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_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 2-kilogram model of a sports car body made of aluminum was created in a 1:8 scale. What is the mass of the actual car body, assuming it is also made entirely of aluminum?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1024 \\, \\text{kg}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A 2-kilogram model of a sports car body made of aluminum was created in a 1:8 scale. What is the mass of the actual car body, assuming it is also made entirely of aluminum? |
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 real numbers $a$ and $b$, the operation is defined as follows: $a*b=\\left\\{\\begin{array}{l}{{a}^{b} \\text{ when } a>b \\text{ and } a\\neq 0}\\\\{{a}^{-b} \\text{ when } a<b \\text{ and } a\\neq 0}\\end{array}\\right.$. For example, $2*3=2^{-3}=\\frac{1}{8}$. Calculate: $\\left[2*\\left(-4\\right)\\right]\\times \\left[\\left(-4\\right)*\\left(-2\\right)\\right]=$____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For real numbers $a$ and $b$, the operation is defined as follows: $a*b=\left\{\begin{array}{l}{{a}^{b} \text{ when } a>b \text{ and } a\neq 0}\\{{a}^{-b} \text{ when } a<b \text{ and } a\neq 0}\end{array}\right.$. For example, $2*3=2^{-3}=\frac{1}{8}$. Calculate: $\left[2*\left(-4\right)\right]\times \left[\left(-4\right)*\left(-2\right)\right]=$____. |
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 circle is inscribed in an isosceles triangle with a base of 12 cm. From this circle, three tangents are drawn, which partition the given triangle into three smaller triangles. The sum of the perimeters of these smaller triangles is 48 cm. Find the lateral side of the given triangle.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "18 \\text{ cm}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A circle is inscribed in an isosceles triangle with a base of 12 cm. From this circle, three tangents are drawn, which partition the given triangle into three smaller triangles. The sum of the perimeters of these smaller triangles is 48 cm. Find the lateral side of the given 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": "996 jin of cotton is to be divided among eight sons as their allowance, with each son receiving a decreasing amount according to their age, such that the difference from eldest to youngest is 17 jin. Determine the amount of cotton the eighth son receives.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "184",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | 996 jin of cotton is to be divided among eight sons as their allowance, with each son receiving a decreasing amount according to their age, such that the difference from eldest to youngest is 17 jin. Determine the amount of cotton the eighth son receives. |
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 random variable \\( X \\), whose possible values belong to the interval \\((0 ; \\pi / 3)\\), is given in this interval by the probability density function \\( f(x) = C \\cdot \\sin 3x \\). Find the coefficient \\( C \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The random variable \( X \), whose possible values belong to the interval \((0 ; \pi / 3)\), is given in this interval by the probability density function \( f(x) = C \cdot \sin 3x \). Find the coefficient \( C \). |
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 shape composed of several cells, what is the minimum number of cells that need to be added to this shape to form a square where all the cells inside are filled without gaps?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "49",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a shape composed of several cells, what is the minimum number of cells that need to be added to this shape to form a square where all the cells inside are filled without gaps? |
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 expansion of $\\frac{1}{x}(x-y)(x+y)^{8}$, the coefficient of $x^{4}y^{4}$ is ______ (provide your answer as a number).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the expansion of $\frac{1}{x}(x-y)(x+y)^{8}$, the coefficient of $x^{4}y^{4}$ is ______ (provide your answer as a number). |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that ray OA rotates 120° counterclockwise around endpoint O to reach position OB, then rotates 270° clockwise to reach position OC, calculate the angle ∠AOC.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-150°",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that ray OA rotates 120° counterclockwise around endpoint O to reach position OB, then rotates 270° clockwise to reach position OC, calculate the angle ∠AOC. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "If from the numbers \\(1, 2, \\cdots, 14\\), we select \\(a_{1}, a_{2}, a_{3}\\) in ascending order such that \\(a_{2} - a_{1} \\geq 3\\) and \\(a_{1} - a_{2} \\geq 3\\) are satisfied, how many different ways are there to make such selections?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "120",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If from the numbers \(1, 2, \cdots, 14\), we select \(a_{1}, a_{2}, a_{3}\) in ascending order such that \(a_{2} - a_{1} \geq 3\) and \(a_{1} - a_{2} \geq 3\) are satisfied, how many different ways are there to make such selections? |
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 real numbers $a$ and $b$ are two real roots of the equation $x^{2}-8x+5=0$, then $a+b=\\_\\_\\_\\_\\_\\_$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If the real numbers $a$ and $b$ are two real roots of the equation $x^{2}-8x+5=0$, then $a+b=\_\_\_\_\_\_$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $(x^2-x-2)^5 = a + a_1x + a_2x^2 + \\ldots + a_{10}x^{10}$, find the value of $a_1 + a_2 + \\ldots + a_9 + a_{10}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $(x^2-x-2)^5 = a + a_1x + a_2x^2 + \ldots + a_{10}x^{10}$, find the value of $a_1 + a_2 + \ldots + a_9 + a_{10}$. |
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 matrix $\\mathbf{M}$ that performs the transformation that sends square $ABCD$ to $A'B'C'D'$ such that the transformation involves a 90-degree clockwise rotation and a scaling by 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\begin{pmatrix} 0 & 3 \\\\ -3 & 0 \\end{pmatrix}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine the matrix $\mathbf{M}$ that performs the transformation that sends square $ABCD$ to $A'B'C'D'$ such that the transformation involves a 90-degree clockwise rotation and a scaling by 3. |
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": "You are given a number composed of three different non-zero digits, 7, 8, and a third digit which is not 7 or 8. Find the minimum value of the quotient of this number divided by the sum of its digits.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "11.125",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | You are given a number composed of three different non-zero digits, 7, 8, and a third digit which is not 7 or 8. Find the minimum value of the quotient of this number divided by the sum of its digits. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In equilateral triangle $ABC$ a point $P$ lies such that $PA = 7$, $PB = 7$, and $PC = 14$. Determine the area of the triangle $ABC$.\n**A)** $49\\sqrt{3}$ \\\\\n**B)** $98\\sqrt{3}$ \\\\\n**C)** $42\\sqrt{3}$ \\\\\n**D)** $21\\sqrt{3}$ \\\\\n**E)** $98$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "49\\sqrt{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In equilateral triangle $ABC$ a point $P$ lies such that $PA = 7$, $PB = 7$, and $PC = 14$. Determine the area of the triangle $ABC$.
**A)** $49\sqrt{3}$ \\
**B)** $98\sqrt{3}$ \\
**C)** $42\sqrt{3}$ \\
**D)** $21\sqrt{3}$ \\
**E)** $98$ |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "If the graph of \\( y = 2 \\sqrt{2t} \\sqrt{x} - 2t \\) passes through the point \\( (a, a) \\), what is the value of \\( a \\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If the graph of \( y = 2 \sqrt{2t} \sqrt{x} - 2t \) passes through the point \( (a, a) \), what is the value of \( a \)? |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "On the blackboard, there is a number: 321321321321. By erasing some digits, we can get a number that is divisible by 9. The largest possible number is $\\qquad$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "3213212121",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | On the blackboard, there is a number: 321321321321. By erasing some digits, we can get a number that is divisible by 9. The largest possible number is $\qquad$. |
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 that the number \\( a \\) satisfies the equation \\( 5 = a + a^{-1} \\). What is the value of \\( a^4 + a^{-4} \\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "527",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Suppose that the number \( a \) satisfies the equation \( 5 = a + a^{-1} \). What is the value of \( a^4 + a^{-4} \)? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The product of three inches and the circumference of a circle, in inches, is equal to twice the circle's area. What is the length of the radius of the circle, in inches?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The product of three inches and the circumference of a circle, in inches, is equal to twice the circle's area. What is the length of the radius of the circle, in inches? |
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 probability that a positive integer less than or equal to 50 is not relatively prime to 50? Express your answer as a common fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the probability that a positive integer less than or equal to 50 is not relatively prime to 50? Express your answer as a common fraction. |
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 cube 5 units on each side is composed of 125 smaller unit cubes. Three faces of the larger cube that share a vertex are painted blue. The cube is then disassembled into 125 unit cubes. Two of the unit cubes are selected at random. What is the probability that one of the two selected unit cubes has exactly three painted faces while the other unit cube has no painted faces?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{9}{2583}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A cube 5 units on each side is composed of 125 smaller unit cubes. Three faces of the larger cube that share a vertex are painted blue. The cube is then disassembled into 125 unit cubes. Two of the unit cubes are selected at random. What is the probability that one of the two selected unit cubes has exactly three painted faces while the other unit cube has no painted faces? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "How many different prime factors does $30!$ have?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many different prime factors does $30!$ have? |
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": "Expand the expression $(17x + 12) \\cdot 3x$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "51x^2 + 36x",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Expand the expression $(17x + 12) \cdot 3x$. |
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 an inscribed quadrilateral $ABCD$ with perpendicular diagonals. On its circumscribing circle, there is a point $E$ diametrically opposite to $D$, such that segments $AB$ and $DE$ do not intersect. Find the ratio of the areas of triangle $BCD$ and quadrilateral $ABED$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given an inscribed quadrilateral $ABCD$ with perpendicular diagonals. On its circumscribing circle, there is a point $E$ diametrically opposite to $D$, such that segments $AB$ and $DE$ do not intersect. Find the ratio of the areas of triangle $BCD$ and quadrilateral $ABED$. |
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 quadratic function $y=(m-1)x^{2}-2mx+3$ is an even function, find the value of $m$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If the quadratic function $y=(m-1)x^{2}-2mx+3$ is an even function, find the value of $m$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The odd function $f(x)$ satisfies the conditions: (1) $f(x)$ is monotonically increasing on the interval $(0, +\\infty)$; (2) $f(1) = 0$. The solution set for the inequality $x \\cdot f(x) < 0$ is _____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(-1, 0) \\cup (0, 1)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The odd function $f(x)$ satisfies the conditions: (1) $f(x)$ is monotonically increasing on the interval $(0, +\infty)$; (2) $f(1) = 0$. The solution set for the inequality $x \cdot f(x) < 0$ 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": "Solve the equation \\[\\frac{3x + 6}{x^2 + 5x - 6} = \\frac{3 - x}{x - 1}.\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Solve the equation \[\frac{3x + 6}{x^2 + 5x - 6} = \frac{3 - x}{x - 1}.\] |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $f(x)= \\frac {2(2-a)}{x}+(a+2)\\ln x-ax-2$. \n(I) When $0 < a < 2$, find the intervals of monotonicity for the function $f(x)$. \n(II) Given $a=1$, the function $g(x)=x^{2}-4bx- \\frac {1}{4}$. If for any $x_{1}\\in(0,e]$, there exists $x_{2}\\in(0,2]$ such that $f(x_{1})\\geqslant g(x_{2})$ holds, find the range of the real number $b$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "b\\geqslant \\frac { \\sqrt {3}}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x)= \frac {2(2-a)}{x}+(a+2)\ln x-ax-2$.
(I) When $0 < a < 2$, find the intervals of monotonicity for the function $f(x)$.
(II) Given $a=1$, the function $g(x)=x^{2}-4bx- \frac {1}{4}$. If for any $x_{1}\in(0,e]$, there exists $x_{2}\in(0,2]$ such that $f(x_{1})\geqslant g(x_{2})$ holds, find the range of the real number $b$. |
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": "Find the maximum value of the function $f(x) = \\cos x - \\cos\\left(x + \\frac{\\pi}{3}\\right)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the maximum value of the function $f(x) = \cos x - \cos\left(x + \frac{\pi}{3}\right)$. |
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": "Given that $x$ and $y$ are both positive real numbers, and $x+2y=xy$, find the minimum value of $xy$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that $x$ and $y$ are both positive real numbers, and $x+2y=xy$, find the minimum value of $xy$. |
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": "At $3:40$ o'clock, calculate the angle formed by the hour and minute hands of a clock.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "130^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | At $3:40$ o'clock, calculate the angle formed by the hour and minute hands of a clock. |
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 $a$ and $b$ are integers such that $$4b = 10 - 3a.$$ How many of the first four positive integers must be divisors of $3b + 9$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Suppose $a$ and $b$ are integers such that $$4b = 10 - 3a.$$ How many of the first four positive integers must be divisors of $3b + 9$? |
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 $F_1$ and $F_2$ are two foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a>0$, $b>0$), and PQ is a chord of the hyperbola passing through $F_1$ and perpendicular to the x-axis. If $\\angle PF_2Q = 90^\\circ$, determine the eccentricity of the hyperbola.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1 + \\sqrt{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $F_1$ and $F_2$ are two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0$, $b>0$), and PQ is a chord of the hyperbola passing through $F_1$ and perpendicular to the x-axis. If $\angle PF_2Q = 90^\circ$, determine the eccentricity of the hyperbola. |
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": "Consider a polynomial of the form\n\\[x^4 + ax^3 + bx^2 + cx + d = 0\\]\nwhere all coefficients are integers. Determine all possible values of $m$, the exact number of distinct integer roots of the polynomial, counting multiplicity. For example, the polynomial $(x - 2)^2(x^2 + x + 1) = 0$ has one distinct integer root, $2$, counted twice.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0, 1, 2, 4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Consider a polynomial of the form
\[x^4 + ax^3 + bx^2 + cx + d = 0\]
where all coefficients are integers. Determine all possible values of $m$, the exact number of distinct integer roots of the polynomial, counting multiplicity. For example, the polynomial $(x - 2)^2(x^2 + x + 1) = 0$ has one distinct integer root, $2$, counted twice. |
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 16 students are selected by stratified sampling from 25 students from Sichuan University and 15 students from the University of Electronic Science and Technology, determine the number of students selected from Sichuan University.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that 16 students are selected by stratified sampling from 25 students from Sichuan University and 15 students from the University of Electronic Science and Technology, determine the number of students selected from Sichuan University. |
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 \\(\\log_{x} a = a\\) (where \\( a \\) is a positive integer greater than 1), find the value of \\( x \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "10^{\\frac{\\lg a}{a}}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given \(\log_{x} a = a\) (where \( a \) is a positive integer greater than 1), find the value of \( x \). |
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": "Two balls are randomly chosen from a box containing 20 balls numbered from 1 to 20. Calculate the probability that the sum of the numbers on the two balls is divisible by 3.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{32}{95}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Two balls are randomly chosen from a box containing 20 balls numbered from 1 to 20. Calculate the probability that the sum of the numbers on the two balls is divisible by 3. |
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": "Under the sunlight, a sphere is placed on a plane. At a certain moment, the furthest point reached by the sphere's shadow is $10 \\mathrm{~m}$ from the point of tangency between the sphere and the plane. At the same moment, the length of the shadow cast by a vertical meter stick on the plane is $2 \\mathrm{~m}$. What is the radius of the sphere (unit: $\\mathrm{m}$)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "10 \\sqrt{5} - 20",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Under the sunlight, a sphere is placed on a plane. At a certain moment, the furthest point reached by the sphere's shadow is $10 \mathrm{~m}$ from the point of tangency between the sphere and the plane. At the same moment, the length of the shadow cast by a vertical meter stick on the plane is $2 \mathrm{~m}$. What is the radius of the sphere (unit: $\mathrm{m}$)? |
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": "In triangle $ABC$, if $4\\sin A + 2\\cos B = 4$ and $\\frac{1}{2}\\sin B + \\cos A = \\frac{\\sqrt{3}}{2}$, then the measure of angle $C$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "C = \\frac{\\pi}{6}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In triangle $ABC$, if $4\sin A + 2\cos B = 4$ and $\frac{1}{2}\sin B + \cos A = \frac{\sqrt{3}}{2}$, then the measure of angle $C$ 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": "If the sum of a and its absolute value is 0, calculate the value of a - |2a|.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "3a",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If the sum of a and its absolute value is 0, calculate the value of a - |2a|. |
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": "There are 160 items in total, among which there are 48 first-grade items, 64 second-grade items, 3 equal-grade items, and 1 substandard item. Determine the relationship between the probabilities of each item being selected in the population using simple random sampling, stratified sampling, and systematic sampling.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | There are 160 items in total, among which there are 48 first-grade items, 64 second-grade items, 3 equal-grade items, and 1 substandard item. Determine the relationship between the probabilities of each item being selected in the population using simple random sampling, stratified sampling, and systematic sampling. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In a three-dimensional rectangular coordinate system, what is the number of units in the distance from the origin to the point (8, -3, 6)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\sqrt{109}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In a three-dimensional rectangular coordinate system, what is the number of units in the distance from the origin to the point (8, -3, 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 the ellipse $\\frac{x^{2}}{10-m} + \\frac{y^{2}}{m-2} = 1$ with its major axis lying on the $y$-axis, and a focal distance of $4$, find the value of $m$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the ellipse $\frac{x^{2}}{10-m} + \frac{y^{2}}{m-2} = 1$ with its major axis lying on the $y$-axis, and a focal distance of $4$, find the value of $m$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the equation of line $l$ as $ax + 2y - 3 = 0$, where $a \\in [-5, 4]$, find the probability that the slope of line $l$ is no less than $1$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the equation of line $l$ as $ax + 2y - 3 = 0$, where $a \in [-5, 4]$, find the probability that the slope of line $l$ is no less than $1$. |
numina_amc_aime | [
{
"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": "Eight people are sitting around a circular table, each holding a fair coin. What is the probability that no two adjacent people will stand after all eight people flip their coins?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{47}{256}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Eight people are sitting around a circular table, each holding a fair coin. What is the probability that no two adjacent people will stand after all eight people flip their coins? |
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 ellipse $C: \\frac{x^{2}}{4}+y^{2}=1$, and a moving line $l$ with a slope of $\\frac{\\sqrt{3}}{2}$. The line $l$ intersects with the ellipse $C$ at two distinct points $A$ and $B$.\n\n1. Let $M$ be the midpoint of the chord $AB$, find the trajectory equation of the moving point $M$.\n2. Let $F\\_1$ and $F\\_2$ be the left and right foci of the ellipse $C$, and let $P$ be a point on the ellipse in the first quadrant such that $\\overrightarrow{PF\\_1} \\cdot \\overrightarrow{PF\\_2} = -\\frac{5}{4}$. Find the maximum area of $\\triangle PAB$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the ellipse $C: \frac{x^{2}}{4}+y^{2}=1$, and a moving line $l$ with a slope of $\frac{\sqrt{3}}{2}$. The line $l$ intersects with the ellipse $C$ at two distinct points $A$ and $B$.
1. Let $M$ be the midpoint of the chord $AB$, find the trajectory equation of the moving point $M$.
2. Let $F\_1$ and $F\_2$ be the left and right foci of the ellipse $C$, and let $P$ be a point on the ellipse in the first quadrant such that $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2} = -\frac{5}{4}$. Find the maximum area of $\triangle PAB$. |
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 measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 7:35?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "17.5^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 7:35? |
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": "There are many fascinating connections between mathematics and literature. For example, there is the palindrome poem, such as \"云边月影沙边雁，水外天光山外树\", which if read backwards, becomes \"树外山光天外水，雁边沙影月边云\", offering a delightful sense and rhyme both ways! Similarly, mathematics features palindromic numbers, such as 88, 454, 7337, 43534, etc., which are the same whether read from left to right or right to left. Such numbers are known as \"palindromic numbers,\" and reading them is quite interesting!\nThe two-digit palindromic numbers are 11, 22, 33, 44, 55, 66, 77, 88, 99, making a total of 9;\nThe three-digit palindromic numbers include 101, 111, 121, 131, ..., 969, 979, 989, 999, making a total of 90;\nSo, how many five-digit palindromic numbers are there in total?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "900",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | There are many fascinating connections between mathematics and literature. For example, there is the palindrome poem, such as "云边月影沙边雁，水外天光山外树", which if read backwards, becomes "树外山光天外水，雁边沙影月边云", offering a delightful sense and rhyme both ways! Similarly, mathematics features palindromic numbers, such as 88, 454, 7337, 43534, etc., which are the same whether read from left to right or right to left. Such numbers are known as "palindromic numbers," and reading them is quite interesting!
The two-digit palindromic numbers are 11, 22, 33, 44, 55, 66, 77, 88, 99, making a total of 9;
The three-digit palindromic numbers include 101, 111, 121, 131, ..., 969, 979, 989, 999, making a total of 90;
So, how many five-digit palindromic numbers are there in total? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "For what base is the representation of $625_{10}$ a five-digit number whose final digit is odd?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For what base is the representation of $625_{10}$ a five-digit number whose final digit is odd? |
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 $\\left|\\frac{24}{x}+4\\right| = 4$, find the product of all possible values of $x$. Express your answer as an improper fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $\left|\frac{24}{x}+4\right| = 4$, find the product of all possible values of $x$. Express your answer as an improper fraction. |
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": "Fill in the digits in the $\\square$ so that the following subtraction equation is correct.\n$$\n\\begin{array}{r}\n\\square 2 \\square 6 \\\\\n-\\quad \\square 97 \\\\\n\\hline 543 \\square\n\\end{array}\n$$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\begin{array}{r}\n6236 \\\\\n- \\quad 797 \\\\\n\\hline \n5439\n\\end{array}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Fill in the digits in the $\square$ so that the following subtraction equation is correct.
$$
\begin{array}{r}
\square 2 \square 6 \\
-\quad \square 97 \\
\hline 543 \square
\end{array}
$$ |
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 man labels the squares of a chessboard from $1$ through $64$. On each square $k$, he places $3^k$ grains of sand. How many more grains of sand are placed on the $12^{th}$ square than on the first $10$ squares combined?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "442869",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A man labels the squares of a chessboard from $1$ through $64$. On each square $k$, he places $3^k$ grains of sand. How many more grains of sand are placed on the $12^{th}$ square than on the first $10$ squares combined? |
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 diagonal of a square is \\( \\sqrt{a^2 + 4b^2} \\), calculate the area of this square.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{a^2 + 4b^2}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If the diagonal of a square is \( \sqrt{a^2 + 4b^2} \), calculate the area of this square. |
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 equation of the hyperbola with foci at the vertices of the ellipse $\\frac{x^2}{16} + \\frac{y^2}{9} = 1$, and vertices at the foci of the same ellipse.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{x^2}{7} - \\frac{y^2}{9} = 1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the equation of the hyperbola with foci at the vertices of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$, and vertices at the foci of the same ellipse. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The purchase price of a product is 90 yuan per unit. When sold at 100 yuan each, 500 units can be sold. If the price of this product increases by 1 yuan, its sales volume will decrease by 10 units. Determine the optimal unit price to maximize profit.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "120",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The purchase price of a product is 90 yuan per unit. When sold at 100 yuan each, 500 units can be sold. If the price of this product increases by 1 yuan, its sales volume will decrease by 10 units. Determine the optimal unit price to maximize profit. |
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, \"p is a sufficient but not necessary condition for q\" is (fill in the serial number):\n\n(1) p: $a=0$, q: $f(x)=x^2+ax$, ($x\\in\\mathbb{R}$) is an even function; \n(2) p: $\\sin\\alpha>\\sin\\beta$, q: $\\alpha>\\beta$; \n(3) p: $\\log_a=\\log_b$, q: $a=b$; \n(4) p: $x\\in M\\cap N$, q: $x\\in M\\cup N$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(3), (4)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Among the following propositions, "p is a sufficient but not necessary condition for q" is (fill in the serial number):
(1) p: $a=0$, q: $f(x)=x^2+ax$, ($x\in\mathbb{R}$) is an even function;
(2) p: $\sin\alpha>\sin\beta$, q: $\alpha>\beta$;
(3) p: $\log_a=\log_b$, q: $a=b$;
(4) p: $x\in M\cap N$, q: $x\in M\cup N$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Fido's leash is tied to a stake at the center of his yard, which is in the shape of a square. His leash is exactly long enough to reach the midpoint of each side of the square. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest form as $\\frac{a}{b}\\pi$, what is the value of $ab$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Fido's leash is tied to a stake at the center of his yard, which is in the shape of a square. His leash is exactly long enough to reach the midpoint of each side of the square. If the fraction of the area of Fido's yard that he is able to reach while on his leash is expressed in simplest form as $\frac{a}{b}\pi$, what is the value of $ab$? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $f(x) = A\\sin(\\omega x + \\phi)$, where $x \\in \\mathbb{R}$, and $(A > 0, \\omega > 0, 0 < \\phi < \\frac{\\pi}{2})$, the period of the function is $\\pi$, and one of the lowest points on its graph is $M(\\frac{2\\pi}{3}, -2)$.\n1. Find the analytical expression of $f(x)$.\n2. When $x \\in [0, \\frac{\\pi}{12}]$, find the maximum and minimum values of $f(x)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x) = A\sin(\omega x + \phi)$, where $x \in \mathbb{R}$, and $(A > 0, \omega > 0, 0 < \phi < \frac{\pi}{2})$, the period of the function is $\pi$, and one of the lowest points on its graph is $M(\frac{2\pi}{3}, -2)$.
1. Find the analytical expression of $f(x)$.
2. When $x \in [0, \frac{\pi}{12}]$, find the maximum and minimum values of $f(x)$. |
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 conference hall with $N$ participants, where $N > 5$, determine the maximum number of participants that could have shaken hands with all other participants.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\text{In a conference hall with N participants, where N > 5, at least two participants have not shaken hands with everyone else in the hall. What is the maximum number of participants that could have shaken hands with all other participants?}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a conference hall with $N$ participants, where $N > 5$, determine the maximum number of participants that could have shaken hands with all other participants. |
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 domain of the function $y= \\sqrt{2+\\log_{\\frac{1}{2}}x}+ \\sqrt{\\tan x}$ is __________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\{ x | 0 < x < \\frac{\\pi}{2} \\text{ or } \\pi \\leqslant x \\leqslant 4 \\}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The domain of the function $y= \sqrt{2+\log_{\frac{1}{2}}x}+ \sqrt{\tan x}$ is __________. |
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": "What is the hundreds digit of $(12! - 8!)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the hundreds digit of $(12! - 8!)$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $y=\\cos^2x+ \\frac{\\sqrt{3}}{2}\\sin 2x- \\frac{1}{2}$, with $x \\in \\left(0, \\frac{\\pi}{2}\\right)$, find the range of this function.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left(-\\frac{1}{2}, 1\\right]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $y=\cos^2x+ \frac{\sqrt{3}}{2}\sin 2x- \frac{1}{2}$, with $x \in \left(0, \frac{\pi}{2}\right)$, find the range of this function. |
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 $y$ is inversely proportional to $x-1$, and when $x=4$, $y=1$. \n$(1)$ Find the functional relationship between $y$ and $x$; \n$(2)$ Determine whether the point $(-2,-1)$ lies on the graph of this function.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\text{Yes, the point } (-2,-1) \\text{ lies on the graph of this function.}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that $y$ is inversely proportional to $x-1$, and when $x=4$, $y=1$.
$(1)$ Find the functional relationship between $y$ and $x$;
$(2)$ Determine whether the point $(-2,-1)$ lies on the graph of this function. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "How many non-congruent squares can be drawn such that their vertices are lattice points on a 6 by 6 grid?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "75",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many non-congruent squares can be drawn such that their vertices are lattice points on a 6 by 6 grid? |
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 line segments $a$, $b$, $c$, $d$ are in proportion, and $a=1cm$, $b=4cm$, $c=2cm$, then find the value of $d$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "8\\,cm",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If line segments $a$, $b$, $c$, $d$ are in proportion, and $a=1cm$, $b=4cm$, $c=2cm$, then find the value of $d$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "How many distinct triangles can be constructed by connecting three different vertices of a regular octahedron, provided that no triangle coincides entirely with any of the faces of the octahedron?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many distinct triangles can be constructed by connecting three different vertices of a regular octahedron, provided that no triangle coincides entirely with any of the faces of the octahedron? |
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 owner of an individual clothing store purchased 30 dresses for $32 each. The selling price of the 30 dresses varies for different customers. Using $47 as the standard price, any excess amount is recorded as positive and any shortfall is recorded as negative. The results are shown in the table below:\n\n| Number Sold | 7 | 6 | 3 | 5 | 4 | 5 |\n|-------------|---|---|---|---|---|---|\n| Price/$ | +3 | +2 | +1 | 0 | -1 | -2 |\n\nAfter selling these 30 dresses, how much money did the clothing store earn?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "472",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The owner of an individual clothing store purchased 30 dresses for $32 each. The selling price of the 30 dresses varies for different customers. Using $47 as the standard price, any excess amount is recorded as positive and any shortfall is recorded as negative. The results are shown in the table below:
| Number Sold | 7 | 6 | 3 | 5 | 4 | 5 |
|-------------|---|---|---|---|---|---|
| Price/$ | +3 | +2 | +1 | 0 | -1 | -2 |
After selling these 30 dresses, how much money did the clothing store earn? |
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: $-1-2=$____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate: $-1-2=$____. |
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": " $g(x):\\mathbb{Z}\\rightarrow\\mathbb{Z}$ is a function that satisfies $$ g(x)+g(y)=g(x+y)-xy. $$ If $g(23)=0$ , what is the sum of all possible values of $g(35)$ ?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "210",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | $g(x):\mathbb{Z}\rightarrow\mathbb{Z}$ is a function that satisfies $$ g(x)+g(y)=g(x+y)-xy. $$ If $g(23)=0$ , what is the sum of all possible values of $g(35)$ ? |
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 natural number greater than 0, if it satisfies the condition that the sum of all its factors equals twice itself, is called a perfect number. For example, 6 has all its factors as $1, 2, 3, 6$ and $1 + 2 + 3 + 6 = 12$. Thus, 6 is the smallest perfect number. Whether there are infinitely many perfect numbers is still one of the great unsolved problems in mathematics. The study of perfect numbers can begin by calculating the sum of all factors of a natural number. The sum of all factors of 81 is $\\qquad$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "121",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A natural number greater than 0, if it satisfies the condition that the sum of all its factors equals twice itself, is called a perfect number. For example, 6 has all its factors as $1, 2, 3, 6$ and $1 + 2 + 3 + 6 = 12$. Thus, 6 is the smallest perfect number. Whether there are infinitely many perfect numbers is still one of the great unsolved problems in mathematics. The study of perfect numbers can begin by calculating the sum of all factors of a natural number. The sum of all factors of 81 is $\qquad$. |
numina_amc_aime | [
{
"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 $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2047",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Simplify the following expression in \\( x \\): \\[ 6x^2 + 4x + 9 - (7 - 5x - 9x^3 + 8x^2) \\]. Express your answer in the form \\( ax^3 + bx^2 + cx + d \\), where \\( a \\), \\( b \\), \\( c \\), and \\( d \\) are constants.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "9x^3 - 2x^2 + 9x + 2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Simplify the following expression in \( x \): \[ 6x^2 + 4x + 9 - (7 - 5x - 9x^3 + 8x^2) \]. Express your answer in the form \( ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants. |
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 graph of the function $f(x)=\\sin(2x+\\varphi)$ is translated to the right by $\\frac{\\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\\left[0, \\frac{\\pi}{4}\\right]$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The graph of the function $f(x)=\sin(2x+\varphi)$ is translated to the right by $\frac{\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\left[0, \frac{\pi}{4}\right]$. |
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) Find the locus of points \\( M \\) such that the sum of their distances to two given lines \\( l_{1} \\) and \\( l_{3} \\) is a known constant \\( a \\).\n\nb) Find the locus of points \\( M \\) such that the difference of their distances to two given lines \\( l_{1} \\) and \\( l_{2} \\) is a known constant \\( a \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\text{The locus of point M is on the bisectors (internal and external) of the angles formed by } l_{1} \\text{ and } l_{2}.",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | a) Find the locus of points \( M \) such that the sum of their distances to two given lines \( l_{1} \) and \( l_{3} \) is a known constant \( a \).
b) Find the locus of points \( M \) such that the difference of their distances to two given lines \( l_{1} \) and \( l_{2} \) is a known constant \( a \). |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the complex plane, the point corresponding to the complex number $\\left(1+i\\right)\\left(a-i\\right)$ is in the third quadrant, calculate the range of the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(-\\infty,-1)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the complex plane, the point corresponding to the complex number $\left(1+i\right)\left(a-i\right)$ is in the third quadrant, calculate the range of the real number $a$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the domain and intervals of monotonicity for the function $f(x) = \\log_{\\frac{1}{3}}(x^2 - 5x + 4)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(4, +\\infty)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the domain and intervals of monotonicity for the function $f(x) = \log_{\frac{1}{3}}(x^2 - 5x + 4)$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In triangle $PQR$, we have $\\angle PRQ = \\angle PQR$ and $\\angle QPR = 24^\\circ$. If $\\overline{QS}$ bisects $\\angle PQR$, determine the measure of $\\angle PSR$.\n[asy]\n/* New problem based on modification */\ndraw((0,0)--(1,1.75)--(2,0)--cycle);\ndraw((2,0)--(0.4,.85));\nlabel(\"$S$\", (0.4,.85), W);\nlabel(\"$P$\", (0,0), SW);\nlabel(\"$R$\", (2,0), SE);\nlabel(\"$Q$\", (1,1.75), N);\n[/asy]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "24^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In triangle $PQR$, we have $\angle PRQ = \angle PQR$ and $\angle QPR = 24^\circ$. If $\overline{QS}$ bisects $\angle PQR$, determine the measure of $\angle PSR$.
[asy]
/* New problem based on modification */
draw((0,0)--(1,1.75)--(2,0)--cycle);
draw((2,0)--(0.4,.85));
label("$S$", (0.4,.85), W);
label("$P$", (0,0), SW);
label("$R$", (2,0), SE);
label("$Q$", (1,1.75), N);
[/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": "Given Alex's car averages 35 miles per gallon of gasoline and Mia's car averages 20 miles per gallon of gasoline, Alex drives 140 miles while Mia drives 100 miles. Find the combined rate of miles per gallon of gasoline.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "26.67",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given Alex's car averages 35 miles per gallon of gasoline and Mia's car averages 20 miles per gallon of gasoline, Alex drives 140 miles while Mia drives 100 miles. Find the combined rate of miles per gallon of gasoline. |
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 $2x - 3y = 18$ and $x + 2y = 8$, what is the value of $x$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{60}{7}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $2x - 3y = 18$ and $x + 2y = 8$, what is the value of $x$? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the sum of $1000_7$ and $666_7$ first, and then subtract $1234_7$ from the result in base $7$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "254_7",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the sum of $1000_7$ and $666_7$ first, and then subtract $1234_7$ from the result in base $7$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "When $x=30$, the value of the algebraic expression $ax^{3}+bx-7$ is $9$. Find the value of the expression $ax^{3}+bx+2$ when $x=-30$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-14",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | When $x=30$, the value of the algebraic expression $ax^{3}+bx-7$ is $9$. Find the value of the expression $ax^{3}+bx+2$ when $x=-30$. |
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 $T$ be a subset of $\\{1, 2, 3, \\ldots, 2023\\}$ such that no two members of $T$ differ by $5$ or $8$. What is the largest number of elements $T$ can have?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "777",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $T$ be a subset of $\{1, 2, 3, \ldots, 2023\}$ such that no two members of $T$ differ by $5$ or $8$. What is the largest number of elements $T$ can have? |
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": "Point $P$ moves on the curve $y= \\frac{1}{3}x^{3}-2x^{2}+3x$. Let the slope angle of the tangent line at point $P$ be $\\alpha$. Find the range of values for angle $\\alpha$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left[0, \\frac{\\pi}{2}\\right) \\cup \\left[ \\frac{3\\pi}{4},\\pi\\right)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Point $P$ moves on the curve $y= \frac{1}{3}x^{3}-2x^{2}+3x$. Let the slope angle of the tangent line at point $P$ be $\alpha$. Find the range of values for angle $\alpha$. |
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 four-digit number $\\overline{abcd}$ has the properties that $a + b + c + d = 26$, the tens digit of $b \\cdot d$ equals $a + c$, and $bd - c^2$ is a multiple of 2. Find this four-digit number (provide justification).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1979",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A four-digit number $\overline{abcd}$ has the properties that $a + b + c + d = 26$, the tens digit of $b \cdot d$ equals $a + c$, and $bd - c^2$ is a multiple of 2. Find this four-digit number (provide justification). |
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": "How many squares with sides one or two units long, each parallel to the axes and having vertices with integer coordinates, are completely contained within the area bounded by $y=\\frac{3}{2}x$, $y=-1$, and $x=6$?\nA) 88\nB) 94\nC) 98\nD) 108\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "B) 94",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many squares with sides one or two units long, each parallel to the axes and having vertices with integer coordinates, are completely contained within the area bounded by $y=\frac{3}{2}x$, $y=-1$, and $x=6$?
A) 88
B) 94
C) 98
D) 108 |
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 vectors $\\overrightarrow{a}=(\\cos 2x,\\sin 2x)$ and $\\overrightarrow{b}=(\\sqrt{3},1)$, and the function $f(x)= \\overrightarrow{a}\\cdot \\overrightarrow{b}+m$.\n$(1)$ Find the smallest positive period of $f(x)$;\n$(2)$ When $x\\in[0, \\frac{\\pi}{2}]$, the minimum value of $f(x)$ is $5$, find the value of $m$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "5+\\sqrt{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given vectors $\overrightarrow{a}=(\cos 2x,\sin 2x)$ and $\overrightarrow{b}=(\sqrt{3},1)$, and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}+m$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ When $x\in[0, \frac{\pi}{2}]$, the minimum value of $f(x)$ is $5$, find the value of $m$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given sets $A=\\{2,3\\}$, $B=\\{x|mx-1=0\\}$, if $A\\cup B=A$, then the value of the real number $m$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left\\{\\frac{1}{2}, \\frac{1}{3}\\right\\}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given sets $A=\{2,3\}$, $B=\{x|mx-1=0\}$, if $A\cup B=A$, then the value of the real number $m$ 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": "In a triangle with sides of lengths \\(a\\), \\(b\\), and \\(c\\),\n\\[(a + b + c)(a + b - c) = 2ab.\\] Find the angle opposite the side of length \\(c\\), in degrees.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "120^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In a triangle with sides of lengths \(a\), \(b\), and \(c\),
\[(a + b + c)(a + b - c) = 2ab.\] Find the angle opposite the side of length \(c\), in degrees. |
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 $R$ varies directly as the square of $S$ and inversely as $T^2$, and when $R = 2$ and $T = 2$, $S = 1$, find $S$ when $R = 50$ and $T = 5$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "12.5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $R$ varies directly as the square of $S$ and inversely as $T^2$, and when $R = 2$ and $T = 2$, $S = 1$, find $S$ when $R = 50$ and $T = 5$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A unique four-sided die is designed to show numbers 1, 2, 3, and 4 with probabilities $p_1 = \\frac{1}{4}$, $p_2 = \\frac{1}{4}$, $p_3 = \\frac{1}{6}$, $p_4 = \\frac{1}{3}$ respectively. \n- If the die shows a 1, you win 2 dollars.\n- If the die shows a 2 and previously has shown an odd number, you lose 3 dollars; otherwise, you lose nothing.\n- If it shows a 3, you win nothing.\n- If it shows a 4, you win 5 dollars.\nAssuming the die is thrown twice, what is the expected return from playing this game? Each throw is independent of the previous.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\$4.18",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A unique four-sided die is designed to show numbers 1, 2, 3, and 4 with probabilities $p_1 = \frac{1}{4}$, $p_2 = \frac{1}{4}$, $p_3 = \frac{1}{6}$, $p_4 = \frac{1}{3}$ respectively.
- If the die shows a 1, you win 2 dollars.
- If the die shows a 2 and previously has shown an odd number, you lose 3 dollars; otherwise, you lose nothing.
- If it shows a 3, you win nothing.
- If it shows a 4, you win 5 dollars.
Assuming the die is thrown twice, what is the expected return from playing this game? Each throw is independent of the previous. |
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 two classes of students, one with 20 students and an average score of 80 points, and the other with 30 students and an average score of 70 points, calculate the average score of all students in both classes.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "74",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given two classes of students, one with 20 students and an average score of 80 points, and the other with 30 students and an average score of 70 points, calculate the average score of all students in both classes. |
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: $99\\times 101=\\_\\_\\_\\_\\_\\_$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "9999",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate: $99\times 101=\_\_\_\_\_\_$. |
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": "Two straight lines intersect the x-axis at different points, creating four angles at the intersection. Considering the lines are not perpendicular to each other and form an obtuse angle at one intersection point, how many obtuse angles are formed in total at the intersection?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Two straight lines intersect the x-axis at different points, creating four angles at the intersection. Considering the lines are not perpendicular to each other and form an obtuse angle at one intersection point, how many obtuse angles are formed in total at the intersection? |
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 all integers \\( x, y \\geq 1 \\) such that \\( 2^x - 5 = 11^y \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(x, y) = (4, 1)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find all integers \( x, y \geq 1 \) such that \( 2^x - 5 = 11^y \). |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A square has a side length of 12 cm. Inside this square, four quarter circles are drawn with centers at the four corners such that each quarter circle has a radius of 6 cm. Calculate the area of the shaded region inside the square but outside of these quarter circles. Express your answer in terms of $\\pi$.\n\n[asy]\nunitsize (1.5 cm);\n\ndraw((-1.5,-1.5)--(1.5,-1.5)--(1.5,1.5)--(-1.5,1.5)--cycle);\nfilldraw(arc((1.5,1.5),1.5,270,180)--arc((-1.5,1.5),1.5,360,270)--arc((-1.5,-1.5),1.5,90,0)--arc((1.5,-1.5),1.5,180,90)--cycle,gray);\n[/asy]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "144 - 36\\pi",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A square has a side length of 12 cm. Inside this square, four quarter circles are drawn with centers at the four corners such that each quarter circle has a radius of 6 cm. Calculate the area of the shaded region inside the square but outside of these quarter circles. Express your answer in terms of $\pi$.
[asy]
unitsize (1.5 cm);
draw((-1.5,-1.5)--(1.5,-1.5)--(1.5,1.5)--(-1.5,1.5)--cycle);
filldraw(arc((1.5,1.5),1.5,270,180)--arc((-1.5,1.5),1.5,360,270)--arc((-1.5,-1.5),1.5,90,0)--arc((1.5,-1.5),1.5,180,90)--cycle,gray);
[/asy] |
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 parabola $y^{2}=2px\\left(p \\gt 0\\right)$ with focus $F$ and a line with slope $\\sqrt{2}$ intersecting the parabola at points $A(x_{1}$,$y_{1})$ and $B(x_{2}$,$y_{2})$ where $x_{1} \\lt x_{2}$ and $|AB|=6$.<br/>$(1)$ Find the equation of the parabola;<br/>$(2)$ The directrix of the parabola intersects the $x$-axis at point $F'$, and a line $l$ passing through point $F'$ intersects the parabola $C$ at points $M$ and $N$. When $\\overrightarrow{{F'}M}•\\overrightarrow{{F'}N}=12$, find the equation of the line $l$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "y=\\pm\\frac{\\sqrt{2}}{2}(x+1)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given a parabola $y^{2}=2px\left(p \gt 0\right)$ with focus $F$ and a line with slope $\sqrt{2}$ intersecting the parabola at points $A(x_{1}$,$y_{1})$ and $B(x_{2}$,$y_{2})$ where $x_{1} \lt x_{2}$ and $|AB|=6$.<br/>$(1)$ Find the equation of the parabola;<br/>$(2)$ The directrix of the parabola intersects the $x$-axis at point $F'$, and a line $l$ passing through point $F'$ intersects the parabola $C$ at points $M$ and $N$. When $\overrightarrow{{F'}M}•\overrightarrow{{F'}N}=12$, find the equation of the line $l$. |
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 a right triangle, one leg is 7, and the lengths of the other two sides are integers. Calculate the possible area of this triangle.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "84",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In a right triangle, one leg is 7, and the lengths of the other two sides are integers. Calculate the possible area of this 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": "Observe the following set of equations about non-zero real numbers $a$ and $b$:\n$a^{2}-b^{2}=(a-b)(a+b)$\n$a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})$\n$a^{4}-b^{4}=(a-b)(a^{3}+a^{2}b+ab^{2}+b^{3})$\n...\nBy inductive reasoning, we can obtain the equation $a^{2015}-b^{2015}=(a-b)(x_{1}+x_{2}+x_{3}+…+x_{2015})$, where $x_{1}$, $x_{2}$, $x_{3}$, ..., $x_{2015}$ form a finite sequence ${x_{n}}$, then the general term formula for this sequence is $x_{n}=$ _______ ($1\\leqslant n\\leqslant 2015$, and $n\\in N^{*}$) (The result is expressed using $a$, $b$, $n$)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "x_{n}=a^{2014}(\\frac{b}{a})^{n-1}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Observe the following set of equations about non-zero real numbers $a$ and $b$:
$a^{2}-b^{2}=(a-b)(a+b)$
$a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})$
$a^{4}-b^{4}=(a-b)(a^{3}+a^{2}b+ab^{2}+b^{3})$
...
By inductive reasoning, we can obtain the equation $a^{2015}-b^{2015}=(a-b)(x_{1}+x_{2}+x_{3}+…+x_{2015})$, where $x_{1}$, $x_{2}$, $x_{3}$, ..., $x_{2015}$ form a finite sequence ${x_{n}}$, then the general term formula for this sequence is $x_{n}=$ _______ ($1\leqslant n\leqslant 2015$, and $n\in N^{*}$) (The result is expressed using $a$, $b$, $n$) |
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": "Let set $A=\\{x|x^2+3a=(a+3)x, a\\in \\mathbb{R}\\}$, and set $B=\\{x|x^2+3=4x\\}$.\n\n(1) If $A\\cap B=A$, find the value of the real number $a$; \n(2) Find $A\\cup B$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "a=1 \\text{ or } a=3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let set $A=\{x|x^2+3a=(a+3)x, a\in \mathbb{R}\}$, and set $B=\{x|x^2+3=4x\}$.
(1) If $A\cap B=A$, find the value of the real number $a$;
(2) Find $A\cup B$. |
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": "Given lines $l_1: ax + 2y + 6 = 0$ and $l_2: x + (a-1)y + a^2 - 1 = 0$, find the value of $a$ such that $l_1 \\perp l_2$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "a = \\frac{2}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given lines $l_1: ax + 2y + 6 = 0$ and $l_2: x + (a-1)y + a^2 - 1 = 0$, find the value of $a$ such that $l_1 \perp l_2$. |
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 product of four consecutive integers is a perfect square. What are these integers?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-1, 0, 1, 2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The product of four consecutive integers is a perfect square. What are these integers? |
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