conversations listlengths 2 2 | question stringlengths 13 5.4k ⌀ | source stringclasses 8
values | id int64 9 50k ⌀ | prompt listlengths 2 2 | answer stringlengths 1 587 ⌀ | 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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... | For a positive integer $n$ , let $\sigma (n)$ be the sum of the divisors of $n$ (for example $\sigma (10) = 1 + 2 + 5 + 10 = 18$ ). For how many $n \in \{1, 2,. .., 100\}$ , do we have $\sigma (n) < n+ \sqrt{n}$ ? | null | null | [
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"content": "You like to solve problems by understanding the problem, writing a plan, executing the plan, then giving an answer. Write a plan that when reasoned over would solve the question then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end witho... | 26 | plan_and_execute_rationalized | [
"First, I'll try to understand the problem better by writing out a plan and go really deep into detail about how I should solve this.\n1. **Define the function**: The function \\(\\sigma(n)\\) represents the sum of all divisors of \\(n\\).\n2. **Set up the inequality**: We need to find the values of \\(n\\) such th... | [
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"value": "A certain car can travel 32 kilometers on a liter of fuel. The fuel tank's contents decrease by 3.9 gallons as the car moves at a constant speed of approximately 52 miles per hour. How long did it take for the fuel tank's contents to decrease by 3.9 gallons? (1 gallon = 3.8 liter... | null | null | null | [
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"content": "You like to solve problems by understanding the problem, writing a plan, executing the plan, then giving an answer. Write a plan that when reasoned over would solve the question then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end witho... | 5.7 \text{ hours} | plan_and_execute_rationalized | [
"First, I'll try to understand the problem better by writing out a plan and go really deep into detail about how I should solve this.\n\n### Step-by-Step Plan\n\n1. **Convert the given units**:\n - Convert the fuel consumption from gallons to liters.\n - Convert the distance traveled from kilometers to miles.\n... | [
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"value": "5. Let point $D$ be on the side of $\\triangle A B C$, and construct the incircles of $\\triangle A B D$ and $\\triangle A C D$. Their common external tangent (other than $B C$) intersects $A D$ at point $K$. Prove that the length of segment $A K$ does not depend on the position ... | 5. Let point $D$ be on the side of $\triangle A B C$, and construct the incircles of $\triangle A B D$ and $\triangle A C D$. Their common external tangent (other than $B C$) intersects $A D$ at point $K$. Prove that the length of segment $A K$ does not depend on the position of point $D$. Question 5 Figure | null | null | [
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"content": "You like to solve problems by understanding the problem, writing a plan, executing the plan, then giving an answer. Write a plan that when reasoned over would solve the question then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end witho... | AK = \frac{AB + AC - BC}{2} | plan_and_execute_rationalized | [
"First, I'll try to understand the problem better by writing out a plan and go really deep into detail about how I should solve this.\n\n1. **Understanding the Geometry**: We have triangle \\(ABC\\) with point \\(D\\) on side \\(BC\\). The incircles of triangles \\(ABD\\) and \\(ACD\\) are constructed, and their co... | [
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"value": "Shift the graph of the function $f(x) = 2\\sin(\\omega x - \\frac{\\pi}{3})$, ($\\omega > 0$) to the left by $\\frac{\\pi}{3\\omega}$ units to obtain the graph of the function $y = g(x)$. If $y = g(x)$ is increasing on the interval $\\left[-\\frac{\\pi}{6}, \\frac{\\pi}{4}\\right... | Shift the graph of the function $f(x) = 2\sin(\omega x - \frac{\pi}{3})$, ($\omega > 0$) to the left by $\frac{\pi}{3\omega}$ units to obtain the graph of the function $y = g(x)$. If $y = g(x)$ is increasing on the interval $\left[-\frac{\pi}{6}, \frac{\pi}{4}\right]$, then the maximum value of $\omega$ is \_\_\_\_\_\_... | null | null | [
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"content": "You like to solve problems by understanding the problem, writing a plan, executing the plan, then giving an answer. Write a plan that when reasoned over would solve the question then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end witho... | 2 | plan_and_execute_rationalized | [
"First, I'll try to understand the problem better by writing out a plan and go really deep into detail about how I should solve this.\n\n1. **Understand the Transformation**: The original function is \\( f(x) = 2\\sin(\\omega x - \\frac{\\pi}{3}) \\). Shifting this function to the left by \\(\\frac{\\pi}{3\\omega}\... | [
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"value": "Let $A B C$ be a triangle and suppose $A B=3, B C=4, C A=5$. What is the distance from $B$ to line $A C$ ?"
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"value": "<think>\nOkay, so I need to find the distance from point B to the line AC in triangle ABC where the sides are AB=3, BC=4, an... | Let $A B C$ be a triangle and suppose $A B=3, B C=4, C A=5$. What is the distance from $B$ to line $A C$ ? | null | null | [
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"content": "You like to solve problems by understanding the problem, writing a plan, executing the plan, then giving an answer. Write a plan that when reasoned over would solve the question then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end witho... | \dfrac{12}{5} | plan_and_execute_rationalized | [
"First, I'll try to understand the problem better by writing out a plan and go really deep into detail about how I should solve this.\n\n### Understanding the Problem:\n- We are given a triangle \\(ABC\\) where \\(AB = 3\\), \\(BC = 4\\), and \\(CA = 5\\).\n- We need to find the perpendicular distance from point \\... | [
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"value": "Ten students are taking a group photo, arranged in two rows with 3 students in the front row and 7 in the back row. The photographer wants to select 2 students from the back row to move to the front row, while maintaining the relative order of the other students. The number of di... | Ten students are taking a group photo, arranged in two rows with 3 students in the front row and 7 in the back row. The photographer wants to select 2 students from the back row to move to the front row, while maintaining the relative order of the other students. The number of different ways to adjust the arrangement i... | null | null | [
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"content": "You like to solve problems by understanding the problem, writing a plan, executing the plan, then giving an answer. Write a plan that when reasoned over would solve the question then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end witho... | C | plan_and_execute_rationalized | [
"First, I'll try to understand the problem better by writing out a plan and go really deep into detail about how I should solve this.\n\n1. **Identify the Problem**: We need to choose 2 students from the back row of 7 and place them in the front row of 3, ensuring their relative order is maintained.\n \n2. **Choo... | [
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"value": "The solution of the equation $\\dfrac{1+2^{x}}{1+x^{-x}}= \\dfrac{1}{4}$ is $x=$ ."
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"value": "<think>\nAlright, let's try to solve this equation: (1 + 2^x)/(1 + x^(-x)) = 1/4. Hmm, okay. So, first, I need to find the value of... | The solution of the equation $\dfrac{1+2^{x}}{1+x^{-x}}= \dfrac{1}{4}$ is $x=$ . | null | null | [
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"content": "You like to solve problems by understanding the problem, writing a plan, executing the plan, then giving an answer. Write a plan that when reasoned over would solve the question then give your answer in \"<answer>\n(your answer)\n</answer>\". You always end with </answer>, you never ever end witho... | -2 | plan_and_execute_rationalized | [
"First, I'll try to understand the problem better by writing out a plan and go really deep into detail about how I should solve this.\n\n1. **Rewrite the Equation**: Start by rewriting the given equation:\n \\[\n \\frac{1 + 2^x}{1 + x^{-x}} = \\frac{1}{4}\n \\]\n\n2. **Isolate Terms**: To simplify, we can cro... | [
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[{"from":"user","value":"Return your final response within \\boxed{}. In a given circle, draw an ins(...TRUNCATED) | "In a given circle, draw an inscribed angle $CAD$ and its angle bisector $AB$. From an arbitrary poi(...TRUNCATED) | null | null | [{"content":"You like to solve problems by understanding the problem, writing a plan, executing the (...TRUNCATED) | \frac{CA + DA}{GE + HE} = \frac{BA}{FE} | plan_and_execute_rationalized | ["First, I'll try to understand the problem better by writing out a plan and go really deep into det(...TRUNCATED) | [
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