data_source
stringclasses 6
values | prompt
listlengths 2
2
| ability
stringclasses 1
value | reward_model
dict | extra_info
dict | problem
stringlengths 17
4.55k
|
|---|---|---|---|---|---|
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the arithmetic sequence ${a_n}$, it is known that $a_5 + a_7 = 10$, and $S_n$ represents the sum of the first $n$ terms of the sequence ${a_n}$. Calculate the value of $S_{11}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "55",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the arithmetic sequence ${a_n}$, it is known that $a_5 + a_7 = 10$, and $S_n$ represents the sum of the first $n$ terms of the sequence ${a_n}$. Calculate the value of $S_{11}$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $f(x) = \\ln x - \\frac{1}{2}a(x-1)$ $(a \\in \\mathbb{R})$.\n(Ⅰ) If $a=-2$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$.\n(Ⅱ) If the inequality $f(x) < 0$ holds for any $x \\in (1, +\\infty)$, determine the range of the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "[2, +\\infty)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x) = \ln x - \frac{1}{2}a(x-1)$ $(a \in \mathbb{R})$.
(Ⅰ) If $a=-2$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$.
(Ⅱ) If the inequality $f(x) < 0$ holds for any $x \in (1, +\infty)$, determine the range of the real number $a$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that an ellipse passes through point $P(2,3)$, and its foci are $F_1(-2,0)$ and $F_2(2,0)$, calculate the eccentricity of this ellipse.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that an ellipse passes through point $P(2,3)$, and its foci are $F_1(-2,0)$ and $F_2(2,0)$, calculate the eccentricity of this ellipse. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given an arithmetic-geometric sequence $\\{a\\_n\\}$ with the sum of its first $n$ terms denoted as $S\\_n$, it satisfies the equation $2S\\_n = 2^{n+1} + λ (λ ∈ R)$.\n\n(I) Find the general term formula for the sequence $\\{a\\_n\\}$.\n(II) If sequence $\\{b\\_n\\}$ satisfies $b\\_n = \\frac{1}{(2n+1)\\log\\_4(a\\_na\\_n+1)}$, find the sum of the first $n$ terms of sequence $\\{b\\_n\\}$, denoted as $T\\_n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{2n}{2n+1}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given an arithmetic-geometric sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, it satisfies the equation $2S\_n = 2^{n+1} + λ (λ ∈ R)$.
(I) Find the general term formula for the sequence $\{a\_n\}$.
(II) If sequence $\{b\_n\}$ satisfies $b\_n = \frac{1}{(2n+1)\log\_4(a\_na\_n+1)}$, find the sum of the first $n$ terms of sequence $\{b\_n\}$, denoted as $T\_n$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Two cars, Car A and Car B, simultaneously depart from locations A and B respectively, traveling towards each other at a constant speed. They meet at a point 60 kilometers from location A. After meeting, they continue to their respective destinations (Car A reaches B and Car B reaches A) and then immediately return. They meet again at a point 50 kilometers from location B. Determine the distance between locations A and B in kilometers.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "130",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Two cars, Car A and Car B, simultaneously depart from locations A and B respectively, traveling towards each other at a constant speed. They meet at a point 60 kilometers from location A. After meeting, they continue to their respective destinations (Car A reaches B and Car B reaches A) and then immediately return. They meet again at a point 50 kilometers from location B. Determine the distance between locations A and B in kilometers. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Compute $\\displaystyle \\sum_{n=2}^\\infty \\sum_{k=1}^{n-1} \\frac{k}{3^{n+k}}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{9}{128}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Compute $\displaystyle \sum_{n=2}^\infty \sum_{k=1}^{n-1} \frac{k}{3^{n+k}}$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the plane rectangular coordinate system $O-xy$, if $A(\\cos\\alpha, \\sin\\alpha)$, $B(\\cos\\beta, \\sin\\beta)$, $C\\left(\\frac{1}{2}, \\frac{\\sqrt{3}}{2}\\right)$, then one possible value of $\\beta$ that satisfies $\\overrightarrow{OC}=\\overrightarrow{OB}-\\overrightarrow{OA}$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{2\\pi}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the plane rectangular coordinate system $O-xy$, if $A(\cos\alpha, \sin\alpha)$, $B(\cos\beta, \sin\beta)$, $C\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, then one possible value of $\beta$ that satisfies $\overrightarrow{OC}=\overrightarrow{OB}-\overrightarrow{OA}$ is ______. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $f(x)=\\frac{1}{2}{\\cos^4}x+\\sqrt{3}\\sin{x}\\cos{x}-\\frac{1}{2}{\\sin^4}x+m$ has a maximum value of $\\frac{3}{2}$, \n$(1)$ Find the value of the constant $m$ and determine the corresponding set of $x$ values when $f(x)$ attains its maximum value; \n$(2)$ Find the intervals where $f(x)$ is monotonically increasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "[k\\pi - \\frac{\\pi}{3}, k\\pi + \\frac{\\pi}{6}], k \\in \\mathbb{Z}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x)=\frac{1}{2}{\cos^4}x+\sqrt{3}\sin{x}\cos{x}-\frac{1}{2}{\sin^4}x+m$ has a maximum value of $\frac{3}{2}$,
$(1)$ Find the value of the constant $m$ and determine the corresponding set of $x$ values when $f(x)$ attains its maximum value;
$(2)$ Find the intervals where $f(x)$ is monotonically increasing. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "If a test consists of 6 multiple-choice questions with the grading rule that each person initially scores 6 points, each correct answer adds 4 points, each incorrect answer deducts 1 point, and no answer gets 0 points, determine the number of students that will have the same score.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If a test consists of 6 multiple-choice questions with the grading rule that each person initially scores 6 points, each correct answer adds 4 points, each incorrect answer deducts 1 point, and no answer gets 0 points, determine the number of students that will have the same score. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Jane is 35 years old. Marty is older than Jane. In $n$ years, where $n$ is a positive integer, Marty's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Marty's age. Let $m$ be Marty's current age. How many ordered pairs of positive integers $(m,n)$ make this possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Jane is 35 years old. Marty is older than Jane. In $n$ years, where $n$ is a positive integer, Marty's age and Jane's age will both be two-digit numbers and will have the property that Jane's age is obtained by interchanging the digits of Marty's age. Let $m$ be Marty's current age. How many ordered pairs of positive integers $(m,n)$ make this possible? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The vertex of the parabola $C$ is at the origin $O$, and the focus is on the $x$-axis. The line $l: x=1$ intersects $C$ at points $P$ and $Q$, where $OP \\perp OQ$. Point $M(2,0)$ is given, and circle $M$ is tangent to line $l$. <br/>$(1)$ Find the equations of $C$ and circle $M$; <br/>$(2)$ Let $A_{1}$, $A_{2}$, $A_{3}$ be three points on $C$, and lines $A_{1}A_{2}$, $A_{1}A_{3}$ are both tangent to circle $M$. Determine the position relationship between line $A_{2}A_{3}$ and circle $M$, and explain the reason.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\text{Line } A_{2}A_{3} \\text{ is tangent to circle } M",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The vertex of the parabola $C$ is at the origin $O$, and the focus is on the $x$-axis. The line $l: x=1$ intersects $C$ at points $P$ and $Q$, where $OP \perp OQ$. Point $M(2,0)$ is given, and circle $M$ is tangent to line $l$. <br/>$(1)$ Find the equations of $C$ and circle $M$; <br/>$(2)$ Let $A_{1}$, $A_{2}$, $A_{3}$ be three points on $C$, and lines $A_{1}A_{2}$, $A_{1}A_{3}$ are both tangent to circle $M$. Determine the position relationship between line $A_{2}A_{3}$ and circle $M$, and explain the reason. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Simplify \\[\\frac{1}{\\dfrac{3}{\\sqrt{5}+2} + \\dfrac{4}{\\sqrt{7}-2}}.\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{3}{9\\sqrt{5} + 4\\sqrt{7} - 10}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Simplify \[\frac{1}{\dfrac{3}{\sqrt{5}+2} + \dfrac{4}{\sqrt{7}-2}}.\] |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Using systematic sampling to select a sample of size $20$ from $160$ students, the $160$ students are numbered from $1$ to $160$ and evenly divided into $20$ groups ($1$ to $8$, $9$ to $16$, ..., $153$ to $160$). If it is assumed that the number drawn from the first group is $3$, then the determined number in the fifth group is $\\_\\_\\_$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "35",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Using systematic sampling to select a sample of size $20$ from $160$ students, the $160$ students are numbered from $1$ to $160$ and evenly divided into $20$ groups ($1$ to $8$, $9$ to $16$, ..., $153$ to $160$). If it is assumed that the number drawn from the first group is $3$, then the determined number in the fifth group is $\_\_\_$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Pete thinks of a number. He triples it, subtracts 5, multiplies by 3, and ends up with 96. What was his original number?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "12.33",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Pete thinks of a number. He triples it, subtracts 5, multiplies by 3, and ends up with 96. What was his original number? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The two squares shown share the same center $O$. The larger square has sides of length $2$, and the smaller square has sides of length $1$. The line $\\overline{AB}$, which is a segment between the larger and smaller squares and one which lies along their shared diagonal path, has a length of $\\frac{3}{4}$. Calculate the area of the octagon $ABCDEFGH$ formed by the intersecting points of the squares' sides, given that the smaller square is rotated $45^\\circ$ with respect to the larger one. Express the area as a fraction $m/n$, where $m$ and $n$ are relatively prime integers, and find $m+n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The two squares shown share the same center $O$. The larger square has sides of length $2$, and the smaller square has sides of length $1$. The line $\overline{AB}$, which is a segment between the larger and smaller squares and one which lies along their shared diagonal path, has a length of $\frac{3}{4}$. Calculate the area of the octagon $ABCDEFGH$ formed by the intersecting points of the squares' sides, given that the smaller square is rotated $45^\circ$ with respect to the larger one. Express the area as a fraction $m/n$, where $m$ and $n$ are relatively prime integers, and find $m+n$. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Maria leaves Kingsford at 7:45 AM heading for Iron Mountain on her bike. She bikes at a uniform rate of 15 miles per hour. Joey leaves Iron Mountain at 8:15 AM heading for Kingsford on his bike. He bikes at a uniform rate of 18 miles per hour. They both bike on the same 75-mile route between Kingsford and Iron Mountain. At what time do they meet.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "10:15",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Maria leaves Kingsford at 7:45 AM heading for Iron Mountain on her bike. She bikes at a uniform rate of 15 miles per hour. Joey leaves Iron Mountain at 8:15 AM heading for Kingsford on his bike. He bikes at a uniform rate of 18 miles per hour. They both bike on the same 75-mile route between Kingsford and Iron Mountain. At what time do they meet. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The graph of the equation $y = \\frac{x}{x^3 + Ax^2 + Bx + C}$, where $A, B, C$ are integers, displays three vertical asymptotes at $x = -3, 0, 4$. Calculate $A + B + C$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-13",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The graph of the equation $y = \frac{x}{x^3 + Ax^2 + Bx + C}$, where $A, B, C$ are integers, displays three vertical asymptotes at $x = -3, 0, 4$. Calculate $A + B + C$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Samantha has an unfair eight-sided die. The probabilities are such that the die shows 8 with a probability of $\\frac{1}{3}$, shows an even number less than 8 with a probability of $\\frac{1}{6}$ each, and shows an odd number with a probability of $\\frac{1}{12}$ each. Calculate the expected value of the number shown when this die is rolled.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{35}{9}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Samantha has an unfair eight-sided die. The probabilities are such that the die shows 8 with a probability of $\frac{1}{3}$, shows an even number less than 8 with a probability of $\frac{1}{6}$ each, and shows an odd number with a probability of $\frac{1}{12}$ each. Calculate the expected value of the number shown when this die is rolled. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=13$ and $AB=DO=OC=15$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ such that $\\angle OPA = 30^\\circ$.\n\nPoint $X$ is the midpoint of $AD$ and point $Y$ is the midpoint of $BC$. When $X$ and $Y$ are joined, the trapezoid is divided into two smaller trapezoids. The ratio of the area of trapezoid $ABYX$ to the area of trapezoid $XYCD$ in simplified form is $p:q$. Find $p+q$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=13$ and $AB=DO=OC=15$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ such that $\angle OPA = 30^\circ$.
Point $X$ is the midpoint of $AD$ and point $Y$ is the midpoint of $BC$. When $X$ and $Y$ are joined, the trapezoid is divided into two smaller trapezoids. The ratio of the area of trapezoid $ABYX$ to the area of trapezoid $XYCD$ in simplified form is $p:q$. Find $p+q$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let set $A = \\{x | x^2 - x - 2 < 0\\}$, and $B = \\{x | a < x < a + 5\\}$. If $A \\subseteq B$, find the range of values for $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-3 \\leq a \\leq -1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let set $A = \{x | x^2 - x - 2 < 0\}$, and $B = \{x | a < x < a + 5\}$. If $A \subseteq B$, find the range of values for $a$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the ellipse \\(\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1\\) with one focus \\(F\\), point \\(P\\) lies on the \\(y\\)-axis. The line \\(PF\\) intersects the ellipse at points \\(M\\) and \\(N\\). Given \\(\\overrightarrow{PM} = \\lambda_{1} \\overrightarrow{MF}\\) and \\(\\overrightarrow{PN} = \\lambda_{2} \\overrightarrow{NF}\\), find the sum of the real numbers \\(\\lambda_{1} + \\lambda_{2}\\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-\\frac{2 a^{2}}{b^{2}}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) with one focus \(F\), point \(P\) lies on the \(y\)-axis. The line \(PF\) intersects the ellipse at points \(M\) and \(N\). Given \(\overrightarrow{PM} = \lambda_{1} \overrightarrow{MF}\) and \(\overrightarrow{PN} = \lambda_{2} \overrightarrow{NF}\), find the sum of the real numbers \(\lambda_{1} + \lambda_{2}\). |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let \\( x \\) and \\( y \\) be real numbers such that \\( \\frac{\\sin x}{\\sin y} = 2 \\) and \\( \\frac{\\cos x}{\\cos (y + \\frac{\\pi}{4})} = 3 \\). Find the value of\n\\[ \\frac{\\sin 2x}{\\sin 2y} + \\frac{\\cos 2x}{\\cos 2y}. \\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "6\\sqrt{2} + \\frac{11}{29}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let \( x \) and \( y \) be real numbers such that \( \frac{\sin x}{\sin y} = 2 \) and \( \frac{\cos x}{\cos (y + \frac{\pi}{4})} = 3 \). Find the value of
\[ \frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}. \] |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Use the Horner's Rule to calculate the value of the polynomial $f(x) = 3x^4 + x^2 + 2x + 4$ when $x = 10$. What is the value of $v_2$ during the process?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "310",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Use the Horner's Rule to calculate the value of the polynomial $f(x) = 3x^4 + x^2 + 2x + 4$ when $x = 10$. What is the value of $v_2$ during the process? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given a point $N(x,y)$ on the circle $x^{2}+y^{2}=1$, find the range of $\\dfrac {y}{x+2}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left[- \\dfrac { \\sqrt {3}}{3}, \\dfrac { \\sqrt {3}}{3}\\right]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a point $N(x,y)$ on the circle $x^{2}+y^{2}=1$, find the range of $\dfrac {y}{x+2}$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given an arithmetic sequence $\\left\\{ a_n \\right\\}$ satisfies: $a_3=7$, $a_5+a_7=26$, and the sum of the first $n$ terms of $\\left\\{ a_n \\right\\}$ is $S_n$,\n\n$(1)$ Find $a_n$ and $S_n$.\n\n$(2)$ Let $b_n=\\frac{1}{a_n^2-1}$ $(n\\in N_+)$, find the sum of the first $n$ terms of the sequence $\\left\\{ b_n \\right\\}$, denoted as $T_n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{n}{4(n+1)}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given an arithmetic sequence $\left\{ a_n \right\}$ satisfies: $a_3=7$, $a_5+a_7=26$, and the sum of the first $n$ terms of $\left\{ a_n \right\}$ is $S_n$,
$(1)$ Find $a_n$ and $S_n$.
$(2)$ Let $b_n=\frac{1}{a_n^2-1}$ $(n\in N_+)$, find the sum of the first $n$ terms of the sequence $\left\{ b_n \right\}$, denoted as $T_n$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "For positive integers $n$, let $g(n)$ return the smallest positive integer $m$ such that $\\frac{1}{m}$ has exactly $n$ digits after the decimal point. Find the number of positive integer divisors of $g(2010)$ if $g(n)=3^n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2011",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For positive integers $n$, let $g(n)$ return the smallest positive integer $m$ such that $\frac{1}{m}$ has exactly $n$ digits after the decimal point. Find the number of positive integer divisors of $g(2010)$ if $g(n)=3^n$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "National fitness is the foundation and guarantee for all people to enhance their physical fitness and live a healthy life. In order to study the fitness situation of citizens in Fuzhou, a research group randomly selected 100 citizens in our city for a survey and obtained the following data in the table:\n\n| Weekly Exercise Frequency | 1 time | 2 times | 3 times | 4 times | 5 times | 6 times and above |\n|---------------------------|--------|---------|---------|---------|---------|-------------------|\n| Male | 4 | 3 | 3 | 3 | 7 | 30 |\n| Female | 6 | 5 | 4 | 7 | 8 | 20 |\n\n$(1)$ If users who exercise more than 3 times a week are considered \"fitness enthusiasts,\" please complete the $2 \\times 2$ contingency table (see answer sheet) and, based on the small probability value $\\alpha = 0.05$, conduct an independence test to determine whether \"fitness enthusiasts\" and \"gender\" are related. \n$(2)$ Users who exercise 6 times or more per week are called \"fitness experts.\" Let the random variable $Y$ be the total amount of rewards. In a yoga studio in Fuzhou, each selected female \"fitness expert\" will be rewarded with a $1000$ yuan fitness card. Calculate the mathematical expectation of $Y$. \nGiven: ${K}^{2} = \\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$, \n| $\\alpha$ | 0.10 | 0.05 | 0.01 | 0.005 | 0.001 |\n|----------|------|------|------|-------|-------|\n| $k_{\\alpha}$ | 2.706 | 3.841 | 6.635 | 7.879 | 10.828 |\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1600 \\text{ yuan}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | National fitness is the foundation and guarantee for all people to enhance their physical fitness and live a healthy life. In order to study the fitness situation of citizens in Fuzhou, a research group randomly selected 100 citizens in our city for a survey and obtained the following data in the table:
| Weekly Exercise Frequency | 1 time | 2 times | 3 times | 4 times | 5 times | 6 times and above |
|---------------------------|--------|---------|---------|---------|---------|-------------------|
| Male | 4 | 3 | 3 | 3 | 7 | 30 |
| Female | 6 | 5 | 4 | 7 | 8 | 20 |
$(1)$ If users who exercise more than 3 times a week are considered "fitness enthusiasts," please complete the $2 \times 2$ contingency table (see answer sheet) and, based on the small probability value $\alpha = 0.05$, conduct an independence test to determine whether "fitness enthusiasts" and "gender" are related.
$(2)$ Users who exercise 6 times or more per week are called "fitness experts." Let the random variable $Y$ be the total amount of rewards. In a yoga studio in Fuzhou, each selected female "fitness expert" will be rewarded with a $1000$ yuan fitness card. Calculate the mathematical expectation of $Y$.
Given: ${K}^{2} = \frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$,
| $\alpha$ | 0.10 | 0.05 | 0.01 | 0.005 | 0.001 |
|----------|------|------|------|-------|-------|
| $k_{\alpha}$ | 2.706 | 3.841 | 6.635 | 7.879 | 10.828 | |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Linda sells three sandwiches and four cookies for $\\$$4.20 and she sells four sandwiches and three cookies for $\\$$4.50. Assuming a fixed price per item, what is the cost, in dollars, of four sandwiches and five cookies? Express your answer as a decimal to the nearest hundredth.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\$5.44",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Linda sells three sandwiches and four cookies for $\$$4.20 and she sells four sandwiches and three cookies for $\$$4.50. Assuming a fixed price per item, what is the cost, in dollars, of four sandwiches and five cookies? Express your answer as a decimal to the nearest hundredth. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In triangle $ABC$, $AB = 12$, $BC = 15$, and $CA = 17$. Let $E$ be a point in the interior of $\\overline{BC}$. Let points $I_B$ and $I_C$ denote the incenters of triangles $ABE$ and $ACE$, respectively. The circumcircles of triangles $BI_BE$ and $CI_CE$ meet at distinct points $Q$ and $E$. Let's find the maximum possible area of $\\triangle BQC$ in a similar reducible form.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "84",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In triangle $ABC$, $AB = 12$, $BC = 15$, and $CA = 17$. Let $E$ be a point in the interior of $\overline{BC}$. Let points $I_B$ and $I_C$ denote the incenters of triangles $ABE$ and $ACE$, respectively. The circumcircles of triangles $BI_BE$ and $CI_CE$ meet at distinct points $Q$ and $E$. Let's find the maximum possible area of $\triangle BQC$ in a similar reducible form. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the largest real number $\\lambda$ such that for the polynomial $f(x) = x^3 + a x^2 + b x + c$ with real coefficients, if all roots are non-negative real numbers, then for all $x \\geq 0$, we have $f(x) \\geq \\lambda (x - a)^3$. Also determine when the equality holds.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\lambda = -\\frac{1}{27}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the largest real number $\lambda$ such that for the polynomial $f(x) = x^3 + a x^2 + b x + c$ with real coefficients, if all roots are non-negative real numbers, then for all $x \geq 0$, we have $f(x) \geq \lambda (x - a)^3$. Also determine when the equality holds. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In triangle $\\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $ccosB+\\frac{{\\sqrt{3}}}{3}bsinC-a=0$. \n$(1)$ Find the measure of angle $C$. \n$(2)$ If $c=3$ and the area of $\\triangle ABC$ is $\\frac{{3\\sqrt{3}}}{4}$, find the value of $a+b$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "a + b = 3\\sqrt{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $ccosB+\frac{{\sqrt{3}}}{3}bsinC-a=0$.
$(1)$ Find the measure of angle $C$.
$(2)$ If $c=3$ and the area of $\triangle ABC$ is $\frac{{3\sqrt{3}}}{4}$, find the value of $a+b$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the Cartesian coordinate system, given point $A(-1,8)$ is symmetric to point $B$ with respect to the $y$-axis, then the coordinates of point $B$ are ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(1,8)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the Cartesian coordinate system, given point $A(-1,8)$ is symmetric to point $B$ with respect to the $y$-axis, then the coordinates of point $B$ are ______. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ be $S_n$, and it satisfies $S_{2014} > 0$, $S_{2015} < 0$. For any positive integer $n$, $|a_n| \\geqslant |a_k|$, determine the value of $k$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1008",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2014} > 0$, $S_{2015} < 0$. For any positive integer $n$, $|a_n| \geqslant |a_k|$, determine the value of $k$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The members of a choir are arranged in a rectangular formation. When they are arranged in 10 rows, there are 4 positions unoccupied in the formation. When they are arranged in 11 rows, there are 5 positions unoccupied. How many members are in the choir if the membership is between 150 and 300?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "226",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The members of a choir are arranged in a rectangular formation. When they are arranged in 10 rows, there are 4 positions unoccupied in the formation. When they are arranged in 11 rows, there are 5 positions unoccupied. How many members are in the choir if the membership is between 150 and 300? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given a sequence $\\{a\\_n\\}$ with the sum of its first $n$ terms denoted as $S\\_n$, where the first term $a\\_1 \\neq 0$ and $2S\\_n = a\\_n^2 + a\\_n$ for all $n \\in \\mathbb{N}^+$.\n\n1. Find the general formula for the $n$-th term of the sequence $\\{a\\_n\\}$.\n2. If $a\\_n > 0$, let $b\\_n = \\frac{4}{a\\_n(a\\_n + 2)}$. The sum of the first $n$ terms of the sequence $\\{b\\_n\\}$ is denoted as $T\\_n$. If $T\\_n < m$ always holds true, where $m \\in \\mathbb{Z}$, find the minimum value of $m$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "m\\_{min} = 3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, where the first term $a\_1 \neq 0$ and $2S\_n = a\_n^2 + a\_n$ for all $n \in \mathbb{N}^+$.
1. Find the general formula for the $n$-th term of the sequence $\{a\_n\}$.
2. If $a\_n > 0$, let $b\_n = \frac{4}{a\_n(a\_n + 2)}$. The sum of the first $n$ terms of the sequence $\{b\_n\}$ is denoted as $T\_n$. If $T\_n < m$ always holds true, where $m \in \mathbb{Z}$, find the minimum value of $m$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $f(x) = 2^x + 1$, calculate $f(2)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $f(x) = 2^x + 1$, calculate $f(2)$. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Consider the function $f(x) = 3x^3 + 2x^2 - 4x - 1$. Find $f(x + h) - f(x)$.\n**A)** $9x^2 + 4x + 2h$\n**B)** $9x^2h + 4xh - 4h$\n**C)** $3h^3 + 9xh^2 + 9x^2h + 4xh + 2h^2$\n**D)** $h(9x^2 + 9xh + 3h^2 + 4x + 2h - 4)$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "h(9x^2 + 9xh + 3h^2 + 4x + 2h - 4)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Consider the function $f(x) = 3x^3 + 2x^2 - 4x - 1$. Find $f(x + h) - f(x)$.
**A)** $9x^2 + 4x + 2h$
**B)** $9x^2h + 4xh - 4h$
**C)** $3h^3 + 9xh^2 + 9x^2h + 4xh + 2h^2$
**D)** $h(9x^2 + 9xh + 3h^2 + 4x + 2h - 4)$ |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The domain of the equation $y = \\frac{4x}{3x^2 - 9x + 6}$ is the set of all real numbers except for the values $x = A$ and $x = B$. What is $A + B$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The domain of the equation $y = \frac{4x}{3x^2 - 9x + 6}$ is the set of all real numbers except for the values $x = A$ and $x = B$. What is $A + B$? |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that there are 10 children and a total of 26 wheels, determine the number of tricycles.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that there are 10 children and a total of 26 wheels, determine the number of tricycles. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A company has 750 employees, among which there are 300 male employees. To conduct a certain survey, it is planned to use stratified sampling to draw a sample of 45. The number of female employees to be drawn is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A company has 750 employees, among which there are 300 male employees. To conduct a certain survey, it is planned to use stratified sampling to draw a sample of 45. The number of female employees to be drawn is ______. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Convex pentagon $FGHIJ$ has side lengths $FG=8$, $GH=HI=IJ=9$, and $JF=10$. The pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Determine the area of $FGHIJ$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "A = 45r",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Convex pentagon $FGHIJ$ has side lengths $FG=8$, $GH=HI=IJ=9$, and $JF=10$. The pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Determine the area of $FGHIJ$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "When two items are randomly selected from a product containing three genuine items and one defective item, calculate the probability of selecting exactly one defective item.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | When two items are randomly selected from a product containing three genuine items and one defective item, calculate the probability of selecting exactly one defective item. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the number $86743.2951$, find the ratio between the place value of the digit 6 and the place value of the digit 5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "100,000",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the number $86743.2951$, find the ratio between the place value of the digit 6 and the place value of the digit 5. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A line passes through the points (3,3,2) and (8,2,-3). Determine the z-coordinate of a point on this line when the x-coordinate is 7.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A line passes through the points (3,3,2) and (8,2,-3). Determine the z-coordinate of a point on this line when the x-coordinate is 7. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the phase shift of the graph of \\( y = 3 \\sin \\left( 4x + \\frac{\\pi}{4} \\right) \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-\\frac{\\pi}{16}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the phase shift of the graph of \( y = 3 \sin \left( 4x + \frac{\pi}{4} \right) \). |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the plane Cartesian coordinate system $xOy$, the radius of circle $C$ is $r=2$, and the center of the circle is the intersection point $C$ of the lines $l_{1}$: $2x-y-4=0$ and $l_{2}$: $x-y-1=0$. <br/>$(1)$ Find the equation of circle $C$; <br/>$(2)$ Determine the positional relationship between the line $l_{3}$: $4x-3y-1=0$ and circle $C$. If they intersect, let the intersection points be $A$ and $B$, and find the length of the chord $|AB|$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the plane Cartesian coordinate system $xOy$, the radius of circle $C$ is $r=2$, and the center of the circle is the intersection point $C$ of the lines $l_{1}$: $2x-y-4=0$ and $l_{2}$: $x-y-1=0$. <br/>$(1)$ Find the equation of circle $C$; <br/>$(2)$ Determine the positional relationship between the line $l_{3}$: $4x-3y-1=0$ and circle $C$. If they intersect, let the intersection points be $A$ and $B$, and find the length of the chord $|AB|$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $(1+x)^{10} = a + a_1(1-x) + a_2(1-x)^2 + \\ldots + a_{10}(1-x)^{10}$, find the value of $a_8$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "180",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $(1+x)^{10} = a + a_1(1-x) + a_2(1-x)^2 + \ldots + a_{10}(1-x)^{10}$, find the value of $a_8$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Express the scientific notation for the predicted prospective resources of methane hydrate in China in terms of a numerical value, where the resources exceed $100$ billion tons of oil equivalent.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1\\times 10^{11}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Express the scientific notation for the predicted prospective resources of methane hydrate in China in terms of a numerical value, where the resources exceed $100$ billion tons of oil equivalent. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Calculate the opposite of $-\\frac{1}{6}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the opposite of $-\frac{1}{6}$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "How many ways are there to put 6 balls in 3 boxes if the balls are distinguishable and the boxes are distinguishable?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "729",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many ways are there to put 6 balls in 3 boxes if the balls are distinguishable and the boxes are distinguishable? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "When $\\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 40 digits after the decimal point?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "72",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | When $\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 40 digits after the decimal point? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the binomial expansion of $\\left(x^{2}- \\frac {1}{x}\\right)^{5}$, the coefficient of the second term is $\\boxed{\\text{(D) } -5}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\text{(D) } -5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the binomial expansion of $\left(x^{2}- \frac {1}{x}\right)^{5}$, the coefficient of the second term is $\boxed{\text{(D) } -5}$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "49",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The opposite of $-7$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The opposite of $-7$ is ______. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the constant $a \\in \\mathbb{R}$, define the sets $A = \\{x | (x - 1)(x - a) \\geq 0\\}$ and $B = \\{x | x \\geq a - 1\\}$. If $A \\cup B = \\mathbb{R}$, determine the range of values for $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(-\\infty, 2]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the constant $a \in \mathbb{R}$, define the sets $A = \{x | (x - 1)(x - a) \geq 0\}$ and $B = \{x | x \geq a - 1\}$. If $A \cup B = \mathbb{R}$, determine the range of values for $a$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A train 18 m long passes a pole in 9 seconds. How long will it take to pass a bridge 36 m long?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "27 \\text{ seconds}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A train 18 m long passes a pole in 9 seconds. How long will it take to pass a bridge 36 m long? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that the sequence $\\{\\frac{{a}_{n}}{n}\\}$ is an arithmetic sequence, $a_{3}-3a_{1}=6$, and $a_{2}=6$. \n$(1)$ Find $a_{n}$; \n$(2)$ Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\\{\\frac{1}{{a}_{n}}\\}$, find $S_{n}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "S_{n} = \\frac{n}{n+1}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the sequence $\{\frac{{a}_{n}}{n}\}$ is an arithmetic sequence, $a_{3}-3a_{1}=6$, and $a_{2}=6$.
$(1)$ Find $a_{n}$;
$(2)$ Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{\frac{1}{{a}_{n}}\}$, find $S_{n}$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given an isosceles triangle with two angles of $\\left(2x-2\\right)^{\\circ}$ and $\\left(3x-5\\right)^{\\circ}$, the degree of the vertex angle of this isosceles triangle is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "172^{\\circ}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given an isosceles triangle with two angles of $\left(2x-2\right)^{\circ}$ and $\left(3x-5\right)^{\circ}$, the degree of the vertex angle of this isosceles triangle is ______. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Express the number 80 as the sum of two prime numbers. In how many ways can this be done? Note that the number one is not a prime number.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "4 \\text{ ways}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Express the number 80 as the sum of two prime numbers. In how many ways can this be done? Note that the number one is not a prime number. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the arithmetic sequence $\\{a_n\\}$, $a_2=6$, $a_3+a_6=27$. \n$(1)$ Find the general formula for the sequence $\\{a_n\\}$. \n$(2)$ If the general formula for the sequence $\\{b_n\\}$ is $b_n=3^{n-1}$, find the sum of the first $n$ terms $T_n$ of the sequence $\\{a_n \\cdot b_n\\}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "T_n= \\dfrac{(2n-1)\\cdot 3^{n+1}+3}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the arithmetic sequence $\{a_n\}$, $a_2=6$, $a_3+a_6=27$.
$(1)$ Find the general formula for the sequence $\{a_n\}$.
$(2)$ If the general formula for the sequence $\{b_n\}$ is $b_n=3^{n-1}$, find the sum of the first $n$ terms $T_n$ of the sequence $\{a_n \cdot b_n\}$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "For how many integers $n$ with $1 \\le n \\le 2016$ is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 2 + e^{4 \\pi i k / n} \\right)^n - 1 \\right) \n\\]\nequal to zero?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "504",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For how many integers $n$ with $1 \le n \le 2016$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 2 + e^{4 \pi i k / n} \right)^n - 1 \right)
\]
equal to zero? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Rewrite the angle $-\\frac{27}{4}\\pi$ in the form of $\\alpha + 2k\\pi$ ($k\\in \\mathbb{Z}$, $0 \\leq \\alpha < 2\\pi$).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{5\\pi}{4} - 8\\pi",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Rewrite the angle $-\frac{27}{4}\pi$ in the form of $\alpha + 2k\pi$ ($k\in \mathbb{Z}$, $0 \leq \alpha < 2\pi$). |
numina_aops_forum | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $\\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$ ,\n(i) $(x + 1)\\star 0 = (0\\star x) + 1$ (ii) $0\\star (y + 1) = (y\\star 0) + 1$ (iii) $(x + 1)\\star (y + 1) = (x\\star y) + 1$ .\nIf $123\\star 456 = 789$ , find $246\\star 135$ .\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "579",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $\star$ be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers $x$ and $y$ ,
(i) $(x + 1)\star 0 = (0\star x) + 1$ (ii) $0\star (y + 1) = (y\star 0) + 1$ (iii) $(x + 1)\star (y + 1) = (x\star y) + 1$ .
If $123\star 456 = 789$ , find $246\star 135$ . |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In how many ways can 9 people be seated in a row of chairs if three of the people, Wilma, Paul, and Alice, refuse to sit next to each other?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "181200",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In how many ways can 9 people be seated in a row of chairs if three of the people, Wilma, Paul, and Alice, refuse to sit next to each other? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the equation of the directrix of the parabola given by the equation $y = 4x^2 + 4.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "y = \\frac{63}{16}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the equation of the directrix of the parabola given by the equation $y = 4x^2 + 4.$ |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "62398, 63298, 62389, 63289\n\n\\_\\_\\_\\_\\_\\_ > \\_\\_\\_\\_\\_\\_ > \\_\\_\\_\\_\\_\\_ > \\_\\_\\_\\_\\_\\_ .\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "63298, 62398, 63289, 62389",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | 62398, 63298, 62389, 63289
\_\_\_\_\_\_ > \_\_\_\_\_\_ > \_\_\_\_\_\_ > \_\_\_\_\_\_ . |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $ABC$ be an equilateral triangle. Extend each side of the triangle beyond each vertex so that $BB' = 2 \\cdot AB$, $CC' = 2 \\cdot BC$, and $AA' = 2 \\cdot CA$. Determine the ratio of the area of triangle $A'B'C'$ to the area of $\\triangle ABC$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "9:1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $ABC$ be an equilateral triangle. Extend each side of the triangle beyond each vertex so that $BB' = 2 \cdot AB$, $CC' = 2 \cdot BC$, and $AA' = 2 \cdot CA$. Determine the ratio of the area of triangle $A'B'C'$ to the area of $\triangle ABC$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "What is the maximum possible value of the greatest common divisor of two consecutive terms of the sequence $b_n = n! + n^2$, where $n \\ge 0$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the maximum possible value of the greatest common divisor of two consecutive terms of the sequence $b_n = n! + n^2$, where $n \ge 0$? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Jessica is tasked with placing four identical, dotless dominoes on a 4 by 5 grid to form a continuous path from the upper left-hand corner \\(C\\) to the lower right-hand corner \\(D\\). The dominoes are shaded 1 by 2 rectangles that must touch each other at their sides, not just at the corners, and cannot be placed diagonally. Each domino covers exactly two of the unit squares on the grid. Determine how many distinct arrangements are possible for Jessica to achieve this, assuming the path only moves right or down.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "35",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Jessica is tasked with placing four identical, dotless dominoes on a 4 by 5 grid to form a continuous path from the upper left-hand corner \(C\) to the lower right-hand corner \(D\). The dominoes are shaded 1 by 2 rectangles that must touch each other at their sides, not just at the corners, and cannot be placed diagonally. Each domino covers exactly two of the unit squares on the grid. Determine how many distinct arrangements are possible for Jessica to achieve this, assuming the path only moves right or down. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $w$ be a complex number satisfying $w^2 - w + 1 = 0.$ Compute\n\\[w^{97} + w^{98} + w^{99} + w^{100} + w^{101} + w^{102}.\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-2 + 2w",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $w$ be a complex number satisfying $w^2 - w + 1 = 0.$ Compute
\[w^{97} + w^{98} + w^{99} + w^{100} + w^{101} + w^{102}.\] |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the minimum value of\n\\[(15 - x)(8 - x)(15 + x)(8 + x) - 200.\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-6680.25",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the minimum value of
\[(15 - x)(8 - x)(15 + x)(8 + x) - 200.\] |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the real numbers \\( (x, y) \\) satisfy the system of equations \\[\n\\begin{cases}\nx^{3} - 6x^{2} + 12x = 13, \\\\\ny^{3} + 3y - 3y^{2} = -4,\n\\end{cases}\n\\]\nfind \\( x + y \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the real numbers \( (x, y) \) satisfy the system of equations \[
\begin{cases}
x^{3} - 6x^{2} + 12x = 13, \\
y^{3} + 3y - 3y^{2} = -4,
\end{cases}
\]
find \( x + y \). |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Suppose there exist nonzero complex numbers \\( p, q, r, s, \\) and \\( t \\) such that \\( k \\) is a root of both the equations \\( px^4 + qx^3 + rx^2 + sx + t = 0 \\) and \\( qx^4 + rx^3 + sx^2 + tx + p = 0 \\). Determine all possible values of \\( k \\) under these conditions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1, \\omega, \\omega^2, \\omega^3, \\omega^4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Suppose there exist nonzero complex numbers \( p, q, r, s, \) and \( t \) such that \( k \) is a root of both the equations \( px^4 + qx^3 + rx^2 + sx + t = 0 \) and \( qx^4 + rx^3 + sx^2 + tx + p = 0 \). Determine all possible values of \( k \) under these conditions. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Three distinct numbers are randomly selected from the set $\\{-3, -2, 0, 1, 2, 5, 6\\}$. Calculate the probability that the product of these three numbers is positive.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{8}{35}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Three distinct numbers are randomly selected from the set $\{-3, -2, 0, 1, 2, 5, 6\}$. Calculate the probability that the product of these three numbers is positive. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Consider positive integers $n$ where $D(n)$ denotes the number of pairs of different adjacent digits in the binary (base two) representation of $n$. Determine the number of positive integers less than or equal to $50$ for which $D(n) = 3$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "11",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Consider positive integers $n$ where $D(n)$ denotes the number of pairs of different adjacent digits in the binary (base two) representation of $n$. Determine the number of positive integers less than or equal to $50$ for which $D(n) = 3$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given in $\\triangle ABC$, $sinA+cosA=\\frac{\\sqrt{5}}{5}$, calculate $\\sin A-\\cos A$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{3\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given in $\triangle ABC$, $sinA+cosA=\frac{\sqrt{5}}{5}$, calculate $\sin A-\cos A$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The algebraic expression \"5-4a\" is expressed in words as ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\text{the difference of 5 and 4 times a}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The algebraic expression "5-4a" is expressed in words as ____. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "For each integer $n\\geq 5$, let $a_n$ denote the base-$n$ number $0.\\overline{133}_n$. The product $a_5a_6 \\dotsm a_{99}$ can be expressed as $\\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "10101",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For each integer $n\geq 5$, let $a_n$ denote the base-$n$ number $0.\overline{133}_n$. The product $a_5a_6 \dotsm a_{99}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "For the one-variable quadratic equation in $x$, $kx^{2}+2x+1=0$, which has two distinct real roots, determine the range of values for $k$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "k < 1 \\text{ and } k \\neq 0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For the one-variable quadratic equation in $x$, $kx^{2}+2x+1=0$, which has two distinct real roots, determine the range of values for $k$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Bushes of two varieties of tomato seedlings are randomly mixed: 9 bushes of the \"Beliy Naliv\" variety and 7 bushes of the \"Verlioka\" variety. Find the probability that the first three consecutively planted bushes are seedlings of the \"Beliy Naliv\" variety.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0.15",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Bushes of two varieties of tomato seedlings are randomly mixed: 9 bushes of the "Beliy Naliv" variety and 7 bushes of the "Verlioka" variety. Find the probability that the first three consecutively planted bushes are seedlings of the "Beliy Naliv" variety. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A certain city started to vigorously develop its tourism industry in the year $2020$. According to statistics, the city's tourism revenue was approximately $2$ billion yuan in $2020$. It is estimated that the tourism revenue will reach around $2.88$ billion yuan in $2022$. Let $x$ be the annual average growth rate of the city's tourism revenue. Write down the equation based on the given information as ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2(1+x)^2 = 2.88",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A certain city started to vigorously develop its tourism industry in the year $2020$. According to statistics, the city's tourism revenue was approximately $2$ billion yuan in $2020$. It is estimated that the tourism revenue will reach around $2.88$ billion yuan in $2022$. Let $x$ be the annual average growth rate of the city's tourism revenue. Write down the equation based on the given information as ____. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Evaluate: \\\\( \\left( \\frac {x^{2}}{x-3} - \\frac {2x}{x-3} \\right) \\div \\frac {x}{x-3} \\\\), where \\\\(x= \\sqrt {7}+1\\\\). Round the result to \\\\(0.01\\\\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1.65",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Evaluate: \\( \left( \frac {x^{2}}{x-3} - \frac {2x}{x-3} \right) \div \frac {x}{x-3} \\), where \\(x= \sqrt {7}+1\\). Round the result to \\(0.01\\). |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $y=\\sin 2x$, determine the horizontal shift required to obtain the graph of the function $y=\\sin (2x-\\frac {\\pi}{4})$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac {\\pi}{8}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $y=\sin 2x$, determine the horizontal shift required to obtain the graph of the function $y=\sin (2x-\frac {\pi}{4})$. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Henry, Linda, and Jack went on a weekend getaway and decided to share the expenses evenly. Throughout their trip, Henry paid $120, Linda paid $150, and Jack paid $210. To balance the costs, Henry gave Jack $h$ dollars, and Linda gave Jack $l$ dollars. Calculate $h-l$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "30",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Henry, Linda, and Jack went on a weekend getaway and decided to share the expenses evenly. Throughout their trip, Henry paid $120, Linda paid $150, and Jack paid $210. To balance the costs, Henry gave Jack $h$ dollars, and Linda gave Jack $l$ dollars. Calculate $h-l$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In a programming workshop, the average age of the 50 members is 20 years. There are 22 girls, 18 boys, and 10 adults. If the average age of the girls is 18 years and the average age of the boys is 19 years, what is the average age of the adults?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "26.2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In a programming workshop, the average age of the 50 members is 20 years. There are 22 girls, 18 boys, and 10 adults. If the average age of the girls is 18 years and the average age of the boys is 19 years, what is the average age of the adults? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given a Cartesian coordinate system $xOy$ with its origin as the pole and the non-negative semi-axis of $x$ as the polar axis, a polar coordinate system is established with the same unit length. Known point $N$ has polar coordinates $( \\sqrt {2}, \\frac {π}{4})$, $M$ is any point on curve $C_{1}$: $ρ=1$, and point $G$ satisfies $\\overrightarrow{OG}= \\overrightarrow{OM}+ \\overrightarrow{ON}$. The trajectory of point $G$ is curve $C_{2}$.\n(1) Find the rectangular coordinate equation of curve $C_{2}$;\n(2) If the parametric equation of line $l$ passing through point $P(2,0)$ is $\\begin{cases} x=2- \\frac {1}{2}t \\\\y= \\frac { \\sqrt {3}}{2}t\\end{cases}(t$ as the parameter $)$, and line $l$ intersects curve $C_{2}$ at points $A$ and $B$, find the value of $\\frac {1}{|PA|}+ \\frac {1}{|PB|}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1+ \\sqrt {3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a Cartesian coordinate system $xOy$ with its origin as the pole and the non-negative semi-axis of $x$ as the polar axis, a polar coordinate system is established with the same unit length. Known point $N$ has polar coordinates $( \sqrt {2}, \frac {π}{4})$, $M$ is any point on curve $C_{1}$: $ρ=1$, and point $G$ satisfies $\overrightarrow{OG}= \overrightarrow{OM}+ \overrightarrow{ON}$. The trajectory of point $G$ is curve $C_{2}$.
(1) Find the rectangular coordinate equation of curve $C_{2}$;
(2) If the parametric equation of line $l$ passing through point $P(2,0)$ is $\begin{cases} x=2- \frac {1}{2}t \\y= \frac { \sqrt {3}}{2}t\end{cases}(t$ as the parameter $)$, and line $l$ intersects curve $C_{2}$ at points $A$ and $B$, find the value of $\frac {1}{|PA|}+ \frac {1}{|PB|}$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given a sequence $\\{a_{n}\\}$ where $a_{n}\\cdot a_{n+1}\\cdot a_{n+2}=-\\frac{1}{2}$, $a_{1}=-2$, and $a_{2}=\\frac{1}{4}$, find the maximum value of the product of the first $n$ terms of $\\{a_{n}\\}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a sequence $\{a_{n}\}$ where $a_{n}\cdot a_{n+1}\cdot a_{n+2}=-\frac{1}{2}$, $a_{1}=-2$, and $a_{2}=\frac{1}{4}$, find the maximum value of the product of the first $n$ terms of $\{a_{n}\}$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In $\\Delta ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and they satisfy $b\\cos A - a\\sin B = 0$.\n\n(1) Find the measure of angle $A$;\n\n(2) Given that $b = \\sqrt{2}$, and the area of $\\Delta ABC$ is $1$, find the length of side $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In $\Delta ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and they satisfy $b\cos A - a\sin B = 0$.
(1) Find the measure of angle $A$;
(2) Given that $b = \sqrt{2}$, and the area of $\Delta ABC$ is $1$, find the length of side $a$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Two buses operate on a circular route, traveling at the same speed with an interval of 21 minutes. What will the interval be if 3 buses operate on the same route at the same speed?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Two buses operate on a circular route, traveling at the same speed with an interval of 21 minutes. What will the interval be if 3 buses operate on the same route at the same speed? |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "From a $6 \\times 6$ grid square, gray triangles were cut out. What is the area of the remaining shape? The side length of each cell is 1 cm. Provide your answer in square centimeters.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | From a $6 \times 6$ grid square, gray triangles were cut out. What is the area of the remaining shape? The side length of each cell is 1 cm. Provide your answer in square centimeters. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "What is the greatest common divisor of $2^{2048} - 1$ and $2^{2035} - 1$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "8191",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the greatest common divisor of $2^{2048} - 1$ and $2^{2035} - 1$? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The diameter $AB$ of a circle has a radius of $3$. This diameter is extended to a point $D$ outside the circle such that $BD=4$. A point $E$ is placed so that $ED=6$ and line $ED$ is perpendicular to line $AD$. Segment $AE$ intersects the circle at a point $C$ between $A$ and $E$. What is the area of $\\triangle ABC$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{28}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The diameter $AB$ of a circle has a radius of $3$. This diameter is extended to a point $D$ outside the circle such that $BD=4$. A point $E$ is placed so that $ED=6$ and line $ED$ is perpendicular to line $AD$. Segment $AE$ intersects the circle at a point $C$ between $A$ and $E$. What is the area of $\triangle ABC$? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Calculate: $-2^2 \\times (-1)^{2015}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate: $-2^2 \times (-1)^{2015}$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The set of points $(x,y)$ as $t$ takes on all real values is given by:\n\\[\nx = t^2 - 3,\n\\]\n\\[\ny = t^3 - 6t + 2.\n\\]\nFind the ordered pair $(x, y)$ where the curve crosses itself.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(3, 2)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The set of points $(x,y)$ as $t$ takes on all real values is given by:
\[
x = t^2 - 3,
\]
\[
y = t^3 - 6t + 2.
\]
Find the ordered pair $(x, y)$ where the curve crosses itself. |
numina_aops_forum | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the set of solutions for $x$ in the inequality $\\dfrac{x+1}{x+2}>\\dfrac{3x+4}{2x+9}$ when $x\\neq 2$ , $x\\neq -\\dfrac{9}{2}$ .\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "x \\in (-\\frac{9}{2}, -2) \\cup \\left(\\frac{1-\\sqrt{5}}{2}, \\frac{1+\\sqrt{5}}{2}\\right)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the set of solutions for $x$ in the inequality $\dfrac{x+1}{x+2}>\dfrac{3x+4}{2x+9}$ when $x\neq 2$ , $x\neq -\dfrac{9}{2}$ . |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Regular octagon $ABCDEFGH$ is the base of right pyramid $PABCDEFGH$. If $PAE$ is an equilateral triangle with side length 10, then what is the volume of the pyramid?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1000 \\sqrt{6}}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Regular octagon $ABCDEFGH$ is the base of right pyramid $PABCDEFGH$. If $PAE$ is an equilateral triangle with side length 10, then what is the volume of the pyramid? |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Two real numbers are selected independently at random from the interval $[-30, 15]$. Calculate the probability that the product of those numbers is greater than zero.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{5}{9}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Two real numbers are selected independently at random from the interval $[-30, 15]$. Calculate the probability that the product of those numbers is greater than zero. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find a four-digit number that leaves a remainder of 112 when divided by 131 and a remainder of 98 when divided by 132.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1946",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find a four-digit number that leaves a remainder of 112 when divided by 131 and a remainder of 98 when divided by 132. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "For the binomial ${( \\frac{x}{2} - \\frac{1}{\\sqrt[3]{x}} )^{12}}$, the constant term in the expansion is ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-\\frac{55}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For the binomial ${( \frac{x}{2} - \frac{1}{\sqrt[3]{x}} )^{12}}$, the constant term in the expansion is ____. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $f(x)=2\\ln x+1-\\frac{a}{x}$, determine the range of real number $a$ for which $f(x)$ is an increasing function on the interval $\\left(1,+\\infty \\right)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left[-2,+\\infty \\right)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x)=2\ln x+1-\frac{a}{x}$, determine the range of real number $a$ for which $f(x)$ is an increasing function on the interval $\left(1,+\infty \right)$. |
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