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": "Among the following propositions: \n① If there are infinitely many points on line $l$ that are not within plane $\\alpha$, then $l$ is parallel to $\\alpha$; \n② If line $l$ is parallel to plane $\\alpha$, then it is parallel to any line within the plane; \n③ If one of two parallel lines is parallel to a plane, then the other line is also parallel to this plane; \n④ If line $l$ is parallel to plane $\\alpha$, then it has no common points with any line within the plane. \nThe correct proposition numbers are __________ (Note: Fill in the numbers of the propositions you think are correct).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Among the following propositions:
① If there are infinitely many points on line $l$ that are not within plane $\alpha$, then $l$ is parallel to $\alpha$;
② If line $l$ is parallel to plane $\alpha$, then it is parallel to any line within the plane;
③ If one of two parallel lines is parallel to a plane, then the other line is also parallel to this plane;
④ If line $l$ is parallel to plane $\alpha$, then it has no common points with any line within the plane.
The correct proposition numbers are __________ (Note: Fill in the numbers of the propositions you think are correct). |
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 \\( k \\) such that the sides \\( a \\), \\( b \\), and \\( c \\) of any triangle satisfy the inequality\n$$\n\\frac{b c}{b+c-a}+\\frac{a c}{a+c-b}+\\frac{a b}{a+b-c} \\geq k(a+b+c).\n$$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the largest real number \( k \) such that the sides \( a \), \( b \), and \( c \) of any triangle satisfy the inequality
$$
\frac{b c}{b+c-a}+\frac{a c}{a+c-b}+\frac{a b}{a+b-c} \geq k(a+b+c).
$$ |
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": "On a Cartesian plane, where each unit represents one meter, a cat is tied to a post at the point $(5, -2)$ with an $8$ meter long rope. Determine the shortest distance the cat can get to the origin.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2.615 \\text{ meters}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | On a Cartesian plane, where each unit represents one meter, a cat is tied to a post at the point $(5, -2)$ with an $8$ meter long rope. Determine the shortest distance the cat can get to the origin. |
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": "Calculate the sum of the infinite series: $\\frac{1}{20} + \\frac{2}{20^2} + \\frac{3}{20^3} + \\dots$, where the nth term is $\\frac{n}{20^n}$.\nA) $\\frac{20}{400}$\nB) $\\frac{20}{361}$\nC) $\\frac{1}{20}$\nD) $\\frac{20}{380}$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{20}{361}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the sum of the infinite series: $\frac{1}{20} + \frac{2}{20^2} + \frac{3}{20^3} + \dots$, where the nth term is $\frac{n}{20^n}$.
A) $\frac{20}{400}$
B) $\frac{20}{361}$
C) $\frac{1}{20}$
D) $\frac{20}{380}$ |
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 three positive numbers \\( a, b, \\mathrm{and} c \\) satisfying \\( a \\leq b+c \\leq 3a \\) and \\( 3b^2 \\leq a(a+c) \\leq 5b^2 \\), what is the minimum value of \\(\\frac{b-2c}{a}\\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-\\frac{18}{5}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given three positive numbers \( a, b, \mathrm{and} c \) satisfying \( a \leq b+c \leq 3a \) and \( 3b^2 \leq a(a+c) \leq 5b^2 \), what is the minimum value of \(\frac{b-2c}{a}\)? |
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 friends, Chris, David, and Emily listen to five different tracks and discuss their preferences. For each two-person combination among the friends, there exists at least one track liked by only those two friends and not by the third. Calculate the number of different ways this liking pattern can be established among the friends.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "88",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Three friends, Chris, David, and Emily listen to five different tracks and discuss their preferences. For each two-person combination among the friends, there exists at least one track liked by only those two friends and not by the third. Calculate the number of different ways this liking pattern can be established among the friends. |
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": "In an isosceles triangle \\( ABC \\) with \\( AB = BC \\), the altitude \\( CD \\) is drawn. The angle \\( BAC \\) is \\( \\alpha \\). The radius of the circle passing through the points \\( A, C, \\) and \\( D \\) is \\( R \\). Find the area of the triangle \\( ABC \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "R^2 \\tan(\\alpha)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In an isosceles triangle \( ABC \) with \( AB = BC \), the altitude \( CD \) is drawn. The angle \( BAC \) is \( \alpha \). The radius of the circle passing through the points \( A, C, \) and \( D \) is \( R \). Find the area of the triangle \( ABC \). |
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": "16 children of different heights stood in a circle, all facing the center. Each of them said: \"My right neighbor is taller than my left neighbor.\" What is the minimum number?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | 16 children of different heights stood in a circle, all facing the center. Each of them said: "My right neighbor is taller than my left neighbor." What is the minimum number? |
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 the height of the mountain range is 800 feet and the grade is reduced from 1.5% to 1%, calculate the additional length of the track needed.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "26667",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the height of the mountain range is 800 feet and the grade is reduced from 1.5% to 1%, calculate the additional length of the track needed. |
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 a sector has a central angle of $\\alpha$ and the radius of the circle is $R$. \n(1) If $\\alpha = 60^\\circ$ and $R = 10$ cm, find the arc length and the area of the sector. \n(2) If the perimeter of the sector is 12 cm, what is the value of $\\alpha$ in radians when the sector has the maximum area, and what is that maximum area?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "9 \\text{ cm}^2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that a sector has a central angle of $\alpha$ and the radius of the circle is $R$.
(1) If $\alpha = 60^\circ$ and $R = 10$ cm, find the arc length and the area of the sector.
(2) If the perimeter of the sector is 12 cm, what is the value of $\alpha$ in radians when the sector has the maximum area, and what is that maximum area? |
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 \\( z \\) and \\( w \\) are complex numbers such that\n\\[\n|z| = |w| = 1 \\quad \\text{and} \\quad z \\overline{w} + \\overline{z} w = 2.\n\\]\nFind the largest possible value of the real part of \\( z + w \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Suppose \( z \) and \( w \) are complex numbers such that
\[
|z| = |w| = 1 \quad \text{and} \quad z \overline{w} + \overline{z} w = 2.
\]
Find the largest possible value of the real part of \( z + w \). |
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 largest digit $M$ for which $4567M$ is divisible by 6?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the largest digit $M$ for which $4567M$ is divisible by 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": "The expression $25x^2 - 135x - 150$ can be written as $(5x + c)(5x + d),$ where $c$ and $d$ are integers. What is $c + 2d$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-59",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The expression $25x^2 - 135x - 150$ can be written as $(5x + c)(5x + d),$ where $c$ and $d$ are integers. What is $c + 2d$? |
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 function $y=f(x)$ is an increasing function on $\\mathbb{R}$, and $f(m+3) \\leq f(5)$, then the range of the real number $m$ is.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(-\\infty, 2]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the function $y=f(x)$ is an increasing function on $\mathbb{R}$, and $f(m+3) \leq f(5)$, then the range of the real number $m$ is. |
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": "6. A triple of integers $(a,b,c)$ is said to be $\\gamma$ *-special* if $a\\le \\gamma(b+c)$ , $b\\le \\gamma(c+a)$ and $c\\le\\gamma(a+b)$ . For each integer triple $(a,b,c)$ such that $1\\le a,b,c \\le 20$ , Kodvick writes down the smallest value of $\\gamma$ such that $(a,b,c)$ is $\\gamma$ -special. How many distinct values does he write down?\n\n*Proposed by winnertakeover*\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\sum_{a=1}^{20} (2\\phi(a) - 1) + 1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | 6. A triple of integers $(a,b,c)$ is said to be $\gamma$ *-special* if $a\le \gamma(b+c)$ , $b\le \gamma(c+a)$ and $c\le\gamma(a+b)$ . For each integer triple $(a,b,c)$ such that $1\le a,b,c \le 20$ , Kodvick writes down the smallest value of $\gamma$ such that $(a,b,c)$ is $\gamma$ -special. How many distinct values does he write down?
*Proposed by winnertakeover* |
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 $\\{a\\_n\\}$ where all terms are positive, it is known that $a\\_4 = 2a\\_2$, and $a\\_1$, $4$, $a\\_4$ form a geometric sequence.\n\n(1) Find the general term formula for the sequence $\\{a\\_n\\}$.\n(2) Find the sum of all $a\\_n$ that satisfy the following conditions: ① $20 ≤ n ≤ 116$; ② $n$ is divisible by $5$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2S_{20} = 2 \\times 1350 = 2700",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given an arithmetic sequence $\{a\_n\}$ where all terms are positive, it is known that $a\_4 = 2a\_2$, and $a\_1$, $4$, $a\_4$ form a geometric sequence.
(1) Find the general term formula for the sequence $\{a\_n\}$.
(2) Find the sum of all $a\_n$ that satisfy the following conditions: ① $20 ≤ n ≤ 116$; ② $n$ is divisible by $5$. |
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 function $f(x)$ satisfies the following conditions:\n1. The domain is $R$;\n2. The range is $[0, 2]$;\n3. $f(x) - f(-x) = 0$.\nFind a functional expression for $f(x) =$ \\_\\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "y = 2|\\cos(x)|",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the function $f(x)$ satisfies the following conditions:
1. The domain is $R$;
2. The range is $[0, 2]$;
3. $f(x) - f(-x) = 0$.
Find a functional expression for $f(x) =$ \_\_\_\_\_\_. |
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 square pyramid has a base side length of $a$, and its slant edge makes an angle $\\alpha$ with the base plane. A cube is inscribed in the pyramid such that four of its vertices lie on the slant heights of the pyramid and four vertices lie on the base of the pyramid. Find the edge length of the cube.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{a \\sin(\\alpha)}{2 \\sin \\left( \\frac{\\pi}{4} + \\alpha \\right) }",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A square pyramid has a base side length of $a$, and its slant edge makes an angle $\alpha$ with the base plane. A cube is inscribed in the pyramid such that four of its vertices lie on the slant heights of the pyramid and four vertices lie on the base of the pyramid. Find the edge length of the cube. |
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": "Supplementary Question: \n(B) The 2006 World Cup was held in Germany, with a total of 32 teams evenly divided into 8 groups to first compete in the group stage. A single round-robin tournament was held within each group (each team played against every other team in its group once), selecting two teams to advance to the round of 16. The total number of matches played in the group stage of this World Cup was ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Supplementary Question:
(B) The 2006 World Cup was held in Germany, with a total of 32 teams evenly divided into 8 groups to first compete in the group stage. A single round-robin tournament was held within each group (each team played against every other team in its group once), selecting two teams to advance to the round of 16. The total number of matches played in the group stage of this World Cup was ____. |
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 rationalized factor of $\\sqrt{x}+1$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\sqrt{x}-1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The rationalized factor of $\sqrt{x}+1$ 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": "Compute the product of the following matrices:\n\\[\n\\begin{pmatrix} 1 & 2 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 4 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 6 \\\\ 0 & 1 \\end{pmatrix} \\dotsm \\begin{pmatrix} 1 & 100 \\\\ 0 & 1 \\end{pmatrix}.\n\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\begin{pmatrix} 1 & 2550 \\\\ 0 & 1 \\end{pmatrix}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Compute the product of the following matrices:
\[
\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix} \dotsm \begin{pmatrix} 1 & 100 \\ 0 & 1 \end{pmatrix}.
\] |
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": "Sides $\\overline{AB}$ and $\\overline{CD}$ of regular hexagon $ABCDEF$ are extended to meet at point $Q$. What is the degree measure of angle $Q$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "60^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Sides $\overline{AB}$ and $\overline{CD}$ of regular hexagon $ABCDEF$ are extended to meet at point $Q$. What is the degree measure of angle $Q$? |
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": "Determine for which values of $x$ the following function is nonnegative:\n\\[\\frac{x-10x^2+35x^3}{9-x^3}\\]\nAnswer in interval notation.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "[3, \\infty)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine for which values of $x$ the following function is nonnegative:
\[\frac{x-10x^2+35x^3}{9-x^3}\]
Answer in interval notation. |
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": "Circle $\\Omega$ is the incircle of $\\triangle DEF$ and is also the circumcircle of $\\triangle PQR$. The point $P$ is on $\\overline{EF}$, the point $Q$ is on $\\overline{DE}$, and the point $R$ is on $\\overline{DF}$. If $\\angle D=50^\\circ$, $\\angle E=70^\\circ$, and $\\angle F=60^\\circ$, what is the measure of $\\angle QRP$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\angle QRP = 60^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Circle $\Omega$ is the incircle of $\triangle DEF$ and is also the circumcircle of $\triangle PQR$. The point $P$ is on $\overline{EF}$, the point $Q$ is on $\overline{DE}$, and the point $R$ is on $\overline{DF}$. If $\angle D=50^\circ$, $\angle E=70^\circ$, and $\angle F=60^\circ$, what is the measure of $\angle QRP$? |
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": "For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\\le y \\le z$ of the equation $$ 8^k=x^3+y^3+z^3-3xyz $$ \n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "k + 1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation $$ 8^k=x^3+y^3+z^3-3xyz $$ |
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 $a$, $b$, and $c$ are digits, not all nine and not all zero, and the repeating decimal $0.\\overline{abc}$ is expressed as a fraction in lowest terms. How many different denominators are possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Suppose $a$, $b$, and $c$ are digits, not all nine and not all zero, and the repeating decimal $0.\overline{abc}$ is expressed as a fraction in lowest terms. How many different denominators are 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": "Given the function $f(x)=x+a\\ln x$ at $x=1$, its tangent line is perpendicular to the line $x+2y=0$, and $g(x)=f(x)+ \\frac {1}{2}x^{2}-bx$.\n(1) Find the value of the real number $a$;\n(2) Let $x_{1}$, $x_{2}(x_{1} < x_{2})$ be the two extreme points of the function $g(x)$, if $|g(x_{1})-g(x_{2})|\\geqslant \\frac {3}{4}-\\ln 2$, find the range of $b$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "b > 1+ \\frac {3 \\sqrt {2}}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x)=x+a\ln x$ at $x=1$, its tangent line is perpendicular to the line $x+2y=0$, and $g(x)=f(x)+ \frac {1}{2}x^{2}-bx$.
(1) Find the value of the real number $a$;
(2) Let $x_{1}$, $x_{2}(x_{1} < x_{2})$ be the two extreme points of the function $g(x)$, if $|g(x_{1})-g(x_{2})|\geqslant \frac {3}{4}-\ln 2$, find the range of $b$. |
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": "Twelve tiles are numbered from 1 to 12. Each of three players randomly selects three tiles each. The probability that all three players obtain an odd sum and the total sum of all their tiles is greater than 36 is $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "206",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Twelve tiles are numbered from 1 to 12. Each of three players randomly selects three tiles each. The probability that all three players obtain an odd sum and the total sum of all their tiles is greater than 36 is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. |
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": "Calculate the standard deviation of the data 5, 7, 7, 8, 10, 11.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the standard deviation of the data 5, 7, 7, 8, 10, 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": "A set of data $12$, $34$, $15$, $24$, $39$, $25$, $31$, $48$, $32$, $36$, $36$, $37$, $42$, $50$ has a $75$th percentile of ____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A set of data $12$, $34$, $15$, $24$, $39$, $25$, $31$, $48$, $32$, $36$, $36$, $37$, $42$, $50$ has a $75$th percentile of ____. |
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": "Between \\(1^{2}, 2^{2}, 3^{2}, \\cdots, 1989^{2}\\), by inserting \"+\" and \"-\" signs, what is the smallest non-negative sum that can be obtained when evaluating the resulting expression?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Between \(1^{2}, 2^{2}, 3^{2}, \cdots, 1989^{2}\), by inserting "+" and "-" signs, what is the smallest non-negative sum that can be obtained when evaluating the resulting expression? |
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": "How many positive integers at most $420$ leave different remainders when divided by each of $5$ , $6$ , and $7$ ?\n\n*Proposed by Milan Haiman.*\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "386",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many positive integers at most $420$ leave different remainders when divided by each of $5$ , $6$ , and $7$ ?
*Proposed by Milan Haiman.* |
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": "The modified Fibonacci sequence starts with 2 and 1, and each subsequent term is the sum of its two predecessors. Which digit from 0 to 9 is the last to appear in the units position of a number in this modified Fibonacci sequence?\nA) 0\nB) 5\nC) 7\nD) 9\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "A) 0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The modified Fibonacci sequence starts with 2 and 1, and each subsequent term is the sum of its two predecessors. Which digit from 0 to 9 is the last to appear in the units position of a number in this modified Fibonacci sequence?
A) 0
B) 5
C) 7
D) 9 |
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": "Segment \\(AB\\) has midpoint \\(C\\), and segment \\(BC\\) has midpoint \\(D\\). Semi-circles are constructed with diameters \\(\\overline{AB}\\) and \\(\\overline{BC}\\) to form the entire region shown. Segment \\(CP\\) splits the region into parts with area ratio \\(2:3\\). If \\(CP\\) starts from \\(C\\) and terminates at the perimeter of the semicircle on \\(\\overline{AB}\\), what is the degree measure of angle \\(ACP\\)? Express your answer as a decimal to the nearest tenth.\n \n[asy]\ndraw((0,0)--10dir(180),linewidth(2));\ndraw((0,0)--10dir(40),linewidth(2)); // Adjust the angle here to approximately show the 2:3 ratio visually\ndraw((0,0)--10dir(0),dashed);\n\ndraw(10dir(180)..10dir(90)..10dir(0),linewidth(2));\n\ndraw((5,0)+3.33dir(180)..(5,0)+3.33dir(-90)..(5,0)+3.33dir(0),linewidth(2));\n\ndot((0,0));\ndot((5,0));\n\nlabel(\"A\",10dir(180),W);\nlabel(\"B\",10dir(0),E);\nlabel(\"C\",(0,0),SW);\nlabel(\"D\",3.33dir(0),NE);\nlabel(\"P\",10dir(40),NE);\n[/asy]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "120.0^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Segment \(AB\) has midpoint \(C\), and segment \(BC\) has midpoint \(D\). Semi-circles are constructed with diameters \(\overline{AB}\) and \(\overline{BC}\) to form the entire region shown. Segment \(CP\) splits the region into parts with area ratio \(2:3\). If \(CP\) starts from \(C\) and terminates at the perimeter of the semicircle on \(\overline{AB}\), what is the degree measure of angle \(ACP\)? Express your answer as a decimal to the nearest tenth.
[asy]
draw((0,0)--10dir(180),linewidth(2));
draw((0,0)--10dir(40),linewidth(2)); // Adjust the angle here to approximately show the 2:3 ratio visually
draw((0,0)--10dir(0),dashed);
draw(10dir(180)..10dir(90)..10dir(0),linewidth(2));
draw((5,0)+3.33dir(180)..(5,0)+3.33dir(-90)..(5,0)+3.33dir(0),linewidth(2));
dot((0,0));
dot((5,0));
label("A",10dir(180),W);
label("B",10dir(0),E);
label("C",(0,0),SW);
label("D",3.33dir(0),NE);
label("P",10dir(40),NE);
[/asy] |
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": "Calculate the value of $5^2 + 5^2 + 5^2 + 5^2$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "100",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the value of $5^2 + 5^2 + 5^2 + 5^2$. |
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": "Given a three-digit number with digits that sum exactly to $14$ and have the first digit equal to the last digit, find the total count of such numbers that are not divisible by $5$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a three-digit number with digits that sum exactly to $14$ and have the first digit equal to the last digit, find the total count of such numbers that are not divisible by $5$. |
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 following variables: ① Smoking, ② Gender, ③ Religious Belief, ④ Nationality \nAmong these, the categorical variables are __________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "②③④",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the following variables: ① Smoking, ② Gender, ③ Religious Belief, ④ Nationality
Among these, the categorical variables are __________. |
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": "Players $A$ and $B$ are playing a chess match under the following conditions: The winner is the first to reach (at least) 2 points; if they reach 2-2 points simultaneously, the match is a draw. (A win earns 1 point, a draw earns $1/2$ point, and a loss earns 0 points).\n\na) What is the expected number of games, given that the probability of a draw in each game is 0.5, and the probability of $A$ winning is 0.3?\n\nb) What is the probability that $B$ wins the match?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0.315",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Players $A$ and $B$ are playing a chess match under the following conditions: The winner is the first to reach (at least) 2 points; if they reach 2-2 points simultaneously, the match is a draw. (A win earns 1 point, a draw earns $1/2$ point, and a loss earns 0 points).
a) What is the expected number of games, given that the probability of a draw in each game is 0.5, and the probability of $A$ winning is 0.3?
b) What is the probability that $B$ wins the match? |
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 Bob requires $3\\frac{1}{3}$ cups of flour and $1\\frac{1}{2}$ cups of milk, calculate the total number of times Bob must fill the measuring cup, which holds $\\frac{1}{3}$ cup of flour.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given Bob requires $3\frac{1}{3}$ cups of flour and $1\frac{1}{2}$ cups of milk, calculate the total number of times Bob must fill the measuring cup, which holds $\frac{1}{3}$ cup of flour. |
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": "The line with equation \\( y = 2x - 6 \\) is translated upwards by 4 units. Find the \\( x \\)-intercept of the resulting line.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The line with equation \( y = 2x - 6 \) is translated upwards by 4 units. Find the \( x \)-intercept of the resulting line. |
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 manager plans to meet three interns at a café. Each of them can arrive randomly between 3:00 and 5:00 p.m. The manager will leave if not all interns are present upon his arrival. Each intern will wait for up to 30 minutes for the others before leaving. What is the probability that the meeting successfully takes place?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0.25",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A manager plans to meet three interns at a café. Each of them can arrive randomly between 3:00 and 5:00 p.m. The manager will leave if not all interns are present upon his arrival. Each intern will wait for up to 30 minutes for the others before leaving. What is the probability that the meeting successfully takes place? |
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": "Variables $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 40, their difference is 8. Find the value of $y$ when $x$ equals 7.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "y = 54\\frac{6}{7}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Variables $x$ and $y$ are inversely proportional. When the sum of $x$ and $y$ is 40, their difference is 8. Find the value of $y$ when $x$ equals 7. |
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 $S$ be a set of $7$ integers taken from $\\{1, 2, \\dots, 15\\}$ with the property that if $a$ and $b$ are elements of $S$ with $a < b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$?\nA) 2\nB) 3\nC) 4 \nD) 5\nE) 6\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "C) 4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $S$ be a set of $7$ integers taken from $\{1, 2, \dots, 15\}$ with the property that if $a$ and $b$ are elements of $S$ with $a < b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$?
A) 2
B) 3
C) 4
D) 5
E) 6 |
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 center of the circle $(x-a)^2+(y-b)^2=r^2$ is the focus of the parabola $y^2=4x$, and it is tangent to the line $3x+4y+2=0$, then the equation of the circle is __________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(x-1)^2+y^2=1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the center of the circle $(x-a)^2+(y-b)^2=r^2$ is the focus of the parabola $y^2=4x$, and it is tangent to the line $3x+4y+2=0$, then the equation of the circle 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": "Homewood Middle School has 1500 students, and 900 of these students attend a spring festival. If three-fourths of the girls in the school and two-fifths of the boys in the school attend the festival, how many girls attend the festival? (Assume that each student in the school is either a boy or a girl.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "643",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Homewood Middle School has 1500 students, and 900 of these students attend a spring festival. If three-fourths of the girls in the school and two-fifths of the boys in the school attend the festival, how many girls attend the festival? (Assume that each student in the school is either a boy or a girl.) |
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 negation of the proposition \"$\\forall x \\in \\mathbb{R}, x^2 > 0$\" is $\\_\\_\\_\\_\\_\\_$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\exists x\\_0 \\in \\mathbb{R}, x\\_0^2 \\leqslant 0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The negation of the proposition "$\forall x \in \mathbb{R}, x^2 > 0$" 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 $f(x)=2x-\\frac{2}{x}-2\\ln x$, determine the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "y = 2x - 2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x)=2x-\frac{2}{x}-2\ln x$, determine the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$. |
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": "A taxi ride costs $\\$2.00$ plus $\\$0.30$ per mile for the first 3 miles and $\\$0.40$ per mile for each additional mile. How much, in dollars, does an 8-mile taxi ride cost?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\$4.90",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A taxi ride costs $\$2.00$ plus $\$0.30$ per mile for the first 3 miles and $\$0.40$ per mile for each additional mile. How much, in dollars, does an 8-mile taxi ride cost? |
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 $x,$ $y,$ $z$ be complex numbers such that\n\\begin{align*}\nxy + 5y &= -20, \\\\\nyz + 5z &= -20, \\\\\nzx + 5x &= -20.\n\\end{align*}\nDetermine all possible values of $xyz.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "xyz = 100",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $x,$ $y,$ $z$ be complex numbers such that
\begin{align*}
xy + 5y &= -20, \\
yz + 5z &= -20, \\
zx + 5x &= -20.
\end{align*}
Determine all possible values of $xyz.$ |
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": "I have 7 books, three of which are identical copies of the same math book, and the rest are different. How many ways can I arrange them on a shelf?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "840",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | I have 7 books, three of which are identical copies of the same math book, and the rest are different. How many ways can I arrange them on a shelf? |
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 distance between two parallel lines $x + 2y -1 = 0$ and $2x + 4y + 3 = 0$ is\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The distance between two parallel lines $x + 2y -1 = 0$ and $2x + 4y + 3 = 0$ 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": "Determine the sides of a triangle, if they are expressed as integers forming an arithmetic progression, and if the perimeter of the triangle is 15.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "5, 5, 5, \\quad 4, 5, 6, \\quad 3, 5, 7",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine the sides of a triangle, if they are expressed as integers forming an arithmetic progression, and if the perimeter of the triangle is 15. |
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 trapezoid $PQRS$, the parallel sides $PQ$ and $RS$ have lengths of 10 and 25 units, respectively, and the altitude is 15 units. Points $T$ and $U$ are the midpoints of sides $PS$ and $QR$, respectively. What is the area of quadrilateral $TURS$ in square units?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "159.375",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In trapezoid $PQRS$, the parallel sides $PQ$ and $RS$ have lengths of 10 and 25 units, respectively, and the altitude is 15 units. Points $T$ and $U$ are the midpoints of sides $PS$ and $QR$, respectively. What is the area of quadrilateral $TURS$ in square units? |
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": "If $a$ and $b$ are reciprocals of each other, then the value of $a^{2}b-\\left(a-2023\\right)$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2023",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $a$ and $b$ are reciprocals of each other, then the value of $a^{2}b-\left(a-2023\right)$ 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": "A logistics company plans to establish a warehouse to store goods. After market research, the following information is obtained: the monthly land occupation fee $y_{1}$ (unit: ten thousand yuan) is inversely proportional to the distance $x$ (unit: km) from the warehouse to the station, and the monthly inventory fee $y_{2}$ (unit: ten thousand yuan) is directly proportional to $x$. If a warehouse is established at a distance of $10$ km from the station, then $y_{1}$ and $y_{2}$ are $4$ ten thousand yuan and $16$ ten thousand yuan, respectively. Find the value of $x$ when the sum of the two fees is minimized.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A logistics company plans to establish a warehouse to store goods. After market research, the following information is obtained: the monthly land occupation fee $y_{1}$ (unit: ten thousand yuan) is inversely proportional to the distance $x$ (unit: km) from the warehouse to the station, and the monthly inventory fee $y_{2}$ (unit: ten thousand yuan) is directly proportional to $x$. If a warehouse is established at a distance of $10$ km from the station, then $y_{1}$ and $y_{2}$ are $4$ ten thousand yuan and $16$ ten thousand yuan, respectively. Find the value of $x$ when the sum of the two fees is minimized. |
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 function $y=f(x)$, where $x \\in [0,+\\infty)$, for a given non-zero real number $a$, there always exists a non-zero constant $T$ such that for any real number $x$ in the domain $[0,+\\infty)$, $af(x)=f(x+T)$ always holds. In this case, $T$ is called the quasi-period of $f(x)$, and the function $y=f(x)$ is an $a$-level quasi-periodic function on $[0,+\\infty)$. If the function $y=f(x)$ is a 2-level quasi-periodic function defined on the interval $[0,+\\infty)$ with $T=2$, and when $x \\in [0,2)$, $f(x)= \\begin{cases} \\frac {1}{2}-2x^{2}, & 0\\leqslant x\\leqslant 1 \\\\ f(2-x), & 1 < x < 2 \\end{cases}$. The function $g(x)=-2\\ln x+ \\frac {1}{2}x^{2}+x+m$. Determine the range of values for the real number $m$ such that there exists $x_{1} \\in [6,8]$ and $x_{2} \\in (0,+\\infty)$ such that $g(x_{2})-f(x_{1})\\leqslant 0$ holds.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(-\\infty, \\frac {13}{2}]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a function $y=f(x)$, where $x \in [0,+\infty)$, for a given non-zero real number $a$, there always exists a non-zero constant $T$ such that for any real number $x$ in the domain $[0,+\infty)$, $af(x)=f(x+T)$ always holds. In this case, $T$ is called the quasi-period of $f(x)$, and the function $y=f(x)$ is an $a$-level quasi-periodic function on $[0,+\infty)$. If the function $y=f(x)$ is a 2-level quasi-periodic function defined on the interval $[0,+\infty)$ with $T=2$, and when $x \in [0,2)$, $f(x)= \begin{cases} \frac {1}{2}-2x^{2}, & 0\leqslant x\leqslant 1 \\ f(2-x), & 1 < x < 2 \end{cases}$. The function $g(x)=-2\ln x+ \frac {1}{2}x^{2}+x+m$. Determine the range of values for the real number $m$ such that there exists $x_{1} \in [6,8]$ and $x_{2} \in (0,+\infty)$ such that $g(x_{2})-f(x_{1})\leqslant 0$ 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": "Given $3\\cos (\\frac{π}{2}-θ) + \\cos (π+θ) = 0$, find the value of $\\cos 2θ$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $3\cos (\frac{π}{2}-θ) + \cos (π+θ) = 0$, find the value of $\cos 2θ$. |
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 $N,O$ be functions such that $N(x) = 3\\sqrt{x}$, and $O(x) = x^2$. What is $N(O(N(O(N(O(4))))))$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "108",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $N,O$ be functions such that $N(x) = 3\sqrt{x}$, and $O(x) = x^2$. What is $N(O(N(O(N(O(4))))))$? |
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": "Integers $x_1,x_2,\\cdots,x_{100}$ satisfy \\[ \\frac {1}{\\sqrt{x_1}} + \\frac {1}{\\sqrt{x_2}} + \\cdots + \\frac {1}{\\sqrt{x_{100}}} = 20. \\]Find $ \\displaystyle\\prod_{i \\ne j} \\left( x_i - x_j \\right) $ .\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Integers $x_1,x_2,\cdots,x_{100}$ satisfy \[ \frac {1}{\sqrt{x_1}} + \frac {1}{\sqrt{x_2}} + \cdots + \frac {1}{\sqrt{x_{100}}} = 20. \]Find $ \displaystyle\prod_{i \ne j} \left( x_i - x_j \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 geometric sequence $\\{a_{n}\\}$, if $a_{n} > 0$, and $a_{3}$, $a_{7}$ are the two roots of the equation $x^{2}-32x+64=0$, find the value of $\\log _{2}a_{1}+\\log _{2}a_{2}+\\log _{2}a_{3}+...+\\log _{2}a_{9}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the geometric sequence $\{a_{n}\}$, if $a_{n} > 0$, and $a_{3}$, $a_{7}$ are the two roots of the equation $x^{2}-32x+64=0$, find the value of $\log _{2}a_{1}+\log _{2}a_{2}+\log _{2}a_{3}+...+\log _{2}a_{9}$. |
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 $f$ is a mapping from set $M = \\{a, b, c, d\\}$ to set $N = \\{0, 1, 2\\}$, with $f(a) + f(b) + f(c) + f(d) = 4$, determine the number of different mappings.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "19",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that $f$ is a mapping from set $M = \{a, b, c, d\}$ to set $N = \{0, 1, 2\}$, with $f(a) + f(b) + f(c) + f(d) = 4$, determine the number of different mappings. |
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": "If the product of two positive numbers is 12 and the reciprocal of one of these numbers is 5 times the reciprocal of the other number, find the sum of these two numbers.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{6\\sqrt{60}}{5}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If the product of two positive numbers is 12 and the reciprocal of one of these numbers is 5 times the reciprocal of the other number, find the sum of these two numbers. |
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 three vertices of $\\triangle ABC$ are $A(2,3)$, $B(-1,-2)$, and $C(-3,4)$, find:\n\n(I) the equation of the line containing the median $AD$ where $D$ is the midpoint of $BC$;\n\n(II) the area of $\\triangle ABC$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the three vertices of $\triangle ABC$ are $A(2,3)$, $B(-1,-2)$, and $C(-3,4)$, find:
(I) the equation of the line containing the median $AD$ where $D$ is the midpoint of $BC$;
(II) 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 (1) $sin60°-\\sqrt{3}cos60°+\\frac{1}{2}tan45°$. (2) Given three numbers $a$, $b$, $c$, where $b$ is the mean proportional between $a$ and $c$. If $a=9$ and $c=4$, find the value of $b$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\pm 6",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate (1) $sin60°-\sqrt{3}cos60°+\frac{1}{2}tan45°$. (2) Given three numbers $a$, $b$, $c$, where $b$ is the mean proportional between $a$ and $c$. If $a=9$ and $c=4$, find the value of $b$. |
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": "Let $M=\\{2,4\\}$, $N=\\{a,b\\}$. If $M=N$, then $\\log_a b =$ ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\log_a b = 2 \\text{ or } \\frac{1}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $M=\{2,4\}$, $N=\{a,b\}$. If $M=N$, then $\log_a b =$ ______. |
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": "If $\\frac{q}{r} = 12$, $\\frac{s}{r} = 8$, and $\\frac{s}{t} = \\frac{1}{3}$, then what is $\\frac{t}{q}$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $\frac{q}{r} = 12$, $\frac{s}{r} = 8$, and $\frac{s}{t} = \frac{1}{3}$, then what is $\frac{t}{q}$? |
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": "Let \\( n \\) be an integer greater than 1. Within a square with side length \\( n \\), we want to place \\( n^2 \\) closed discs of unit diameter without any overlap. Is there an \\( n \\) for which this is possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let \( n \) be an integer greater than 1. Within a square with side length \( n \), we want to place \( n^2 \) closed discs of unit diameter without any overlap. Is there an \( n \) for which this is 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": "A certain TV station broadcasts 6 advertisements in a row, including three different commercial advertisements, two different Asian Games promotional advertisements, and one public service advertisement. The last advertisement cannot be a commercial advertisement, and the Asian Games promotional advertisements and the public service advertisement cannot be played consecutively. Also, the two Asian Games promotional advertisements cannot be played consecutively. Find the total number of different broadcasting methods.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "108",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A certain TV station broadcasts 6 advertisements in a row, including three different commercial advertisements, two different Asian Games promotional advertisements, and one public service advertisement. The last advertisement cannot be a commercial advertisement, and the Asian Games promotional advertisements and the public service advertisement cannot be played consecutively. Also, the two Asian Games promotional advertisements cannot be played consecutively. Find the total number of different broadcasting methods. |
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 complex number $z$ satisfying $(3+4i)z=1$ (where $i$ is the imaginary unit), find the modulus of $z$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a complex number $z$ satisfying $(3+4i)z=1$ (where $i$ is the imaginary unit), find the modulus of $z$. |
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 a prism with base \\(ABC\\), draw a line \\(EA \\parallel B_1C_1\\) in the plane of the base \\(ABC\\). Let \\(EA = MC_1\\), \\(ME \\parallel AC_1\\), and \\(ME\\) lie in the plane of the section. In the plane of the base \\(ABC\\), draw a line connecting point \\(E\\) to the midpoint \\(D\\) of side \\(AB\\). Point \\(K\\) is the intersection of this line with side \\(BC\\).\n\nIn the plane of the base \\(A_1B_1C_1\\), draw a line \\(MN \\parallel DK\\) such that point \\(N\\) is the intersection of line \\(MN\\) with side \\(A_1B_1\\). The quadrilateral \\(DKMN\\) is the required section.\n\nProject the section onto the base of the prism, denote the side of the base \\(a\\). Then \\(BK = \\frac{3a}{4}\\), \\(BD = \\frac{a}{2}\\), and \\(BQ = \\frac{a}{4}\\) where \\(Q\\) is the projection of point \\(M\\) on the base \\(ABC\\). Furthermore, let \\(G\\) be the projection of point \\(N\\) on the base \\(ABC\\). Given \\(GQ \\parallel DK\\), deduce that \\(\\frac{BQ}{BK} = \\frac{BG}{BD}\\), leading to \\(BG = \\frac{a}{6}\\). This establishes \\(B_1N:BD = B_1M:BK = NM:DK = 1:3\\).\n\nFind the cosine of the angle \\(\\alpha\\) of the inclination of the section plane to the base plane. The distance \\(d\\) from point \\(C\\) to the section plane is one-third of the distance from point \\(B\\) to the section plane, given \\(BK = 3 CK\\) and \\(K\\) belonging to the section plane.\n\nConstruct plane \\(BHL\\) through point \\(B\\) perpendicular to the intersection line \\(DK\\). Then \\(BH \\perp DK\\), and \\(LH \\perp DK\\). Draw \\(BP \\perp LH\\), such that the distance \\(3d\\) equals \\(BP\\). Therefore, the angle of inclination of the section plane to the base plane equals angle \\(BHL\\).\n\nCalculate:\n1. \\(\\cos \\alpha = \\sqrt{\\frac{2}{3}}\\)\n2. Area of the quadrilateral \\(LBKDK\\)\n3. Volume of the section \\(V_{LB, NM}\\)\n4. Total volume \\(V_{ABC, A_1B_1C_1}\\)\n5. Volume of the section \\(V_{ACKD, ACMN}\\)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{91 \\sqrt{2}}{3}, \\frac{413 \\sqrt{2}}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a prism with base \(ABC\), draw a line \(EA \parallel B_1C_1\) in the plane of the base \(ABC\). Let \(EA = MC_1\), \(ME \parallel AC_1\), and \(ME\) lie in the plane of the section. In the plane of the base \(ABC\), draw a line connecting point \(E\) to the midpoint \(D\) of side \(AB\). Point \(K\) is the intersection of this line with side \(BC\).
In the plane of the base \(A_1B_1C_1\), draw a line \(MN \parallel DK\) such that point \(N\) is the intersection of line \(MN\) with side \(A_1B_1\). The quadrilateral \(DKMN\) is the required section.
Project the section onto the base of the prism, denote the side of the base \(a\). Then \(BK = \frac{3a}{4}\), \(BD = \frac{a}{2}\), and \(BQ = \frac{a}{4}\) where \(Q\) is the projection of point \(M\) on the base \(ABC\). Furthermore, let \(G\) be the projection of point \(N\) on the base \(ABC\). Given \(GQ \parallel DK\), deduce that \(\frac{BQ}{BK} = \frac{BG}{BD}\), leading to \(BG = \frac{a}{6}\). This establishes \(B_1N:BD = B_1M:BK = NM:DK = 1:3\).
Find the cosine of the angle \(\alpha\) of the inclination of the section plane to the base plane. The distance \(d\) from point \(C\) to the section plane is one-third of the distance from point \(B\) to the section plane, given \(BK = 3 CK\) and \(K\) belonging to the section plane.
Construct plane \(BHL\) through point \(B\) perpendicular to the intersection line \(DK\). Then \(BH \perp DK\), and \(LH \perp DK\). Draw \(BP \perp LH\), such that the distance \(3d\) equals \(BP\). Therefore, the angle of inclination of the section plane to the base plane equals angle \(BHL\).
Calculate:
1. \(\cos \alpha = \sqrt{\frac{2}{3}}\)
2. Area of the quadrilateral \(LBKDK\)
3. Volume of the section \(V_{LB, NM}\)
4. Total volume \(V_{ABC, A_1B_1C_1}\)
5. Volume of the section \(V_{ACKD, ACMN}\) |
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": "Determine the volume in cubic inches of a right, rectangular prism where the areas of the side, front, and bottom faces are 20 square inches, 12 square inches, and 8 square inches, respectively.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "8\\sqrt{30}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine the volume in cubic inches of a right, rectangular prism where the areas of the side, front, and bottom faces are 20 square inches, 12 square inches, and 8 square inches, respectively. |
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": "Xiao Min plans to buy a bouquet of lilies and carnations to give to her mother on Mother's Day. It is known that buying 1 lily and 3 carnations costs $23$, and the price of 2 lilies is $4$ more than the price of 1 carnation.\n$(1)$ Find the cost of buying one carnation and one lily each.\n$(2)$ Xiao Min plans to buy a total of 12 lilies and carnations, with at least 5 carnations. Let the cost of buying this bouquet of flowers be $w$ dollars, and let the number of lilies be $x$. Find the functional relationship between $w$ and $x$, and design a flower-buying plan to minimize the cost, stating the minimum cost.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "65",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Xiao Min plans to buy a bouquet of lilies and carnations to give to her mother on Mother's Day. It is known that buying 1 lily and 3 carnations costs $23$, and the price of 2 lilies is $4$ more than the price of 1 carnation.
$(1)$ Find the cost of buying one carnation and one lily each.
$(2)$ Xiao Min plans to buy a total of 12 lilies and carnations, with at least 5 carnations. Let the cost of buying this bouquet of flowers be $w$ dollars, and let the number of lilies be $x$. Find the functional relationship between $w$ and $x$, and design a flower-buying plan to minimize the cost, stating the minimum cost. |
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 R be the area enclosed by the x-axis, the line x=10, and the curve defined by the piecewise function f={(x,y) | y=2x when 0 ≤ x ≤ 6, y=3x-9 when 6 ≤ x ≤ 10}. Calculate the area R enclosed by the x-axis, the line x=10, and the curve.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "102",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let R be the area enclosed by the x-axis, the line x=10, and the curve defined by the piecewise function f={(x,y) | y=2x when 0 ≤ x ≤ 6, y=3x-9 when 6 ≤ x ≤ 10}. Calculate the area R enclosed by the x-axis, the line x=10, and the curve. |
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": "A circle with center \\( O \\) and radius \\( r \\) and a point \\( P \\) inside this circle are given. Point \\( P \\) is located a distance \\( \\frac{r}{2} \\) from \\( O \\). Determine the locus of the midpoint \\( M \\) of the segment \\( PQ \\) as point \\( Q \\) varies over the circle.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\text{Circle with center at }\\left(\\frac{O_x + P_x}{2}, \\frac{O_y + P_y}{2}\\right) \\text{ and radius }\\frac{r}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A circle with center \( O \) and radius \( r \) and a point \( P \) inside this circle are given. Point \( P \) is located a distance \( \frac{r}{2} \) from \( O \). Determine the locus of the midpoint \( M \) of the segment \( PQ \) as point \( Q \) varies over the circle. |
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 of the four endpoints of the axes of an ellipse are, in some order, \\[(1, 5), \\; (4, -3), \\; (9, 5).\\] Find the distance between the foci of the ellipse.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Three of the four endpoints of the axes of an ellipse are, in some order, \[(1, 5), \; (4, -3), \; (9, 5).\] Find the distance between the foci of the ellipse. |
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 general form of the plane equation \\( 2x - 2y + z - 20 = 0 \\).\n\nTransform this equation into: \n1) the intercept form of the plane equation;\n2) the normal form of the plane equation.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-\\frac{2}{3} x + \\frac{2}{3} y - \\frac{1}{3} z + \\frac{20}{3} = 0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the general form of the plane equation \( 2x - 2y + z - 20 = 0 \).
Transform this equation into:
1) the intercept form of the plane equation;
2) the normal form of the plane equation. |
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 club has $4$ members from each of $4$ different schools, totaling $16$ members. Determine the number of ways to arrange a presidency meeting under the following conditions:\n\ni. The club must choose one of the $4$ schools at which to host the meeting.\n\nii. The host school sends $3$ representatives to the meeting, and each of the other three schools sends $1$ representative.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1024",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A club has $4$ members from each of $4$ different schools, totaling $16$ members. Determine the number of ways to arrange a presidency meeting under the following conditions:
i. The club must choose one of the $4$ schools at which to host the meeting.
ii. The host school sends $3$ representatives to the meeting, and each of the other three schools sends $1$ representative. |
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 mode and median of the data set 5, 4, 6, 8, 7, 7 can be determined.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "7, 6.5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The mode and median of the data set 5, 4, 6, 8, 7, 7 can be determined. |
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": "Simplify: $\\arctan \\frac{1+|x|-\\sqrt{1-x^{2}}}{1+|x|+\\sqrt{1-x^{2}}}+\\frac{1}{2} \\arccos |x|(-1 \\leqslant x \\leqslant 1)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Simplify: $\arctan \frac{1+|x|-\sqrt{1-x^{2}}}{1+|x|+\sqrt{1-x^{2}}}+\frac{1}{2} \arccos |x|(-1 \leqslant x \leqslant 1)$. |
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": "A right triangle has leg lengths 6 cm and 8 cm. The hypotenuse of a similar right triangle is 20 cm. What is the perimeter of the larger triangle?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "48 \\text{ cm}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A right triangle has leg lengths 6 cm and 8 cm. The hypotenuse of a similar right triangle is 20 cm. What is the perimeter of the larger triangle? |
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 set $M=\\{y|y=2\\sin x, x \\in [-5,5]\\}$, and the set $N=\\{x|y=\\log_{2}(x-1)\\}$. Find the intersection $M \\cap N$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(1,2]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let the set $M=\{y|y=2\sin x, x \in [-5,5]\}$, and the set $N=\{x|y=\log_{2}(x-1)\}$. Find the intersection $M \cap N$. |
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": "Given that triangle \\( ABC \\) has all side lengths as positive integers, \\(\\angle A = 2 \\angle B\\), and \\(CA = 9\\), what is the minimum possible value of \\( BC \\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that triangle \( ABC \) has all side lengths as positive integers, \(\angle A = 2 \angle B\), and \(CA = 9\), what is the minimum possible value of \( BC \)? |
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": "If $n > 0$, find the minimum value of $n + \\frac{32}{n^{2}}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $n > 0$, find the minimum value of $n + \frac{32}{n^{2}}$. |
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 that the one-variable quadratic equation in $x$ is $\\left(k-3\\right)x^{2}+6x+k^{2}-k=0$ and one of its roots is $-1$, find the value of $k$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the one-variable quadratic equation in $x$ is $\left(k-3\right)x^{2}+6x+k^{2}-k=0$ and one of its roots is $-1$, find 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": "In the diagram, the square has a perimeter of $64$ and the triangle has a height of $32.$ The triangle's base is equal to the side length of the square, and the square and the triangle have the same area. What is the value of $x?$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the diagram, the square has a perimeter of $64$ and the triangle has a height of $32.$ The triangle's base is equal to the side length of the square, and the square and the triangle have the same area. What is the value of $x?$ |
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 $\\{a_n\\}$ is a geometric sequence, and $a_{1}a_{13}+2 a_{ 7 }^{ 2 }=4π$, find the value of $\\tan({a_{2}a_{12}})$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the sequence $\{a_n\}$ is a geometric sequence, and $a_{1}a_{13}+2 a_{ 7 }^{ 2 }=4π$, find the value of $\tan({a_{2}a_{12}})$. |
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 $a, b \\in \\mathbb{R}$, if $(a+3i) + (2-i) = 5+bi$, then calculate the value of $ab$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $a, b \in \mathbb{R}$, if $(a+3i) + (2-i) = 5+bi$, then calculate the value of $ab$. |
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": "Given a set $S$ of $n$ variables, a binary operation $\\times$ on $S$ is called *simple* if it satisfies $(x \\times y) \\times z = x \\times (y \\times z)$ for all $x,y,z \\in S$ and $x \\times y \\in \\{x,y\\}$ for all $x,y \\in S$ . Given a simple operation $\\times$ on $S$ , any string of elements in $S$ can be reduced to a single element, such as $xyz \\to x \\times (y \\times z)$ . A string of variables in $S$ is called*full*if it contains each variable in $S$ at least once, and two strings are *equivalent* if they evaluate to the same variable regardless of which simple $\\times$ is chosen. For example $xxx$ , $xx$ , and $x$ are equivalent, but these are only full if $n=1$ . Suppose $T$ is a set of strings such that any full string is equivalent to exactly one element of $T$ . Determine the number of elements of $T$ .\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "n!^2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a set $S$ of $n$ variables, a binary operation $\times$ on $S$ is called *simple* if it satisfies $(x \times y) \times z = x \times (y \times z)$ for all $x,y,z \in S$ and $x \times y \in \{x,y\}$ for all $x,y \in S$ . Given a simple operation $\times$ on $S$ , any string of elements in $S$ can be reduced to a single element, such as $xyz \to x \times (y \times z)$ . A string of variables in $S$ is called*full*if it contains each variable in $S$ at least once, and two strings are *equivalent* if they evaluate to the same variable regardless of which simple $\times$ is chosen. For example $xxx$ , $xx$ , and $x$ are equivalent, but these are only full if $n=1$ . Suppose $T$ is a set of strings such that any full string is equivalent to exactly one element of $T$ . Determine the number of elements of $T$ . |
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": "Coach Dexter is preparing a 6-person starting lineup for his basketball team, the Dexters. There are 15 players on the team. Three of them, Alpha, Bravo, and Charlie, have been unanimously voted as season All-Stars and will definitely be in the starting lineup. How many different starting lineups are possible?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "220",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Coach Dexter is preparing a 6-person starting lineup for his basketball team, the Dexters. There are 15 players on the team. Three of them, Alpha, Bravo, and Charlie, have been unanimously voted as season All-Stars and will definitely be in the starting lineup. How many different starting lineups are 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": "Find the value of the 50th term of the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ….\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the value of the 50th term of the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 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": "A high school's senior class has 295 students, numbered 1, 2, …, 295. To understand the students' learning situation, a sample needs to be drawn at a 1:5 ratio using systematic sampling. Describe the process.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "59",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A high school's senior class has 295 students, numbered 1, 2, …, 295. To understand the students' learning situation, a sample needs to be drawn at a 1:5 ratio using systematic sampling. Describe 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": "Determine the interval(s) where the function $y = x - \\ln x$ is monotonically decreasing.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(0, 1)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine the interval(s) where the function $y = x - \ln x$ is monotonically decreasing. |
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 $\\frac{3}{7}$ divided by 4, then added to $\\frac{2}{7}$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{11}{28}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is $\frac{3}{7}$ divided by 4, then added to $\frac{2}{7}$? |
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": "A town hall clock chimes every 18 minutes, a library bell rings every 24 minutes, and a railway station clock chimes every 30 minutes. If all three chime together at 9:00 AM, at what time will they next all chime together?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "03\\!:\\!00",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A town hall clock chimes every 18 minutes, a library bell rings every 24 minutes, and a railway station clock chimes every 30 minutes. If all three chime together at 9:00 AM, at what time will they next all chime together? |
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 $f(x) = \\begin{cases} \\sin(\\pi x) + 1, & x \\leqslant 0 \\\\ \\log_{2}(3x^{2}-12x+15), & x > 0 \\end{cases}$, find the sum of all the zeros of the function $y=f(x)-1$ in the interval $[-3,3]$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x) = \begin{cases} \sin(\pi x) + 1, & x \leqslant 0 \\ \log_{2}(3x^{2}-12x+15), & x > 0 \end{cases}$, find the sum of all the zeros of the function $y=f(x)-1$ in the interval $[-3,3]$. |
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 the complex number $z=1-i$ (where $i$ is the imaginary unit), the conjugate of $\\frac{2}{z} -z^{2}$ is \\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1-3i",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the complex number $z=1-i$ (where $i$ is the imaginary unit), the conjugate of $\frac{2}{z} -z^{2}$ 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": "Factor $(x^2 + 2x + 1)(x^2 + 8x + 15) + (x^2 + 6x + 5)$ as the product of two non-constant polynomials.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(x + 1)(x + 5)(x + 2)^2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Factor $(x^2 + 2x + 1)(x^2 + 8x + 15) + (x^2 + 6x + 5)$ as the product of two non-constant polynomials. |
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": "If the complex number $z = m + 1 + (m - 1)i$ is purely imaginary, then the real number $m =$ ___.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If the complex number $z = m + 1 + (m - 1)i$ is purely imaginary, then the real number $m =$ ___. |
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 any $x > 0$, the inequality $a - x - |\\ln x| \\leqslant 0$ holds true. Determine the range of values for the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(0, 1]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For any $x > 0$, the inequality $a - x - |\ln x| \leqslant 0$ holds true. Determine the range of values for the real number $a$. |
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 least five-digit whole number that is both a perfect square and a perfect cube?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "15625",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the least five-digit whole number that is both a perfect square and a perfect cube? |
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