instruction_seed stringlengths 42 6.11k | source stringclasses 4
values | to_be_used float64 95 100 | classifier_reasoning stringlengths 226 912 | __original_row_idx int64 0 2.5k | r1_distill_70b_response stringlengths 647 76.5k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|
Richard has 360 feet of fencing to build a rectangular backyard. The fencing will enclose three sides of the backyard, while the fourth side will be the border of Richard's house. What is the largest possible area of this backyard? | tiger_math | 95 | The question is structured similarly to the positive examples, as it involves a mathematical problem that requires optimization (finding the largest possible area) under given constraints (amount of fencing). It asks for a solution involving geometry and area calculation, much like the positive examples that involve sp... | 11 | <think>
I start with Richard's 360 feet of fencing. He needs to enclose three sides of his rectangular backyard, with the fourth side being his house. I'll let the length parallel to the house be \( y \) and the two widths (perpendicular to the house) be \( x \) each.
So, the total fencing used is \( x + x + y = 2x +... | [
{
"from": "human",
"value": "Richard has 360 feet of fencing to build a rectangular backyard. The fencing will enclose three sides of the backyard, while the fourth side will be the border of Richard's house. What is the largest possible area of this backyard?"
},
{
"from": "gpt",
"value": "<thi... |
Find the number of distinct terms in the expansion of \((x_1 + x_2 + \ldots + x_n)^m\), where \(m\) is a positive integer greater than 1 and \(n = 2m\). | nvidia_math | 95 | The question is a combinatorial problem that involves finding distinct terms in the expansion of a polynomial, which is mathematically rich and requires analytical reasoning similar to the positive examples. It asks for an analytical result and involves mathematical notation and understanding of combinations, making it... | 7 | <think>
Alright, I have this problem where I need to find the number of distinct terms in the expansion of \((x_1 + x_2 + \ldots + x_n)^m\), with \(m > 1\) and \(n = 2m\). Hmm, okay. Let me try to break this down step by step.
First, I remember that when you expand a multinomial like \((x_1 + x_2 + \ldots + x_n)^m\), ... | [
{
"from": "human",
"value": "Find the number of distinct terms in the expansion of \\((x_1 + x_2 + \\ldots + x_n)^m\\), where \\(m\\) is a positive integer greater than 1 and \\(n = 2m\\)."
},
{
"from": "gpt",
"value": "<think>\nAlright, I have this problem where I need to find the number of dis... |
What is the maximum value of \( x + y \) for the objective function subject to the constraints \( x + 2y \leq 70 \), \( 2x + y \leq 95 \), and \( x, y \geq 0 \)? Provide the point at which this maximum occurs. | automath | 95 | The question is similar to the positive examples because it involves finding the maximum value of an objective function under given constraints, which is a common type of optimization problem in mathematics. Unlike the negative examples, which either involve direct calculations, coding problems, or basic fact-checking,... | 1 | <think>
First, I recognize that I need to maximize the expression \( x + y \) under the given constraints \( x + 2y \leq 70 \), \( 2x + y \leq 95 \), and \( x, y \geq 0 \).
To solve this linear programming problem, I'll follow these steps:
1. **Identify the Constraints**:
- \( x + 2y \leq 70 \)
- \( 2x + y \leq... | [
{
"from": "human",
"value": "What is the maximum value of \\( x + y \\) for the objective function subject to the constraints \\( x + 2y \\leq 70 \\), \\( 2x + y \\leq 95 \\), and \\( x, y \\geq 0 \\)? Provide the point at which this maximum occurs."
},
{
"from": "gpt",
"value": "<think>\nFirst,... |
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 5,$ and $\|\mathbf{a} \times \mathbf{b}\| = 8.$ Find $|\mathbf{a} \cdot \mathbf{b}|.$ | tiger_math | 98 | The question about vectors is very similar to the positive examples as it presents a mathematical problem involving vectors, norms, and vector operations (cross product and dot product), reminiscent of example 4 focusing on mathematical expectations. The language is formal and requires analytical thought similar to the... | 17 | <think>
Okay, so I have this problem where there are two vectors, a and b. The magnitudes are given: the norm of vector a is 2, and the norm of vector b is 5. Also, the magnitude of their cross product is 8. I need to find the absolute value of their dot product. Hmm, okay.
First, I remember that for two vectors, the ... | [
{
"from": "human",
"value": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be vectors such that $\\|\\mathbf{a}\\| = 2,$ $\\|\\mathbf{b}\\| = 5,$ and $\\|\\mathbf{a} \\times \\mathbf{b}\\| = 8.$ Find $|\\mathbf{a} \\cdot \\mathbf{b}|.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem ... |
Consider an infinite ladder network of capacitors where each unit consists of a $1\mu F$ capacitor in parallel with a series combination of a $2\mu F$ capacitor and the equivalent capacitance of the next unit. If we denote the equivalent capacitance of the network as \( C_{eq} \), find the value of \( C_{eq} \) in micr... | automath | 95 | The question about the infinite ladder network of capacitors involves advanced concepts in circuit theory, similar to the positive examples which require mathematical reasoning and problem-solving skills in areas such as physics, probability, and computer science. It asks for a calculation related to equivalent capacit... | 3 | <think>
Alright, so I've got this problem about an infinite ladder network of capacitors. It sounds a bit intimidating at first, but let me try to break it down step by step.
First, the problem says each unit consists of a 1 μF capacitor in parallel with a series combination of a 2 μF capacitor and the equivalent capa... | [
{
"from": "human",
"value": "Consider an infinite ladder network of capacitors where each unit consists of a $1\\mu F$ capacitor in parallel with a series combination of a $2\\mu F$ capacitor and the equivalent capacitance of the next unit. If we denote the equivalent capacitance of the network as \\( C_{eq... |
Consider the series defined by
\[
\sum_{n=1}^{\infty} \frac{1}{3n + 2\cos n}.
\]
Does this series converge or diverge? Please apply an appropriate convergence test and justify your reasoning. | automath | 95 | The question involves analyzing the convergence of an infinite series, which requires understanding of mathematical concepts similar to those in the positive examples. The question calls for application of a convergence test and justification, aligning closely with the analytical and problem-solving nature of positive ... | 10 | <think>
Okay, so I have to figure out whether the series ∑_{n=1}^∞ 1/(3n + 2cos n) converges or diverges. Hmm, let me start by recalling some convergence tests I know. There's the comparison test, the limit comparison test, the ratio test, the root test, the integral test, and so on.
Looking at the general term of th... | [
{
"from": "human",
"value": "Consider the series defined by \n\n\\[\n\\sum_{n=1}^{\\infty} \\frac{1}{3n + 2\\cos n}.\n\\]\n\nDoes this series converge or diverge? Please apply an appropriate convergence test and justify your reasoning."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have to f... |
For each positive integer $n$, the sum of the first $n$ terms of a sequence is $2n^2$. What is the 2010th term of the sequence? | nvidia_math | 100 | The question involves a mathematical sequence, specifically asking for the nth term based on the given sum formula, which clearly aligns with the pattern of the positive examples provided. It requires logical reasoning and mathematical calculation, similar to the positive questions about points of intersection, expecta... | 19 | <think>
Okay, so I have this problem here that says: For each positive integer \( n \), the sum of the first \( n \) terms of a sequence is \( 2n^2 \). I need to find the 2010th term of the sequence. Hmm, let's try to figure this out.
First off, I remember that the sum of the first \( n \) terms of a sequence is rela... | [
{
"from": "human",
"value": "For each positive integer $n$, the sum of the first $n$ terms of a sequence is $2n^2$. What is the 2010th term of the sequence?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here that says: For each positive integer \\( n \\), the sum of the fi... |
What are all the complex numbers \( z \) such that \( z^4 = 1 \), and can you express these solutions in both rectangular form \( a + bi \) and exponential form \( re^{i\theta} \)? | automath | 95 | The given question asks for the solutions to the equation z^4 = 1 and specifically requests the solutions in both rectangular and exponential form. This question is closely related to the properties of complex numbers, which is a mathematical concept that aligns well with the positive examples that also involve advance... | 16 | <think>
Okay, so I have this problem where I need to find all the complex numbers \( z \) such that \( z^4 = 1 \). Hmm, I remember that complex numbers can be roots of equations just like real numbers, but they can also have multiple roots. I think in this case, since it's a fourth-degree equation, there should be four... | [
{
"from": "human",
"value": "What are all the complex numbers \\( z \\) such that \\( z^4 = 1 \\), and can you express these solutions in both rectangular form \\( a + bi \\) and exponential form \\( re^{i\\theta} \\)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem where I ... |
Coral reads 30 pages of a book on night 1, and 2 less than twice that on night 2. Night 3 Coral reads 3 more pages than the sum of the first two nights. How many total pages did Coral read in the 3 nights? | nvidia_math | 95 | The question about Coral reading pages requires solving a problem with a systematic approach to find a total based on the provided conditions. This is similar to the positive examples, where a specific mathematical or logical problem is posed that must be solved. Unlike the negative examples, which are either unrelated... | 55 | <think>
First, I note that Coral reads 30 pages on night 1.
On night 2, she reads 2 less than twice the number of pages she read on night 1. So, twice 30 is 60, and subtracting 2 gives her 58 pages on night 2.
By adding the pages from the first two nights, Coral has read a total of 88 pages. On night 3, she reads 3 m... | [
{
"from": "human",
"value": "Coral reads 30 pages of a book on night 1, and 2 less than twice that on night 2. Night 3 Coral reads 3 more pages than the sum of the first two nights. How many total pages did Coral read in the 3 nights?"
},
{
"from": "gpt",
"value": "<think>\nFirst, I note that ... |
Suppose that $\{a_n\}$ is an arithmetic sequence with $$
a_2+a_3+ \cdots +a_{101}=150 \quad \text{and} \quad
a_{102}+a_{103}+ \cdots + a_{201}=300.
$$What is the value of $a_3 - a_2$? Express your answer as a common fraction. | tiger_math | 95 | The question is about an arithmetic sequence and involves a mathematical problem that requires reasoning about summation and properties of sequences, similar to the computations and logic found in the positive examples. The structure of the question is mathematical and seeks a specific value related to an arithmetic pr... | 18 | <think>
Okay, so I have this problem about an arithmetic sequence. Let me see if I can figure it out step by step.
First, what do I know about arithmetic sequences? Well, an arithmetic sequence is a sequence where each term is the previous term plus a constant difference. So, in general, the nth term can be written as... | [
{
"from": "human",
"value": "Suppose that $\\{a_n\\}$ is an arithmetic sequence with $$\na_2+a_3+ \\cdots +a_{101}=150 \\quad \\text{and} \\quad\na_{102}+a_{103}+ \\cdots + a_{201}=300.\n$$What is the value of $a_3 - a_2$? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": ... |
Given the equations \( y = 3x \) and \( 2y^2 - xy = 15 \), find all points of intersection between the line and the hyperbola, and specifically calculate the coordinates of these intersection points. | automath | 95 | The question provided is similar to the positive examples in that it involves mathematics and seeks to solve a problem by finding points of intersection between a linear equation and a hyperbola. Like the positive examples, it poses a specific mathematical challenge that requires reasoning and solution finding. However... | 37 | <think>
Okay, so I need to find the points where the line \( y = 3x \) intersects with the hyperbola given by \( 2y^2 - xy = 15 \). Hmm, let's think about this. Since one equation is already solved for y in terms of x, maybe I can substitute that into the second equation. That should give me an equation with only x ter... | [
{
"from": "human",
"value": "Given the equations \\( y = 3x \\) and \\( 2y^2 - xy = 15 \\), find all points of intersection between the line and the hyperbola, and specifically calculate the coordinates of these intersection points."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find... |
What is the result of applying the operation \( (X \cdot R1(\theta))^2 \) in terms of a matrix, and what global phase does it represent when expressed in terms of \(\theta\)? | automath | 95 | The question involves applying a mathematical operation related to quantum mechanics, which fits the style and complexity of the positive questions that often require a deep understanding of specific mathematical constructs or principles. Unlike the negative examples, which tend to focus on simpler problems or direct c... | 14 | <think>
Okay, so I have this quantum operation given by \( (X \cdot R1(\theta))^2 \), and I need to figure out two things: first, what the result is as a matrix, and second, what global phase it represents in terms of \( \theta \). Hmm, quantum operations can sometimes be a bit tricky, especially when they involve rota... | [
{
"from": "human",
"value": "What is the result of applying the operation \\( (X \\cdot R1(\\theta))^2 \\) in terms of a matrix, and what global phase does it represent when expressed in terms of \\(\\theta\\)?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this quantum operation given ... |
\begin{align*}
2a + 3b + 5c + 7d &= 34 \\
3(d+c) &= b \\
3b + c &= a \\
c &= d + 1 \\
\end{align*} Given the above system of equations, find $a \cdot b \cdot c \cdot d$. | tiger_math | 95 | The provided question involves a system of equations that must be solved for variables a, b, c, and d, which aligns closely with the mathematical reasoning and problem-solving nature of the positive examples. However, it is slightly less complex than some of the positive examples, which explore deeper mathematical conc... | 40 | <think>
Okay, so I have this system of equations to solve. Let me write them down again so I can see everything clearly:
1. \(2a + 3b + 5c + 7d = 34\)
2. \(3(d + c) = b\)
3. \(3b + c = a\)
4. \(c = d + 1\)
Hmm, four equations and four variables: a, b, c, d. Should be solvable by substitution or elimination. Let me se... | [
{
"from": "human",
"value": "\\begin{align*}\n2a + 3b + 5c + 7d &= 34 \\\\\n3(d+c) &= b \\\\\n3b + c &= a \\\\\nc &= d + 1 \\\\\n\\end{align*} Given the above system of equations, find $a \\cdot b \\cdot c \\cdot d$."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this system of equation... |
Calculate the root mean square of the function \( f(x) = x \) over the interval \( [1, 2] \) and determine if \( x_{rms} = \sqrt{\frac{1}{2-1} \int_{1}^{2} x^2 dx} \) yields the same result as \( x_{rms} = \sqrt{\frac{1}{2} (1^2 + 2^2)} \). What is the correct value of \( x_{rms} \) in each case, and do they match? | automath | 100 | The question provided involves a mathematical calculation of the root mean square (RMS) of a function over a specified interval. It requires the computation of an integral and the evaluation of an expression, similar to the types of mathematical problem-solving and conceptual understanding found in the positive example... | 41 | <think>
Okay, so I need to calculate the root mean square (RMS) of the function \( f(x) = x \) over the interval [1, 2]. Then, I have to see if two different methods give the same result. The first method is using the formula \( x_{rms} = \sqrt{\frac{1}{2-1} \int_{1}^{2} x^2 dx} \), and the second is using \( x_{rms} =... | [
{
"from": "human",
"value": "Calculate the root mean square of the function \\( f(x) = x \\) over the interval \\( [1, 2] \\) and determine if \\( x_{rms} = \\sqrt{\\frac{1}{2-1} \\int_{1}^{2} x^2 dx} \\) yields the same result as \\( x_{rms} = \\sqrt{\\frac{1}{2} (1^2 + 2^2)} \\). What is the correct value... |
If the least common multiple of $A$ and $B$ is $720$, and the ratio of $A$ to $B$ is $5:8$, then what is their greatest common divisor? | tiger_math | 95 | The provided question involves determining the greatest common divisor (GCD) of two numbers based on their least common multiple (LCM) and their ratio, which fits well within the mathematical problem-solving context of the positive examples. It requires applying knowledge of number theory, similar to how the first exam... | 44 | <think>
Okay, so I have this problem here: the least common multiple (LCM) of two numbers A and B is 720, and the ratio of A to B is 5:8. I need to find their greatest common divisor (GCD). Hmm, let me think about how to approach this.
First, I remember that for any two numbers, there's a relationship between their LC... | [
{
"from": "human",
"value": "If the least common multiple of $A$ and $B$ is $720$, and the ratio of $A$ to $B$ is $5:8$, then what is their greatest common divisor?"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this problem here: the least common multiple (LCM) of two numbers A and B i... |
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