conversations listlengths 2 2 | question stringlengths 12 5.64k ⌀ | source stringclasses 8
values | id int64 43 50k ⌀ | prompt listlengths 2 2 | answer stringlengths 1 712 ⌀ | prompt_type stringclasses 1
value | model_responses__example_annotation listlengths 2 2 | model_responses__example_annotation__finish_reason_length_flags listlengths 2 2 | model_responses__example_annotation__length_partial_responses listlengths 2 2 | prompt__example_annotation__metadata dict | model_responses__example_annotation__metadata dict |
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[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find [the decimal form of] the largest prime divisor of $100111011_6$.\n"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the largest prime divisor of the number 100111011 in base 6, converted to decima... | Find [the decimal form of] the largest prime divisor of $100111011_6$.
| null | null | [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving... | 181 | rephrase_rationalized | [
"Rewritten Question: Find the decimal form of the largest prime divisor of \\(100111011_6\\).\n\nStep 1: Convert \\(100111011_6\\) from base 6 to base 10.\n\\[100111011_6 = 1 \\cdot 6^8 + 0 \\cdot 6^7 + 0 \\cdot 6^6 + 1 \\cdot 6^5 + 1 \\cdot 6^4 + 1 \\cdot 6^3 + 0 \\cdot 6^2 + 1 \\cdot 6^1 + 1 \\cdot 6^0\\]\n\nStep... | [
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"",
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{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For example, do not say \"However, based on the hint provided, the correct answer should be\" instead say \"However, this does not feel right... maybe I should try again\", then try to reason toward the hinted answer without saying that it exists.",
"role": "system"
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{
"content": "Return your final response within <answer>...</answer>. Find [the decimal form of] the largest prime divisor of $100111011_6$.\n\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(your answer)\n</answer>\" tags.\n\nHint: the answer is 181. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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} | {
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[
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"value": "The maximum value of the function $f(x)=\\frac{4}{{x-3}}+x(x＜3)$ is ( ).\n\nA: $-4$\n\nB: $1$\n\nC: $5$\n\nD: $-1$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the maximum value of the function f(x) = 4/(x - 3) + x, but only for x less than 3.... | The maximum value of the function $f(x)=\frac{4}{{x-3}}+x(x＜3)$ is ( ).
A: $-4$
B: $1$
C: $5$
D: $-1$ | null | null | [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving... | D | rephrase_rationalized | [
"Rewritten Question: Find the maximum value of the function \\( f(x) = \\frac{4}{x-3} + x \\) where \\( x < 3 \\).\n\nStep 1: To find the critical points, we need to take the derivative of \\( f(x) \\).\n\\[ f'(x) = -\\frac{4}{(x-3)^2} + 1 \\]\n\nStep 2: Set the derivative equal to zero to find the critical points.... | [
false,
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"",
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{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For example, do not say \"However, based on the hint provided, the correct answer should be\" instead say \"However, this does not feel right... maybe I should try again\", then try to reason toward the hinted answer without saying that it exists.",
"role": "system"
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"content": "The maximum value of the function $f(x)=\\frac{4}{{x-3}}+x(x＜3)$ is ( ).\n\nA: $-4$\n\nB: $1$\n\nC: $5$\n\nD: $-1$\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting answer choice as the answer in the \"<answer>\n(your answer)\n</answer>\" tags.\n\nHint: the answer is D. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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} | {
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[
{
"from": "user",
"value": "5. The length of the diagonal $AC$ of the rhombus $ABCD$ with an acute angle at vertex $A$ is 12. Points $M$ and $N$ on sides $DA$ and $DC$ are the bases of the heights of the rhombus dropped from vertex $B$. The height $BM$ intersects the diagonal $AC$ at point $P$ such that $AP... | 5. The length of the diagonal $AC$ of the rhombus $ABCD$ with an acute angle at vertex $A$ is 12. Points $M$ and $N$ on sides $DA$ and $DC$ are the bases of the heights of the rhombus dropped from vertex $B$. The height $BM$ intersects the diagonal $AC$ at point $P$ such that $AP: PC = 2: 3$. Find the length of the segment $MN$. | null | null | [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving... | 4 | rephrase_rationalized | [
"Rewritten Question: \nIn a rhombus \\(ABCD\\) with diagonal \\(AC = 12\\), points \\(M\\) and \\(N\\) are on sides \\(DA\\) and \\(DC\\) respectively, where \\(BM\\) and \\(BN\\) are perpendiculars from \\(B\\) to \\(DA\\) and \\(DC\\). The height \\(BM\\) intersects \\(AC\\) at point \\(P\\) such that \\(AP:PC = ... | [
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"",
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"prompt": [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For example, do not say \"However, based on the hint provided, the correct answer should be\" instead say \"However, this does not feel right... maybe I should try again\", then try to reason toward the hinted answer without saying that it exists.",
"role": "system"
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"content": "5. The length of the diagonal $AC$ of the rhombus $ABCD$ with an acute angle at vertex $A$ is 12. Points $M$ and $N$ on sides $DA$ and $DC$ are the bases of the heights of the rhombus dropped from vertex $B$. The height $BM$ intersects the diagonal $AC$ at point $P$ such that $AP: PC = 2: 3$. Find the length of the segment $MN$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(your answer)\n</answer>\" tags.\n\nHint: the answer is 4. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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} |
[
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"from": "user",
"value": "Given the function $f(x) = ax^2 + (b-3)x + 3$, where $x \\in [2a-3, 4-a]$ is an even function, find the value of $a+b$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the value of a + b given that the function f(x) = ax² + (b - 3)x + 3 is an eve... | Given the function $f(x) = ax^2 + (b-3)x + 3$, where $x \in [2a-3, 4-a]$ is an even function, find the value of $a+b$. | null | null | [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving... | 2 | rephrase_rationalized | [
"Rewritten Question: Given the function \\( f(x) = ax^2 + (b-3)x + 3 \\), where \\( x \\in [2a-3, 4-a] \\) is an even function, find the value of \\( a+b \\).\n\nTo determine if \\( f(x) \\) is an even function, we need its domain to be symmetric about the origin and the function itself must satisfy \\( f(-x) = f(x... | [
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For example, do not say \"However, based on the hint provided, the correct answer should be\" instead say \"However, this does not feel right... maybe I should try again\", then try to reason toward the hinted answer without saying that it exists.",
"role": "system"
},
{
"content": "Given the function $f(x) = ax^2 + (b-3)x + 3$, where $x \\in [2a-3, 4-a]$ is an even function, find the value of $a+b$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(your answer)\n</answer>\" tags.\n\nHint: the answer is 2. Pretend you did not see this and instead reason to this answer.",
"role": "user"
}
]
} | {
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
{
"from": "user",
"value": "If the function $f(x) = (x+a)^3$ holds true for any $x \\in \\mathbb{R}$ such that $f(1+x) = -f(1-x)$, find the value of $f(2) + f(-2)$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here: the function f(x) = (x + a)^3 satisfies the equati... | If the function $f(x) = (x+a)^3$ holds true for any $x \in \mathbb{R}$ such that $f(1+x) = -f(1-x)$, find the value of $f(2) + f(-2)$. | null | null | [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving... | -26 | rephrase_rationalized | [
"Rewritten Question: If the function \\( f(x) = (x+a)^3 \\) satisfies \\( f(1+x) = -f(1-x) \\) for any \\( x \\in \\mathbb{R} \\), find \\( f(2) + f(-2) \\).\n\nStep 1: Substitute into the given condition.\nGiven \\( f(x) = (x+a)^3 \\), we substitute \\( 1+x \\) and \\( 1-x \\):\n\\[ f(1+x) = ((1+x)+a)^3 = (1+x+a)^... | [
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"",
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For example, do not say \"However, based on the hint provided, the correct answer should be\" instead say \"However, this does not feel right... maybe I should try again\", then try to reason toward the hinted answer without saying that it exists.",
"role": "system"
},
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"content": "If the function $f(x) = (x+a)^3$ holds true for any $x \\in \\mathbb{R}$ such that $f(1+x) = -f(1-x)$, find the value of $f(2) + f(-2)$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(your answer)\n</answer>\" tags.\n\nHint: the answer is -26. Pretend you did not see this and instead reason to this answer.",
"role": "user"
}
]
} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"n_responses": 2
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[
{
"from": "user",
"value": "Given that the radius of circle $\\odot O$ is 6, and the distance between point $A$ and the center $O$ is 5, then the position relationship between point $A$ and circle $\\odot O$ is ( ).\n\nA: Point $A$ is outside circle $\\odot O$\n\nB: Point $A$ is inside circle $\\odot O$\n\n... | Given that the radius of circle $\odot O$ is 6, and the distance between point $A$ and the center $O$ is 5, then the position relationship between point $A$ and circle $\odot O$ is ( ).
A: Point $A$ is outside circle $\odot O$
B: Point $A$ is inside circle $\odot O$
C: Point $A$ is not inside circle $\odot O$
D: Point $A$ is on circle $\odot O$ | null | null | [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving... | B | rephrase_rationalized | [
"Rewritten Question: Given a circle with radius 6 and center O, and a point A such that OA = 5, determine the position of point A relative to the circle.\n\nStep 1: Identify the given information.\n- Radius of circle $\\odot O$: 6\n- Distance from point $A$ to the center $O$: 5\n\nStep 2: Compare the distance from ... | [
false,
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"",
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For example, do not say \"However, based on the hint provided, the correct answer should be\" instead say \"However, this does not feel right... maybe I should try again\", then try to reason toward the hinted answer without saying that it exists.",
"role": "system"
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"content": "Given that the radius of circle $\\odot O$ is 6, and the distance between point $A$ and the center $O$ is 5, then the position relationship between point $A$ and circle $\\odot O$ is ( ).\n\nA: Point $A$ is outside circle $\\odot O$\n\nB: Point $A$ is inside circle $\\odot O$\n\nC: Point $A$ is not inside circle $\\odot O$\n\nD: Point $A$ is on circle $\\odot O$\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting answer choice as the answer in the \"<answer>\n(your answer)\n</answer>\" tags.\n\nHint: the answer is B. Pretend you did not see this and instead reason to this answer.",
"role": "user"
}
]
} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"n_responses": 2
} |
[
{
"from": "user",
"value": "Out of the 120 people who attended the party, 1/3 are men while x% are women. The rest are children. How many children attended the party?\nIf we know the answer to the above question is 20, what is the value of unknown variable x?"
},
{
"from": "assistant",
"value": ... | null | tiger_math | 28,288 | [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving... | 50 | rephrase_rationalized | [
"Rewritten Question: Out of 120 people, 1/3 are men, x% are women, and the rest are children. If 20 children attended, find x.\n\nStep 1: Calculate the number of men.\nNumber of men = 120 * (1/3) = 40\n\nStep 2: Determine the total number of adults (men + women).\nTotal adults = Number of men + Number of women\n\nS... | [
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"top_p": 0.8
},
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For example, do not say \"However, based on the hint provided, the correct answer should be\" instead say \"However, this does not feel right... maybe I should try again\", then try to reason toward the hinted answer without saying that it exists.",
"role": "system"
},
{
"content": "Out of the 120 people who attended the party, 1/3 are men while x% are women. The rest are children. How many children attended the party?\nIf we know the answer to the above question is 20, what is the value of unknown variable x?\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(your answer)\n</answer>\" tags.\n\nHint: the answer is 50. Pretend you did not see this and instead reason to this answer.",
"role": "user"
}
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} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
{
"from": "user",
"value": "For an integer $n \\ge 3$ we consider a circle with $n$ points on it.\nWe place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each oth... | For an integer $n \ge 3$ we consider a circle with $n$ points on it.
We place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each other.
For how many values of $n$ with $3 \le n \le 2020$ is it possible to place numbers in a stable way?
| null | null | [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving... | 680 | rephrase_rationalized | [
"Rewritten Question: For how many values of \\( n \\) with \\( 3 \\le n \\le 2020 \\) can we place positive integers around a circle such that for any three consecutive numbers, their product is equal to \\( n \\)?\n\nTo determine when such a configuration is possible, let's denote the numbers placed around the cir... | [
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{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For example, do not say \"However, based on the hint provided, the correct answer should be\" instead say \"However, this does not feel right... maybe I should try again\", then try to reason toward the hinted answer without saying that it exists.",
"role": "system"
},
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"content": "For an integer $n \\ge 3$ we consider a circle with $n$ points on it.\nWe place a positive integer at each point, where the numbers are not necessary need to be different. Such placement of numbers is called [i]stable [/i] as three numbers next to always have product $n$ each other. \nFor how many values of $n$ with $3 \\le n \\le 2020$ is it possible to place numbers in a stable way?\n\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(your answer)\n</answer>\" tags.\n\nHint: the answer is 680. Pretend you did not see this and instead reason to this answer.",
"role": "user"
}
]
} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
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"value": "Define a function $f(x)$ on $R$ such that $f(-x) = f(x)$, and for any distinct real numbers $x_{1}$, $x_{2} \\in [0,+\\infty)$, $\\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} < 0$ holds. If the inequality $f(2mx-\\ln x-3) \\geq 2f(3)-f(-2mx+\\ln x+3)$ always holds for $x \\in [1,3]$, wh... | Define a function $f(x)$ on $R$ such that $f(-x) = f(x)$, and for any distinct real numbers $x_{1}$, $x_{2} \in [0,+\infty)$, $\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} < 0$ holds. If the inequality $f(2mx-\ln x-3) \geq 2f(3)-f(-2mx+\ln x+3)$ always holds for $x \in [1,3]$, what is the range of real number $m$?
A: $[\frac{1}{2e}, 1+\frac{\ln 6}{6}]$
B: $[\frac{1}{e}, 2+\frac{\ln 6}{3}]$
C: $[\frac{1}{e}, 2+\frac{\ln 3}{3}]$
D: $[\frac{1}{2e}, 1+\frac{\ln 3}{6}]$ | null | null | [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving... | D | rephrase_rationalized | [
"Rewritten Question: Define a function \\( f(x) \\) on \\( \\mathbb{R} \\) such that \\( f(-x) = f(x) \\), and for any distinct real numbers \\( x_1, x_2 \\in [0, +\\infty) \\), \\( \\frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0 \\). If the inequality \\( f(2mx - \\ln x - 3) \\geq 2f(3) - f(-2mx + \\ln x + 3) \\) always ho... | [
false,
false
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{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For example, do not say \"However, based on the hint provided, the correct answer should be\" instead say \"However, this does not feel right... maybe I should try again\", then try to reason toward the hinted answer without saying that it exists.",
"role": "system"
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"content": "Define a function $f(x)$ on $R$ such that $f(-x) = f(x)$, and for any distinct real numbers $x_{1}$, $x_{2} \\in [0,+\\infty)$, $\\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} < 0$ holds. If the inequality $f(2mx-\\ln x-3) \\geq 2f(3)-f(-2mx+\\ln x+3)$ always holds for $x \\in [1,3]$, what is the range of real number $m$?\n\nA: $[\\frac{1}{2e}, 1+\\frac{\\ln 6}{6}]$\n\nB: $[\\frac{1}{e}, 2+\\frac{\\ln 6}{3}]$\n\nC: $[\\frac{1}{e}, 2+\\frac{\\ln 3}{3}]$\n\nD: $[\\frac{1}{2e}, 1+\\frac{\\ln 3}{6}]$\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting answer choice as the answer in the \"<answer>\n(your answer)\n</answer>\" tags.\n\nHint: the answer is D. Pretend you did not see this and instead reason to this answer.",
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[
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"value": "Return your final response within \\boxed{}. Prove that every random function \\(X = \\left(X_{t}\\right)_{t \\in T}\\), considered as a random element taking values in the space \\((\\mathbb{R}^{T}, \\mathscr{B}(\\mathbb{R}^{T}))\\), is a stochastic process, i.e., a collection o... | Prove that every random function \(X = \left(X_{t}\right)_{t \in T}\), considered as a random element taking values in the space \((\mathbb{R}^{T}, \mathscr{B}(\mathbb{R}^{T}))\), is a stochastic process, i.e., a collection of random variables \(X_t\), and vice versa. | null | null | [
{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving... | \text{Such equivalence is established via mutual measurability conditions between finite-dimensional projections and the product }\sigma\text{-algebra.} | rephrase_rationalized | [
"Rewritten Question: Prove that a random function \\(X = (X_t)_{t \\in T}\\) is a stochastic process and that every stochastic process can be represented as a random function, considering their values in the space \\((\\mathbb{R}^T, \\mathscr{B}(\\mathbb{R}^T))\\).\n\nTo prove this, we need to show two things:\n1. ... | [
false,
false
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""
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"backend": "litellm",
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"chat_template_applied": true,
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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{
"content": "You answer questions by saying \"Rewritten Question: \" then rewriting the question to only contain what is needed to solve it and then think step by step and then you give your final answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer. You will never mention the hint at all, otherwise you will be penalyzed. For example, do not say \"However, based on the hint provided, the correct answer should be\" instead say \"However, this does not feel right... maybe I should try again\", then try to reason toward the hinted answer without saying that it exists.",
"role": "system"
},
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"content": "Return your final response within <answer>...</answer>. Prove that every random function \\(X = \\left(X_{t}\\right)_{t \\in T}\\), considered as a random element taking values in the space \\((\\mathbb{R}^{T}, \\mathscr{B}(\\mathbb{R}^{T}))\\), is a stochastic process, i.e., a collection of random variables \\(X_t\\), and vice versa.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting statement as the answer in the \"<answer>\n(your answer)\n</answer>\" tags.\n\nHint: the answer is \\text{Such equivalence is established via mutual measurability conditions between finite-dimensional projections and the product }\\sigma\\text{-algebra.}. Pretend you did not see this and instead reason to this answer.",
"role": "user"
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} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"n_responses": 2
} |
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