conversations listlengths 2 2 | question stringlengths 12 5.4k ⌀ | source stringclasses 8 values | id int64 21 50k ⌀ | prompt listlengths 2 2 | answer stringlengths 1 560 ⌀ | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|
[
{
"from": "user",
"value": "Given the line $ax+y+1=0$ always passes through a fixed point, the coordinates of this fixed point are \\_\\_\\_\\_\\_\\_."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The problem says that the line ax + y + 1 = 0 always passes through a fixed point,... | Given the line $ax+y+1=0$ always passes through a fixed point, the coordinates of this fixed point are \_\_\_\_\_\_. | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given the line $ax+y+1=0$ alway... | (0, -1) | original | [
"To determine the fixed point that the line \\( ax + y + 1 = 0 \\) always passes through, we need to find a point \\((x_0, y_0)\\) such that substituting these coordinates into the equation results in a true statement regardless of the value of \\(a\\).\n\nLet's substitute \\((x_0, y_0)\\) into the equation:\n\\[ a... | [
false,
false
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"",
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{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given the line $ax+y+1=0$ always passes through a fixed point, the coordinates of this fixed point are \\_\\_\\_\\_\\_\\_.",
"role": "user"
}
]
} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
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"value": "Given $x\\_1=\\log \\_{\\frac {1}{3}}2$, $x\\_2=2^{-\\frac {1}{2}}$, and $(\\frac {1}{3})^{x3}=\\log \\_{3}x\\_{3}$, determine the correct relationship among $x\\_1$, $x\\_2$, and $x\\_3$.\n\nA: $x\\_1 < x\\_3 < x\\_2$\nB: $x\\_2 < x\\_1 < x\\_3$\nC: $x\\_1 < x\\_2 < x\\_3$\nD: $... | Given $x\_1=\log \_{\frac {1}{3}}2$, $x\_2=2^{-\frac {1}{2}}$, and $(\frac {1}{3})^{x3}=\log \_{3}x\_{3}$, determine the correct relationship among $x\_1$, $x\_2$, and $x\_3$.
A: $x\_1 < x\_3 < x\_2$
B: $x\_2 < x\_1 < x\_3$
C: $x\_1 < x\_2 < x\_3$
D: $x\_3 < x\_1 < x\_2$ | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given $x\\_1=\\log \\_{\\frac {... | C | original | [
"To determine the correct relationship among \\(x_1\\), \\(x_2\\), and \\(x_3\\), we need to evaluate each of these expressions individually.\n\nFirst, let's find \\(x_1\\):\n\\[ x_1 = \\log_{\\frac{1}{3}} 2 \\]\nUsing the change of base formula for logarithms:\n\\[ x_1 = \\frac{\\log 2}{\\log \\frac{1}{3}} = \\fra... | [
false,
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"",
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"prompt": [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given $x\\_1=\\log \\_{\\frac {1}{3}}2$, $x\\_2=2^{-\\frac {1}{2}}$, and $(\\frac {1}{3})^{x3}=\\log \\_{3}x\\_{3}$, determine the correct relationship among $x\\_1$, $x\\_2$, and $x\\_3$.\n\nA: $x\\_1 < x\\_3 < x\\_2$\nB: $x\\_2 < x\\_1 < x\\_3$\nC: $x\\_1 < x\\_2 < x\\_3$\nD: $x\\_3 < x\\_1 < x\\_2$",
"role": "user"
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[
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"from": "user",
"value": "1. Given sets\n$$\n\\begin{array}{l}\nS_{1}=\\left\\{(x, y) \\mid \\lg \\left(1+x^{2}+y^{2}\\right) \\leqslant 1+\\lg (x+y)\\right\\}, \\\\\nS_{2}=\\left\\{(x, y) \\mid \\lg \\left(2+x^{2}+y^{2}\\right) \\leqslant 2+\\lg (x+y)\\right\\} .\n\\end{array}\n$$\n\nThen the ratio of th... | 1. Given sets
$$
\begin{array}{l}
S_{1}=\left\{(x, y) \mid \lg \left(1+x^{2}+y^{2}\right) \leqslant 1+\lg (x+y)\right\}, \\
S_{2}=\left\{(x, y) \mid \lg \left(2+x^{2}+y^{2}\right) \leqslant 2+\lg (x+y)\right\} .
\end{array}
$$
Then the ratio of the area of $S_{2}$ to the area of $S_{1}$ is ( ).
(A) 99
(B) 100
(C) 101
(D) 102 | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "1. Given sets\n$$\n\\begin{arra... | D | original | [
"To determine the ratio of the areas of \\( S_2 \\) and \\( S_1 \\), we start by simplifying the given inequalities.\n\nFirst, consider the inequality for \\( S_1 \\):\n\\[\n\\lg \\left(1 + x^2 + y^2\\right) \\leq 1 + \\lg (x + y).\n\\]\nRewriting this using properties of logarithms, we get:\n\\[\n\\lg \\left(1 + x... | [
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false
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"",
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "1. Given sets\n$$\n\\begin{array}{l}\nS_{1}=\\left\\{(x, y) \\mid \\lg \\left(1+x^{2}+y^{2}\\right) \\leqslant 1+\\lg (x+y)\\right\\}, \\\\\nS_{2}=\\left\\{(x, y) \\mid \\lg \\left(2+x^{2}+y^{2}\\right) \\leqslant 2+\\lg (x+y)\\right\\} .\n\\end{array}\n$$\n\nThen the ratio of the area of $S_{2}$ to the area of $S_{1}$ is ( ).\n(A) 99\n(B) 100\n(C) 101\n(D) 102",
"role": "user"
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} | {
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
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"value": "Given that function $f(x)$ is an even function defined on $\\mathbb{R}$ and is monotonically increasing in the interval $[0,+\\infty)$. If the real number $a$ satisfies $f(\\log_{2}a)+f(\\log_{\\frac{1}{2}}a) < 2f(1)$, determine the range of $a$.\n\nA: $[1,2]$\nB: $(0, \\frac{1}{... | Given that function $f(x)$ is an even function defined on $\mathbb{R}$ and is monotonically increasing in the interval $[0,+\infty)$. If the real number $a$ satisfies $f(\log_{2}a)+f(\log_{\frac{1}{2}}a) < 2f(1)$, determine the range of $a$.
A: $[1,2]$
B: $(0, \frac{1}{2}]$
C: $(\frac{1}{2},2)$
D: $(0,2]$ | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given that function $f(x)$ is a... | C | original | [
"To solve the problem, we start by analyzing the given inequality \\( f(\\log_{2}a) + f(\\log_{\\frac{1}{2}}a) < 2f(1) \\).\n\nFirst, note that \\(\\log_{\\frac{1}{2}}a = -\\log_{2}a\\). This allows us to rewrite the inequality as:\n\\[ f(\\log_{2}a) + f(-\\log_{2}a) < 2f(1). \\]\n\nSince \\( f(x) \\) is an even fu... | [
false,
false
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"",
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given that function $f(x)$ is an even function defined on $\\mathbb{R}$ and is monotonically increasing in the interval $[0,+\\infty)$. If the real number $a$ satisfies $f(\\log_{2}a)+f(\\log_{\\frac{1}{2}}a) < 2f(1)$, determine the range of $a$.\n\nA: $[1,2]$\nB: $(0, \\frac{1}{2}]$\nC: $(\\frac{1}{2},2)$\nD: $(0,2]$",
"role": "user"
}
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} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
{
"from": "user",
"value": "The organizing committee of the sports meeting needs to select four volunteers from Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang to take on four different tasks: translation, tour guide, etiquette, and driver. If Xiao Zhang and Xiao Zhao can only take on the first two ... | The organizing committee of the sports meeting needs to select four volunteers from Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang to take on four different tasks: translation, tour guide, etiquette, and driver. If Xiao Zhang and Xiao Zhao can only take on the first two tasks, while the other three can take on any of the four tasks, then the total number of different dispatch plans is \_\_\_\_\_\_ (The result should be expressed in numbers). | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "The organizing committee of the... | 36 | original | [
"To determine the total number of different dispatch plans for the volunteers, we need to consider the constraints given: Xiao Zhang and Xiao Zhao can only take on the first two tasks (translation or tour guide), while Xiao Li, Xiao Luo, and Xiao Wang can take on any of the four tasks.\n\nLet's break down the probl... | [
false,
false
] | [
"",
""
] | {
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "The organizing committee of the sports meeting needs to select four volunteers from Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang to take on four different tasks: translation, tour guide, etiquette, and driver. If Xiao Zhang and Xiao Zhao can only take on the first two tasks, while the other three can take on any of the four tasks, then the total number of different dispatch plans is \\_\\_\\_\\_\\_\\_ (The result should be expressed in numbers).",
"role": "user"
}
]
} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"n_responses": 2
} |
[
{
"from": "user",
"value": "Given that point $Q(a-1,a+2)$ lies on the $x$-axis, what are the coordinates of point $Q$?\n\nA: $(-3,0)$\n\nB: $(3,0)$\n\nC: $(0,3)$\n\nD: $(0,-3)$"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The problem is about finding the coordinates of point Q,... | Given that point $Q(a-1,a+2)$ lies on the $x$-axis, what are the coordinates of point $Q$?
A: $(-3,0)$
B: $(3,0)$
C: $(0,3)$
D: $(0,-3)$ | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given that point $Q(a-1,a+2)$ l... | A | original | [
"To determine the coordinates of point \\( Q \\) given that it lies on the \\( x \\)-axis, we need to understand a key property of points on the \\( x \\)-axis: their \\( y \\)-coordinate is always 0.\n\nThe coordinates of point \\( Q \\) are given as \\( (a-1, a+2) \\). Since \\( Q \\) lies on the \\( x \\)-axis, ... | [
false,
false
] | [
"",
""
] | {
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given that point $Q(a-1,a+2)$ lies on the $x$-axis, what are the coordinates of point $Q$?\n\nA: $(-3,0)$\n\nB: $(3,0)$\n\nC: $(0,3)$\n\nD: $(0,-3)$",
"role": "user"
}
]
} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
{
"from": "user",
"value": "## Task B-2.3.\n\nCalculate:\n\n$$\n\\frac{\\left(1+\\frac{1}{3}\\right)\\left(1+\\frac{1}{5}\\right) \\cdot \\ldots \\cdot\\left(1+\\frac{1}{2 n+1}\\right) \\cdot \\ldots \\cdot\\left(1+\\frac{1}{4047}\\right)}{\\sqrt{\\left(1-\\frac{1}{3^{2}}\\right)\\left(1-\\frac{1}{5^{2}}\\r... | ## Task B-2.3.
Calculate:
$$
\frac{\left(1+\frac{1}{3}\right)\left(1+\frac{1}{5}\right) \cdot \ldots \cdot\left(1+\frac{1}{2 n+1}\right) \cdot \ldots \cdot\left(1+\frac{1}{4047}\right)}{\sqrt{\left(1-\frac{1}{3^{2}}\right)\left(1-\frac{1}{5^{2}}\right) \cdot \ldots \cdot\left(1-\frac{1}{(2 n+1)^{2}}\right) \cdot \ldots \cdot\left(1-\frac{1}{4047^{2}}\right)}}
$$ | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "## Task B-2.3.\n\nCalculate:\n\... | \sqrt{2024} | original | [
"To solve the given problem, we need to simplify both the numerator and the denominator of the expression separately before combining them.\n\nThe expression is:\n\\[\n\\frac{\\left(1+\\frac{1}{3}\\right)\\left(1+\\frac{1}{5}\\right) \\cdot \\ldots \\cdot\\left(1+\\frac{1}{2 n+1}\\right) \\cdot \\ldots \\cdot\\left... | [
false,
false
] | [
"",
""
] | {
"api_url": "http://10.32.36.231:9080/v1",
"backend": "litellm",
"chat_template_applied": true,
"generation_params": {
"chat_template_applied": true,
"max_tokens": 4096,
"n": 2,
"repetition_penalty": 1.1,
"temperature": 0.7,
"top_k": 20,
"top_p": 0.8
},
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "## Task B-2.3.\n\nCalculate:\n\n$$\n\\frac{\\left(1+\\frac{1}{3}\\right)\\left(1+\\frac{1}{5}\\right) \\cdot \\ldots \\cdot\\left(1+\\frac{1}{2 n+1}\\right) \\cdot \\ldots \\cdot\\left(1+\\frac{1}{4047}\\right)}{\\sqrt{\\left(1-\\frac{1}{3^{2}}\\right)\\left(1-\\frac{1}{5^{2}}\\right) \\cdot \\ldots \\cdot\\left(1-\\frac{1}{(2 n+1)^{2}}\\right) \\cdot \\ldots \\cdot\\left(1-\\frac{1}{4047^{2}}\\right)}}\n$$",
"role": "user"
}
]
} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"n_responses": 2
} |
[
{
"from": "user",
"value": "Solve the following system of equations:\n\n$$\n\\begin{gathered}\n\\frac{10}{2 x+3 y-29}+\\frac{9}{7 x-8 y+24}=8 \\\\\n\\frac{2 x+3 y-29}{2}=\\frac{7 x-8 y}{3}+8\n\\end{gathered}\n$$"
},
{
"from": "assistant",
"value": "<think>\nAlright, so I need to solve this syste... | Solve the following system of equations:
$$
\begin{gathered}
\frac{10}{2 x+3 y-29}+\frac{9}{7 x-8 y+24}=8 \\
\frac{2 x+3 y-29}{2}=\frac{7 x-8 y}{3}+8
\end{gathered}
$$ | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Solve the following system of e... | (5, 7) | original | [
"Let's denote \\(a = 2x + 3y - 29\\) and \\(b = 7x - 8y + 24\\). The given system can be rewritten as:\n\\[ \\frac{10}{a} + \\frac{9}{b} = 8 \\]\n\\[ a = \\frac{2(7x - 8y) + 16}{3} = \\frac{14x - 16y + 16}{3} \\]\n\\[ b = 2a - 52 \\]\n\nFirst, let's simplify the second equation:\n\\[ a = \\frac{14x - 16y + 16}{3} \... | [
false,
false
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"",
""
] | {
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"chat_template_applied": true,
"generation_params": {
"chat_template_applied": true,
"max_tokens": 4096,
"n": 2,
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"temperature": 0.7,
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"top_p": 0.8
},
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Solve the following system of equations:\n\n$$\n\\begin{gathered}\n\\frac{10}{2 x+3 y-29}+\\frac{9}{7 x-8 y+24}=8 \\\\\n\\frac{2 x+3 y-29}{2}=\\frac{7 x-8 y}{3}+8\n\\end{gathered}\n$$",
"role": "user"
}
]
} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"n_responses": 2
} |
[
{
"from": "user",
"value": "the length of each side of square a is increased by 100 percent to make square b . if the length of the side of square b is increased by 80 percent to make square c , by what percent is the area of square c greater than the sum of the areas of squares a and b ? Please respond by ... | null | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "the length of each side of squa... | 159.2\% | original | [
"To solve this problem, let's break it down into steps:\n\n1. Let the side length of square A be \\( s \\).\n2. The area of square A is \\( s^2 \\).\n\n3. When we increase the side length of square A by 100%, the new side length becomes \\( 2s \\) (since 100% more means doubling). This gives us square B.\n4. The ar... | [
false,
false
] | [
"",
""
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"backend": "litellm",
"chat_template_applied": true,
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"chat_template_applied": true,
"max_tokens": 4096,
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"top_k": 20,
"top_p": 0.8
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "the length of each side of square a is increased by 100 percent to make square b . if the length of the side of square b is increased by 80 percent to make square c , by what percent is the area of square c greater than the sum of the areas of squares a and b ? Please respond by writing a program in Python.",
"role": "user"
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} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"n_responses": 2
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[
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"value": "Given the function $f(2^x)$ has a domain of $[1, 2]$, find the domain of $f(\\log_2 x)$."
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"value": "<think>\nOkay, let's see. The problem says that the function f(2^x) has a domain of [1, 2], and we need to find the domain of f(log₂ x). Hmm, ... | Given the function $f(2^x)$ has a domain of $[1, 2]$, find the domain of $f(\log_2 x)$. | null | null | [
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"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
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"content": "Given the function $f(2^x)$ has... | [4, 16] | original | [
"To determine the domain of \\( f(\\log_2 x) \\), we first need to understand the relationship between the domains of \\( f(2^x) \\) and \\( f(\\log_2 x) \\).\n\nThe given information states that the domain of \\( f(2^x) \\) is \\([1, 2]\\). This means:\n\\[ 1 \\leq x \\leq 2 \\]\n\nFor this interval, the argument ... | [
false,
false
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"",
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"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
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"content": "Given the function $f(2^x)$ has a domain of $[1, 2]$, find the domain of $f(\\log_2 x)$.",
"role": "user"
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"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
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"value": "Given that the functions $f(x)$ and $g(x)$ defined on $\\mathbb{R}$ satisfy $\\frac{f(x)}{g(x)} = b^x$, and $f'(x)g(x) < f(x)g'(x)$, $\\frac{f(1)}{g(1)} + \\frac{f(-1)}{g(-1)} = \\frac{5}{2}$. If $\\{a_n\\}$ is a positive geometric sequence and $a_5a_7 + 2a_6a_8 + a_4a_{12} = \\f... | Given that the functions $f(x)$ and $g(x)$ defined on $\mathbb{R}$ satisfy $\frac{f(x)}{g(x)} = b^x$, and $f'(x)g(x) < f(x)g'(x)$, $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}$. If $\{a_n\}$ is a positive geometric sequence and $a_5a_7 + 2a_6a_8 + a_4a_{12} = \frac{f(4)}{g(4)}$, find the value of $a_6 + a_8$. | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given that the functions $f(x)$... | \dfrac{1}{4} | original | [
"We start by analyzing the given conditions. The first condition is:\n\n\\[\n\\frac{f(x)}{g(x)} = b^x.\n\\]\n\nThis implies:\n\n\\[\nf(x) = g(x) b^x.\n\\]\n\nNext, we differentiate both sides of \\( f(x) = g(x) b^x \\):\n\n\\[\nf'(x) = g'(x) b^x + g(x) b^x \\ln(b).\n\\]\n\nUsing the second condition \\( f'(x)g(x) <... | [
false,
false
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"",
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"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Given that the functions $f(x)$ and $g(x)$ defined on $\\mathbb{R}$ satisfy $\\frac{f(x)}{g(x)} = b^x$, and $f'(x)g(x) < f(x)g'(x)$, $\\frac{f(1)}{g(1)} + \\frac{f(-1)}{g(-1)} = \\frac{5}{2}$. If $\\{a_n\\}$ is a positive geometric sequence and $a_5a_7 + 2a_6a_8 + a_4a_{12} = \\frac{f(4)}{g(4)}$, find the value of $a_6 + a_8$.",
"role": "user"
}
]
} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
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[
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"from": "user",
"value": "Example 5 Let $a, b, c$ be the lengths of the sides of a right triangle, and $a \\leqslant b < c$. Find the maximum constant $k$, such that $a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b) \\geqslant k a b c$ holds for all right triangles, and determine when equality occurs."
},
{
"from... | Example 5 Let $a, b, c$ be the lengths of the sides of a right triangle, and $a \leqslant b < c$. Find the maximum constant $k$, such that $a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b) \geqslant k a b c$ holds for all right triangles, and determine when equality occurs. | null | null | [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Example 5 Let $a, b, c$ be the ... | 3\sqrt{2} + 2 | original | [
"To find the maximum constant \\( k \\) such that the inequality \\( a^2(b+c) + b^2(c+a) + c^2(a+b) \\geq kabc \\) holds for all right triangles with sides \\( a \\leq b < c \\), we start by noting that since it is a right triangle, we have \\( c = \\sqrt{a^2 + b^2} \\).\n\nFirst, let's substitute \\( c = \\sqrt{a^... | [
false,
false
] | [
"",
""
] | {
"api_url": "http://10.32.36.231:9080/v1",
"backend": "litellm",
"chat_template_applied": true,
"generation_params": {
"chat_template_applied": true,
"max_tokens": 4096,
"n": 2,
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},
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"prompt": [
{
"content": "You like to solve problems by thinking step by step. Write out your reasoning then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end without giving an answer.",
"role": "system"
},
{
"content": "Example 5 Let $a, b, c$ be the lengths of the sides of a right triangle, and $a \\leqslant b < c$. Find the maximum constant $k$, such that $a^{2}(b+c)+b^{2}(c+a)+c^{2}(a+b) \\geqslant k a b c$ holds for all right triangles, and determine when equality occurs.",
"role": "user"
}
]
} | {
"backend": "litellm",
"model_name": "hosted_vllm/Qwen/Qwen2.5-7B-Instruct",
"n_responses": 2
} |
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