instruction_seed stringlengths 18 2k | source stringclasses 1
value | r1_distill_70b_response sequencelengths 3 9 | __original_row_idx int64 1 6k | _majority_responses sequencelengths 1 1 | verified_r1_distill_70b_response stringlengths 365 62.8k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|
The areas of two similar regular hexagons are in the ratio $49:81$. What is the ratio of their perimeters? Express your answer in the form $a:b$. | tiger_math | [
"<think>\nOkay, so I'm trying to solve this problem where the areas of two similar regular hexagons are in the ratio 49:81, and I need to find the ratio of their perimeters. Hmm, I remember that when dealing with similar figures, especially regular polygons, the ratios of their corresponding linear measures (like s... | 8 | [
"<think>\nAlright, so I've got this problem here about two similar regular hexagons. The areas are in the ratio 49:81, and I need to find the ratio of their perimeters, expressing it as a:b. Hmm, okay, so similar figures, hexagons in this case. Hmm.\n\nFirst off, I remember that when two figures are similar, their ... | <think>
Alright, so I've got this problem here about two similar regular hexagons. The areas are in the ratio 49:81, and I need to find the ratio of their perimeters, expressing it as a:b. Hmm, okay, so similar figures, hexagons in this case. Hmm.
First off, I remember that when two figures are similar, their correspo... | [
{
"from": "human",
"value": "The areas of two similar regular hexagons are in the ratio $49:81$. What is the ratio of their perimeters? Express your answer in the form $a:b$."
},
{
"from": "gpt",
"value": "<think>\nAlright, so I've got this problem here about two similar regular hexagons. The ar... |
Convert the point $(\rho,\theta,\phi) = \left( 2, \frac{2 \pi}{3}, \frac{\pi}{4} \right)$ in spherical coordinates to rectangular coordinates. | tiger_math | [
"<think>\nOkay, so I need to convert a point from spherical coordinates to rectangular coordinates. Hmm, I remember spherical coordinates have three components: rho (ρ), theta (θ), and phi (φ). Right, and rectangular coordinates are just the regular x, y, z that I'm used to. \n\nFirst, let me recall what each of th... | 1 | [
"<think>\nOkay, so I need to convert a point from spherical coordinates to rectangular coordinates. Hmm, I remember spherical coordinates have three components: rho (ρ), theta (θ), and phi (φ). Right, and rectangular coordinates are just the regular x, y, z that I'm used to. \n\nFirst, let me recall what each of th... | <think>
Okay, so I need to convert a point from spherical coordinates to rectangular coordinates. Hmm, I remember spherical coordinates have three components: rho (ρ), theta (θ), and phi (φ). Right, and rectangular coordinates are just the regular x, y, z that I'm used to.
First, let me recall what each of these comp... | [
{
"from": "human",
"value": "Convert the point $(\\rho,\\theta,\\phi) = \\left( 2, \\frac{2 \\pi}{3}, \\frac{\\pi}{4} \\right)$ in spherical coordinates to rectangular coordinates."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to convert a point from spherical coordinates to rectangula... |
Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | tiger_math | [
"<think>\nOkay, so I have this quadratic function y = q(x) = ax² + bx + c, and there's a graph showing part of it. The graph passes through (0, -3), which means when x=0, y is -3. That should help me find one of the coefficients.\n\nLooking at the Asymptote code, they defined the function as f1(x) = 4 - (x - 8)*(x ... | 12 | [
"<think>\nOkay, so I have this quadratic function y = q(x) = ax² + bx + c, and there's a graph showing part of it. The graph passes through (0, -3), which means when x=0, y is -3. That should help me find one of the coefficients.\n\nLooking at the Asymptote code, they defined the function as f1(x) = 4 - (x - 8)*(x ... | <think>
Okay, so I have this quadratic function y = q(x) = ax² + bx + c, and there's a graph showing part of it. The graph passes through (0, -3), which means when x=0, y is -3. That should help me find one of the coefficients.
Looking at the Asymptote code, they defined the function as f1(x) = 4 - (x - 8)*(x - 7)/8. ... | [
{
"from": "human",
"value": "Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:\n\n[asy]\nimport graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;\n\npen cqcqcq=rgb(0.75,0.75,0.75);\n/*g... |
If $y = \displaystyle\frac{1}{3x+1}$, what is the value of $x$ when $y = 2$? | tiger_math | [
"<think>\nTo solve for \\( x \\) when \\( y = 2 \\) in the equation \\( y = \\frac{1}{3x + 1} \\), I start by substituting 2 for \\( y \\). This gives me \\( 2 = \\frac{1}{3x + 1} \\).\n\nNext, I multiply both sides of the equation by \\( 3x + 1 \\) to eliminate the denominator, resulting in \\( 2(3x + 1) = 1 \\).\... | 25 | [
"<think>\nTo solve for \\( x \\) when \\( y = 2 \\), I start with the equation \\( 2 = \\frac{1}{3x + 1} \\).\n\nNext, I isolate \\( 3x + 1 \\) by taking the reciprocal of both sides, which gives \\( 3x + 1 = \\frac{1}{2} \\).\n\nThen, I subtract 1 from both sides to obtain \\( 3x = \\frac{1}{2} - 1 \\), which simp... | <think>
To solve for \( x \) when \( y = 2 \), I start with the equation \( 2 = \frac{1}{3x + 1} \).
Next, I isolate \( 3x + 1 \) by taking the reciprocal of both sides, which gives \( 3x + 1 = \frac{1}{2} \).
Then, I subtract 1 from both sides to obtain \( 3x = \frac{1}{2} - 1 \), which simplifies to \( 3x = -\frac{... | [
{
"from": "human",
"value": "If $y = \\displaystyle\\frac{1}{3x+1}$, what is the value of $x$ when $y = 2$?"
},
{
"from": "gpt",
"value": "<think>\nTo solve for \\( x \\) when \\( y = 2 \\), I start with the equation \\( 2 = \\frac{1}{3x + 1} \\).\n\nNext, I isolate \\( 3x + 1 \\) by taking the ... |
How many factors of 8000 are perfect cubes? | tiger_math | [
"<think>\nOkay, so I need to figure out how many factors of 8000 are perfect cubes. Hmm, I remember that to find factors with certain properties, prime factorization is usually a good starting point. Let me try to recall how that works.\n\nFirst, I think I should factorize 8000 into its prime components. I know tha... | 19 | [
"<think>\nAlright, so I need to figure out how many factors of 8000 are perfect cubes. Hmm, okay, let me start by understanding the problem. \n\nFirst, what exactly is a factor? A factor of a number is another number that divides into it without leaving a remainder. For example, factors of 6 are 1, 2, 3, and 6 beca... | <think>
Alright, so I need to figure out how many factors of 8000 are perfect cubes. Hmm, okay, let me start by understanding the problem.
First, what exactly is a factor? A factor of a number is another number that divides into it without leaving a remainder. For example, factors of 6 are 1, 2, 3, and 6 because each... | [
{
"from": "human",
"value": "How many factors of 8000 are perfect cubes?"
},
{
"from": "gpt",
"value": "<think>\nAlright, so I need to figure out how many factors of 8000 are perfect cubes. Hmm, okay, let me start by understanding the problem. \n\nFirst, what exactly is a factor? A factor of a n... |
For real numbers $x > 4,$ find the minimum value of
\[\frac{x + 10}{\sqrt{x - 4}}.\] | tiger_math | ["<think>\nOkay, so I've got this problem: For real numbers \\( x > 4 \\), find the minimum value of(...TRUNCATED) | 23 | ["<think>\nOkay, so I have this problem here: For real numbers x > 4, find the minimum value of (x +(...TRUNCATED) | "<think>\nOkay, so I have this problem here: For real numbers x > 4, find the minimum value of (x + (...TRUNCATED) | [{"from":"human","value":"For real numbers $x > 4,$ find the minimum value of\n\\[\\frac{x + 10}{\\s(...TRUNCATED) |
What number, when divided by 3, gives the same result as when 3 is subtracted from it? | tiger_math | ["<think>\nFirst, let \\( x \\) be the unknown number.\n\nWhen \\( x \\) is divided by 3, the result(...TRUNCATED) | 34 | ["<think>\nOkay, let me try to figure this problem out. So, the question is: What number, when divid(...TRUNCATED) | "<think>\nOkay, let me try to figure this problem out. So, the question is: What number, when divide(...TRUNCATED) | [{"from":"human","value":"What number, when divided by 3, gives the same result as when 3 is subtrac(...TRUNCATED) |
"Two sectors of a circle of radius $10$ overlap as shown, with $P$ and $R$ as the centers of the res(...TRUNCATED) | tiger_math | ["<think>\nOkay, so I have this problem with two overlapping sectors of a circle, both with radius 1(...TRUNCATED) | 4 | ["<think>\nOkay, so I have this problem with two overlapping sectors of a circle, both with radius 1(...TRUNCATED) | "<think>\nOkay, so I have this problem with two overlapping sectors of a circle, both with radius 10(...TRUNCATED) | [{"from":"human","value":"Two sectors of a circle of radius $10$ overlap as shown, with $P$ and $R$ (...TRUNCATED) |
Factor the quadratic $9y^2-30y+25$ into the form $(ay+b)^2$. | tiger_math | ["<think>\nOkay, so I need to factor the quadratic expression \\(9y^2 - 30y + 25\\) into the form \\(...TRUNCATED) | 53 | ["<think>\nFirst, I notice that the given quadratic is \\(9y^2 - 30y + 25\\), and it needs to be fac(...TRUNCATED) | "<think>\nFirst, I notice that the given quadratic is \\(9y^2 - 30y + 25\\), and it needs to be fact(...TRUNCATED) | [{"from":"human","value":"Factor the quadratic $9y^2-30y+25$ into the form $(ay+b)^2$."},{"from":"gp(...TRUNCATED) |
"Angles $P$ and $Q$ are supplementary. If the measure of angle $P$ is $5$ times angle $Q$ minus $10$(...TRUNCATED) | tiger_math | ["<think>\nOkay, so I have this problem where angles P and Q are supplementary. Hmm, supplementary a(...TRUNCATED) | 37 | ["<think>\nOkay, so I've got this problem here: Angles P and Q are supplementary. The measure of ang(...TRUNCATED) | "<think>\nOkay, so I've got this problem here: Angles P and Q are supplementary. The measure of angl(...TRUNCATED) | [{"from":"human","value":"Angles $P$ and $Q$ are supplementary. If the measure of angle $P$ is $5$ t(...TRUNCATED) |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 9