sums:
\[
12 + 17 + 19 = 48
\]
Since each number is counted twice in this sum, we have:
\[
2(x + y + z) = 48 \implies x + y + ... | null | OpenMathInstruct-AIME | <think>
Okay, so I need to find the middle number when three whole numbers are taken in pairs to get the sums 12, 17, and 19. Hmm. Let me think. Let's call the three numbers a, b, and c. Then, the sums taken in pairs would be a+b, a+c, and b+c. The problem states that these sums are 12, 17, and 19. But I don't know whi... | [
{
"from": "human",
"value": "The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?\n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 5\\qquad\\textbf{(C)}\\ 6\\qquad\\textbf{(D)}\\ 7\\qquad\\textbf{(E)}\\ 8$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need ... |
Given $\tan \alpha =3$, then $\sin \alpha \cos \alpha =$ ? | null | Okay, so I need to find sin α times cos α, given that tan α is 3. Hmm, let me think. I remember that tan α is sin α over cos α, right? So tan α = sin α / cos α. Since tan α is 3, that means sin α divided by cos α equals 3. Maybe I can express sin α in terms of cos α or vice versa.
Let me write that down: tan α = 3 = s... | Given \(\tan \alpha = 3\), we need to find \(\sin \alpha \cos \alpha\).
First, we know that \(\tan \alpha = \frac{\sin \alpha}{\cos \alpha}\). Given \(\tan \alpha = 3\), we have \(\sin \alpha = 3 \cos \alpha\).
Using the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\), we substitute \(\sin \alpha = 3 \cos ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find sin α times cos α, given that tan α is 3. Hmm, let me think. I remember that tan α is sin α over cos α, right? So tan α = sin α / cos α. Since tan α is 3, that means sin α divided by cos α equals 3. Maybe I can express sin α in terms of cos α or vice versa.
Let me write that down: tan α... | [
{
"from": "human",
"value": "Given $\\tan \\alpha =3$, then $\\sin \\alpha \\cos \\alpha =$ ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find sin α times cos α, given that tan α is 3. Hmm, let me think. I remember that tan α is sin α over cos α, right? So tan α = sin α / cos α. S... |
Given the function $y=x+\cos x$, consider the following propositions:
$(1) f(x)$'s domain is $(2k\pi, 2k\pi + 2\pi)$;
$(2) f(x)$'s range is $\mathbb{R}$;
$(3) f(x)$ is an odd function;
$(4) The graph of $f(x)$ intersects with the line $y=x$ at a point where the x-coordinate is $\frac{\pi}{2}$,
The correct number... | null | Okay, so I have this problem here about the function y = x + cos x, and there are four propositions given. I need to figure out which ones are correct and then choose how many are correct from the options A to D. Let me go through each proposition one by one.
First, proposition (1): "f(x)'s domain is (2kπ, 2kπ + 2π)"... | The function given is \( y = x + \cos x \). We need to evaluate the four propositions about this function.
1. **Domain of \( f(x) \):** The function \( f(x) = x + \cos x \) is defined for all real numbers because both \( x \) and \( \cos x \) are defined for all real numbers. Therefore, the domain is \( \mathbb{R} \),... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem here about the function y = x + cos x, and there are four propositions given. I need to figure out which ones are correct and then choose how many are correct from the options A to D. Let me go through each proposition one by one.
First, proposition (1): "f(x)'s domain is (2kπ, 2k... | [
{
"from": "human",
"value": "Given the function $y=x+\\cos x$, consider the following propositions:\n\n$(1) f(x)$'s domain is $(2k\\pi, 2k\\pi + 2\\pi)$; \n$(2) f(x)$'s range is $\\mathbb{R}$;\n\n$(3) f(x)$ is an odd function; \n$(4) The graph of $f(x)$ intersects with the line $y=x$ at a point where the ... |
What is the relationship between the limiting behavior of Gaussian multiplicative chaos measures for planar Brownian motion and the random measures proposed by Bass, Burdzy, and Khoshnevisan in their work on thick points? In particular, could you explain how the measures converge in the context of arbitrary domains? | Okay, so I need to figure out the relationship between Gaussian multiplicative chaos (GMC) measures for planar Brownian motion and the random measures introduced by Bass, Burdzy, and Khoshnevisan (BBK) in their work on thick points. The question also asks about how these measures converge in arbitrary domains. Hmm, let... | The relationship between Gaussian multiplicative chaos (GMC) measures and the random measures constructed by Bass, Burdzy, and Khoshnevisan (BBK) for thick points of planar Brownian motion lies in their shared structure of exponentiating log-correlated fields and renormalizing to obtain non-trivial limiting measures. H... | 30,574 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the relationship between Gaussian multiplicative chaos (GMC) measures for planar Brownian motion and the random measures introduced by Bass, Burdzy, and Khoshnevisan (BBK) in their work on thick points. The question also asks about how these measures converge in arbitrary domains. ... | [
{
"from": "human",
"value": "What is the relationship between the limiting behavior of Gaussian multiplicative chaos measures for planar Brownian motion and the random measures proposed by Bass, Burdzy, and Khoshnevisan in their work on thick points? In particular, could you explain how the measures converg... | |
Let $a_n$ be the number of permutations $(x_1, x_2, \dots, x_n)$ of the numbers $(1,2,\dots, n)$ such that the $n$ ratios $\frac{x_k}{k}$ for $1\le k\le n$ are all distinct. Prove that $a_n$ is odd for all $n\ge 1.$ | null | Okay, so I need to prove that the number of permutations of (1, 2, ..., n), called a_n, where all the ratios x_k / k are distinct, is always odd. The problem states that this holds for all n ≥ 1. Hmm, interesting.
First, let me make sure I understand the problem correctly. We are looking at permutations (x_1, x_2, ..... | To prove that the number \(a_n\) of permutations \((x_1, x_2, \dots, x_n)\) of \((1, 2, \dots, n)\) such that all ratios \(\frac{x_k}{k}\) are distinct is odd for all \(n \geq 1\), we can use the concept of involutions and fixed points.
1. **Involution Concept**: An involution is a permutation that is its own inverse.... | null | OpenMathInstruct-AIME | <think>
Okay, so I need to prove that the number of permutations of (1, 2, ..., n), called a_n, where all the ratios x_k / k are distinct, is always odd. The problem states that this holds for all n ≥ 1. Hmm, interesting.
First, let me make sure I understand the problem correctly. We are looking at permutations (x_1,... | [
{
"from": "human",
"value": "Let $a_n$ be the number of permutations $(x_1, x_2, \\dots, x_n)$ of the numbers $(1,2,\\dots, n)$ such that the $n$ ratios $\\frac{x_k}{k}$ for $1\\le k\\le n$ are all distinct. Prove that $a_n$ is odd for all $n\\ge 1.$"
},
{
"from": "gpt",
"value": "<think>\nOkay,... |
Let $S$ be a set of $6$ integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}... | null | Okay, let me try to figure out this problem. So, we need to pick a set S of 6 integers from the numbers 1 to 12. The key condition is that for any two elements a and b in S, where a < b, b is not a multiple of a. We need to find the least possible value of an element in such a set S. The answer choices are 2, 3, 4, 5, ... | To determine the least possible value of an element in a set \( S \) of 6 integers taken from \(\{1, 2, \ldots, 12\}\) such that no element is a multiple of another, we need to check each possible minimum value starting from the smallest.
1. **Considering 2 as the minimum element**:
- Including 2 requires excluding... | null | OpenMathInstruct-AIME | <think>
Okay, let me try to figure out this problem. So, we need to pick a set S of 6 integers from the numbers 1 to 12. The key condition is that for any two elements a and b in S, where a < b, b is not a multiple of a. We need to find the least possible value of an element in such a set S. The answer choices are 2, 3... | [
{
"from": "human",
"value": "Let $S$ be a set of $6$ integers taken from $\\{1,2,\\dots,12\\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$?\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\t... |
Estimate to determine which of the following numbers is closest to $\frac{401}{.205}$.
$\text{(A)}\ .2 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 200 \qquad \text{(E)}\ 2000$ | null | Okay, so I need to figure out which of the options A through E is closest to the value of 401 divided by 0.205. Let me start by understanding the problem. The question is asking for an estimate, so I don't have to calculate the exact value, just approximate it enough to pick the closest option. The options are 0.2, 2, ... | To determine which of the given numbers is closest to \(\frac{401}{0.205}\), we start by recognizing that dividing by a decimal less than 1 results in a larger number.
First, approximate \(0.205\) as \(0.2\). Dividing by \(0.2\) is equivalent to multiplying by 5:
\[
\frac{401}{0.2} = 401 \times 5 = 2005
\]
Since \(0... | null | OpenMathInstruct-AIME | <think>
Okay, so I need to figure out which of the options A through E is closest to the value of 401 divided by 0.205. Let me start by understanding the problem. The question is asking for an estimate, so I don't have to calculate the exact value, just approximate it enough to pick the closest option. The options are ... | [
{
"from": "human",
"value": "Estimate to determine which of the following numbers is closest to $\\frac{401}{.205}$.\n$\\text{(A)}\\ .2 \\qquad \\text{(B)}\\ 2 \\qquad \\text{(C)}\\ 20 \\qquad \\text{(D)}\\ 200 \\qquad \\text{(E)}\\ 2000$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need t... |
Given a continuous-time linear switching system described by matrices from a compact set \(\mathcal{A}\) with real spectra, find the largest discretization step size \(\tau > 0\) such that the condition \(\rho(I + \tau \mathcal{A}) < 1\) implies the Lyapunov exponent \(\sigma(\mathcal{A}) < -\epsilon\) for a given accu... | Okay, so I have this problem about continuous-time linear switching systems. The system is described by matrices from a compact set 𝒜, and all these matrices have real spectra. The question is about finding the largest discretization step size τ > 0 such that if the spectral radius ρ(I + τ𝒜) is less than 1, then the ... | Given a continuous-time linear switching system described by matrices from a compact set \(\mathcal{A}\) with real spectra, we need to find the largest discretization step size \(\tau > 0\) such that the condition \(\rho(I + \tau \mathcal{A}) < 1\) implies the Lyapunov exponent \(\sigma(\mathcal{A}) < -\epsilon\) for a... | 21,515 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I have this problem about continuous-time linear switching systems. The system is described by matrices from a compact set 𝒜, and all these matrices have real spectra. The question is about finding the largest discretization step size τ > 0 such that if the spectral radius ρ(I + τ𝒜) is less than 1, t... | [
{
"from": "human",
"value": "Given a continuous-time linear switching system described by matrices from a compact set \\(\\mathcal{A}\\) with real spectra, find the largest discretization step size \\(\\tau > 0\\) such that the condition \\(\\rho(I + \\tau \\mathcal{A}) < 1\\) implies the Lyapunov exponent ... | |
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than the common value. Which of the following is a correct description of $S$?
$\textbf{(A)}\ \text{a single point} \qquad\textbf{(B)}\ \... | null | Okay, let's try to figure out this problem step by step. The question is about describing the set S of points (x, y) where two of the three quantities 3, x+2, and y-4 are equal, and the third quantity is no greater than the common value. The answer choices suggest it could be a single point, two lines, three lines with... | To determine the set \( S \) of points \((x, y)\) such that two of the three quantities \( 3 \), \( x + 2 \), and \( y - 4 \) are equal and the third is no greater than the common value, we analyze three cases:
1. **Case 1: \( 3 = x + 2 \) and \( y - 4 \leq 3 \)**
- Solving \( 3 = x + 2 \) gives \( x = 1 \).
- T... | null | OpenMathInstruct-AIME | <think>
Okay, let's try to figure out this problem step by step. The question is about describing the set S of points (x, y) where two of the three quantities 3, x+2, and y-4 are equal, and the third quantity is no greater than the common value. The answer choices suggest it could be a single point, two lines, three li... | [
{
"from": "human",
"value": "Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than the common value. Which of the following is a correct description of $S$?\n$\\textbf{(A)}\\... |
Simplify the expression: $[(3+4+5+6)\div2] + [(3\cdot6+9)\div4]$ | Simplify the expression: $[(3+4+5+6)\div2] + [(3\cdot6+9)\div4]$ | Okay, let's see. I need to simplify this expression: [(3 + 4 + 5 + 6) ÷ 2] + [(3·6 + 9) ÷ 4]. Hmm, let me break this down step by step.
First, I'll handle the left part of the addition, which is [(3 + 4 + 5 + 6) ÷ 2]. Let me compute the sum inside the brackets first. So, 3 + 4 is 7, then 5 + 6 is 11. Adding those tog... | First, simplify the left part of the expression:
\[
(3 + 4 + 5 + 6) \div 2
\]
Adding the numbers inside the brackets:
\[
3 + 4 + 5 + 6 = 18
\]
Dividing by 2:
\[
18 \div 2 = 9
\]
Next, simplify the right part of the expression:
\[
(3 \cdot 6 + 9) \div 4
\]
Performing the multiplication first:
\[
3 \cdot 6 = 18
\]
Then... | 16,649 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to simplify this expression: [(3 + 4 + 5 + 6) ÷ 2] + [(3·6 + 9) ÷ 4]. Hmm, let me break this down step by step.
First, I'll handle the left part of the addition, which is [(3 + 4 + 5 + 6) ÷ 2]. Let me compute the sum inside the brackets first. So, 3 + 4 is 7, then 5 + 6 is 11. Adding t... | [
{
"from": "human",
"value": "Simplify the expression: $[(3+4+5+6)\\div2] + [(3\\cdot6+9)\\div4]$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to simplify this expression: [(3 + 4 + 5 + 6) ÷ 2] + [(3·6 + 9) ÷ 4]. Hmm, let me break this down step by step. \n\nFirst, I'll handle ... |
Given a network of \(N\) agents, each connected in a star topology, and each agent \(i\) has a utility function defined by their benefit from neighbors' sharing actions and the cost of sharing their own information, determine the conditions under which the price of anarchy (PoA) for the proposed rating protocol equals ... | Okay, let me try to figure out this problem. So, we have a network of N agents connected in a star topology. Each agent has a utility function that depends on the benefit they get from their neighbors' sharing actions and the cost of sharing their own information. We need to determine the conditions under which the Pri... | To determine the conditions under which the Price of Anarchy (PoA) equals one in a star topology network, we need to ensure that the Nash equilibrium aligns with the social optimum. The key insight is that for PoA=1, the connectivity degree \(d\) of each agent must be equal to the average degree \(\bar{d}\) of the netw... | 23,835 | mlfoundations-dev/gair_mathpile | <think>
Okay, let me try to figure out this problem. So, we have a network of N agents connected in a star topology. Each agent has a utility function that depends on the benefit they get from their neighbors' sharing actions and the cost of sharing their own information. We need to determine the conditions under which... | [
{
"from": "human",
"value": "Given a network of \\(N\\) agents, each connected in a star topology, and each agent \\(i\\) has a utility function defined by their benefit from neighbors' sharing actions and the cost of sharing their own information, determine the conditions under which the price of anarchy (... | |
Given the context of cardinality estimation using sketches, if one were to achieve a 1% standard error using the \textsf{Martingale} \textsf{Fishmonger} sketch, what would be the minimum required size of the sketch in bits? | Okay, so I need to figure out the minimum required size of the Martingale Fishmonger sketch in bits to achieve a 1% standard error for cardinality estimation. Hmm, first, let me recall what I know about cardinality estimation and sketches.
Cardinality estimation is about determining the number of distinct elements in ... | To determine the minimum required size of the Martingale Fishmonger sketch in bits for achieving a 1% standard error in cardinality estimation, we start by understanding the relationship between the standard error and the number of registers (or buckets) in the sketch.
The Martingale Fishmonger sketch is an improveme... | 39,175 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the minimum required size of the Martingale Fishmonger sketch in bits to achieve a 1% standard error for cardinality estimation. Hmm, first, let me recall what I know about cardinality estimation and sketches.
Cardinality estimation is about determining the number of distinct elem... | [
{
"from": "human",
"value": "Given the context of cardinality estimation using sketches, if one were to achieve a 1% standard error using the \\textsf{Martingale} \\textsf{Fishmonger} sketch, what would be the minimum required size of the sketch in bits?"
},
{
"from": "gpt",
"value": "<think>\nO... | |
Three vertices of parallelogram $PQRS$ are $P(-3,-2), Q(1,-5), R(9,1)$ with $P$ and $R$ diagonally opposite.
The sum of the coordinates of vertex $S$ is:
$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 9$ | null | Okay, so we have a problem here where three vertices of a parallelogram PQRS are given: P(-3,-2), Q(1,-5), R(9,1), and P and R are diagonally opposite. We need to find the sum of the coordinates of vertex S. The answer choices are from 9 to 13. Hmm, let's see.
First, let me recall that in a parallelogram, the diagona... | Given three vertices of parallelogram \(PQRS\) are \(P(-3, -2)\), \(Q(1, -5)\), \(R(9, 1)\) with \(P\) and \(R\) diagonally opposite, we need to find the sum of the coordinates of vertex \(S\).
1. **Midpoint Method**:
- The midpoint of diagonal \(PR\) is calculated as:
\[
\left( \frac{-3 + 9}{2}, \frac{-2... | null | OpenMathInstruct-AIME | <think>
Okay, so we have a problem here where three vertices of a parallelogram PQRS are given: P(-3,-2), Q(1,-5), R(9,1), and P and R are diagonally opposite. We need to find the sum of the coordinates of vertex S. The answer choices are from 9 to 13. Hmm, let's see.
First, let me recall that in a parallelogram, the... | [
{
"from": "human",
"value": "Three vertices of parallelogram $PQRS$ are $P(-3,-2), Q(1,-5), R(9,1)$ with $P$ and $R$ diagonally opposite. \nThe sum of the coordinates of vertex $S$ is:\n$\\textbf{(A)}\\ 13 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ 11 \\qquad \\textbf{(D)}\\ 10 \\qquad \\textbf{(E)}... |
Find the product of all values of $x$ for which the expression $\frac{x-2}{x^2+3x-10}$ is undefined. | Find the product of all values of $x$ for which the expression $\frac{x-2}{x^2+3x-10}$ is undefined. | Okay, so I need to find the product of all values of x where the expression (x - 2)/(x² + 3x - 10) is undefined. Hmm, let me think. I remember that a rational expression is undefined when the denominator is zero because you can't divide by zero. So first, I need to find the values of x that make the denominator x² + 3x... | To determine the product of all values of \( x \) for which the expression \(\frac{x-2}{x^2+3x-10}\) is undefined, we need to find the values of \( x \) that make the denominator zero.
The denominator is \( x^2 + 3x - 10 \). Factoring this quadratic expression, we look for two numbers that multiply to \(-10\) and add... | 10,893 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the product of all values of x where the expression (x - 2)/(x² + 3x - 10) is undefined. Hmm, let me think. I remember that a rational expression is undefined when the denominator is zero because you can't divide by zero. So first, I need to find the values of x that make the denominator... | [
{
"from": "human",
"value": "Find the product of all values of $x$ for which the expression $\\frac{x-2}{x^2+3x-10}$ is undefined."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the product of all values of x where the expression (x - 2)/(x² + 3x - 10) is undefined. Hmm, let me ... |
Given that $x = 2y - 5$, evaluate the expression $(3y - x + 2)$ when $y = 4$. | Given that $x = 2y - 5$, evaluate the expression $(3y - x + 2)$ when $y = 4$. | Okay, let's see. I need to evaluate the expression (3y - x + 2) when y = 4. But wait, x is given in terms of y: x = 2y - 5. Hmm, so first, maybe I should find out what x is when y is 4, and then plug both y and x into the expression?
Let me start by substituting y = 4 into the equation for x. So, x = 2*(4) - 5. Let me... | Given the equation \( x = 2y - 5 \), we need to evaluate the expression \( 3y - x + 2 \) when \( y = 4 \).
First, substitute \( y = 4 \) into the equation for \( x \):
\[
x = 2(4) - 5 = 8 - 5 = 3
\]
Next, substitute \( y = 4 \) and \( x = 3 \) into the expression \( 3y - x + 2 \):
\[
3(4) - 3 + 2 = 12 - 3 + 2 = 9 + 2... | 7,751 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to evaluate the expression (3y - x + 2) when y = 4. But wait, x is given in terms of y: x = 2y - 5. Hmm, so first, maybe I should find out what x is when y is 4, and then plug both y and x into the expression?
Let me start by substituting y = 4 into the equation for x. So, x = 2*(4) - 5... | [
{
"from": "human",
"value": "Given that $x = 2y - 5$, evaluate the expression $(3y - x + 2)$ when $y = 4$."
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to evaluate the expression (3y - x + 2) when y = 4. But wait, x is given in terms of y: x = 2y - 5. Hmm, so first, maybe I sh... |
Given a finite solvable group \( G \) and a block \( b \) of \( G \) with a defect group \( P \) that is normalized by \( G \) and stabilized by a subgroup \( A \) of \( G \) with coprime order, if \( H \) is an \( A \)-stable normal subgroup of \( G \), what condition must be satisfied for the extensions of the block ... | Okay, so I need to figure out the condition required for the extensions of the block algebra \( kG b \) and its Glauberman correspondent \( kG^A w(b) \) to be Morita equivalent. The setup is a finite solvable group \( G \), a block \( b \) with defect group \( P \), which is normalized by \( G \) and stabilized by a su... | Given a finite solvable group \( G \) and a block \( b \) of \( G \) with a defect group \( P \) that is normalized by \( G \) and stabilized by a subgroup \( A \) of \( G \) with coprime order, we need to determine the condition under which the extensions of the block algebra \( kG b \) and the corresponding Glauberma... | 18,468 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the condition required for the extensions of the block algebra \( kG b \) and its Glauberman correspondent \( kG^A w(b) \) to be Morita equivalent. The setup is a finite solvable group \( G \), a block \( b \) with defect group \( P \), which is normalized by \( G \) and stabilized... | [
{
"from": "human",
"value": "Given a finite solvable group \\( G \\) and a block \\( b \\) of \\( G \\) with a defect group \\( P \\) that is normalized by \\( G \\) and stabilized by a subgroup \\( A \\) of \\( G \\) with coprime order, if \\( H \\) is an \\( A \\)-stable normal subgroup of \\( G \\), what... | |
Consider a bounded simply connected open set \( \Omega \subset \mathbb{R}^n \) with smooth boundary. Assume the torsion function \( u \) defined by the Robin boundary value problem
\[
\begin{cases}
-\Delta u = 1 & \text{in } \Omega \\
\frac{\partial u}{\partial \nu} + \beta u = 0 & \text{on } \partial \Omega
\end{... | Okay, so I need to figure out the necessary conditions on the geometry of Ω such that the ball B_R is a critical shape for the functional M (which is the L^∞-norm of the torsion function u) under volume-preserving perturbations. The problem mentions that the shape derivative M'(B_R, V) is zero for any volume-preserving... | To determine the necessary conditions on the geometry of \(\Omega\) such that the ball \(B_R\) remains a critical shape for the functional \(M\) (the \(L^\infty\)-norm of the torsion function \(u\)) under volume-preserving perturbations, we analyze the shape derivative of the functional.
The torsion function \(u\) so... | 35,579 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the necessary conditions on the geometry of Ω such that the ball B_R is a critical shape for the functional M (which is the L^∞-norm of the torsion function u) under volume-preserving perturbations. The problem mentions that the shape derivative M'(B_R, V) is zero for any volume-pr... | [
{
"from": "human",
"value": "Consider a bounded simply connected open set \\( \\Omega \\subset \\mathbb{R}^n \\) with smooth boundary. Assume the torsion function \\( u \\) defined by the Robin boundary value problem \n\n\\[\n\\begin{cases} \n-\\Delta u = 1 & \\text{in } \\Omega \\\\ \n\\frac{\\partial u}{\... | |
Given a quantum state that allows for a precision scaling of \( \frac{1}{N^{2-\varepsilon}} \) with \( \varepsilon > 0 \) and with both the relative size of the largest entangled block and the geometric measure of entanglement approaching zero as \( N \to \infty \), what is the maximum achievable limit for the Quantum ... | Okay, so I need to figure out the maximum achievable limit for the Quantum Fisher Information (QFI) \( F_Q \) of a quantum state that has a precision scaling of \( \frac{1}{N^{2-\varepsilon}} \) with \( \varepsilon > 0 \). Also, both the relative size of the largest entangled block and the geometric measure of entangle... | Given a quantum state that allows for a precision scaling of \( \frac{1}{N^{2-\varepsilon}} \) with \( \varepsilon > 0 \) and with both the relative size of the largest entangled block and the geometric measure of entanglement (GME) approaching zero as \( N \to \infty \), we need to determine the maximum achievable lim... | 39,030 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the maximum achievable limit for the Quantum Fisher Information (QFI) \( F_Q \) of a quantum state that has a precision scaling of \( \frac{1}{N^{2-\varepsilon}} \) with \( \varepsilon > 0 \). Also, both the relative size of the largest entangled block and the geometric measure of ... | [
{
"from": "human",
"value": "Given a quantum state that allows for a precision scaling of \\( \\frac{1}{N^{2-\\varepsilon}} \\) with \\( \\varepsilon > 0 \\) and with both the relative size of the largest entangled block and the geometric measure of entanglement approaching zero as \\( N \\to \\infty \\), w... | |
Given a semisimple finite-dimensional Lie algebra \(\mathfrak{g}\) over an algebraically closed field \(\mathbb{F}\) of characteristic 0, consider a nilpotent element \(e\) that can be expressed in terms of the structure of an \( \mathfrak{sl}_2 \)-triple \(\{e,h,f\}\) and its corresponding eigenspace decomposition \(\... | Okay, so I need to figure out the conditions under which a nilpotent element e in a semisimple Lie algebra can be classified as nilpotent, semisimple, or mixed type based on its depth d and the structure of its cyclic elements e + F where F is in the -d eigenspace. Hmm. Let me start by recalling some basics.
First, in... | To classify a nilpotent element \( e \) in a semisimple Lie algebra \( \mathfrak{g} \) based on its depth \( d \) and cyclic elements \( e + F \) (for \( F \in \mathfrak{g}_{-d} \)), we analyze the interplay between the grading induced by an \( \mathfrak{sl}_2 \)-triple and the structure of cyclic elements:
### Condit... | 38,904 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the conditions under which a nilpotent element e in a semisimple Lie algebra can be classified as nilpotent, semisimple, or mixed type based on its depth d and the structure of its cyclic elements e + F where F is in the -d eigenspace. Hmm. Let me start by recalling some basics.
F... | [
{
"from": "human",
"value": "Given a semisimple finite-dimensional Lie algebra \\(\\mathfrak{g}\\) over an algebraically closed field \\(\\mathbb{F}\\) of characteristic 0, consider a nilpotent element \\(e\\) that can be expressed in terms of the structure of an \\( \\mathfrak{sl}_2 \\)-triple \\(\\{e,h,f\... | |
For which values of \( n \) does the simple alternating group \( A_n \) fail to be recognizable by spectrum, specifically confirming that \( n = 6 \) and \( n = 10 \) are the only exceptions in the range \( n \geq 5 \)? Additionally, can you provide a general statement regarding the recognizability of the groups \( A_n... | Okay, so I need to figure out for which values of \( n \) the alternating group \( A_n \) isn't recognizable by its spectrum, and confirm that only \( n = 6 \) and \( n = 10 \) are exceptions when \( n \geq 5 \). Also, I need to provide a general statement about the recognizability of \( A_n \) otherwise. Hmm, let me s... | The spectrum of a group \( G \), denoted \( \omega(G) \), is the set of element orders in \( G \). A group \( G \) is recognizable by spectrum if every group \( H \) with \( \omega(H) = \omega(G) \) is isomorphic to \( G \). For alternating groups \( A_n \), recognizability by spectrum is confirmed for all \( n \geq 5 ... | 20,059 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out for which values of \( n \) the alternating group \( A_n \) isn't recognizable by its spectrum, and confirm that only \( n = 6 \) and \( n = 10 \) are exceptions when \( n \geq 5 \). Also, I need to provide a general statement about the recognizability of \( A_n \) otherwise. Hmm, ... | [
{
"from": "human",
"value": "For which values of \\( n \\) does the simple alternating group \\( A_n \\) fail to be recognizable by spectrum, specifically confirming that \\( n = 6 \\) and \\( n = 10 \\) are the only exceptions in the range \\( n \\geq 5 \\)? Additionally, can you provide a general statemen... | |
A library charges a monthly membership fee of $10, and an additional $0.25 per book borrowed.
A book club decides to pay for each member's borrowings,
and to cover the costs, each member is charged a flat annual fee.
However, members who borrow more than 20 books per year will be charged an additional $0.10 per book... | A library charges a monthly membership fee of $10, and an additional $0.25 per book borrowed.
A book club decides to pay for each member's borrowings,
and to cover the costs, each member is charged a flat annual fee.
However, members who borrow more than 20 books per year will be charged an additional $0.10 per book... | Okay, let's see. The library charges a monthly membership fee of $10, so that's $10 per month. Since there are 12 months in a year, the annual membership fee would be 12 times $10, which is $120. Then, there's an additional $0.25 per book borrowed.
The book club is paying for each member's borrowings, but to cover th... | The library charges a monthly membership fee of $10, which amounts to $120 annually. Additionally, there is a fee of $0.25 per book borrowed. The book club wants to cover these costs by charging a flat annual fee and an additional $0.10 per book for members who borrow more than 20 books per year.
To break even, the fl... | 4,160 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The library charges a monthly membership fee of $10, so that's $10 per month. Since there are 12 months in a year, the annual membership fee would be 12 times $10, which is $120. Then, there's an additional $0.25 per book borrowed.
The book club is paying for each member's borrowings, but to ... | [
{
"from": "human",
"value": "A library charges a monthly membership fee of $10, and an additional $0.25 per book borrowed. \nA book club decides to pay for each member's borrowings, \nand to cover the costs, each member is charged a flat annual fee. \nHowever, members who borrow more than 20 books per year ... |
[ Projections of the bases, sides, or vertices of a trapezoid ]
The height of an isosceles trapezoid, dropped from the vertex of the smaller base, divides the larger base in the ratio $1: 3$. Find the ratio of the bases of the trapezoid. | null | Okay, so I have this problem about an isosceles trapezoid. The height dropped from the vertex of the smaller base divides the larger base in the ratio 1:3. I need to find the ratio of the bases of the trapezoid. Hmm, let me visualize this first.
An isosceles trapezoid has two parallel bases, the larger one and the sma... | Given an isosceles trapezoid with the larger base \( AB \) and the smaller base \( CD \), we know that the height dropped from a vertex of the smaller base divides the larger base in the ratio \( 1:3 \). We need to find the ratio of the bases \( AB \) and \( CD \).
1. Denote the larger base \( AB \) as \( B \) and the... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this problem about an isosceles trapezoid. The height dropped from the vertex of the smaller base divides the larger base in the ratio 1:3. I need to find the ratio of the bases of the trapezoid. Hmm, let me visualize this first.
An isosceles trapezoid has two parallel bases, the larger one and... | [
{
"from": "human",
"value": "[ Projections of the bases, sides, or vertices of a trapezoid ]\n\nThe height of an isosceles trapezoid, dropped from the vertex of the smaller base, divides the larger base in the ratio $1: 3$. Find the ratio of the bases of the trapezoid."
},
{
"from": "gpt",
"valu... |
A department store has a promotion where customers can get a 10% discount on their total purchase if they pay in cash. However, if they use a credit card, they will be charged a 2% processing fee. A customer purchases a TV and a sound system, with a total price of $\$800$. If they pay in cash, how much will they pay? I... | A department store has a promotion where customers can get a 10% discount on their total purchase if they pay in cash. However, if they use a credit card, they will be charged a 2% processing fee. A customer purchases a TV and a sound system, with a total price of $\$800$. If they pay in cash, how much will they pay? I... | Okay, let's tackle this problem step by step. So, the customer is buying a TV and a sound system that total $800. The store has a promotion where if you pay in cash, you get a 10% discount. But if you use a credit card, they charge a 2% processing fee instead. We need to find out how much the customer pays in each case... | If the customer pays in cash, they receive a 10% discount on the total purchase of \$800. The discount is calculated as:
\[ 0.10 \times 800 = 80 \]
Subtracting this discount from the original total:
\[ 800 - 80 = 720 \]
**Cash payment: \$720**
If the customer uses a credit card, a 2% processing fee is added to... | 129 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's tackle this problem step by step. So, the customer is buying a TV and a sound system that total $800. The store has a promotion where if you pay in cash, you get a 10% discount. But if you use a credit card, they charge a 2% processing fee instead. We need to find out how much the customer pays in e... | [
{
"from": "human",
"value": "A department store has a promotion where customers can get a 10% discount on their total purchase if they pay in cash. However, if they use a credit card, they will be charged a 2% processing fee. A customer purchases a TV and a sound system, with a total price of $\\$800$. If t... |
Let $a$, $b$, $c$ be positive real numbers. Prove that
\[\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2).\] | null | Okay, so I have this inequality to prove: \[\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2).\] Hmm, let's see. The variables a, b, c are positive real numbers, so I don't have to worry about division by zero or negative numbers here. The left side has ... | To prove the inequality
\[
\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2),
\]
we start by noting that the inequality is homogeneous, allowing us to normalize variables if needed. We use the Cauchy-Schwarz inequality to bound the left-hand side.
First... | null | OpenMathInstruct-AIME | <think>
Okay, so I have this inequality to prove: \[\frac{a^3 + 3b^3}{5a + b} + \frac{b^3 + 3c^3}{5b + c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2).\] Hmm, let's see. The variables a, b, c are positive real numbers, so I don't have to worry about division by zero or negative numbers here. The left s... | [
{
"from": "human",
"value": "Let $a$, $b$, $c$ be positive real numbers. Prove that\n\\[\\frac{a^3 + 3b^3}{5a + b} + \\frac{b^3 + 3c^3}{5b + c} + \\frac{c^3 + 3a^3}{5c + a} \\ge \\frac{2}{3} (a^2 + b^2 + c^2).\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this inequality to prove: ... |
A circle is inscribed in a triangle whose sides have lengths 8, 15 and 17 units. Find the radius of the inscribed circle. | A circle is inscribed in a triangle whose sides have lengths 8, 15 and 17 units. Find the radius of the inscribed circle. | Okay, so I need to find the radius of the inscribed circle in a triangle with sides 8, 15, and 17 units. Hmm, inscribed circle... right, the radius of the incircle. I remember there's a formula for the radius of the incircle, something involving the area of the triangle and the semi-perimeter. Let me recall.
First, m... | To find the radius of the inscribed circle in a triangle with sides 8, 15, and 17 units, we first verify if the triangle is a right triangle. Checking the Pythagorean theorem:
\[
8^2 + 15^2 = 64 + 225 = 289 = 17^2
\]
Thus, it is a right triangle with legs 8 and 15, and hypotenuse 17. The area \(A\) of the triangle is... | 10,763 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the radius of the inscribed circle in a triangle with sides 8, 15, and 17 units. Hmm, inscribed circle... right, the radius of the incircle. I remember there's a formula for the radius of the incircle, something involving the area of the triangle and the semi-perimeter. Let me recall.
... | [
{
"from": "human",
"value": "A circle is inscribed in a triangle whose sides have lengths 8, 15 and 17 units. Find the radius of the inscribed circle."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the radius of the inscribed circle in a triangle with sides 8, 15, and 17 units. ... |
The function $y=\sin \frac {πx}{3}$ takes at least $2$ maximum values in the interval $[0,n]$. What is the minimum value of the positive integer $n$? | null | Okay, so the problem is: The function y = sin(πx/3) takes at least 2 maximum values in the interval [0, n]. What is the minimum value of the positive integer n?
Hmm. Let me try to figure this out. First, I need to understand when the function sin(πx/3) reaches its maximum. The sine function usually has maximums at π/2... | To determine the minimum value of the positive integer \( n \) such that the function \( y = \sin \frac{\pi x}{3} \) takes at least 2 maximum values in the interval \([0, n]\), we first identify the points where the maximum values occur.
The sine function \(\sin(\theta)\) reaches its maximum at \(\theta = \frac{\pi}{... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so the problem is: The function y = sin(πx/3) takes at least 2 maximum values in the interval [0, n]. What is the minimum value of the positive integer n?
Hmm. Let me try to figure this out. First, I need to understand when the function sin(πx/3) reaches its maximum. The sine function usually has maximum... | [
{
"from": "human",
"value": "The function $y=\\sin \\frac {πx}{3}$ takes at least $2$ maximum values in the interval $[0,n]$. What is the minimum value of the positive integer $n$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so the problem is: The function y = sin(πx/3) takes at least 2 maximu... |
The value of (0.02)^2 + (0.52)^2 + (0.035)^2 divided by (0.002)^2 + a certain number squared + (0.0035)^2 is 100. What is the number? | null | Okay, let's see. I need to find a certain number, let's call it x, such that when I calculate the sum of (0.02)^2 + (0.52)^2 + (0.035)^2 divided by the sum of (0.002)^2 + x^2 + (0.0035)^2, the result is 100. Hmm, that sounds a bit complicated, but let's break it down step by step.
First, let me write down the equation... | Given the equation:
\[
\frac{(0.02)^2 + (0.52)^2 + (0.035)^2}{(0.002)^2 + x^2 + (0.0035)^2} = 100
\]
we need to find the value of \( x \).
First, calculate the numerator:
\[
(0.02)^2 = 0.0004
\]
\[
(0.52)^2 = 0.2704
\]
\[
(0.035)^2 = 0.001225
\]
Summing these values:
\[
0.0004 + 0.2704 + 0.001225 = 0.272025
\]
N... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to find a certain number, let's call it x, such that when I calculate the sum of (0.02)^2 + (0.52)^2 + (0.035)^2 divided by the sum of (0.002)^2 + x^2 + (0.0035)^2, the result is 100. Hmm, that sounds a bit complicated, but let's break it down step by step.
First, let me write down the ... | [
{
"from": "human",
"value": "The value of (0.02)^2 + (0.52)^2 + (0.035)^2 divided by (0.002)^2 + a certain number squared + (0.0035)^2 is 100. What is the number?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find a certain number, let's call it x, such that when I calculate... |
What is the domain of the function $f(x)=\log_2(\log_7(\log_5x))$? | null | Okay, so I need to find the domain of the function f(x) = log₂(log₇(log₅x)). Hmm, the domain of a logarithmic function is where the argument is positive, right? Because you can't take the logarithm of a non-positive number. So, since this function has multiple logarithms nested inside each other, I need to make sure th... | To find the domain of the function \( f(x) = \log_2(\log_7(\log_5 x)) \), we need to ensure that the arguments of all logarithmic functions are positive.
1. The innermost function is \( \log_5 x \). For this to be defined, \( x \) must be positive:
\[
x > 0
\]
2. The next function is \( \log_7(\log_5 x) \). ... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to find the domain of the function f(x) = log₂(log₇(log₅x)). Hmm, the domain of a logarithmic function is where the argument is positive, right? Because you can't take the logarithm of a non-positive number. So, since this function has multiple logarithms nested inside each other, I need to make... | [
{
"from": "human",
"value": "What is the domain of the function $f(x)=\\log_2(\\log_7(\\log_5x))$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the domain of the function f(x) = log₂(log₇(log₅x)). Hmm, the domain of a logarithmic function is where the argument is positive, rig... |
Apples are sold at the farmer's market at a rate of $\$4$ per five pounds. According to this rate, how many dollars does it cost to buy 15 pounds of apples? | Apples are sold at the farmer's market at a rate of $\$4$ per five pounds. According to this rate, how many dollars does it cost to buy 15 pounds of apples? | Okay, let's see here. The problem is about figuring out how much 15 pounds of apples cost if they're sold at $4 per five pounds. Hmm, so first, I need to understand the rate given. It says $4 for five pounds. That means for every five pounds of apples, it costs $4. So, I need to find out how much 15 pounds would cost. ... | Apples are sold at a rate of $4 per five pounds. To find the cost of 15 pounds of apples, we can use the following methods:
1. **Cost per pound method**:
- The cost per pound is calculated as:
\[
\text{Cost per pound} = \frac{\$4}{5 \text{ pounds}} = \$0.80 \text{ per pound}
\]
- The total cost fo... | 6,642 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see here. The problem is about figuring out how much 15 pounds of apples cost if they're sold at $4 per five pounds. Hmm, so first, I need to understand the rate given. It says $4 for five pounds. That means for every five pounds of apples, it costs $4. So, I need to find out how much 15 pounds woul... | [
{
"from": "human",
"value": "Apples are sold at the farmer's market at a rate of $\\$4$ per five pounds. According to this rate, how many dollars does it cost to buy 15 pounds of apples?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see here. The problem is about figuring out how much 15 p... |
Determine the smallest value of $x$ such that
\[\frac{15x^2 - 40x + 18}{4x - 3} + 4x = 8x - 3.\] | null | Okay, let's see. I need to find the smallest value of x that satisfies this equation:
(15x² - 40x + 18)/(4x - 3) + 4x = 8x - 3.
Hmm. Alright, let me start by writing down the equation again to make sure I have it right:
(15x² - 40x + 18)/(4x - 3) + 4x = 8x - 3.
First, maybe I should simplify this equation. Let me s... | To determine the smallest value of \( x \) that satisfies the equation
\[
\frac{15x^2 - 40x + 18}{4x - 3} + 4x = 8x - 3,
\]
we start by eliminating the denominator. Multiply both sides by \( 4x - 3 \):
\[
15x^2 - 40x + 18 + 4x(4x - 3) = (8x - 3)(4x - 3).
\]
Expanding and simplifying both sides:
Left side:
\[
15x^2... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to find the smallest value of x that satisfies this equation:
(15x² - 40x + 18)/(4x - 3) + 4x = 8x - 3.
Hmm. Alright, let me start by writing down the equation again to make sure I have it right:
(15x² - 40x + 18)/(4x - 3) + 4x = 8x - 3.
First, maybe I should simplify this equation. ... | [
{
"from": "human",
"value": "Determine the smallest value of $x$ such that\n\\[\\frac{15x^2 - 40x + 18}{4x - 3} + 4x = 8x - 3.\\]"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the smallest value of x that satisfies this equation:\n\n(15x² - 40x + 18)/(4x - 3) + 4x = 8x ... |
Jordan wants to divide his 24 chocolate bars into 8 boxes of equal weight. Each box can hold only whole bars, so some boxes may have more bars than others. If he gives one of these boxes to his friend Shaina, what is the maximum number of bars Shaina can get? | Jordan wants to divide his 24 chocolate bars into 8 boxes of equal weight. Each box can hold only whole bars, so some boxes may have more bars than others. If he gives one of these boxes to his friend Shaina, what is the maximum number of bars Shaina can get? | Okay, so Jordan has 24 chocolate bars, and he wants to divide them into 8 boxes with equal weight. Each box can only hold whole bars, so some boxes might have more bars than others. The question is, if he gives one of these boxes to his friend Shaina, what's the maximum number of bars Shaina can get?
Hmm, let's break ... | Jordan wants to divide his 24 chocolate bars into 8 boxes of equal weight, where each box can only hold whole bars. The goal is to determine the maximum number of bars Shaina can get if she receives one of these boxes.
Key considerations:
1. The chocolate bars can have different weights.
2. To maximize the number of b... | 30,781 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Jordan has 24 chocolate bars, and he wants to divide them into 8 boxes with equal weight. Each box can only hold whole bars, so some boxes might have more bars than others. The question is, if he gives one of these boxes to his friend Shaina, what's the maximum number of bars Shaina can get?
Hmm, let'... | [
{
"from": "human",
"value": "Jordan wants to divide his 24 chocolate bars into 8 boxes of equal weight. Each box can hold only whole bars, so some boxes may have more bars than others. If he gives one of these boxes to his friend Shaina, what is the maximum number of bars Shaina can get?"
},
{
"from... |
Example 2 Calculate $(1+2 i) \div(3-4 i)$.
.
The translation maintains the original text's line breaks and format, as requested. | null | Okay, so I need to figure out how to divide these two complex numbers: (1 + 2i) divided by (3 - 4i). Hmm, complex numbers can be tricky when you first start working with them, especially division. Let me remember what the teacher said in class.
I think the key here is to get rid of the imaginary unit in the denominat... | To calculate \((1 + 2i) \div (3 - 4i)\), we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(3 + 4i\):
\[
\frac{1 + 2i}{3 - 4i} \cdot \frac{3 + 4i}{3 + 4i}
\]
First, we expand the numerator:
\[
(1 + 2i)(3 + 4i) = 1 \cdot 3 + 1 \cdot 4i + 2i \cdot 3 + 2i \cdot 4i
\]
\[
=... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to figure out how to divide these two complex numbers: (1 + 2i) divided by (3 - 4i). Hmm, complex numbers can be tricky when you first start working with them, especially division. Let me remember what the teacher said in class.
I think the key here is to get rid of the imaginary unit in the d... | [
{
"from": "human",
"value": "Example 2 Calculate $(1+2 i) \\div(3-4 i)$.\n\n. \n\nThe translation maintains the original text's line breaks and format, as requested."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out how to divide these two complex numbers: (1 + 2i) divided by... |
A line is parameterized by $(x,y) = (\cos t + 3, \sin t - 2)$, where $t$ is a real number. Find the equation of the line. Enter the equation in the form "$y = mx + b$". | A line is parameterized by $(x,y) = (\cos t + 3, \sin t - 2)$, where $t$ is a real number. Find the equation of the line. Enter the equation in the form "$y = mx + b$". | Okay, let me try to figure out the equation of this line. The problem says it's parameterized by (x, y) = (cos t + 3, sin t - 2), where t is a real number. Hmm, parameterized lines usually involve a direction vector and a point, but here they're using cosine and sine. That makes me think of a circle, actually, because ... | The given parameterization is \((x, y) = (\cos t + 3, \sin t - 2)\). To determine the equation of the line, we start by expressing \(\cos t\) and \(\sin t\) in terms of \(x\) and \(y\):
\[
\cos t = x - 3
\]
\[
\sin t = y + 2
\]
Using the Pythagorean identity \(\cos^2 t + \sin^2 t = 1\), we substitute the expressions ... | 22,243 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let me try to figure out the equation of this line. The problem says it's parameterized by (x, y) = (cos t + 3, sin t - 2), where t is a real number. Hmm, parameterized lines usually involve a direction vector and a point, but here they're using cosine and sine. That makes me think of a circle, actually, ... | [
{
"from": "human",
"value": "A line is parameterized by $(x,y) = (\\cos t + 3, \\sin t - 2)$, where $t$ is a real number. Find the equation of the line. Enter the equation in the form \"$y = mx + b$\"."
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out the equation of this ... |
13. List the natural numbers that can be divided by both 5 and 7 in ascending order, starting from 105, and take the first 2013 numbers. The remainder when the sum of these 2013 numbers is divided by 12 is $\qquad$ _. | null | Okay, let me see. The problem is asking for the natural numbers divisible by both 5 and 7, starting from 105, and we need to take the first 2013 of these numbers. Then, we have to find the sum of these 2013 numbers and compute the remainder when that sum is divided by 12. Alright, step by step.
First, numbers divisibl... | The natural numbers divisible by both 5 and 7 are multiples of their least common multiple (LCM), which is 35. Starting from 105 (which is \(35 \times 3\)), the sequence is \(35 \times 3, 35 \times 4, 35 \times 5, \ldots\). The first 2013 numbers in this sequence are \(35 \times 3, 35 \times 4, \ldots, 35 \times 2015\)... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me see. The problem is asking for the natural numbers divisible by both 5 and 7, starting from 105, and we need to take the first 2013 of these numbers. Then, we have to find the sum of these 2013 numbers and compute the remainder when that sum is divided by 12. Alright, step by step.
First, numbers ... | [
{
"from": "human",
"value": "13. List the natural numbers that can be divided by both 5 and 7 in ascending order, starting from 105, and take the first 2013 numbers. The remainder when the sum of these 2013 numbers is divided by 12 is $\\qquad$ _."
},
{
"from": "gpt",
"value": "<think>\nOkay, le... |
In a bounded Petri game with a single environment player and \( n \) system players, how can one decide the existence of a winning strategy for the system players, and what is the computational complexity of this decision problem? | Okay, so I need to figure out how to decide the existence of a winning strategy for the system players in a bounded Petri game with a single environment player and n system players. Also, what's the computational complexity of this problem? Hmm, let's start by recalling what a Petri game is.
From what I remember, Pet... | To determine the existence of a winning strategy for the system players in a bounded Petri game with a single environment player and \( n \) system players, we start by noting that boundedness ensures a finite state space. This allows us to model the game as a finite game graph where nodes represent markings (states) a... | 21,145 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out how to decide the existence of a winning strategy for the system players in a bounded Petri game with a single environment player and n system players. Also, what's the computational complexity of this problem? Hmm, let's start by recalling what a Petri game is.
From what I remem... | [
{
"from": "human",
"value": "In a bounded Petri game with a single environment player and \\( n \\) system players, how can one decide the existence of a winning strategy for the system players, and what is the computational complexity of this decision problem?"
},
{
"from": "gpt",
"value": "<th... | |
A deck of 52 playing cards is shuffled, and three cards are drawn at random without replacement. What is the probability that the third card drawn is a spade, given that the first two cards drawn are not spades? Express your answer as a common fraction. | A deck of 52 playing cards is shuffled, and three cards are drawn at random without replacement. What is the probability that the third card drawn is a spade, given that the first two cards drawn are not spades? Express your answer as a common fraction. | Okay, so I need to find the probability that the third card drawn is a spade, given that the first two cards drawn are not spades. Hmm, let's think about this step by step.
First, the problem involves conditional probability. The formula for conditional probability is P(A|B) = P(A and B) / P(B). But in this case, sinc... | Given a deck of 52 playing cards, we need to find the probability that the third card drawn is a spade, given that the first two cards drawn are not spades.
1. **Initial Consideration**:
- The deck originally has 52 cards: 13 spades and 39 non-spades.
- After drawing two non-spades, the deck has 50 cards remaini... | 35,705 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the probability that the third card drawn is a spade, given that the first two cards drawn are not spades. Hmm, let's think about this step by step.
First, the problem involves conditional probability. The formula for conditional probability is P(A|B) = P(A and B) / P(B). But in this ca... | [
{
"from": "human",
"value": "A deck of 52 playing cards is shuffled, and three cards are drawn at random without replacement. What is the probability that the third card drawn is a spade, given that the first two cards drawn are not spades? Express your answer as a common fraction."
},
{
"from": "gp... |
29.45. Let $t>0$. Prove that
$$
t-\frac{t^{2}}{2}+\frac{t^{3}}{3}-\frac{t^{4}}{4}+\ldots-\frac{t^{2 n}}{2 n}<\ln (1+t)<t-\frac{t^{2}}{2}+\frac{t^{3}}{3}-\frac{t^{4}}{4}+\ldots+\frac{t^{2 n+1}}{2 n+1}
$$ | null | Okay, so I need to prove this inequality involving the natural logarithm and an alternating series. Let me read it again to make sure I understand. The problem states that for any t > 0, the alternating series up to -t^{2n}/(2n) is less than ln(1+t), which in turn is less than the series up to +t^{2n+1}/(2n+1).
Hmm, ... | To prove the inequality involving the natural logarithm and an alternating series for \( t > 0 \):
\[
t - \frac{t^{2}}{2} + \frac{t^{3}}{3} - \frac{t^{4}}{4} + \ldots - \frac{t^{2n}}{2n} < \ln(1+t) < t - \frac{t^{2}}{2} + \frac{t^{3}}{3} - \frac{t^{4}}{4} + \ldots + \frac{t^{2n+1}}{2n+1}
\]
we consider the Taylor ser... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove this inequality involving the natural logarithm and an alternating series. Let me read it again to make sure I understand. The problem states that for any t > 0, the alternating series up to -t^{2n}/(2n) is less than ln(1+t), which in turn is less than the series up to +t^{2n+1}/(2n+1).... | [
{
"from": "human",
"value": "29.45. Let $t>0$. Prove that\n\n$$\nt-\\frac{t^{2}}{2}+\\frac{t^{3}}{3}-\\frac{t^{4}}{4}+\\ldots-\\frac{t^{2 n}}{2 n}<\\ln (1+t)<t-\\frac{t^{2}}{2}+\\frac{t^{3}}{3}-\\frac{t^{4}}{4}+\\ldots+\\frac{t^{2 n+1}}{2 n+1}\n$$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so... |
Given the function $f(x) = \frac{x}{4} + \frac{a}{x} - \ln x - \frac{3}{2}$, where $a \in \mathbb{R}$, and the curve $y = f(x)$ has a tangent line at the point $(1, f(1))$ that is perpendicular to the line $y = \frac{1}{2}x$.
(1) Find the value of $a$;
(2) Determine the intervals where the function $f(x)$ is monotonic... | null | Okay, so I need to solve this problem where there's a function f(x) = x/4 + a/x - ln x - 3/2, and they want me to find the value of 'a' such that at the point (1, f(1)), the tangent line to the curve is perpendicular to the line y = (1/2)x. Then, part (2) is to determine where the function is monotonic. Let me start wi... | Given the function \( f(x) = \frac{x}{4} + \frac{a}{x} - \ln x - \frac{3}{2} \), where \( a \in \mathbb{R} \), and the curve \( y = f(x) \) has a tangent line at the point \( (1, f(1)) \) that is perpendicular to the line \( y = \frac{1}{2}x \).
(1) To find the value of \( a \):
1. The slope of the line \( y = \frac{... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to solve this problem where there's a function f(x) = x/4 + a/x - ln x - 3/2, and they want me to find the value of 'a' such that at the point (1, f(1)), the tangent line to the curve is perpendicular to the line y = (1/2)x. Then, part (2) is to determine where the function is monotonic. Let me ... | [
{
"from": "human",
"value": "Given the function $f(x) = \\frac{x}{4} + \\frac{a}{x} - \\ln x - \\frac{3}{2}$, where $a \\in \\mathbb{R}$, and the curve $y = f(x)$ has a tangent line at the point $(1, f(1))$ that is perpendicular to the line $y = \\frac{1}{2}x$.\n\n(1) Find the value of $a$;\n(2) Determine t... |
9.2 On the blackboard, there are 100 fractions. In their numerators, the integers 1 to 100 each appear exactly once; in their denominators, the integers 1 to 100 each appear exactly once. If the sum of these 100 fractions can be simplified to a fraction with a denominator of 2. Prove: It is possible to swap the numerat... | null | Alright, let's tackle this problem. So, we have 100 fractions on the blackboard. Each numerator is a unique integer from 1 to 100, and each denominator is also a unique integer from 1 to 100. The sum of these fractions can be simplified to a fraction with a denominator of 2. We need to prove that by swapping the numera... | Given 100 fractions where each numerator and denominator is a unique integer from 1 to 100, and the sum of these fractions simplifies to a fraction with a denominator of 2, we need to prove that swapping the numerators of two fractions can result in a sum that simplifies to a fraction with an odd denominator.
1. **Ini... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, let's tackle this problem. So, we have 100 fractions on the blackboard. Each numerator is a unique integer from 1 to 100, and each denominator is also a unique integer from 1 to 100. The sum of these fractions can be simplified to a fraction with a denominator of 2. We need to prove that by swapping th... | [
{
"from": "human",
"value": "9.2 On the blackboard, there are 100 fractions. In their numerators, the integers 1 to 100 each appear exactly once; in their denominators, the integers 1 to 100 each appear exactly once. If the sum of these 100 fractions can be simplified to a fraction with a denominator of 2. ... |
Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is
$\text{(A)}\ 40\text{ dollars} \qquad \text{(B)}\ 50\text{ dollars} \qquad \text{(C)}\ 80\text{ dollars} \qquad \text{(D)}\ 100\text{ dollars} \qquad ... | null | Okay, so the problem here is about figuring out how much each person in the U.S. would have to pay if the $20$ billion dollar cost to send someone to Mars is split equally among all 250 million people. The answer choices are given, and I need to pick the correct one. Let me think through this step by step.
First, let'... | To determine each person's share of the $20$ billion dollar cost to send a person to Mars, we need to divide the total cost by the number of people in the U.S.
First, convert the numbers to the same units. The total cost is $20$ billion dollars, which is equivalent to $20,000$ million dollars. The population is 250 m... | null | OpenMathInstruct-AIME | <think>
Okay, so the problem here is about figuring out how much each person in the U.S. would have to pay if the $20$ billion dollar cost to send someone to Mars is split equally among all 250 million people. The answer choices are given, and I need to pick the correct one. Let me think through this step by step.
Fir... | [
{
"from": "human",
"value": "Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is \n$\\text{(A)}\\ 40\\text{ dollars} \\qquad \\text{(B)}\\ 50\\text{ dollars} \\qquad \\text{(C)}\\ 80\\text{ d... |
What is the sum of the reciprocals of the first five positive integers? | What is the sum of the reciprocals of the first five positive integers? | Okay, so I need to find the sum of the reciprocals of the first five positive integers. Let me think... The first five positive integers are 1, 2, 3, 4, and 5. Their reciprocals would be 1/1, 1/2, 1/3, 1/4, and 1/5. So the problem is asking me to add these fractions together.
Hmm, adding fractions. I remember that to... | The first five positive integers are 1, 2, 3, 4, and 5. Their reciprocals are \( \frac{1}{1} \), \( \frac{1}{2} \), \( \frac{1}{3} \), \( \frac{1}{4} \), and \( \frac{1}{5} \). To find the sum of these reciprocals, we need a common denominator.
The least common multiple (LCM) of the denominators 1, 2, 3, 4, and 5 is ... | 9,193 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the sum of the reciprocals of the first five positive integers. Let me think... The first five positive integers are 1, 2, 3, 4, and 5. Their reciprocals would be 1/1, 1/2, 1/3, 1/4, and 1/5. So the problem is asking me to add these fractions together.
Hmm, adding fractions. I remember... | [
{
"from": "human",
"value": "What is the sum of the reciprocals of the first five positive integers?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the sum of the reciprocals of the first five positive integers. Let me think... The first five positive integers are 1, 2, 3, 4, an... |
Eco-Friendly Water Bottle Company charges a $\$75$ set-up fee plus $\$15$ for each bottle produced. Green Bottle Company has no set-up fee but charges $\$20$ per bottle. What is the minimum number of bottles for which a customer saves money by using Eco-Friendly Water Bottle Company? | Eco-Friendly Water Bottle Company charges a $\$75$ set-up fee plus $\$15$ for each bottle produced. Green Bottle Company has no set-up fee but charges $\$20$ per bottle. What is the minimum number of bottles for which a customer saves money by using Eco-Friendly Water Bottle Company? | Okay, so I need to figure out the minimum number of bottles where Eco-Friendly Water Bottle Company becomes cheaper than Green Bottle Company. Let me start by understanding the problem again.
Eco-Friendly has a setup fee of $75 and charges $15 per bottle. Green Bottle has no setup fee but charges $20 per bottle. I nee... | To determine the minimum number of bottles for which a customer saves money by using Eco-Friendly Water Bottle Company instead of Green Bottle Company, we need to compare the total costs from both companies.
The total cost for Eco-Friendly Water Bottle Company is given by:
\[ 75 + 15n \]
where \( n \) is the number of... | 15,408 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to figure out the minimum number of bottles where Eco-Friendly Water Bottle Company becomes cheaper than Green Bottle Company. Let me start by understanding the problem again.
Eco-Friendly has a setup fee of $75 and charges $15 per bottle. Green Bottle has no setup fee but charges $20 per bottl... | [
{
"from": "human",
"value": "Eco-Friendly Water Bottle Company charges a $\\$75$ set-up fee plus $\\$15$ for each bottle produced. Green Bottle Company has no set-up fee but charges $\\$20$ per bottle. What is the minimum number of bottles for which a customer saves money by using Eco-Friendly Water Bottle ... |
Two positive integers, $a$ and $b$, have a greatest common divisor of 12 and a least common multiple of $ab/12$. If the difference between $a$ and $b$ is 60, what is the smallest possible value of the larger integer? | Two positive integers, $a$ and $b$, have a greatest common divisor of 12 and a least common multiple of $ab/12$. If the difference between $a$ and $b$ is 60, what is the smallest possible value of the larger integer? | Okay, so I need to find two positive integers, a and b, such that their greatest common divisor (GCD) is 12, their least common multiple (LCM) is ab divided by 12, and the difference between them is 60. The question is asking for the smallest possible value of the larger integer. Hmm, let me break this down step by ste... | Given two positive integers \(a\) and \(b\) such that their greatest common divisor (GCD) is 12, their least common multiple (LCM) is \(\frac{ab}{12}\), and the difference between them is 60, we need to find the smallest possible value of the larger integer.
1. **GCD and LCM Relationship**: Since the GCD of \(a\) and ... | 23,592 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find two positive integers, a and b, such that their greatest common divisor (GCD) is 12, their least common multiple (LCM) is ab divided by 12, and the difference between them is 60. The question is asking for the smallest possible value of the larger integer. Hmm, let me break this down ste... | [
{
"from": "human",
"value": "Two positive integers, $a$ and $b$, have a greatest common divisor of 12 and a least common multiple of $ab/12$. If the difference between $a$ and $b$ is 60, what is the smallest possible value of the larger integer?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I... |
Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters ... | null | Okay, so Keiko walks around a track every day at a constant speed. The track has straight sides and semicircular ends. It's 6 meters wide. She takes 36 seconds longer to walk around the outside edge compared to the inside edge. We need to find her speed in meters per second. The answer choices are multiples of pi, so t... | Keiko walks around a track with straight sides and semicircular ends, which is 6 meters wide. The difference in time taken to walk around the outside edge compared to the inside edge is 36 seconds. We need to find her speed.
1. **Track Dimensions**:
- The track has a width of 6 meters, meaning the radius of the out... | null | OpenMathInstruct-AIME | <think>
Okay, so Keiko walks around a track every day at a constant speed. The track has straight sides and semicircular ends. It's 6 meters wide. She takes 36 seconds longer to walk around the outside edge compared to the inside edge. We need to find her speed in meters per second. The answer choices are multiples of ... | [
{
"from": "human",
"value": "Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the insid... |
Find the distance between the points (3,4) and (-2,1) in a coordinate system where the x-axis is measured in inches and the y-axis is measured in feet. | Find the distance between the points (3,4) and (-2,1) in a coordinate system where the x-axis is measured in inches and the y-axis is measured in feet. | Okay, so I need to find the distance between the points (3,4) and (-2,1) in a coordinate system where the x-axis is in inches and the y-axis is in feet. Hmm, this seems a bit tricky because the units are different for each axis. Normally, if both axes are in the same units, I can just use the distance formula, right? T... | To find the distance between the points (3, 4) and (-2, 1) in a coordinate system where the x-axis is measured in inches and the y-axis is measured in feet, we need to convert the units to be consistent.
1. Convert the y-coordinates from feet to inches:
- The point (3, 4) has a y-coordinate of 4 feet, which conver... | 11,197 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the distance between the points (3,4) and (-2,1) in a coordinate system where the x-axis is in inches and the y-axis is in feet. Hmm, this seems a bit tricky because the units are different for each axis. Normally, if both axes are in the same units, I can just use the distance formula, ... | [
{
"from": "human",
"value": "Find the distance between the points (3,4) and (-2,1) in a coordinate system where the x-axis is measured in inches and the y-axis is measured in feet."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the distance between the points (3,4) and (-2,1) in... |
A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement,
with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents?
$\text{(A)} \ \frac{37}{924} \qquad \text{(B)} \ \frac{91}{924} \qquad \text{(C)}... | null | Okay, so we have a problem here where we need to find the probability that the value of six coins drawn from a box is at least 50 cents. The box contains 2 pennies, 4 nickels, and 6 dimes. Let me start by understanding the problem step by step.
First, let's note the total number of coins. There are 2 + 4 + 6 = 12 coin... | To find the probability that the value of six coins drawn from a box containing 2 pennies, 4 nickels, and 6 dimes is at least 50 cents, we need to consider the combinations of coins that meet this criterion.
1. **Total number of ways to draw 6 coins from 12:**
\[
\binom{12}{6} = 924
\]
2. **Case analysis for... | null | OpenMathInstruct-AIME | <think>
Okay, so we have a problem here where we need to find the probability that the value of six coins drawn from a box is at least 50 cents. The box contains 2 pennies, 4 nickels, and 6 dimes. Let me start by understanding the problem step by step.
First, let's note the total number of coins. There are 2 + 4 + 6 =... | [
{
"from": "human",
"value": "A box contains $2$ pennies, $4$ nickels, and $6$ dimes. Six coins are drawn without replacement, \nwith each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least $50$ cents? \n$\\text{(A)} \\ \\frac{37}{924} \\qquad ... |
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they d... | null | Okay, let me try to figure this problem out. It's about percentages of students who like or dislike dancing and what they actually say. Hmm, seems like a conditional probability question. The question is asking: What fraction of students who say they dislike dancing actually like it? So, we need to find the proportion ... | Let's assume there are 100 students at Typico High School.
1. **Students who like dancing**: 60% of 100 = 60 students.
- 80% of 60 students say they like dancing: \(0.8 \times 60 = 48\) students.
- 20% of 60 students say they dislike dancing: \(0.2 \times 60 = 12\) students.
2. **Students who dislike dancing**:... | null | OpenMathInstruct-AIME | <think>
Okay, let me try to figure this problem out. It's about percentages of students who like or dislike dancing and what they actually say. Hmm, seems like a conditional probability question. The question is asking: What fraction of students who say they dislike dancing actually like it? So, we need to find the pro... | [
{
"from": "human",
"value": "At Typico High School, $60\\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\\%$ say that they dislike it, and the rest say that they like ... |
Given a quantum process that consists of a unitary rotation of \(\frac{\pi}{4}\) followed by a non-unitary damping, represented as \(S = d\,D + (1-d)\,E\) with \(d = \frac{1}{2}\), calculate the output state for the input state \(|0\rangle\) after the transformation. Specify the expected form of the output state based ... | Okay, so I have this quantum process to work through. Let me start by understanding the problem step by step. The input state is |0⟩, and the process involves two parts: first, a unitary rotation of π/4, and then a non-unitary damping represented by S = d D + (1-d) E with d = 1/2. I need to calculate the output state a... | Given a quantum process involving a unitary rotation of \(\frac{\pi}{4}\) around the Y-axis followed by a non-unitary damping channel \(S = d\,D + (1-d)\,E\) with \(d = \frac{1}{2}\), we calculate the output state for the input state \(|0\rangle\).
1. **Unitary Rotation**:
- The input state \(|0\rangle\) is rotated... | 25,100 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I have this quantum process to work through. Let me start by understanding the problem step by step. The input state is |0⟩, and the process involves two parts: first, a unitary rotation of π/4, and then a non-unitary damping represented by S = d D + (1-d) E with d = 1/2. I need to calculate the output... | [
{
"from": "human",
"value": "Given a quantum process that consists of a unitary rotation of \\(\\frac{\\pi}{4}\\) followed by a non-unitary damping, represented as \\(S = d\\,D + (1-d)\\,E\\) with \\(d = \\frac{1}{2}\\), calculate the output state for the input state \\(|0\\rangle\\) after the transformatio... | |
A bookstore has three shelves, each containing a different type of book: fiction, non-fiction, and mystery. If 1/5 of the fiction books, 1/3 of the non-fiction books, and 3/5 of the mystery books are sold, and the remaining books fill 2/3 of the total shelf space, what fraction of the books were sold? | A bookstore has three shelves, each containing a different type of book: fiction, non-fiction, and mystery. If 1/5 of the fiction books, 1/3 of the non-fiction books, and 3/5 of the mystery books are sold, and the remaining books fill 2/3 of the total shelf space, what fraction of the books were sold? | Okay, let's see. The problem is about a bookstore with three shelves: fiction, non-fiction, and mystery. Each shelf has a different type of book. Then, fractions of each type are sold: 1/5 of fiction, 1/3 of non-fiction, and 3/5 of mystery. After those sales, the remaining books take up 2/3 of the total shelf space. Th... | Let \( F \) be the number of fiction books, \( N \) be the number of non-fiction books, and \( M \) be the number of mystery books.
When the fractions of each type of book are sold, the remaining books are:
- Fiction: \( \frac{4}{5}F \)
- Non-fiction: \( \frac{2}{3}N \)
- Mystery: \( \frac{2}{5}M \)
The total remain... | 11,682 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. The problem is about a bookstore with three shelves: fiction, non-fiction, and mystery. Each shelf has a different type of book. Then, fractions of each type are sold: 1/5 of fiction, 1/3 of non-fiction, and 3/5 of mystery. After those sales, the remaining books take up 2/3 of the total shelf s... | [
{
"from": "human",
"value": "A bookstore has three shelves, each containing a different type of book: fiction, non-fiction, and mystery. If 1/5 of the fiction books, 1/3 of the non-fiction books, and 3/5 of the mystery books are sold, and the remaining books fill 2/3 of the total shelf space, what fraction ... |
Given a Gromov hyperbolic manifold \( M \) of bounded geometry and a second-order elliptic operator \( L \) defined on \( M \), determine whether the following statement is true or false:
For any two positive \( L \)-superharmonic functions \( u \) and \( v \) that are \( L \)-harmonic and \( L \)-vanishing on a unif... | Okay, so I need to figure out whether the given statement about Gromov hyperbolic manifolds with bounded geometry and certain elliptic operators is true or false. Let me start by breaking down the problem and recalling some relevant concepts.
First, the manifold \( M \) is Gromov hyperbolic and has bounded geometry. G... | Given a Gromov hyperbolic manifold \( M \) with bounded geometry and a second-order elliptic operator \( L \), we need to determine whether the ratio of two positive \( L \)-superharmonic functions \( u \) and \( v \), which are \( L \)-harmonic and \( L \)-vanishing on a uniform neighborhood \( \mathcal{N}^\delta \) o... | 39,083 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out whether the given statement about Gromov hyperbolic manifolds with bounded geometry and certain elliptic operators is true or false. Let me start by breaking down the problem and recalling some relevant concepts.
First, the manifold \( M \) is Gromov hyperbolic and has bounded geo... | [
{
"from": "human",
"value": "Given a Gromov hyperbolic manifold \\( M \\) of bounded geometry and a second-order elliptic operator \\( L \\) defined on \\( M \\), determine whether the following statement is true or false: \n\nFor any two positive \\( L \\)-superharmonic functions \\( u \\) and \\( v \\) th... | |
5. In the acute triangle $\triangle ABC$, $\angle C > \angle B$. Point $D$ is a point on side $BC$ such that $\angle ADB$ is obtuse. $H$ is the orthocenter of $\triangle ABD$, and point $F$ is inside $\triangle ABC$ and lies on the circumcircle of $\triangle ABD$. Prove that point $F$ is the orthocenter of $\triangle A... | null | Okay, so I need to prove that point F is the orthocenter of triangle ABC if and only if two conditions are met: HD is parallel to CF, and H lies on the circumcircle of triangle ABC. Hmm, let's start by understanding all the components here.
First, we have an acute triangle ABC where angle C is greater than angle B. So... | To prove that point \( F \) is the orthocenter of \(\triangle ABC\) if and only if \( HD \) is parallel to \( CF \) and \( H \) lies on the circumcircle of \(\triangle ABC\), we need to show both directions of the implication.
### Forward Direction:
Assume \( F \) is the orthocenter of \(\triangle ABC\).
1. **HD || C... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to prove that point F is the orthocenter of triangle ABC if and only if two conditions are met: HD is parallel to CF, and H lies on the circumcircle of triangle ABC. Hmm, let's start by understanding all the components here.
First, we have an acute triangle ABC where angle C is greater than ang... | [
{
"from": "human",
"value": "5. In the acute triangle $\\triangle ABC$, $\\angle C > \\angle B$. Point $D$ is a point on side $BC$ such that $\\angle ADB$ is obtuse. $H$ is the orthocenter of $\\triangle ABD$, and point $F$ is inside $\\triangle ABC$ and lies on the circumcircle of $\\triangle ABD$. Prove t... |
Letters $A,B,C,$ and $D$ represent four different digits selected from $0,1,2,\ldots ,9.$ If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$?
$\mathrm{(A) \ }13 \qquad \mathrm{(B) \ }14 \qquad \mathrm{(C) \ } 15\qquad \mathrm{(D) \ }16 \qquad \mathrm{(E) \ } 17$ | null | Okay, so the problem is about four different digits A, B, C, D selected from 0 to 9. The goal is to make the fraction (A+B)/(C+D) as large as possible, and this fraction has to be an integer. Then we need to find the value of A+B from the given options. Let me try to break this down step by step.
First, let's understa... | To solve the problem, we need to maximize the integer value of \(\frac{A+B}{C+D}\) where \(A, B, C,\) and \(D\) are distinct digits from 0 to 9.
1. **Maximize \(A + B\)**: The largest possible sum of two distinct digits is \(9 + 8 = 17\).
2. **Minimize \(C + D\)**: The smallest possible sum of two distinct digits is ... | null | OpenMathInstruct-AIME | <think>
Okay, so the problem is about four different digits A, B, C, D selected from 0 to 9. The goal is to make the fraction (A+B)/(C+D) as large as possible, and this fraction has to be an integer. Then we need to find the value of A+B from the given options. Let me try to break this down step by step.
First, let's ... | [
{
"from": "human",
"value": "Letters $A,B,C,$ and $D$ represent four different digits selected from $0,1,2,\\ldots ,9.$ If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$?\n$\\mathrm{(A) \\ }13 \\qquad \\mathrm{(B) \\ }14 \\qquad \\mathrm{(C) \\ } 15\\qquad \\mathrm{(... |
Among the following statements:
① All equilateral triangles are congruent;
② Right-angled triangles with equal hypotenuses are congruent;
③ Isosceles triangles with equal vertex angles and equal sides are congruent;
④ Right-angled triangles with two equal acute angles are congruent.
The correct number of statements is ... | null | Okay, let's take a look at these four statements about congruent triangles and figure out which ones are correct. I need to check each one step by step. Let's start with statement ①: "All equilateral triangles are congruent." Hmm, equilateral triangles all have three sides of equal length and all angles equal to 60 deg... | Let's analyze each statement one by one:
1. **Statement ①**: "All equilateral triangles are congruent."
Equilateral triangles have all sides equal and all angles equal to 60 degrees. However, they can have different side lengths, making them similar but not necessarily congruent. This statement is false.
2. **St... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's take a look at these four statements about congruent triangles and figure out which ones are correct. I need to check each one step by step. Let's start with statement ①: "All equilateral triangles are congruent." Hmm, equilateral triangles all have three sides of equal length and all angles equal t... | [
{
"from": "human",
"value": "Among the following statements:\n① All equilateral triangles are congruent;\n② Right-angled triangles with equal hypotenuses are congruent;\n③ Isosceles triangles with equal vertex angles and equal sides are congruent;\n④ Right-angled triangles with two equal acute angles are co... |
Given vectors $\overrightarrow{a}= (2, 3, 1)$ and $\overrightarrow{b}= (1, 2, 0)$, find the magnitude of $\overrightarrow{a} - \overrightarrow{b}$.
A: 1
B: $\sqrt{3}$
C: 3
D: 9 | null | Okay, let's see. I need to find the magnitude of the vector a minus vector b. Vector a is (2, 3, 1) and vector b is (1, 2, 0). Hmm, first I should probably subtract the two vectors component-wise, right? So, subtract each corresponding component of b from a.
Let me write that out. For the x-component, it's 2 minus 1,... | To find the magnitude of the vector \(\overrightarrow{a} - \overrightarrow{b}\), we first subtract the vectors component-wise. Given \(\overrightarrow{a} = (2, 3, 1)\) and \(\overrightarrow{b} = (1, 2, 0)\):
\[
\overrightarrow{a} - \overrightarrow{b} = (2 - 1, 3 - 2, 1 - 0) = (1, 1, 1)
\]
Next, we calculate the magni... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's see. I need to find the magnitude of the vector a minus vector b. Vector a is (2, 3, 1) and vector b is (1, 2, 0). Hmm, first I should probably subtract the two vectors component-wise, right? So, subtract each corresponding component of b from a.
Let me write that out. For the x-component, it's 2 ... | [
{
"from": "human",
"value": "Given vectors $\\overrightarrow{a}= (2, 3, 1)$ and $\\overrightarrow{b}= (1, 2, 0)$, find the magnitude of $\\overrightarrow{a} - \\overrightarrow{b}$.\nA: 1\nB: $\\sqrt{3}$\nC: 3\nD: 9"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find the magni... |
What are the necessary and sufficient conditions for a relation \( R \) to be definable in Relational Type Theory (RelTT) based on the relational semantics introduced in the paper? | Okay, so I need to figure out the necessary and sufficient conditions for a relation \( R \) to be definable in Relational Type Theory (RelTT) based on the relational semantics from the paper. Hmm, first, I should recall what RelTT is. From what I remember, RelTT is a framework for studying relations in a typed setting... | The necessary and sufficient conditions for a relation \( R \) to be definable in Relational Type Theory (RelTT) based on relational semantics are as follows:
1. **Closure Under Basic Relations**: \( R \) must include or be constructible from the primitive relations provided by RelTT, such as identity relations for ea... | 25,946 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the necessary and sufficient conditions for a relation \( R \) to be definable in Relational Type Theory (RelTT) based on the relational semantics from the paper. Hmm, first, I should recall what RelTT is. From what I remember, RelTT is a framework for studying relations in a typed... | [
{
"from": "human",
"value": "What are the necessary and sufficient conditions for a relation \\( R \\) to be definable in Relational Type Theory (RelTT) based on the relational semantics introduced in the paper?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to figure out the necessary ... | |
Determine the range of the function $$H(x) = |x^2-4|-|x^2-9|.$$ Express your answer in interval notation. | Determine the range of the function $$H(x) = |x^2-4|-|x^2-9|.$$ Express your answer in interval notation. | Okay, so I need to find the range of the function H(x) = |x² - 4| - |x² - 9|. Hmm, let's see. The function involves absolute values of quadratic expressions. Since absolute value functions can sometimes be tricky, maybe I should break it down piece by piece. Let me start by recalling that the absolute value function |A... | To determine the range of the function \( H(x) = |x^2 - 4| - |x^2 - 9| \), we consider different cases based on the value of \( x^2 \):
1. **Case 1: \( x^2 < 4 \) (i.e., \( |x| < 2 \))**
- Both \( x^2 - 4 \) and \( x^2 - 9 \) are negative.
- \( H(x) = (-x^2 + 4) - (-x^2 + 9) = -5 \).
2. **Case 2: \( 4 \leq x^2 ... | 14,849 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the range of the function H(x) = |x² - 4| - |x² - 9|. Hmm, let's see. The function involves absolute values of quadratic expressions. Since absolute value functions can sometimes be tricky, maybe I should break it down piece by piece. Let me start by recalling that the absolute value fun... | [
{
"from": "human",
"value": "Determine the range of the function $$H(x) = |x^2-4|-|x^2-9|.$$ Express your answer in interval notation."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the range of the function H(x) = |x² - 4| - |x² - 9|. Hmm, let's see. The function involves absol... |
Lily purchased a new laptop and paid $15\%$ of the cost upfront, which was $\$200$. What was the price of the laptop? | Lily purchased a new laptop and paid $15\%$ of the cost upfront, which was $\$200$. What was the price of the laptop? | Okay, so Lily bought a new laptop and paid 15% of the cost upfront, and that amount was $200. The question is asking for the total price of the laptop. Hmm, let's see. I need to figure out the total price based on the fact that 15% of it is $200.
Alright, let me start by recalling some basic percentage concepts. If 1... | Lily paid 15% of the laptop's cost upfront, which was $200. To find the total price of the laptop, we set up the equation where 15% of the total price \( P \) equals $200:
\[
0.15 \times P = 200
\]
To isolate \( P \), we divide both sides by 0.15:
\[
P = \frac{200}{0.15}
\]
Converting the division into a fraction f... | 18,666 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so Lily bought a new laptop and paid 15% of the cost upfront, and that amount was $200. The question is asking for the total price of the laptop. Hmm, let's see. I need to figure out the total price based on the fact that 15% of it is $200.
Alright, let me start by recalling some basic percentage concep... | [
{
"from": "human",
"value": "Lily purchased a new laptop and paid $15\\%$ of the cost upfront, which was $\\$200$. What was the price of the laptop?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so Lily bought a new laptop and paid 15% of the cost upfront, and that amount was $200. The question ... |
Consider the group \( Q \) constructed as the semidirect product of a hyperbolic, residually finite group \( K \) with a locally finite group \( P \) of permutations of \( K \) that move only finitely many elements. Show that if \( Q \) is a finitely generated sofic group, then what properties must \( K \) and \( P \) ... | Okay, let me try to work through this problem step by step. The question is about showing that a certain semidirect product group Q (of a hyperbolic residually finite group K and a locally finite permutation group P acting on K with finitely many moved elements) is non-residually amenable, given that Q is a finitely ge... | To address the problem, we need to show that the semidirect product \( Q = K \rtimes P \) of a hyperbolic, residually finite group \( K \) with a locally finite group \( P \) of permutations of \( K \) that move only finitely many elements is non-residually amenable while being sofic.
### Key Properties:
1. **Hyperbo... | 13,912 | mlfoundations-dev/gair_mathpile | <think>
Okay, let me try to work through this problem step by step. The question is about showing that a certain semidirect product group Q (of a hyperbolic residually finite group K and a locally finite permutation group P acting on K with finitely many moved elements) is non-residually amenable, given that Q is a fin... | [
{
"from": "human",
"value": "Consider the group \\( Q \\) constructed as the semidirect product of a hyperbolic, residually finite group \\( K \\) with a locally finite group \\( P \\) of permutations of \\( K \\) that move only finitely many elements. Show that if \\( Q \\) is a finitely generated sofic gr... | |
Given the function defined by the Clausen's Hypergeometric Function
\[
z\,_3F_2(a,b,c;b+1,c+1;z) = z + \sum_{n=2}^{\infty} \frac{(a)_{n-1}(b)_{n-1}(c)_{n-1}}{(b+1)_{n-1}(c+1)_{n-1}(1)_{n-1}} a_n z^n,
\]
determine the sufficient condition such that this function belongs to the class \(\mathcal{S}^{*}_{\lambda}\) for ... | Okay, so I need to figure out the sufficient condition for the function defined by Clausen's Hypergeometric Function to belong to the class S*_λ, where λ is between 0 and 1. Alright, first, let me recall what the class S*_λ is.
From what I remember, S*_λ is the class of starlike functions of order λ. A function f(z) ... | To determine the sufficient condition for the function defined by Clausen's Hypergeometric Function
\[
z\,_3F_2(a,b,c;b+1,c+1;z) = z + \sum_{n=2}^{\infty} \frac{(a)_{n-1}(b)_{n-1}(c)_{n-1}}{(b+1)_{n-1}(c+1)_{n-1}(1)_{n-1}} z^n,
\]
to belong to the class \(\mathcal{S}^{*}_{\lambda}\) for some \(\lambda \in (0, 1]\), ... | 37,832 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the sufficient condition for the function defined by Clausen's Hypergeometric Function to belong to the class S*_λ, where λ is between 0 and 1. Alright, first, let me recall what the class S*_λ is.
From what I remember, S*_λ is the class of starlike functions of order λ. A functi... | [
{
"from": "human",
"value": "Given the function defined by the Clausen's Hypergeometric Function \n\n\\[\nz\\,_3F_2(a,b,c;b+1,c+1;z) = z + \\sum_{n=2}^{\\infty} \\frac{(a)_{n-1}(b)_{n-1}(c)_{n-1}}{(b+1)_{n-1}(c+1)_{n-1}(1)_{n-1}} a_n z^n,\n\\]\n\ndetermine the sufficient condition such that this function be... | |
Given a non-ballistic random walk in an i.i.d. random environment in $\mathbb{Z}^d$ for $d \geq 3$, assume that the transition probabilities of the environment are balanced in one coordinate direction and $\varepsilon$-close to those of simple random walk. Define the mean sojourn time of the random walk in a ball of ra... | Okay, so I need to figure out what value D should converge to in terms of known properties of Brownian motion. Let me start by understanding the problem statement step by step.
We have a non-ballistic random walk in an i.i.d. random environment on ℤ^d for d ≥ 3. The transition probabilities are balanced in one coordin... | Given a non-ballistic random walk in an i.i.d. random environment on \(\mathbb{Z}^d\) for \(d \geq 3\), where the transition probabilities are balanced in one coordinate direction and \(\varepsilon\)-close to those of simple random walk, we need to determine the value \(D\) such that
\[
\lim_{L\rightarrow\infty} \frac... | 39,277 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out what value D should converge to in terms of known properties of Brownian motion. Let me start by understanding the problem statement step by step.
We have a non-ballistic random walk in an i.i.d. random environment on ℤ^d for d ≥ 3. The transition probabilities are balanced in one... | [
{
"from": "human",
"value": "Given a non-ballistic random walk in an i.i.d. random environment in $\\mathbb{Z}^d$ for $d \\geq 3$, assume that the transition probabilities of the environment are balanced in one coordinate direction and $\\varepsilon$-close to those of simple random walk. Define the mean soj... | |
Determine the value of the expression $(\log_{3}3^{ \frac {1}{2}})^{2}+\log_{0.25} \frac {1}{4}+9\log_{5} \sqrt {5}-\log_{ \sqrt {3}}1$. | null | Okay, so I have this expression to evaluate: (log base 3 of 3^(1/2)) squared plus log base 0.25 of 1/4 plus 9 times log base 5 of sqrt(5) minus log base sqrt(3) of 1. Hmm, let me break this down term by term.
First, let's look at the first term: (log₃3^(1/2))². I know that log base a of a^b is just b. So log base 3 o... | To determine the value of the expression \((\log_{3}3^{ \frac {1}{2}})^{2}+\log_{0.25} \frac {1}{4}+9\log_{5} \sqrt {5}-\log_{ \sqrt {3}}1\), we break it down term by term:
1. **First term**: \((\log_{3}3^{ \frac {1}{2}})^{2}\)
- \(\log_{3}3^{ \frac {1}{2}} = \frac{1}{2}\) because the logarithm of a number to its o... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this expression to evaluate: (log base 3 of 3^(1/2)) squared plus log base 0.25 of 1/4 plus 9 times log base 5 of sqrt(5) minus log base sqrt(3) of 1. Hmm, let me break this down term by term.
First, let's look at the first term: (log₃3^(1/2))². I know that log base a of a^b is just b. So log ... | [
{
"from": "human",
"value": "Determine the value of the expression $(\\log_{3}3^{ \\frac {1}{2}})^{2}+\\log_{0.25} \\frac {1}{4}+9\\log_{5} \\sqrt {5}-\\log_{ \\sqrt {3}}1$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this expression to evaluate: (log base 3 of 3^(1/2)) squared plus ... |
Suppose $x$ and $y$ are positive real numbers such that
\[x^y=2^{64}\text{ and }(\log_2{x})^{\log_2{y}}=2^{7}.\]
What is the greatest possible value of $\log_2{y}$?
$\textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }3+\sqrt{2} \qquad \textbf{(D) }4+\sqrt{3} \qquad \textbf{(E) }7$ | null | Okay, so I need to solve this problem where we have two positive real numbers x and y with the conditions:
1. x^y = 2^64
2. (log base 2 of x) raised to the power of (log base 2 of y) equals 2^7.
And we need to find the greatest possible value of log base 2 of y, with the answer choices given.
Let me start by writing... | Given the equations \(x^y = 2^{64}\) and \((\log_2 x)^{\log_2 y} = 2^7\), we need to find the greatest possible value of \(\log_2 y\).
First, we set \(a = \log_2 x\) and \(b = \log_2 y\). This transforms the given equations into:
1. \(a \cdot 2^b = 64\)
2. \(a^b = 128\)
From the first equation, we solve for \(a\):
\[... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to solve this problem where we have two positive real numbers x and y with the conditions:
1. x^y = 2^64
2. (log base 2 of x) raised to the power of (log base 2 of y) equals 2^7.
And we need to find the greatest possible value of log base 2 of y, with the answer choices given.
Let me start by... | [
{
"from": "human",
"value": "Suppose $x$ and $y$ are positive real numbers such that\n\\[x^y=2^{64}\\text{ and }(\\log_2{x})^{\\log_2{y}}=2^{7}.\\]\nWhat is the greatest possible value of $\\log_2{y}$?\n$\\textbf{(A) }3 \\qquad \\textbf{(B) }4 \\qquad \\textbf{(C) }3+\\sqrt{2} \\qquad \\textbf{(D) }4+\\sqrt... |
Show that the equation $5 x^{2}+3 y^{2}=1$ has no solution in the rational numbers. | null | Alright, so I need to show that the equation \(5x^2 + 3y^2 = 1\) has no solutions in the rational numbers. Hmm, okay. Let me start by thinking about what it means for a solution to exist in the rationals. If there are rational numbers \(x = \frac{a}{b}\) and \(y = \frac{c}{d}\) where \(a, b, c, d\) are integers with no... | To show that the equation \(5x^2 + 3y^2 = 1\) has no solution in the rational numbers, we start by considering the equation in integers. If there were a rational solution, there would exist integers \(m\), \(n\), and \(N\) such that \(5m^2 + 3n^2 = N^2\).
First, we analyze the equation modulo 3:
\[
5m^2 + 3n^2 \equiv ... | null | AI-MO/NuminaMath-1.5 | <think>
Alright, so I need to show that the equation \(5x^2 + 3y^2 = 1\) has no solutions in the rational numbers. Hmm, okay. Let me start by thinking about what it means for a solution to exist in the rationals. If there are rational numbers \(x = \frac{a}{b}\) and \(y = \frac{c}{d}\) where \(a, b, c, d\) are integers... | [
{
"from": "human",
"value": "Show that the equation $5 x^{2}+3 y^{2}=1$ has no solution in the rational numbers."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to show that the equation \\(5x^2 + 3y^2 = 1\\) has no solutions in the rational numbers. Hmm, okay. Let me start by thinkin... |
Consider a graph \( G \) with \( n \) vertices and \( m \) edges. Define the conductance \( \phi(S) \) and volume of a subset \( S \subseteq V \) of the graph. If the graph is \( \varepsilon \)-far from any \( (k, \phi) \)-expander, what conditions can you deduce about specific subsets of the graph? Specifically, how w... | Okay, so I need to figure out how to construct a testing algorithm using random walks to verify if a graph G is a (k, φ)-expander, especially when it's ε-far from being one. Let me start by recalling the definitions.
First, the conductance φ(S) of a subset S of vertices is defined as the ratio of the number of edges l... | To determine if a graph \( G \) is a \( (k, \phi) \)-expander or \( \epsilon \)-far from being one, we analyze subsets \( S \subseteq V \) with \( |S| \leq k \) and their conductance \( \phi(S) \). If \( G \) is \( \epsilon \)-far, it contains sufficiently many subsets violating the conductance \( \phi \), necessitatin... | 5,093 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out how to construct a testing algorithm using random walks to verify if a graph G is a (k, φ)-expander, especially when it's ε-far from being one. Let me start by recalling the definitions.
First, the conductance φ(S) of a subset S of vertices is defined as the ratio of the number of... | [
{
"from": "human",
"value": "Consider a graph \\( G \\) with \\( n \\) vertices and \\( m \\) edges. Define the conductance \\( \\phi(S) \\) and volume of a subset \\( S \\subseteq V \\) of the graph. If the graph is \\( \\varepsilon \\)-far from any \\( (k, \\phi) \\)-expander, what conditions can you dedu... | |
Given a list of complex numbers \(\Lambda = \{ \lambda_1, \lambda_2, \ldots, \lambda_n \}\) such that \(\lambda_1 \geq 0\) and \(\sum_{i=1}^{n} \lambda_i \geq 0\), where \(\lambda_2, \ldots, \lambda_n\) are constrained to lie in the region defined by \(Rez \leq 0\) and \(\sqrt{3} |Rez| \geq |Imz|\). What condition must... | Okay, let's tackle this problem step by step. The question is about determining the condition under which a given list of complex numbers Λ can be universally realized as the spectrum of a nonnegative matrix, considering every possible Jordan canonical form allowed by Λ.
First, let's parse the problem statement again... | Given a list of complex numbers \(\Lambda = \{ \lambda_1, \lambda_2, \ldots, \lambda_n \}\) where \(\lambda_1 \geq 0\) and \(\sum_{i=1}^{n} \lambda_i \geq 0\), with \(\lambda_2, \ldots, \lambda_n\) constrained to lie in the region defined by \( \text{Re}(z) \leq 0 \) and \(\sqrt{3} |\text{Re}(z)| \geq |\text{Im}(z)|\),... | 4,717 | mlfoundations-dev/gair_mathpile | <think>
Okay, let's tackle this problem step by step. The question is about determining the condition under which a given list of complex numbers Λ can be universally realized as the spectrum of a nonnegative matrix, considering every possible Jordan canonical form allowed by Λ.
First, let's parse the problem stateme... | [
{
"from": "human",
"value": "Given a list of complex numbers \\(\\Lambda = \\{ \\lambda_1, \\lambda_2, \\ldots, \\lambda_n \\}\\) such that \\(\\lambda_1 \\geq 0\\) and \\(\\sum_{i=1}^{n} \\lambda_i \\geq 0\\), where \\(\\lambda_2, \\ldots, \\lambda_n\\) are constrained to lie in the region defined by \\(Re... | |
There are constants $a$ and $b$ such that $\frac{x-a}{x+b} = \frac{x^2-120x+4554}{x^2+65x-2700}$. What is $a+b$? | There are constants $a$ and $b$ such that $\frac{x-a}{x+b} = \frac{x^2-120x+4554}{x^2+65x-2700}$. What is $a+b$? | Okay, so I need to find constants a and b such that (x - a)/(x + b) is equal to (x² - 120x + 4554)/(x² + 65x - 2700). Then, after finding a and b, I need to compute a + b. Hmm, let me think.
First, I remember that if two rational expressions are equal for all x (except where the denominator is zero), then their numer... | To find the constants \(a\) and \(b\) such that \(\frac{x-a}{x+b} = \frac{x^2 - 120x + 4554}{x^2 + 65x - 2700}\), we start by cross-multiplying and equating the coefficients of the resulting polynomials.
Cross-multiplying gives:
\[
(x - a)(x^2 + 65x - 2700) = (x + b)(x^2 - 120x + 4554)
\]
Expanding both sides, we get... | 26,981 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find constants a and b such that (x - a)/(x + b) is equal to (x² - 120x + 4554)/(x² + 65x - 2700). Then, after finding a and b, I need to compute a + b. Hmm, let me think.
First, I remember that if two rational expressions are equal for all x (except where the denominator is zero), then the... | [
{
"from": "human",
"value": "There are constants $a$ and $b$ such that $\\frac{x-a}{x+b} = \\frac{x^2-120x+4554}{x^2+65x-2700}$. What is $a+b$?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find constants a and b such that (x - a)/(x + b) is equal to (x² - 120x + 4554)/(x² + 65x - 2... |
Let \( \Omega \subset \mathbb{R}^n \) be a bounded domain such that the boundary \( \partial \Omega \) is \( C^{4+\alpha} \) for \( 0 < \alpha < 1 \) and assume that the \( (m-1) \)-curvature of the boundary, \( \mathbf{k}_{m-1}[\partial \Omega] \), is positive. How many distinct nontrivial solutions exist for the Diri... | Okay, so I need to figure out the number of distinct nontrivial solutions to the Dirichlet problem \( T_m[w] = 1 \) with \( w|_{\partial \Omega} = 0 \) where \( 0 \leq l < m \leq n \). The domain \( \Omega \) is bounded with a \( C^{4+\alpha} \) boundary, and the \( (m-1) \)-curvature of the boundary is positive. Hmm, ... | Given a bounded domain \( \Omega \subset \mathbb{R}^n \) with a \( C^{4+\alpha} \) boundary \( \partial \Omega \) where the \( (m-1) \)-curvature \( \mathbf{k}_{m-1}[\partial \Omega] \) is positive, we need to determine the number of distinct nontrivial solutions for the Dirichlet problem
\[
T_m[w] = 1, \quad w|_{\pa... | 6,288 | mlfoundations-dev/gair_mathpile | <think>
Okay, so I need to figure out the number of distinct nontrivial solutions to the Dirichlet problem \( T_m[w] = 1 \) with \( w|_{\partial \Omega} = 0 \) where \( 0 \leq l < m \leq n \). The domain \( \Omega \) is bounded with a \( C^{4+\alpha} \) boundary, and the \( (m-1) \)-curvature of the boundary is positiv... | [
{
"from": "human",
"value": "Let \\( \\Omega \\subset \\mathbb{R}^n \\) be a bounded domain such that the boundary \\( \\partial \\Omega \\) is \\( C^{4+\\alpha} \\) for \\( 0 < \\alpha < 1 \\) and assume that the \\( (m-1) \\)-curvature of the boundary, \\( \\mathbf{k}_{m-1}[\\partial \\Omega] \\), is posi... | |
Given proposition $p$: $\exists x \in \mathbb{R}$, such that $\sin x - \cos x = \sqrt{3}$, and proposition $q$: the set $\{x | x^2 - 2x + 1 = 0, x \in \mathbb{R}\}$ has $2$ subsets. The following conclusions are drawn:
- $(1)$ The proposition "$p \land q$" is true.
- $(2)$ The proposition "$p \land \neg q$" is false.
-... | null | Okay, let's try to figure out this problem step by step. We have two propositions, p and q, and three conclusions based on them. We need to determine which of the conclusions (1), (2), and (3) are correct. The answer choices are A through D, with D meaning all three are correct. Let's break it down.
First, let's under... | First, we analyze proposition \( p \): \(\exists x \in \mathbb{R}\) such that \(\sin x - \cos x = \sqrt{3}\). By rewriting \(\sin x - \cos x\) using a trigonometric identity, we get \(\sqrt{2} \sin(x - \pi/4)\). The equation \(\sqrt{2} \sin(x - \pi/4) = \sqrt{3}\) simplifies to \(\sin(x - \pi/4) = \sqrt{3/2}\), which i... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let's try to figure out this problem step by step. We have two propositions, p and q, and three conclusions based on them. We need to determine which of the conclusions (1), (2), and (3) are correct. The answer choices are A through D, with D meaning all three are correct. Let's break it down.
First, let... | [
{
"from": "human",
"value": "Given proposition $p$: $\\exists x \\in \\mathbb{R}$, such that $\\sin x - \\cos x = \\sqrt{3}$, and proposition $q$: the set $\\{x | x^2 - 2x + 1 = 0, x \\in \\mathbb{R}\\}$ has $2$ subsets. The following conclusions are drawn:\n- $(1)$ The proposition \"$p \\land q$\" is true.... |
Simplify the following expression: $(3x+2)\cdot(6x^8-3x^7+2x^6+x^5)$.
Express the answer as a polynomial with the degrees of the terms in decreasing order. | Simplify the following expression: $(3x+2)\cdot(6x^8-3x^7+2x^6+x^5)$.
Express the answer as a polynomial with the degrees of the terms in decreasing order. | Okay, so I need to simplify the expression (3x + 2) multiplied by (6x^8 - 3x^7 + 2x^6 + x^5). Hmm, let me think. Alright, this is a polynomial multiplication problem. I remember that to multiply two polynomials, I have to use the distributive property, right? That means I need to multiply each term in the first polynom... | To simplify the expression \((3x + 2) \cdot (6x^8 - 3x^7 + 2x^6 + x^5)\), we use the distributive property (also known as the FOIL method for polynomials) and then combine like terms.
First, we distribute each term in the polynomial \(3x + 2\) across each term in the polynomial \(6x^8 - 3x^7 + 2x^6 + x^5\):
1. Multip... | 5,973 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to simplify the expression (3x + 2) multiplied by (6x^8 - 3x^7 + 2x^6 + x^5). Hmm, let me think. Alright, this is a polynomial multiplication problem. I remember that to multiply two polynomials, I have to use the distributive property, right? That means I need to multiply each term in the first... | [
{
"from": "human",
"value": "Simplify the following expression: $(3x+2)\\cdot(6x^8-3x^7+2x^6+x^5)$.\nExpress the answer as a polynomial with the degrees of the terms in decreasing order."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to simplify the expression (3x + 2) multiplied by (6x... |
For any positive integer $n$, let
$f(n) =\left\{\begin{matrix}\log_{8}{n}, &\text{if }\log_{8}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right.$
What is $\sum_{n = 1}^{1997}{f(n)}$?
$\textbf{(A)}\ \log_{8}{2047}\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ \frac{55}{3}\qquad\textbf{(D)}\ \frac{58}{3}\qquad... | null | Okay, so I need to figure out the sum of f(n) from n = 1 to 1997. The function f(n) is defined as log base 8 of n if that logarithm is rational, otherwise it's 0. Hmm, okay. So essentially, I only need to sum log₈(n) for all n between 1 and 1997 where log₈(n) is rational, and ignore all the other n where it's irrationa... | To determine the sum \(\sum_{n=1}^{1997} f(n)\) where \(f(n)\) is defined as \(\log_8 n\) if \(\log_8 n\) is rational and 0 otherwise, we need to identify the values of \(n\) for which \(\log_8 n\) is rational.
A rational logarithm \(\log_8 n\) implies that \(n\) can be expressed as a power of 2. This is because \(\l... | null | OpenMathInstruct-AIME | <think>
Okay, so I need to figure out the sum of f(n) from n = 1 to 1997. The function f(n) is defined as log base 8 of n if that logarithm is rational, otherwise it's 0. Hmm, okay. So essentially, I only need to sum log₈(n) for all n between 1 and 1997 where log₈(n) is rational, and ignore all the other n where it's i... | [
{
"from": "human",
"value": "For any positive integer $n$, let \n$f(n) =\\left\\{\\begin{matrix}\\log_{8}{n}, &\\text{if }\\log_{8}{n}\\text{ is rational,}\\\\ 0, &\\text{otherwise.}\\end{matrix}\\right.$\nWhat is $\\sum_{n = 1}^{1997}{f(n)}$?\n$\\textbf{(A)}\\ \\log_{8}{2047}\\qquad\\textbf{(B)}\\ 6\\qquad... |
A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is
$\text{(A)}\ 8 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 18$ | null | Okay, so there's this problem about a gumball machine with different colored gumballs: 9 red, 7 white, and 8 blue. The question is asking for the least number of gumballs a person must buy to be sure of getting four of the same color. The answer choices are from 8 to 18, with some options in between. Hmm, let me think.... | To determine the least number of gumballs a person must buy to be sure of getting four gumballs of the same color, we use the pigeonhole principle. The gumball machine contains 9 red, 7 white, and 8 blue gumballs.
The worst-case scenario is when a person picks the maximum number of gumballs without getting four of an... | null | OpenMathInstruct-AIME | <think>
Okay, so there's this problem about a gumball machine with different colored gumballs: 9 red, 7 white, and 8 blue. The question is asking for the least number of gumballs a person must buy to be sure of getting four of the same color. The answer choices are from 8 to 18, with some options in between. Hmm, let m... | [
{
"from": "human",
"value": "A gumball machine contains $9$ red, $7$ white, and $8$ blue gumballs. The least number of gumballs a person must buy to be sure of getting four gumballs of the same color is\n$\\text{(A)}\\ 8 \\qquad \\text{(B)}\\ 9 \\qquad \\text{(C)}\\ 10 \\qquad \\text{(D)}\\ 12 \\qquad \\te... |
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