chosen listlengths 2 2 | rejected listlengths 2 2 |
|---|---|
[
{
"content": "A right circular cone has a volume of $12\\pi$ cubic centimeters. The height of the cone is 4 cm. How many centimeters is the circumference of the base of the cone, in terms of $\\pi$?",
"role": "user"
},
{
"content": " To find the circumference of the base of the cone, I'll first use ... | [
{
"content": "A right circular cone has a volume of $12\\pi$ cubic centimeters. The height of the cone is 4 cm. How many centimeters is the circumference of the base of the cone, in terms of $\\pi$?",
"role": "user"
},
{
"content": " Hmm, okay, so I have this problem about a right circular cone. Let... |
[
{
"content": "Determine the smallest positive real number $r$ such that there exist differentiable functions $f\\colon \\mathbb{R} \\to \\mathbb{R}$ and $g\\colon \\mathbb{R} \\to \\mathbb{R}$ satisfying \\begin{enumerate} \\item[(a)] $f(0) > 0$, \\item[(b)] $g(0) = 0$, \\item[(c)] $|f'(x)| \\leq |g(x)|$ for al... | [
{
"content": "Determine the smallest positive real number $r$ such that there exist differentiable functions $f\\colon \\mathbb{R} \\to \\mathbb{R}$ and $g\\colon \\mathbb{R} \\to \\mathbb{R}$ satisfying \\begin{enumerate} \\item[(a)] $f(0) > 0$, \\item[(b)] $g(0) = 0$, \\item[(c)] $|f'(x)| \\leq |g(x)|$ for al... |
[
{
"content": "In the quadrilateral $ABCD$ , $AB = BC = CD$ and $\\angle BMC = 90^\\circ$ , where $M$ is the midpoint of $AD$ . Determine the acute angle between the lines $AC$ and $BD$ .",
"role": "user"
},
{
"content": " To solve this problem, we need to analyze the given quadrilateral \\... | [
{
"content": "In the quadrilateral $ABCD$ , $AB = BC = CD$ and $\\angle BMC = 90^\\circ$ , where $M$ is the midpoint of $AD$ . Determine the acute angle between the lines $AC$ and $BD$ .",
"role": "user"
},
{
"content": " \nAlright, so I have this geometry problem here. Let me try to under... |
[
{
"content": "Let $x_1=97$, and for $n>1$, let $x_n=\\frac{n}{x_{n-1}}$. Calculate the product $x_1x_2x_3x_4x_5x_6x_7x_8$.",
"role": "user"
},
{
"content": " To solve the problem, I'll start by understanding the recursive sequence given. \n\nWe have \\( x_1 = 97 \\) and for each \\( n > 1 \\), \\( x... | [
{
"content": "Let $x_1=97$, and for $n>1$, let $x_n=\\frac{n}{x_{n-1}}$. Calculate the product $x_1x_2x_3x_4x_5x_6x_7x_8$.",
"role": "user"
},
{
"content": " \nAlright, let me try to figure out this problem. So, we have a sequence where x₁ is 97, and for each n greater than 1, xₙ is equal to n divid... |
[
{
"content": "Using the digits $0$, $1$, $2$, $3$, $4$ to form a four-digit number without repeating any digit, determine the total number of four-digit numbers less than $2340$.",
"role": "user"
},
{
"content": " To determine the total number of four-digit numbers less than 2340 using the digits 0,... | [
{
"content": "Using the digits $0$, $1$, $2$, $3$, $4$ to form a four-digit number without repeating any digit, determine the total number of four-digit numbers less than $2340$.",
"role": "user"
},
{
"content": " To determine the total number of four-digit numbers less than 2340 using the digits 0,... |
[
{
"content": "Let $a_n=6^{n}+8^{n}$. Determine the remainder upon dividing $a_ {83}$ by $49$.\n",
"role": "user"
},
{
"content": " To solve the problem, I need to find the remainder when \\( a_{83} = 6^{83} + 8^{83} \\) is divided by 49. I'll start by examining each term separately modulo 49 and the... | [
{
"content": "Let $a_n=6^{n}+8^{n}$. Determine the remainder upon dividing $a_ {83}$ by $49$.\n",
"role": "user"
},
{
"content": " If you think the answer is 0, write \\boxed{0}.\n\nAlright, so I have this problem here: I need to find the remainder when \\( a_{83} = 6^{83} + 8^{83} \\) is divided by... |
[
{
"content": "If $2^{10} \\cdot 2^{15}$ is expressed as some integer to the fifth power, what is that integer?",
"role": "user"
},
{
"content": " To solve the problem \\(2^{10} \\cdot 2^{15}\\) and express it as some integer to the fifth power, I start by combining the exponents since the bases are ... | [
{
"content": "If $2^{10} \\cdot 2^{15}$ is expressed as some integer to the fifth power, what is that integer?",
"role": "user"
},
{
"content": " Let's see, I have to figure out what 2 to the power of 10 multiplied by 2 to the power of 15 is, and then express that result as some integer raised to th... |
[
{
"content": "Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total duration of his bus... | [
{
"content": "Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total duration of his bus... |
[
{
"content": "Simplify the expression \\[\\sqrt{37-20\\sqrt3}.\\]",
"role": "user"
},
{
"content": " To simplify the expression \\(\\sqrt{37 - 20\\sqrt{3}}\\), I start by assuming it can be written in the form \\(\\sqrt{a} - \\sqrt{b}\\), where \\(a\\) and \\(b\\) are positive integers.\n\nNext, I s... | [
{
"content": "Simplify the expression \\[\\sqrt{37-20\\sqrt3}.\\]",
"role": "user"
},
{
"content": "**\nAlright, so I've got this expression to simplify: \\(\\sqrt{37 - 20\\sqrt{3}}\\). Hmm, it looks a bit intimidating at first glance, but I remember from my math classes that expressions like this c... |
[{"content":"(12 points) Using the six digits 0, 1, 2, 3, 4, 5, complete the following three questio(...TRUNCATED) | [{"content":"(12 points) Using the six digits 0, 1, 2, 3, 4, 5, complete the following three questio(...TRUNCATED) |
End of preview. Expand in Data Studio
- This dataset is generated by deepseek-ai/DeepSeek-R1-Distill-Qwen-7B with problem set agentica-org/DeepScaleR-Preview-Dataset
- For each problem, the model generates 10 completions with temp=0.9 and max_gen_length=8000
- Amoung the correct completions, chose the shortest answer as the chosen; Among all completions, choose the longest one as the rejected.
- If none of the completions were correct, skip this problem. This means some of the very hard problems are not inluded in this dataset
- Downloads last month
- 12