conversations listlengths 2 2 | question stringlengths 12 4.04k ⌀ | source stringclasses 8
values | id int64 21 50k ⌀ | prompt listlengths 2 2 | answer stringlengths 1 811 ⌀ | 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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[
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"value": "Example 2.41. $I=\\int_{0}^{1} \\ln x d x$."
},
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"value": "<think>\nOkay, let me try to figure out how to solve the integral I = ∫ from 0 to 1 of ln(x) dx. Hmm, so the integral of the natural logarithm of x from 0 to 1. I remember that integratin... | Example 2.41. $I=\int_{0}^{1} \ln x d x$. | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on 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... | -1 | alternative_rationalized | [
"To solve the integral \\( I = \\int_{0}^{1} \\ln x \\, dx \\), let's approach it step-by-step and consider different methods:\n\n**Method 1:** Integration by Parts\n\nLet \\( u = \\ln x \\) and \\( dv = dx \\). Then, \\( du = \\frac{1}{x} dx \\) and \\( v = x \\).\n\nUsing integration by parts formula:\n\\[\n\\int... | [
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[
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"value": "Example 8 (40th IMO National Team Selection Test) For non-negative real numbers $x_{1}, x_{2}, \\cdots, x_{n}$ satisfying the condition $\\sum_{i=1}^{n} x_{i}=1$, find the maximum value of $\\sum_{i=1}^{n}\\left(x_{j}^{4}-x_{j}^{5}\\right)$."
},
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... | Example 8 (40th IMO National Team Selection Test) For non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying the condition $\sum_{i=1}^{n} x_{i}=1$, find the maximum value of $\sum_{i=1}^{n}\left(x_{j}^{4}-x_{j}^{5}\right)$. | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on 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... | \dfrac{1}{12} | alternative_rationalized | [
"To solve the problem, we need to maximize the expression \\(\\sum_{i=1}^{n} \\left( x_i^4 - x_i^5 \\right)\\) given that \\(\\sum_{i=1}^{n} x_i = 1\\).\n\n### Step-by-Step Analysis\n\n#### Alternative 1: Consider Equal Distribution\nIf \\(x_1 = x_2 = \\cdots = x_n = \\frac{1}{n}\\), then:\n\\[ \\sum_{i=1}^{n} x_i ... | [
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"",
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[
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"value": "## 157. Math Puzzle $6 / 78$\n\nAt a class party, in addition to the milkshake drinks, various soda bottles were also emptied. There were more than 20 and fewer than 25, and specifically five times as many Astoria bottles as Cola, half as many Lemonade bottles as Astoria, and thr... | ## 157. Math Puzzle $6 / 78$
At a class party, in addition to the milkshake drinks, various soda bottles were also emptied. There were more than 20 and fewer than 25, and specifically five times as many Astoria bottles as Cola, half as many Lemonade bottles as Astoria, and three fewer Selters bottles than Astoria. How... | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on 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... | 24 | alternative_rationalized | [
"Let's denote the number of Cola bottles as \\( C \\), Astoria bottles as \\( A \\), Lemonade bottles as \\( L \\), and Selters bottles as \\( S \\).\n\nFrom the problem statement:\n- \\( A = 5C \\) (five times as many Astoria bottles as Cola)\n- \\( L = \\frac{A}{2} \\) (half as many Lemonade bottles as Astoria)\n... | [
false,
false
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"",
""
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[
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"value": "Return your final response within \\boxed{}. Let $S$ be the set of points on the rays forming the sides of a $120^{\\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ i... | Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$ . (Points $Q$ and $R$ may be on the same ray, and switching the names of $... | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on 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... | B | alternative_rationalized | [
"To solve this problem, we need to consider the geometric properties of the given setup. We have a \\(120^\\circ\\) angle with its vertex at point \\(P\\) and two rays forming the sides of the angle. Points \\(Q\\) and \\(R\\) lie on these rays, and we need to form equilateral triangles \\(PQR\\) where \\(P\\) is t... | [
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[
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"value": "2. Find the minimum value of the expression for $a, b>0$\n\n$$\n\\frac{(2 a+2 a b-b(b+1))^{2}+\\left(b-4 a^{2}+2 a(b+1)\\right)^{2}}{4 a^{2}+b^{2}}\n$$"
},
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"value": "<think>\nOkay, so I need to find the minimum value of this expression:\n\n$$\n\... | 2. Find the minimum value of the expression for $a, b>0$
$$
\frac{(2 a+2 a b-b(b+1))^{2}+\left(b-4 a^{2}+2 a(b+1)\right)^{2}}{4 a^{2}+b^{2}}
$$ | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on 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... | 1 | alternative_rationalized | [
"First, let's simplify the numerator:\n\nThe given expression is:\n\\[\n\\frac{(2a + 2ab - b(b+1))^2 + (b - 4a^2 + 2a(b+1))^2}{4a^2 + b^2}\n\\]\n\nSimplify the terms inside the parentheses first:\n1. \\(2a + 2ab - b(b+1) = 2a + 2ab - b^2 - b\\)\n2. \\(b - 4a^2 + 2a(b+1) = b - 4a^2 + 2ab + 2a\\)\n\nSo, the expressio... | [
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[
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"value": "Convert the base-7 number $305_{(7)}$ to a base-5 number."
},
{
"from": "assistant",
"value": "<think>\nAlright, so I need to convert the base-7 number 305_{(7)} to a base-5 number. Hmm, okay. Let's think about how to approach this. I remember that when converting num... | Convert the base-7 number $305_{(7)}$ to a base-5 number. | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on 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... | 1102_{(5)} | alternative_rationalized | [
"First, let's convert \\(305_{(7)}\\) to a decimal (base-10) number. The number \\(305_{(7)}\\) can be expanded as follows:\n\\[3 \\cdot 7^2 + 0 \\cdot 7^1 + 5 \\cdot 7^0 = 3 \\cdot 49 + 0 \\cdot 7 + 5 \\cdot 1 = 147 + 0 + 5 = 152\\]\nSo, \\(305_{(7)} = 152_{(10)}\\).\n\nNext, we need to convert \\(152_{(10)}\\) to... | [
false,
false
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"",
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"value": "Brandon has a collection of 20 baseball cards. Malcom has 8 more cards than Brandon. However, then Malcom gives x% of his cards to his friend Mark. How many cards does Malcom have left?\nIf we know the answer to the above question is 14, what is the value of unknown variable x... | null | null | null | [
{
"content": "You like to find multiple answers for a question then deliberate over them saying \"Alternatively,\" between each answer you are deliberating on 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... | 50 | alternative_rationalized | [
"First, let's determine how many cards Malcom originally had before he gave any away. Brandon has 20 cards, and Malcom has 8 more than Brandon, so:\n\nMalcom initially = 20 + 8 = 28 cards\n\nNow, let's consider different values for \\(x\\), the percentage of cards Malcom gives to his friend Mark, and see which one ... | [
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[{"from":"user","value":"Given that the line $ax + by - 3 = 0$ has no common points with the circle (...TRUNCATED) | "Given that the line $ax + by - 3 = 0$ has no common points with the circle $x^2 + y^2 = 3$, let the(...TRUNCATED) | null | null | [{"content":"You like to find multiple answers for a question then deliberate over them saying \"Alt(...TRUNCATED) | C | alternative_rationalized | ["Firstly, consider the condition given about the line \\( ax + by - 3 = 0 \\) having no common poin(...TRUNCATED) | [
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[{"from":"user","value":"Given the vectors $\\overrightarrow{a}=(1,2)$ and $2\\overrightarrow{a}+ \\(...TRUNCATED) | "Given the vectors $\\overrightarrow{a}=(1,2)$ and $2\\overrightarrow{a}+ \\overrightarrow{b}=(3,2)$(...TRUNCATED) | null | null | [{"content":"You like to find multiple answers for a question then deliberate over them saying \"Alt(...TRUNCATED) | B | alternative_rationalized | ["Firstly, let's start by finding the vector \\(2\\overrightarrow{a}\\). Given \\(\\overrightarrow{a(...TRUNCATED) | [
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[{"from":"user","value":"Prove that $1 + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\ldots + \\fr(...TRUNCATED) | "Prove that $1 + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\ldots + \\frac{1}{2^{n}-1} > \\frac{(...TRUNCATED) | null | null | [{"content":"You like to find multiple answers for a question then deliberate over them saying \"Alt(...TRUNCATED) | 2^k | alternative_rationalized | ["To prove the inequality \\(1 + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\ldots + \\frac{1}{2^(...TRUNCATED) | [
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