data_source
stringclasses 6
values | prompt
listlengths 2
2
| ability
stringclasses 1
value | reward_model
dict | extra_info
dict | problem
stringlengths 17
4.55k
|
|---|---|---|---|---|---|
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let the function $f(x) = 1 - x\\sin x$ attain an extremum at $x = x_0$. Find the value of $(1 + x_0^2)(1 + \\cos 2x_0)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let the function $f(x) = 1 - x\sin x$ attain an extremum at $x = x_0$. Find the value of $(1 + x_0^2)(1 + \cos 2x_0)$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A regular octahedron has a sphere inscribed within it and a sphere circumscribed about it. For each of the eight faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $Q$ is selected at random inside the circumscribed sphere. Determine the probability that $Q$ lies inside one of the nine small spheres.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A regular octahedron has a sphere inscribed within it and a sphere circumscribed about it. For each of the eight faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $Q$ is selected at random inside the circumscribed sphere. Determine the probability that $Q$ lies inside one of the nine small spheres. |
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": "Several people completed the task of planting 2013 trees, each person planting the same number of trees. If 5 people do not participate in planting, the remaining people each plant 2 more trees and cannot complete the task, but if each plants 3 more trees, they can exceed the task. How many people participated in planting the trees?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "61",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Several people completed the task of planting 2013 trees, each person planting the same number of trees. If 5 people do not participate in planting, the remaining people each plant 2 more trees and cannot complete the task, but if each plants 3 more trees, they can exceed the task. How many people participated in planting the trees? |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let the function \\( f(x)=\\frac{x^{2}+x+16}{x} \\) where \\( 2 \\leqslant x \\leqslant a \\) and the real number \\( a > 2 \\). If the range of \\( f(x) \\) is \\([9, 11]\\), then find the range of values for \\( a \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "[4, 8]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let the function \( f(x)=\frac{x^{2}+x+16}{x} \) where \( 2 \leqslant x \leqslant a \) and the real number \( a > 2 \). If the range of \( f(x) \) is \([9, 11]\), then find the range of values for \( a \). |
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": "If $-l$, $a$, $b$, $c$, and $-9$ form a geometric sequence, then find the value of $b$ and $ac$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "ac = 9",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $-l$, $a$, $b$, $c$, and $-9$ form a geometric sequence, then find the value of $b$ and $ac$. |
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": "Inside a sphere, there is a fixed point \\(P\\). On the surface of the sphere, there are three moving points \\(A\\), \\(B\\), and \\(C\\). Given that \\(\\angle BPA = \\angle CPA = \\angle CPB = 90^\\circ\\), a parallelepiped is formed using \\(PA\\), \\(PB\\), and \\(PC\\) as its edges. Point \\(Q\\) is the vertex of this parallelepiped that is diagonally opposite to \\(P\\). Find the locus of point \\(Q\\) as \\(A\\), \\(B\\), and \\(C\\) move on the surface of the sphere.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\text{The locus of point } Q \\text{ is a sphere centered at } O \\text{ with radius } \\sqrt{3R^2 - 2OP^2}.",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Inside a sphere, there is a fixed point \(P\). On the surface of the sphere, there are three moving points \(A\), \(B\), and \(C\). Given that \(\angle BPA = \angle CPA = \angle CPB = 90^\circ\), a parallelepiped is formed using \(PA\), \(PB\), and \(PC\) as its edges. Point \(Q\) is the vertex of this parallelepiped that is diagonally opposite to \(P\). Find the locus of point \(Q\) as \(A\), \(B\), and \(C\) move on the surface of the sphere. |
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": "Triangles with a perimeter less than 15, and with all sides of integral lengths, can be formed. Determine the total number of such triangles.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Triangles with a perimeter less than 15, and with all sides of integral lengths, can be formed. Determine the total number of such triangles. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $x$, $y$, and $z$ be real numbers such that\n\\[\\cos 2x + \\cos 2y + \\cos 2z = \\sin 2x + \\sin 2y + \\sin 2z = 0.\\]\nFind the sum of all possible values of $\\cos 4x + \\cos 4y + \\cos 4z.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $x$, $y$, and $z$ be real numbers such that
\[\cos 2x + \cos 2y + \cos 2z = \sin 2x + \sin 2y + \sin 2z = 0.\]
Find the sum of all possible values of $\cos 4x + \cos 4y + \cos 4z.$ |
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 $f(x) = 1 + x - \\frac{x^2}{2} + \\frac{x^3}{3} - \\frac{x^4}{4} + \\ldots + \\frac{x^{2015}}{2015}$, and $g(x) = 1 - x + \\frac{x^2}{2} - \\frac{x^3}{3} + \\frac{x^4}{4} + \\ldots - \\frac{x^{2015}}{2015}$. Let the function $F(x) = f(x+3) \\cdot g(x-4)$, determine the minimum value of $b-a$, where $a$ and $b$ are the endpoints of the interval containing all the zeros of the function $F(x)$, both endpoints being integers.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x) = 1 + x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots + \frac{x^{2015}}{2015}$, and $g(x) = 1 - x + \frac{x^2}{2} - \frac{x^3}{3} + \frac{x^4}{4} + \ldots - \frac{x^{2015}}{2015}$. Let the function $F(x) = f(x+3) \cdot g(x-4)$, determine the minimum value of $b-a$, where $a$ and $b$ are the endpoints of the interval containing all the zeros of the function $F(x)$, both endpoints being integers. |
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 function \\( G \\) is defined on integers and takes integer values, and for each integer \\( c \\) there exists an integer \\( x \\) such that \\( G(x) \\neq c \\). We call a number \\( a \\) non-standard if for any integer \\( x \\) it holds that \\( G(x) = G(a - x) \\). Can both of the numbers 267 and 269 be non-standard?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\text{No, neither 267 nor 269 can be non-standard numbers.}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The function \( G \) is defined on integers and takes integer values, and for each integer \( c \) there exists an integer \( x \) such that \( G(x) \neq c \). We call a number \( a \) non-standard if for any integer \( x \) it holds that \( G(x) = G(a - x) \). Can both of the numbers 267 and 269 be non-standard? |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A parallelogram $ABCD$ has area $500,\\!000$. Vertex $A$ is at $(0,0)$, and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = 2x$ and $y = mx$ for some integer $m > 2$, respectively. Determine the number of such parallelograms.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "420",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A parallelogram $ABCD$ has area $500,\!000$. Vertex $A$ is at $(0,0)$, and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = 2x$ and $y = mx$ for some integer $m > 2$, respectively. Determine the number of such parallelograms. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Express the decimal number $0.000000022$ in scientific notation.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2.2 \\times 10^{-8}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Express the decimal number $0.000000022$ in scientific notation. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "What is $a - 2b$, where $a = 5 - 3i$ and $b = 4 - i$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-3 - i",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is $a - 2b$, where $a = 5 - 3i$ and $b = 4 - i$? |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $f(x) = x^2 - 2017x + 8052 + |x^2 - 2017x + 8052|$, then find the value of $f(1) + f(2) + f(3) + \\ldots + f(2013)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $f(x) = x^2 - 2017x + 8052 + |x^2 - 2017x + 8052|$, then find the value of $f(1) + f(2) + f(3) + \ldots + f(2013)$. |
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": "Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=5$, and $DC=12$. Calculate the area of $\\triangle ABC$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "17\\sqrt{15}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=5$, and $DC=12$. Calculate the area of $\triangle ABC$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Determine the highest power of 3 that divides the product of all integers from 1 to 2000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "3^{996}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine the highest power of 3 that divides the product of all integers from 1 to 2000. |
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 $(1+x)(a-x)^6=a_0+a_1x+\\ldots+a_7x^7$, if $a_0+a_1+\\ldots+a_7=0$, then the value of $a_3$ is $\\_\\_\\_\\_\\_\\_$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that $(1+x)(a-x)^6=a_0+a_1x+\ldots+a_7x^7$, if $a_0+a_1+\ldots+a_7=0$, then the value of $a_3$ 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": "After the implementation of the \"double reduction\" policy, schools have attached importance to extended services and increased the emphasis on sports activities. Seizing this opportunity, a sports equipment store plans to purchase a total of 300 sets of table tennis rackets and badminton rackets for sale. The purchase prices and selling prices are as follows:\n\n| | Purchase Price (元/set) | Selling Price (元/set) |\n|------------|-------------------------|------------------------|\n| Table Tennis Racket | $a$ | $55$ |\n| Badminton Racket | $b$ | $50$ |\n\nIt is known that purchasing 2 sets of table tennis rackets and 1 set of badminton rackets costs $120$, and purchasing 4 sets of table tennis rackets and 3 sets of badminton rackets costs $270$.\n\n$(1)$ Find the values of $a$ and $b$.\n\n$(2)$ Based on past sales experience, the sports equipment store decides to purchase table tennis rackets in a quantity not less than $\\frac{1}{3}$ of the quantity of badminton rackets. If all these sports equipment can be sold out, how should they purchase the goods to maximize profit? What is the maximum profit?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "5250",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | After the implementation of the "double reduction" policy, schools have attached importance to extended services and increased the emphasis on sports activities. Seizing this opportunity, a sports equipment store plans to purchase a total of 300 sets of table tennis rackets and badminton rackets for sale. The purchase prices and selling prices are as follows:
| | Purchase Price (元/set) | Selling Price (元/set) |
|------------|-------------------------|------------------------|
| Table Tennis Racket | $a$ | $55$ |
| Badminton Racket | $b$ | $50$ |
It is known that purchasing 2 sets of table tennis rackets and 1 set of badminton rackets costs $120$, and purchasing 4 sets of table tennis rackets and 3 sets of badminton rackets costs $270$.
$(1)$ Find the values of $a$ and $b$.
$(2)$ Based on past sales experience, the sports equipment store decides to purchase table tennis rackets in a quantity not less than $\frac{1}{3}$ of the quantity of badminton rackets. If all these sports equipment can be sold out, how should they purchase the goods to maximize profit? What is the maximum profit? |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find the pairs of integers \\((a, n) \\in \\mathbb{N}^{2}\\) that satisfy\n\n\\[ \\frac{(a+1)^{n} - a^{n}}{n} \\in \\mathbb{N} \\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(a, 1)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the pairs of integers \((a, n) \in \mathbb{N}^{2}\) that satisfy
\[ \frac{(a+1)^{n} - a^{n}}{n} \in \mathbb{N} \] |
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": "Find the matrix that corresponds to projecting onto the vector $\\begin{pmatrix} 3 \\\\ -2 \\end{pmatrix}.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\begin{pmatrix} \\frac{9}{13} & \\frac{-6}{13} \\\\ \\frac{-6}{13} & \\frac{4}{13} \\end{pmatrix}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the matrix that corresponds to projecting onto the vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}.$ |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "If $\\tan \\theta = \\frac{3}{4},$ then find $\\tan 3 \\theta.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-12.5",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $\tan \theta = \frac{3}{4},$ then find $\tan 3 \theta.$ |
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 ball travels on a parabolic path where the height (in meters) is given by the expression $-5t^2 + 20t + 10$, where $t$ is the time after launch. What is the maximum height of the ball, in meters?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "30 \\text{ meters}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A ball travels on a parabolic path where the height (in meters) is given by the expression $-5t^2 + 20t + 10$, where $t$ is the time after launch. What is the maximum height of the ball, in meters? |
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": "What is the maximum value that the sum of the first \\(n\\) terms of the arithmetic progression \\(113, 109, 105, \\ldots\\) can take?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1653",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the maximum value that the sum of the first \(n\) terms of the arithmetic progression \(113, 109, 105, \ldots\) can take? |
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": "On the parabola $y=2\\left(x-1\\right)^{2}+2$, there are three points $A(-1$,$y_{1})$, $B(\\sqrt{2},y_2)$, $C(2$,$y_{3})$. Determine the relationship between $y_{1}$, $y_{2}$, and $y_{3}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "y_{1} > y_{3} > y_{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | On the parabola $y=2\left(x-1\right)^{2}+2$, there are three points $A(-1$,$y_{1})$, $B(\sqrt{2},y_2)$, $C(2$,$y_{3})$. Determine the relationship between $y_{1}$, $y_{2}$, and $y_{3}$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Point $P$ is located in the third quadrant and the distance to the $x$-axis and $y$-axis are $2$ and $3$ respectively. The coordinates of point $P$ are ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left(-3,-2\\right)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Point $P$ is located in the third quadrant and the distance to the $x$-axis and $y$-axis are $2$ and $3$ respectively. The coordinates of point $P$ are ______. |
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": "Find the sum of two consecutive perfect squares that totals 181.\nA) $\\{64, 81\\}$ \nB) $\\{81, 100\\}$ \nC) $\\{100, 121\\}$ \nD) $\\{121, 144\\}$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\{81, 100\\}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the sum of two consecutive perfect squares that totals 181.
A) $\{64, 81\}$
B) $\{81, 100\}$
C) $\{100, 121\}$
D) $\{121, 144\}$ |
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 integer $n$, $0 \\le n \\le 180$, such that $\\cos n^\\circ = \\cos 331^\\circ$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "29",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the integer $n$, $0 \le n \le 180$, such that $\cos n^\circ = \cos 331^\circ$. |
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 units digit in the product of all natural numbers from 1 to 50, inclusive?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | What is the units digit in the product of all natural numbers from 1 to 50, inclusive? |
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 $a$ and $b$ be the two real values of $x$ for which \\[\\sqrt[3]{x} + \\sqrt[3]{16 - x} = 2.\\] The larger of the two values can be expressed as $r + \\sqrt{s}$, where $r$ and $s$ are integers. Compute $r + s$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $a$ and $b$ be the two real values of $x$ for which \[\sqrt[3]{x} + \sqrt[3]{16 - x} = 2.\] The larger of the two values can be expressed as $r + \sqrt{s}$, where $r$ and $s$ are integers. Compute $r + s$. |
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": "Alice, Bob, Dave, and Carol take turns tossing a die. The order is always Alice, then Bob, then Dave, and finally Carol, and this sequence repeats indefinitely. Determine the probability that Carol will be the first among them to toss a six. (Assume the probability of rolling a six on any single toss remains $\\frac{1}{6}$.)\nA) $\\frac{125}{671}$\nB) $\\frac{125}{1296}$\nC) $\\frac{625}{1296}$\nD) $\\frac{25}{91}$\nE) $\\frac{5}{18}$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{125}{671}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Alice, Bob, Dave, and Carol take turns tossing a die. The order is always Alice, then Bob, then Dave, and finally Carol, and this sequence repeats indefinitely. Determine the probability that Carol will be the first among them to toss a six. (Assume the probability of rolling a six on any single toss remains $\frac{1}{6}$.)
A) $\frac{125}{671}$
B) $\frac{125}{1296}$
C) $\frac{625}{1296}$
D) $\frac{25}{91}$
E) $\frac{5}{18}$ |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Determine the value of the real constant $a$ given the circle $C$: $(x-a)^2+(y-a-1)^2=9$.\n1. If the chord intercepted by the line $l: x + y - 3 = 0$ on circle $C$ has a length of 2, find the value of $a$.\n2. Let point $A(3,0)$ and the origin $O$ be given. If there exists a point $M$ on circle $C$ such that $|MA| = 2|MO|$, determine the range of values for $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left[-1 - \\frac{5\\sqrt{2}}{2}, -1 - \\frac{\\sqrt{2}}{2}\\right] \\cup \\left[-1 + \\frac{\\sqrt{2}}{2}, -1 + \\frac{5\\sqrt{2}}{2}\\right]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine the value of the real constant $a$ given the circle $C$: $(x-a)^2+(y-a-1)^2=9$.
1. If the chord intercepted by the line $l: x + y - 3 = 0$ on circle $C$ has a length of 2, find the value of $a$.
2. Let point $A(3,0)$ and the origin $O$ be given. If there exists a point $M$ on circle $C$ such that $|MA| = 2|MO|$, determine the range of values for $a$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that the plane vectors $\\boldsymbol{\\alpha}$ and $\\boldsymbol{\\beta}$ satisfy $|\\boldsymbol{\\alpha} + 2\\boldsymbol{\\beta}| = 3$ and $|2\\boldsymbol{\\alpha} + 3\\boldsymbol{\\beta}| = 4$, find the minimum value of $\\boldsymbol{\\alpha} \\cdot \\boldsymbol{\\beta}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-170",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the plane vectors $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ satisfy $|\boldsymbol{\alpha} + 2\boldsymbol{\beta}| = 3$ and $|2\boldsymbol{\alpha} + 3\boldsymbol{\beta}| = 4$, find the minimum value of $\boldsymbol{\alpha} \cdot \boldsymbol{\beta}$. |
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": "After fully charging his phone, Alex used his phone for $30$ minutes. When he is not using his phone, the battery lasts for $18$ hours, and if he uses it constantly, the battery will only last for $2$ hours. The phone has been on for $6$ hours in total since the last charge. If Alex doesn't use his phone anymore but keeps it on, how many more hours will the battery last?\nA) 6 hours\nB) 7 hours\nC) 8 hours\nD) 9 hours\nE) 10 hours\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "C) 8 \\text{ hours}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | After fully charging his phone, Alex used his phone for $30$ minutes. When he is not using his phone, the battery lasts for $18$ hours, and if he uses it constantly, the battery will only last for $2$ hours. The phone has been on for $6$ hours in total since the last charge. If Alex doesn't use his phone anymore but keeps it on, how many more hours will the battery last?
A) 6 hours
B) 7 hours
C) 8 hours
D) 9 hours
E) 10 hours |
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 $a,$ $b,$ and $t$ be real numbers such that $a + b = t.$ Find, in terms of $t,$ the minimum value of $a^3 + b^3.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{t^3}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $a,$ $b,$ and $t$ be real numbers such that $a + b = t.$ Find, in terms of $t,$ the minimum value of $a^3 + b^3.$ |
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 $f(5)=2$ and $f(3x)=3f(x)$ for all $x$, find $f^{-1}(486)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1215",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If $f(5)=2$ and $f(3x)=3f(x)$ for all $x$, find $f^{-1}(486)$. |
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 tangent line drawn to the point \\( A \\) on the cubic parabola \\( y = ax^3 \\) intersects the \\( Y \\)-axis at point \\( B \\) and intersects the curve again at another point \\( C \\). Determine the geometric locus of the midpoints of the distances \\( AB \\) and \\( BC \\), considering that the location of point \\( A \\) varies.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "y = 5 a x^3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The tangent line drawn to the point \( A \) on the cubic parabola \( y = ax^3 \) intersects the \( Y \)-axis at point \( B \) and intersects the curve again at another point \( C \). Determine the geometric locus of the midpoints of the distances \( AB \) and \( BC \), considering that the location of point \( A \) varies. |
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 cube is constructed from 27 dice.\n\na) Find the probability that exactly 25 of the faces showing on the surface of the cube have the number six.\n\nb) Find the probability that at least one of the faces showing on the surface of the cube has the number one.\n\nc) Find the expected number of faces showing the number six on the surface of the cube.\n\nd) Find the expected sum of the numbers on the faces showing on the surface of the cube.\n\ne) Find the expected value of the random variable: \"The number of different digits appearing on the surfaces of the dice.\"\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "6- \\frac{5^{6}}{2 \\cdot 3^{17}} \\approx 5.99",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A cube is constructed from 27 dice.
a) Find the probability that exactly 25 of the faces showing on the surface of the cube have the number six.
b) Find the probability that at least one of the faces showing on the surface of the cube has the number one.
c) Find the expected number of faces showing the number six on the surface of the cube.
d) Find the expected sum of the numbers on the faces showing on the surface of the cube.
e) Find the expected value of the random variable: "The number of different digits appearing on the surfaces of the dice." |
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) Describe all the different towers of height $10 \\mathrm{~cm}$ that Caroba can make with three pieces.\n\n(b) With 12 pieces, being 4 of each color, Caroba managed to build 3 towers of height $10 \\mathrm{~cm}$, having 2 pieces of $3 \\mathrm{~cm}$ left over. Describe how Caroba can build 7 towers of $10 \\mathrm{~cm}$ if she has 27 pieces, being 9 of each color.\n\n(c) Explain why Caroba will not be able to build 8 towers of $10 \\mathrm{~cm}$ if she has 27 pieces, being 9 of each color.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(4, 4, 2),(4, 4, 2),(4, 4, 2),(4, 4, 2),(4, 3, 3),(3, 3, 2, 2),(3,3,2,2)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | (a) Describe all the different towers of height $10 \mathrm{~cm}$ that Caroba can make with three pieces.
(b) With 12 pieces, being 4 of each color, Caroba managed to build 3 towers of height $10 \mathrm{~cm}$, having 2 pieces of $3 \mathrm{~cm}$ left over. Describe how Caroba can build 7 towers of $10 \mathrm{~cm}$ if she has 27 pieces, being 9 of each color.
(c) Explain why Caroba will not be able to build 8 towers of $10 \mathrm{~cm}$ if she has 27 pieces, being 9 of each color. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A square with sides of length 1 cm is given. There are many different ways to cut the square into four rectangles. Let \\( S \\) be the sum of the four rectangles' perimeters. Describe all possible values of \\( S \\) with justification.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "6 < S \\leq 10",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A square with sides of length 1 cm is given. There are many different ways to cut the square into four rectangles. Let \( S \) be the sum of the four rectangles' perimeters. Describe all possible values of \( S \) with justification. |
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 positive integers can be represented as a difference of two distinct members of the set $\\{2, 3, 4, \\ldots, 18, 19\\}$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "17",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many different positive integers can be represented as a difference of two distinct members of the set $\{2, 3, 4, \ldots, 18, 19\}$? |
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 probability it will rain on Friday is $40\\%$, on Saturday it is $50\\%$, and on Sunday it is $30\\%$. If the probability it will rain on any given day is independent of the weather on any other day, except that if it rains on both Friday and Saturday, the probability of rain on Sunday increases to $60\\%$. What is the probability it will rain on all three days?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "12\\%",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The probability it will rain on Friday is $40\%$, on Saturday it is $50\%$, and on Sunday it is $30\%$. If the probability it will rain on any given day is independent of the weather on any other day, except that if it rains on both Friday and Saturday, the probability of rain on Sunday increases to $60\%$. What is the probability it will rain on all three days? |
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 a certain month, the price of a product was $150$ at the beginning. During the month, the price increased by $30\\%$, then decreased by $25\\%$, and finally increased by $x\\%$. At the end of the month, the price was $195.75$. Find the value of $x$.\nA) 32%\nB) 33%\nC) 34%\nD) 35%\nE) 36%\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\textbf{(C)}\\ 34\\%",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In a certain month, the price of a product was $150$ at the beginning. During the month, the price increased by $30\%$, then decreased by $25\%$, and finally increased by $x\%$. At the end of the month, the price was $195.75$. Find the value of $x$.
A) 32%
B) 33%
C) 34%
D) 35%
E) 36% |
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 $f(x)=e^{x}+2f'(0)x+3$, find $f(1)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "e+1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x)=e^{x}+2f'(0)x+3$, find $f(1)$. |
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 $f(x) = ax + \\frac{a-1}{x}$ ($a \\in \\mathbb{R}$), and $g(x) = \\ln x$.\n1. When $a=2$, find the minimum value of the function $h(x) = f(x) - g(x)$;\n2. When $a > 0$, for any $x \\geq 1$, the inequality $f(x) - g(x) \\geq 1$ always holds, find the range of the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "[1, +\\infty)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x) = ax + \frac{a-1}{x}$ ($a \in \mathbb{R}$), and $g(x) = \ln x$.
1. When $a=2$, find the minimum value of the function $h(x) = f(x) - g(x)$;
2. When $a > 0$, for any $x \geq 1$, the inequality $f(x) - g(x) \geq 1$ always holds, find the range of the real number $a$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given an ellipse with its center at the origin \\( O \\), foci on the \\( x \\)-axis, and eccentricity \\( \\frac{\\sqrt{3}}{2} \\) which passes through the point \\(\\left(\\sqrt{2}, \\frac{\\sqrt{2}}{2}\\right)\\). Suppose a line \\( l \\) that does not pass through the origin \\( O \\) intersects the ellipse at points \\( P \\) and \\( Q \\), and the slopes of lines \\( OP \\), \\( PQ \\), and \\( OQ \\) form a geometric sequence. Find the possible range of values for the area of triangle \\( \\triangle OPQ \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(0, 1)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given an ellipse with its center at the origin \( O \), foci on the \( x \)-axis, and eccentricity \( \frac{\sqrt{3}}{2} \) which passes through the point \(\left(\sqrt{2}, \frac{\sqrt{2}}{2}\right)\). Suppose a line \( l \) that does not pass through the origin \( O \) intersects the ellipse at points \( P \) and \( Q \), and the slopes of lines \( OP \), \( PQ \), and \( OQ \) form a geometric sequence. Find the possible range of values for the area of triangle \( \triangle OPQ \). |
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 expression \\( x + \\frac{1}{x} \\) has a maximum for \\( x < 0 \\) and a minimum for \\( x > 0 \\). Find the area of the rectangle whose sides are parallel to the axes and two of whose vertices are the maximum and minimum values of \\( x + \\frac{1}{x} \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The expression \( x + \frac{1}{x} \) has a maximum for \( x < 0 \) and a minimum for \( x > 0 \). Find the area of the rectangle whose sides are parallel to the axes and two of whose vertices are the maximum and minimum values of \( x + \frac{1}{x} \). |
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": "Calculate the value of $500 \\times 3986 \\times 0.3986 \\times 20$.\nA) \\( 3986 \\times 10^4 \\)\nB) \\( (3986)^2 \\)\nC) \\( 1993 \\times 3986 \\)\nD) \\( 500 \\times 20 \\times 3986 \\)\nE) \\( 1.993 \\times 10^4 \\)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\textbf{(B)}\\ (3986)^2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the value of $500 \times 3986 \times 0.3986 \times 20$.
A) \( 3986 \times 10^4 \)
B) \( (3986)^2 \)
C) \( 1993 \times 3986 \)
D) \( 500 \times 20 \times 3986 \)
E) \( 1.993 \times 10^4 \) |
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": "Calculate the limit of the function:\n\n$$\\lim _{x \\rightarrow \\pi} \\frac{\\operatorname{tg}\\left(3^{\\pi / x}-3\\right)}{3^{\\cos (3 x / 2)}-1}$$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{-2}{\\pi}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate the limit of the function:
$$\lim _{x \rightarrow \pi} \frac{\operatorname{tg}\left(3^{\pi / x}-3\right)}{3^{\cos (3 x / 2)}-1}$$ |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A circle centered at $O$ is circumscribed about $\\triangle XYZ$. The central angles are given as $\\angle XOY = 120^\\circ$ and $\\angle YOZ = 140^\\circ$. Determine the measure of $\\angle XYZ$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "60^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A circle centered at $O$ is circumscribed about $\triangle XYZ$. The central angles are given as $\angle XOY = 120^\circ$ and $\angle YOZ = 140^\circ$. Determine the measure of $\angle XYZ$. |
numina_aops_forum | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$ . Give all the possibilities.\n\nRemember that the number $1$ is not prime.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(167, 5, 2), (71, 11, 2), (23, 13, 2), (71, 2, 5)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$ . Give all the possibilities.
Remember that the number $1$ is not prime. |
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": "Square $IJKL$ is inside square $WXYZ$ such that each side of $IJKL$ can be extended to pass through a vertex of $WXYZ$. Square $WXYZ$ has side length $10$, and $WI = 3$. What is the area of the inner square $IJKL$?\nA) $88$\nB) $82$\nC) $91$\nD) $(\\sqrt{91} - 3)^2$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(\\sqrt{91} - 3)^2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Square $IJKL$ is inside square $WXYZ$ such that each side of $IJKL$ can be extended to pass through a vertex of $WXYZ$. Square $WXYZ$ has side length $10$, and $WI = 3$. What is the area of the inner square $IJKL$?
A) $88$
B) $82$
C) $91$
D) $(\sqrt{91} - 3)^2$ |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The roots of the equation\n\\[ z^5 = \\frac{1}{\\sqrt{2}} + \\frac{i}{\\sqrt{2}} \\]\nare $\\text{cis } \\phi_1$, $\\text{cis } \\phi_2$, $\\dots$, $\\text{cis } \\phi_5$, where $0^\\circ \\le \\phi_k < 360^\\circ$ for all $1 \\le k \\le 5$. Find $\\phi_1 + \\phi_2 + \\dots + \\phi_5$. Give your answer in degrees.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "450^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The roots of the equation
\[ z^5 = \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \]
are $\text{cis } \phi_1$, $\text{cis } \phi_2$, $\dots$, $\text{cis } \phi_5$, where $0^\circ \le \phi_k < 360^\circ$ for all $1 \le k \le 5$. Find $\phi_1 + \phi_2 + \dots + \phi_5$. Give your answer in degrees. |
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 points $A(m+1, n-1, 3)$, $B(2m, n, m-2n)$, and $C(m+3, n-3, 9)$ are collinear, find the value of $m+n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "m+n = 3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given points $A(m+1, n-1, 3)$, $B(2m, n, m-2n)$, and $C(m+3, n-3, 9)$ are collinear, find the value of $m+n$. |
numina_olympiads | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "On the complex plane, the area of the region corresponding to the points \\(z\\) such that \\(\\left\\{\\begin{array}{l}0 \\leqslant \\arg (z-1) \\leqslant \\frac{\\pi}{4}, \\\\ \\operatorname{Re}(z) \\leqslant 2\\end{array}\\right.\\) is \\(\\qquad\\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | On the complex plane, the area of the region corresponding to the points \(z\) such that \(\left\{\begin{array}{l}0 \leqslant \arg (z-1) \leqslant \frac{\pi}{4}, \\ \operatorname{Re}(z) \leqslant 2\end{array}\right.\) 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": "If two sides of a triangle measure 5 cm and 11 cm, what is the greatest integer number of centimeters that could be the third side of this triangle?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If two sides of a triangle measure 5 cm and 11 cm, what is the greatest integer number of centimeters that could be the third side of this triangle? |
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": "Altitudes $\\overline{AX}$ and $\\overline{BY}$ of triangle $ABC$ intersect at $H$. Assume that triangle $ABC$ is scalene with $\\angle BAC = 58^\\circ$ and $\\angle ABC = 108^\\circ$. Calculate $\\angle CHY$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "76^\\circ",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Altitudes $\overline{AX}$ and $\overline{BY}$ of triangle $ABC$ intersect at $H$. Assume that triangle $ABC$ is scalene with $\angle BAC = 58^\circ$ and $\angle ABC = 108^\circ$. Calculate $\angle CHY$. |
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 $c$ and $d$ be the solutions of the equation $3x^2 + 9x - 21 = 0$. What is the value of $(3c-4)(6d-8)$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "122",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $c$ and $d$ be the solutions of the equation $3x^2 + 9x - 21 = 0$. What is the value of $(3c-4)(6d-8)$? |
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 two planar vectors $\\overrightarrow{a}=(x_1,y_1)$ and $\\overrightarrow{b}=(x_2,y_2)$. If $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=3$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-6$, find the value of $\\frac{x_1 + y_1}{x_2 + y_2}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-\\frac{2}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given two planar vectors $\overrightarrow{a}=(x_1,y_1)$ and $\overrightarrow{b}=(x_2,y_2)$. If $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-6$, find the value of $\frac{x_1 + y_1}{x_2 + y_2}$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The polynomial $g(x) = x^4 + px^3 + qx^2 + rx + s$ has real coefficients, and $g(3i) = g(1+2i) = 0$. What is $p+q+r+s$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "39",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The polynomial $g(x) = x^4 + px^3 + qx^2 + rx + s$ has real coefficients, and $g(3i) = g(1+2i) = 0$. What is $p+q+r+s$? |
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 $f(x)=a+b\\cos x+c\\sin x$ whose graph passes through points $A(0,1)$ and $B(\\frac{\\pi}{2},1)$.\n(1) Given $b > 0$, find the interval(s) where $f(x)$ is monotonically decreasing;\n(2) Given $x\\in(0,\\frac{\\pi}{2})$ and $|f(x)|\\leqslant 2$ always holds true, find the range of values for the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "[-\\sqrt{2}, 4+3\\sqrt{2}]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the function $f(x)=a+b\cos x+c\sin x$ whose graph passes through points $A(0,1)$ and $B(\frac{\pi}{2},1)$.
(1) Given $b > 0$, find the interval(s) where $f(x)$ is monotonically decreasing;
(2) Given $x\in(0,\frac{\pi}{2})$ and $|f(x)|\leqslant 2$ always holds true, find the range of values for the real number $a$. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that $x=y+z$ where $y>z>0$, evaluate the expression $\\frac{(x+y)^z + (x+z)^y}{y^z + z^y}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "2^y + 2^z",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that $x=y+z$ where $y>z>0$, evaluate the expression $\frac{(x+y)^z + (x+z)^y}{y^z + z^y}$. |
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, the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that $$\\frac {2c-b}{cosB}= \\frac {a}{cosA}$$.\n\n(I) Find angle A.\n(II) If $$a= \\sqrt {7}, 2b=3c$$, find the area of triangle ABC.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "S_{△ABC} = \\frac{3 \\sqrt{3}}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Given that $$\frac {2c-b}{cosB}= \frac {a}{cosA}$$.
(I) Find angle A.
(II) If $$a= \sqrt {7}, 2b=3c$$, find the area of triangle ABC. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Calculate $[-3.7]$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Calculate $[-3.7]$. |
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": "The positive integer solution of $3x-5 \\lt 1$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The positive integer solution of $3x-5 \lt 1$ is ______. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A conical frustum has a top radius of 2 inches, a bottom radius of 3 inches, and a height of 5 inches. What is the number of inches in the radius of a sphere that has the same volume as this frustum?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\sqrt[3]{95}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A conical frustum has a top radius of 2 inches, a bottom radius of 3 inches, and a height of 5 inches. What is the number of inches in the radius of a sphere that has the same volume as this frustum? |
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": "Given the system of equations:\n\\[\n4x + 2y = b,\n\\]\n\\[\n3x + 7y = 3b,\n\\]\nfind the value of $b$ if the solution exists when $x = -1$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-22",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given the system of equations:
\[
4x + 2y = b,
\]
\[
3x + 7y = 3b,
\]
find the value of $b$ if the solution exists when $x = -1$. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A triangle has sides of length 7 and 9 units. The length of the third side is $y$ units, where $y$ is an integer. What is the largest possible perimeter of the triangle?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "31\\text{ units}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A triangle has sides of length 7 and 9 units. The length of the third side is $y$ units, where $y$ is an integer. What is the largest possible perimeter of the triangle? |
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 largest integer less than 100 that leaves a remainder of 5 when divided by 8.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "93",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find the largest integer less than 100 that leaves a remainder of 5 when divided by 8. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Simplify $\\frac{1}{1+\\sqrt{3}}\\cdot\\frac{1}{1-\\sqrt[3]{3}}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1}{1 - \\sqrt[3]{3} + \\sqrt{3} - \\sqrt[3]{3\\sqrt{3}}}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Simplify $\frac{1}{1+\sqrt{3}}\cdot\frac{1}{1-\sqrt[3]{3}}$. |
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": "Determine the smallest possible value of the expression\n\n$$\n4 + x^{2} y^{4} + x^{4} y^{2} - 3 x^{2} y^{2}\n$$\n\nfor real number pairs \\((x, y)\\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Determine the smallest possible value of the expression
$$
4 + x^{2} y^{4} + x^{4} y^{2} - 3 x^{2} y^{2}
$$
for real number pairs \((x, y)\). |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find all $x$ such that $\\lfloor \\lfloor 3x \\rfloor - 1/3 \\rfloor = \\lfloor x + 3 \\rfloor.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left[ \\frac{5}{3}, 3 \\right)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Find all $x$ such that $\lfloor \lfloor 3x \rfloor - 1/3 \rfloor = \lfloor x + 3 \rfloor.$ |
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": "Observing the following expressions:<br/>$1-\\frac{1}{{2}^{2}}=1-\\frac{1}{4}=\\frac{3}{4}=\\frac{1}{2}\\times \\frac{3}{2}$;<br/>$1-\\frac{1}{{3}^{2}}=1-\\frac{1}{9}=\\frac{8}{9}=\\frac{2}{3}\\times \\frac{4}{3}$;<br/>$1-\\frac{1}{{4}^{2}}=1-\\frac{1}{16}=\\frac{15}{16}=\\frac{3}{4}\\times \\frac{5}{4}$;<br/>$1-\\frac{1}{{5}^{2}}=1-\\frac{1}{25}=\\frac{24}{25}=\\frac{4}{5}\\times \\frac{6}{5}$;<br/>$\\ldots $<br/>$(1)$ Fill in the blanks according to the pattern you discovered: $1-\\frac{1}{{6}^{2}}=\\_\\_\\_\\_\\_\\_\\times \\_\\_\\_\\_\\_\\_$, $1-\\frac{1}{1{0}^{2}}=\\_\\_\\_\\_\\_\\_\\times \\_\\_\\_\\_\\_\\_$;<br/>$(2)$ Calculate using the pattern you discovered:<br/>$(1-\\frac{1}{{2}^{2}})\\times (1-\\frac{1}{{3}^{2}})\\times (1-\\frac{1}{{4}^{2}})\\times \\ldots \\times (1-\\frac{1}{202{0}^{2}})\\times (1-\\frac{1}{202{1}^{2}})$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1011}{2021}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Observing the following expressions:<br/>$1-\frac{1}{{2}^{2}}=1-\frac{1}{4}=\frac{3}{4}=\frac{1}{2}\times \frac{3}{2}$;<br/>$1-\frac{1}{{3}^{2}}=1-\frac{1}{9}=\frac{8}{9}=\frac{2}{3}\times \frac{4}{3}$;<br/>$1-\frac{1}{{4}^{2}}=1-\frac{1}{16}=\frac{15}{16}=\frac{3}{4}\times \frac{5}{4}$;<br/>$1-\frac{1}{{5}^{2}}=1-\frac{1}{25}=\frac{24}{25}=\frac{4}{5}\times \frac{6}{5}$;<br/>$\ldots $<br/>$(1)$ Fill in the blanks according to the pattern you discovered: $1-\frac{1}{{6}^{2}}=\_\_\_\_\_\_\times \_\_\_\_\_\_$, $1-\frac{1}{1{0}^{2}}=\_\_\_\_\_\_\times \_\_\_\_\_\_$;<br/>$(2)$ Calculate using the pattern you discovered:<br/>$(1-\frac{1}{{2}^{2}})\times (1-\frac{1}{{3}^{2}})\times (1-\frac{1}{{4}^{2}})\times \ldots \times (1-\frac{1}{202{0}^{2}})\times (1-\frac{1}{202{1}^{2}})$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the Cartesian coordinate system xOy, given points A(0, 1) and B(-3, 4), if point C is on the bisector of $\\angle AOB$ and $|\\overrightarrow{OC}|=2$, then $\\overrightarrow{OC}=$ ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left(- \\frac{\\sqrt{10}}{5}, \\frac{3\\sqrt{10}}{5}\\right)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the Cartesian coordinate system xOy, given points A(0, 1) and B(-3, 4), if point C is on the bisector of $\angle AOB$ and $|\overrightarrow{OC}|=2$, then $\overrightarrow{OC}=$ ______. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A rectangular solid has three adjacent faces with areas of $1$, $2$, and $2$, respectively. All the vertices of the rectangular solid are located on the same sphere. Find the volume of this sphere.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\sqrt{6}\\pi",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A rectangular solid has three adjacent faces with areas of $1$, $2$, and $2$, respectively. All the vertices of the rectangular solid are located on the same sphere. Find the volume of this sphere. |
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 Star lists the whole numbers from $1$ to $50$ and Emilio copies these numbers, replacing each occurrence of the digits $2$ and $3$ by the digit $1$, find how much larger Star's sum is than Emilio's.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "310",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given Star lists the whole numbers from $1$ to $50$ and Emilio copies these numbers, replacing each occurrence of the digits $2$ and $3$ by the digit $1$, find how much larger Star's sum is than Emilio's. |
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 circles with radii \\( r_{1} \\) and \\( r_{2} \\) touch each other externally. The segment of their common external tangent that lies between the points of tangency is rotated around the line through the centers of the circles. Express the lateral surface area of the resulting truncated cone in terms of \\( r_{1} \\) and \\( r_{2} \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "4\\pi r_1 r_2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The circles with radii \( r_{1} \) and \( r_{2} \) touch each other externally. The segment of their common external tangent that lies between the points of tangency is rotated around the line through the centers of the circles. Express the lateral surface area of the resulting truncated cone in terms of \( r_{1} \) and \( r_{2} \). |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that the radius of the base of a cone is $\\sqrt{2} cm$ and the slant height is $3\\sqrt{2} cm$, find the lateral surface area of the cone.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "6\\pi \\, \\text{cm}^2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the radius of the base of a cone is $\sqrt{2} cm$ and the slant height is $3\sqrt{2} cm$, find the lateral surface area of the cone. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A series of $2023$ positive integers has a unique mode, which occurs exactly $15$ times. What is the minimum number of distinct values that can occur in this list?\nA) 143\nB) 144\nC) 145\nD) 146\nE) 147\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\textbf{(C)}\\ 145",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A series of $2023$ positive integers has a unique mode, which occurs exactly $15$ times. What is the minimum number of distinct values that can occur in this list?
A) 143
B) 144
C) 145
D) 146
E) 147 |
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 number of centimeters in 3.5 meters is approximately how many?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "350",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The number of centimeters in 3.5 meters is approximately how many? |
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 whole numbers are between $\\sqrt[4]{50}$ and $\\sqrt[4]{1000}$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | How many whole numbers are between $\sqrt[4]{50}$ and $\sqrt[4]{1000}$? |
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": "For which values of \\( n \\) can a square be divided into \\( n \\) squares?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\{1\\} \\cup \\{2k + 2, 2k + 5 \\text{ pour } k \\geq 1\\}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For which values of \( n \) can a square be divided into \( n \) squares? |
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 $x > 0, y > 0$, and $2x + 8y = xy$, find the minimum value of $x + y$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "18",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that $x > 0, y > 0$, and $2x + 8y = xy$, find the minimum value of $x + y$. |
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 $\\triangle ABC$ with $AB=2AC$ and $BC=4$, find the maximum area of $\\triangle ABC$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{16}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given $\triangle ABC$ with $AB=2AC$ and $BC=4$, find the maximum area of $\triangle ABC$. |
numina_cn_k12 | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "For $x > 1$, the inequality $x^2 \\geq a$ is always true. Determine the range of values for the real number $a$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(-\\infty, 1]",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | For $x > 1$, the inequality $x^2 \geq a$ is always true. Determine the range of values for 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": "A polygon has an interior angle sum that is 180° more than 4 times its exterior angle sum. Find the number of edges of this polygon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "n=11",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A polygon has an interior angle sum that is 180° more than 4 times its exterior angle sum. Find the number of edges of this polygon. |
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 arithmetic sequence 1, -1, -3, -5, …, -89 has how many terms?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "46",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The arithmetic sequence 1, -1, -3, -5, …, -89 has how many terms? |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the coordinate plane, the curve $xy = 2$ intersects a circle at four points, three of which are $\\left( 4, \\frac{1}{2} \\right),$ $\\left( -2, -1 \\right),$ and $\\left( \\frac{2}{3}, 3 \\right).$ Find the fourth point of intersection.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left(-\\frac{3}{4}, -\\frac{8}{3}\\right)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In the coordinate plane, the curve $xy = 2$ intersects a circle at four points, three of which are $\left( 4, \frac{1}{2} \right),$ $\left( -2, -1 \right),$ and $\left( \frac{2}{3}, 3 \right).$ Find the fourth point of intersection. |
numina_synthetic_amc | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that the probability that a boy is chosen is $\\frac{3}{5}$ of the probability that a girl is chosen, find the ratio of the number of boys to the total number of boys and girls.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given that the probability that a boy is chosen is $\frac{3}{5}$ of the probability that a girl is chosen, find the ratio of the number of boys to the total number of boys and girls. |
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": "Compare $x^{2}+1$ and $2x$. \n$(1)$ Try (fill in with \"$<$\", \"$=$\", or \"$>$\"): \n① When $x=1$, $x^{2}+1 \\_\\_\\_\\_\\_\\_2x$; \n② When $x=0$, $x^{2}+1 \\_\\_\\_\\_\\_\\_2x$; \n③ When $x=-2$, $x^{2}+1 \\_\\_\\_\\_\\_\\_2x.$ \n$(2)$ Induction: For any real number $x$, what is the relationship between $x^{2}+1$ and $2x$? Please explain the reason.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "x^{2}+1 \\geqslant 2x",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Compare $x^{2}+1$ and $2x$.
$(1)$ Try (fill in with "$<$", "$=$", or "$>$"):
① When $x=1$, $x^{2}+1 \_\_\_\_\_\_2x$;
② When $x=0$, $x^{2}+1 \_\_\_\_\_\_2x$;
③ When $x=-2$, $x^{2}+1 \_\_\_\_\_\_2x.$
$(2)$ Induction: For any real number $x$, what is the relationship between $x^{2}+1$ and $2x$? Please explain the reason. |
numina_synthetic_math | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A hyperbola has its two foci at $(3, 6)$ and $(9, 10)$. Find the coordinates of its center.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "(6, 8)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | A hyperbola has its two foci at $(3, 6)$ and $(9, 10)$. Find the coordinates of its center. |
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": "An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common fraction.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common 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": "Let \\( A, B, C \\) be distinct points on the parabola \\( y = x^2 \\). \\( R \\) is the radius of the circumcircle of triangle \\( \\triangle ABC \\). Find the range of values for \\( R \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\left( \\frac{1}{2}, +\\infty \\right)",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let \( A, B, C \) be distinct points on the parabola \( y = x^2 \). \( R \) is the radius of the circumcircle of triangle \( \triangle ABC \). Find the range of values for \( R \). |
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 $\\overrightarrow{a}=(4,2)$ and $\\overrightarrow{b}=(m,3)$, if there exists a real number $\\lambda$ such that $\\overrightarrow{a}=\\lambda\\overrightarrow{b}$, then the value of the real number $m$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Given vectors $\overrightarrow{a}=(4,2)$ and $\overrightarrow{b}=(m,3)$, if there exists a real number $\lambda$ such that $\overrightarrow{a}=\lambda\overrightarrow{b}$, 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": "Three points $(1, -3)$, $(4a + 1, 5)$, and $(-3a + 2, -1)$ lie on the same line. What is the value of $a$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Three points $(1, -3)$, $(4a + 1, 5)$, and $(-3a + 2, -1)$ lie on the same line. What is the value of $a$? |
numina_aops_forum | [
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$ \n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "y = x^2 + bx + c - \\frac{1}{4}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$ |
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 a meeting, every two participants shook hands exactly once. It was recorded that there were a total of 91 handshakes. How many people attended the meeting?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | In a meeting, every two participants shook hands exactly once. It was recorded that there were a total of 91 handshakes. How many people attended the meeting? |
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 30% of a number is the same as 40% of 40, what is the number?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\frac{160}{3}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | If 30% of a number is the same as 40% of 40, what is the number? |
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 following experiment is conducted. In a room, there are two identical-looking urns. The left one contains $a$ balls, and the right one contains $b$ balls. People enter the room one by one, and each transfers a ball from the left urn to the right urn or from the right urn to the left urn. It is assumed that the probability of transferring a ball from the left urn to the right urn is equal to the probability of transferring a ball from the right urn to the left urn, which is $\\frac{1}{2}$. The experiment continues until one of the urns is empty. What is the probability that the left urn is emptied? What is the probability that the right urn is emptied? What is the probability that the experiment never ends?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The following experiment is conducted. In a room, there are two identical-looking urns. The left one contains $a$ balls, and the right one contains $b$ balls. People enter the room one by one, and each transfers a ball from the left urn to the right urn or from the right urn to the left urn. It is assumed that the probability of transferring a ball from the left urn to the right urn is equal to the probability of transferring a ball from the right urn to the left urn, which is $\frac{1}{2}$. The experiment continues until one of the urns is empty. What is the probability that the left urn is emptied? What is the probability that the right urn is emptied? What is the probability that the experiment never ends? |
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": "Evaluate $\\left\\lfloor -\\frac{7}{4}\\right\\rfloor.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | Evaluate $\left\lfloor -\frac{7}{4}\right\rfloor.$ |
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 solution set of the system of inequalities $$\\begin{cases} |x|-1＜0 \\\\ x^{2}-3x＜0\\end{cases}$$ is the set of values of x satisfying both $|x|-1＜0$ and $x^{2}-3x＜0$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
]
| math | {
"ground_truth": "\\{x|0＜x＜1\\}",
"style": "rule"
} | {
"index": 0,
"split": "dummy"
} | The solution set of the system of inequalities $$\begin{cases} |x|-1＜0 \\ x^{2}-3x＜0\end{cases}$$ is the set of values of x satisfying both $|x|-1＜0$ and $x^{2}-3x＜0$. |
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