data_source
stringclasses 6
values | prompt
listlengths 2
2
| ability
stringclasses 1
value | reward_model
dict | extra_info
dict | problem
stringlengths 22
2.03k
| prompt_messages
listlengths 2
2
| gt
stringlengths 1
343
|
|---|---|---|---|---|---|---|---|
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 polynomial $\\frac{1}{5}{x}^{m+1}y^{2}+xy-4x^{3}+1$ is a sixth-degree polynomial, and the monomial $\\frac{1}{8}x^{2n}y^{5-m}$ has the same degree as the polynomial, find the value of $\\left(-m\\right)^{3}+2n$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-23",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that the polynomial $\frac{1}{5}{x}^{m+1}y^{2}+xy-4x^{3}+1$ is a sixth-degree polynomial, and the monomial $\frac{1}{8}x^{2n}y^{5-m}$ has the same degree as the polynomial, find the value of $\left(-m\right)^{3}+2n$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that the polynomial $\\frac{1}{5}{x}^{m+1}y^{2}+xy-4x^{3}+1$ is a sixth-degree polynomial, and the monomial $\\frac{1}{8}x^{2n}y^{5-m}$ has the same degree as the polynomial, find the value of $\\left(-m\\right)^{3}+2n$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-23
|
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": "Factorization: $a^{2}-1=$____.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(a+1)(a-1)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Factorization: $a^{2}-1=$____.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Factorization: $a^{2}-1=$____. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(a+1)(a-1)
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find all natural numbers \\( n \\) such that the number \\( n^2 + 77n \\) is a perfect square of a natural number.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4, 99, 175, 1444",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find all natural numbers \( n \) such that the number \( n^2 + 77n \) is a perfect square of a natural number.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find all natural numbers \\( n \\) such that the number \\( n^2 + 77n \\) is a perfect square of a natural number. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
4, 99, 175, 1444
|
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 solution to the equation $2\\left(x-6\\right)=-16$ is also a solution to the equation $a(x+3)=\\frac{1}{2}a+x$, find the value of $a^2-\\frac{a}{2}+1$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "19",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that the solution to the equation $2\left(x-6\right)=-16$ is also a solution to the equation $a(x+3)=\frac{1}{2}a+x$, find the value of $a^2-\frac{a}{2}+1$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that the solution to the equation $2\\left(x-6\\right)=-16$ is also a solution to the equation $a(x+3)=\\frac{1}{2}a+x$, find the value of $a^2-\\frac{a}{2}+1$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
19
|
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": "It is well known that 777-headed dragons have either 9-headed or 13-headed necks. Two dragons are identical if they have the same number of 9-headed necks. How many different 777-headed dragons are there?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "7",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
It is well known that 777-headed dragons have either 9-headed or 13-headed necks. Two dragons are identical if they have the same number of 9-headed necks. How many different 777-headed dragons are there?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "It is well known that 777-headed dragons have either 9-headed or 13-headed necks. Two dragons are identical if they have the same number of 9-headed necks. How many different 777-headed dragons are there? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
7
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In polar coordinates, the point $\\left( -3, \\frac{\\pi}{6} \\right)$ corresponds to which point in the standard polar coordinate representation? Enter your answer as $(r,\\theta)$, where $r > 0$ and $0 \\leq \\theta < 2\\pi$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(3, \\frac{7\\pi}{6})",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In polar coordinates, the point $\left( -3, \frac{\pi}{6} \right)$ corresponds to which point in the standard polar coordinate representation? Enter your answer as $(r,\theta)$, where $r > 0$ and $0 \leq \theta < 2\pi$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In polar coordinates, the point $\\left( -3, \\frac{\\pi}{6} \\right)$ corresponds to which point in the standard polar coordinate representation? Enter your answer as $(r,\\theta)$, where $r > 0$ and $0 \\leq \\theta < 2\\pi$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(3, \frac{7\pi}{6})
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The length of the real axis of the hyperbola $\\dfrac{x^2}{4} - y^2 = 1$ is _______, and the distance from the foci to the asymptote is _______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The length of the real axis of the hyperbola $\dfrac{x^2}{4} - y^2 = 1$ is _______, and the distance from the foci to the asymptote is _______.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The length of the real axis of the hyperbola $\\dfrac{x^2}{4} - y^2 = 1$ is _______, and the distance from the foci to the asymptote is _______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
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": "If $(x+a)(x-2)=x^2+bx-6$, then $a=\\ $; $b=\\ $.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "3, 1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
If $(x+a)(x-2)=x^2+bx-6$, then $a=\ $; $b=\ $.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "If $(x+a)(x-2)=x^2+bx-6$, then $a=\\ $; $b=\\ $. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
3, 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": "Write 640,080,000 in scientific notation, and then convert it to a number using \"billion\" as the unit.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "0.64008 \\text{ billion}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Write 640,080,000 in scientific notation, and then convert it to a number using "billion" as the unit.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Write 640,080,000 in scientific notation, and then convert it to a number using \"billion\" as the unit. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
0.64008 \text{ billion}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Suppose $m$ is a positive integer such that in base $5$, it can be expressed as $\\overline{XYZ}_5$, and in base $9$, it can be expressed as $\\overline{ZYX}_9$. Find the largest possible value of $m$ in base $10$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "121",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Suppose $m$ is a positive integer such that in base $5$, it can be expressed as $\overline{XYZ}_5$, and in base $9$, it can be expressed as $\overline{ZYX}_9$. Find the largest possible value of $m$ in base $10$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Suppose $m$ is a positive integer such that in base $5$, it can be expressed as $\\overline{XYZ}_5$, and in base $9$, it can be expressed as $\\overline{ZYX}_9$. Find the largest possible value of $m$ in base $10$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
121
|
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": "Eight students participate in a cookie eating contest. The graph shows the number of cookies eaten by each participating student. Alice ate the most cookies and Tom ate the fewest. How many more cookies than Tom did Alice eat?\n\n[asy]\ndefaultpen(linewidth(1pt)+fontsize(10pt));\npair[] yaxis = new pair[10];\nfor( int i = 0 ; i < 10 ; ++i ){\n\nyaxis[i] = (0,i);\n\ndraw(yaxis[i]--yaxis[i]+(17,0));\n}\ndraw((0,0)--(0,9));\ndraw((17,9)--(17,0));\n\nfill((1,0)--(1,4)--(2,4)--(2,0)--cycle,grey);\nfill((3,0)--(3,5)--(4,5)--(4,0)--cycle,grey);\nfill((5,0)--(5,8)--(6,8)--(6,0)--cycle,grey);\nfill((7,0)--(7,3)--(8,3)--(8,0)--cycle,grey);\nfill((9,0)--(9,6)--(10,6)--(10,0)--cycle,grey);\nfill((11,0)--(11,7)--(12,7)--(12,0)--cycle,grey);\nfill((13,0)--(13,2)--(14,2)--(14,0)--cycle,grey);\nfill((15,0)--(15,1)--(16,1)--(16,0)--cycle,grey);\nlabel(\"0\",yaxis[0],W);\nlabel(\"1\",yaxis[1],W);\nlabel(\"2\",yaxis[2],W);\nlabel(\"3\",yaxis[3],W);\nlabel(\"4\",yaxis[4],W);\nlabel(\"5\",yaxis[5],W);\nlabel(\"6\",yaxis[6],W);\nlabel(\"7\",yaxis[7],W);\nlabel(\"8\",yaxis[8],W);\nlabel(\"9\",yaxis[9],W);\nlabel(\"Students\",(8.5,0),S);\nlabel(\"Results of a Cookie Eating Contest\",(8.5,9),N);\nlabel(rotate(90)*\"$\\#$ of Cookies Eaten\",(-1,4.5),W);\n[/asy]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "7",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Eight students participate in a cookie eating contest. The graph shows the number of cookies eaten by each participating student. Alice ate the most cookies and Tom ate the fewest. How many more cookies than Tom did Alice eat?
[asy]
defaultpen(linewidth(1pt)+fontsize(10pt));
pair[] yaxis = new pair[10];
for( int i = 0 ; i < 10 ; ++i ){
yaxis[i] = (0,i);
draw(yaxis[i]--yaxis[i]+(17,0));
}
draw((0,0)--(0,9));
draw((17,9)--(17,0));
fill((1,0)--(1,4)--(2,4)--(2,0)--cycle,grey);
fill((3,0)--(3,5)--(4,5)--(4,0)--cycle,grey);
fill((5,0)--(5,8)--(6,8)--(6,0)--cycle,grey);
fill((7,0)--(7,3)--(8,3)--(8,0)--cycle,grey);
fill((9,0)--(9,6)--(10,6)--(10,0)--cycle,grey);
fill((11,0)--(11,7)--(12,7)--(12,0)--cycle,grey);
fill((13,0)--(13,2)--(14,2)--(14,0)--cycle,grey);
fill((15,0)--(15,1)--(16,1)--(16,0)--cycle,grey);
label("0",yaxis[0],W);
label("1",yaxis[1],W);
label("2",yaxis[2],W);
label("3",yaxis[3],W);
label("4",yaxis[4],W);
label("5",yaxis[5],W);
label("6",yaxis[6],W);
label("7",yaxis[7],W);
label("8",yaxis[8],W);
label("9",yaxis[9],W);
label("Students",(8.5,0),S);
label("Results of a Cookie Eating Contest",(8.5,9),N);
label(rotate(90)*"$\#$ of Cookies Eaten",(-1,4.5),W);
[/asy]
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Eight students participate in a cookie eating contest. The graph shows the number of cookies eaten by each participating student. Alice ate the most cookies and Tom ate the fewest. How many more cookies than Tom did Alice eat?\n\n[asy]\ndefaultpen(linewidth(1pt)+fontsize(10pt));\npair[] yaxis = new pair[10];\nfor( int i = 0 ; i < 10 ; ++i ){\n\nyaxis[i] = (0,i);\n\ndraw(yaxis[i]--yaxis[i]+(17,0));\n}\ndraw((0,0)--(0,9));\ndraw((17,9)--(17,0));\n\nfill((1,0)--(1,4)--(2,4)--(2,0)--cycle,grey);\nfill((3,0)--(3,5)--(4,5)--(4,0)--cycle,grey);\nfill((5,0)--(5,8)--(6,8)--(6,0)--cycle,grey);\nfill((7,0)--(7,3)--(8,3)--(8,0)--cycle,grey);\nfill((9,0)--(9,6)--(10,6)--(10,0)--cycle,grey);\nfill((11,0)--(11,7)--(12,7)--(12,0)--cycle,grey);\nfill((13,0)--(13,2)--(14,2)--(14,0)--cycle,grey);\nfill((15,0)--(15,1)--(16,1)--(16,0)--cycle,grey);\nlabel(\"0\",yaxis[0],W);\nlabel(\"1\",yaxis[1],W);\nlabel(\"2\",yaxis[2],W);\nlabel(\"3\",yaxis[3],W);\nlabel(\"4\",yaxis[4],W);\nlabel(\"5\",yaxis[5],W);\nlabel(\"6\",yaxis[6],W);\nlabel(\"7\",yaxis[7],W);\nlabel(\"8\",yaxis[8],W);\nlabel(\"9\",yaxis[9],W);\nlabel(\"Students\",(8.5,0),S);\nlabel(\"Results of a Cookie Eating Contest\",(8.5,9),N);\nlabel(rotate(90)*\"$\\#$ of Cookies Eaten\",(-1,4.5),W);\n[/asy] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
7
|
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": "Kolya is older than Tolya, and the age of each of them is a whole number less than 100. If you reverse the digits of Kolya's age, you get Tolya's age. Moreover, the difference between the squares of their ages is a square of an integer. How old are they?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\text{Kolya is 65 years old and Tolya is 56 years old.}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Kolya is older than Tolya, and the age of each of them is a whole number less than 100. If you reverse the digits of Kolya's age, you get Tolya's age. Moreover, the difference between the squares of their ages is a square of an integer. How old are they?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Kolya is older than Tolya, and the age of each of them is a whole number less than 100. If you reverse the digits of Kolya's age, you get Tolya's age. Moreover, the difference between the squares of their ages is a square of an integer. How old are they? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\text{Kolya is 65 years old and Tolya is 56 years old.}
|
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 $\\overrightarrow {m}$=(sinα-2，-cosα), $\\overrightarrow {n}$=(-sinα，cosα), where α∈R.\n1. If $\\overrightarrow {m}$⊥$\\overrightarrow {n}$, find angle α;\n2. If $|\\overrightarrow {m}$-$\\overrightarrow {n}$|=$\\sqrt {2}$, find the value of cos2α.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac {1}{8}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given vectors $\overrightarrow {m}$=(sinα-2，-cosα), $\overrightarrow {n}$=(-sinα，cosα), where α∈R.
1. If $\overrightarrow {m}$⊥$\overrightarrow {n}$, find angle α;
2. If $|\overrightarrow {m}$-$\overrightarrow {n}$|=$\sqrt {2}$, find the value of cos2α.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given vectors $\\overrightarrow {m}$=(sinα-2，-cosα), $\\overrightarrow {n}$=(-sinα，cosα), where α∈R.\n1. If $\\overrightarrow {m}$⊥$\\overrightarrow {n}$, find angle α;\n2. If $|\\overrightarrow {m}$-$\\overrightarrow {n}$|=$\\sqrt {2}$, find the value of cos2α. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {1}{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": "What perfect-square integer is closest to 342?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "324",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What perfect-square integer is closest to 342?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What perfect-square integer is closest to 342? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
324
|
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 museum is organizing a quiz where participants must guess the year a certain artifact was created. Clues given: the year uses each of the digits: 1, 2, 2, 5, 5, 9 exactly once, and the year must start with a prime digit.\n\nHow many different possible years could be guessed based on these clues?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "120",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A museum is organizing a quiz where participants must guess the year a certain artifact was created. Clues given: the year uses each of the digits: 1, 2, 2, 5, 5, 9 exactly once, and the year must start with a prime digit.
How many different possible years could be guessed based on these clues?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A museum is organizing a quiz where participants must guess the year a certain artifact was created. Clues given: the year uses each of the digits: 1, 2, 2, 5, 5, 9 exactly once, and the year must start with a prime digit.\n\nHow many different possible years could be guessed based on these clues? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
120
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A circle constructed on the side \\( AD \\) of parallelogram \\( ABCD \\) as its diameter passes through vertex \\( B \\) and the midpoint of side \\( BC \\). Find the angles of the parallelogram.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\angle BAD = 60^\\circ \\text{ and } \\angle ABC = 120^\\circ",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A circle constructed on the side \( AD \) of parallelogram \( ABCD \) as its diameter passes through vertex \( B \) and the midpoint of side \( BC \). Find the angles of the parallelogram.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A circle constructed on the side \\( AD \\) of parallelogram \\( ABCD \\) as its diameter passes through vertex \\( B \\) and the midpoint of side \\( BC \\). Find the angles of the parallelogram. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\angle BAD = 60^\circ \text{ and } \angle ABC = 120^\circ
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In triangle $\\triangle ABC$, $a=3\\sqrt{2}$, $c=3$, $C=30^{\\circ}$, find the measure of angle $A$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "135^{\\circ}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In triangle $\triangle ABC$, $a=3\sqrt{2}$, $c=3$, $C=30^{\circ}$, find the measure of angle $A$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In triangle $\\triangle ABC$, $a=3\\sqrt{2}$, $c=3$, $C=30^{\\circ}$, find the measure of angle $A$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
135^{\circ}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $A=\\{x|x^{2}-2x-3 < 0\\}$, $B=\\{x||x-1| < a\\}$.\n(1) Find the range of real number $a$ if $A⊊B$;\n(2) Find the range of real number $a$ if $B⊊A$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(-\\infty,2)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $A=\{x|x^{2}-2x-3 < 0\}$, $B=\{x||x-1| < a\}$.
(1) Find the range of real number $a$ if $A⊊B$;
(2) Find the range of real number $a$ if $B⊊A$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given $A=\\{x|x^{2}-2x-3 < 0\\}$, $B=\\{x||x-1| < a\\}$.\n(1) Find the range of real number $a$ if $A⊊B$;\n(2) Find the range of real number $a$ if $B⊊A$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(-\infty,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 $\\triangle ABC$, if $b = 2a$ and $B = A + 60^{\\circ}$, find the measure of $\\angle A$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "A = 30^{\\circ}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In $\triangle ABC$, if $b = 2a$ and $B = A + 60^{\circ}$, find the measure of $\angle A$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, if $b = 2a$ and $B = A + 60^{\\circ}$, find the measure of $\\angle A$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
A = 30^{\circ}
|
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 increment of the argument and the function \\( y = x^2 \\) if the argument \\( x \\) changes from 2 to 2.5.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2.25",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Determine the increment of the argument and the function \( y = x^2 \) if the argument \( x \) changes from 2 to 2.5.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Determine the increment of the argument and the function \\( y = x^2 \\) if the argument \\( x \\) changes from 2 to 2.5. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
2.25
|
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 Jaime's mathematics test, there were 100 problems: 20 arithmetic, 40 algebra, and 40 geometry problems. Jaime answered $60\\%$ of the arithmetic, $50\\%$ of the algebra, and $70\\%$ of the geometry problems correctly. The passing grade is set at $65\\%$. How many more questions would Jaime have needed to answer correctly to pass the test?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "5",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In Jaime's mathematics test, there were 100 problems: 20 arithmetic, 40 algebra, and 40 geometry problems. Jaime answered $60\%$ of the arithmetic, $50\%$ of the algebra, and $70\%$ of the geometry problems correctly. The passing grade is set at $65\%$. How many more questions would Jaime have needed to answer correctly to pass the test?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In Jaime's mathematics test, there were 100 problems: 20 arithmetic, 40 algebra, and 40 geometry problems. Jaime answered $60\\%$ of the arithmetic, $50\\%$ of the algebra, and $70\\%$ of the geometry problems correctly. The passing grade is set at $65\\%$. How many more questions would Jaime have needed to answer correctly to pass the test? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
5
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The inequality $|x+3|+|x-1| \\geq a^2 - 3a$ holds for any real number $x$. Then, the range of the real number $a$ is.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "[-1, 4]",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The inequality $|x+3|+|x-1| \geq a^2 - 3a$ holds for any real number $x$. Then, the range of the real number $a$ is.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The inequality $|x+3|+|x-1| \\geq a^2 - 3a$ holds for any real number $x$. Then, the range of the real number $a$ is. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[-1, 4]
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Define an ordered triple $(A, B, C)$ of sets to be specifically intersecting if $|A \\cap B| = 2$, $|B \\cap C| = |C \\cap A| = 1$ and $A \\cap B \\cap C = \\emptyset$. Each set is a subset of $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$. Let $M$ be the number of such specifically intersecting ordered triples $(A, B, C)$. Find the remainder when $M$ is divided by 1000.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "288",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Define an ordered triple $(A, B, C)$ of sets to be specifically intersecting if $|A \cap B| = 2$, $|B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. Each set is a subset of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Let $M$ be the number of such specifically intersecting ordered triples $(A, B, C)$. Find the remainder when $M$ is divided by 1000.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Define an ordered triple $(A, B, C)$ of sets to be specifically intersecting if $|A \\cap B| = 2$, $|B \\cap C| = |C \\cap A| = 1$ and $A \\cap B \\cap C = \\emptyset$. Each set is a subset of $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$. Let $M$ be the number of such specifically intersecting ordered triples $(A, B, C)$. Find the remainder when $M$ is divided by 1000. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
288
|
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 smallest natural number \\( n \\) such that there exist \\( n \\) different natural numbers \\( s_{1}, s_{2}, \\ldots, s_{n} \\) for which \n\n\\[\n\\left(1-\\frac{1}{s_{1}}\\right)\\left(1-\\frac{1}{s_{2}}\\right) \\ldots\\left(1-\\frac{1}{s_{n}}\\right)=\\frac{7}{66} ?\n\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "9",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What is the smallest natural number \( n \) such that there exist \( n \) different natural numbers \( s_{1}, s_{2}, \ldots, s_{n} \) for which
\[
\left(1-\frac{1}{s_{1}}\right)\left(1-\frac{1}{s_{2}}\right) \ldots\left(1-\frac{1}{s_{n}}\right)=\frac{7}{66} ?
\]
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the smallest natural number \\( n \\) such that there exist \\( n \\) different natural numbers \\( s_{1}, s_{2}, \\ldots, s_{n} \\) for which \n\n\\[\n\\left(1-\\frac{1}{s_{1}}\\right)\\left(1-\\frac{1}{s_{2}}\\right) \\ldots\\left(1-\\frac{1}{s_{n}}\\right)=\\frac{7}{66} ?\n\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
9
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "If we draw all the rectangles with equal area, a common center, and parallel sides, what shape will the shaded region form on the paper?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "xy = \\frac{T}{4} \\text{ and } xy = -\\frac{T}{4}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
If we draw all the rectangles with equal area, a common center, and parallel sides, what shape will the shaded region form on the paper?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "If we draw all the rectangles with equal area, a common center, and parallel sides, what shape will the shaded region form on the paper? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
xy = \frac{T}{4} \text{ and } xy = -\frac{T}{4}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $M=\\{1, 3, 5, 7, 9\\}$ and $N=\\{2, 4, 6, 8\\}$. Define the set $M \\oplus N = \\{(a, b) | a = a_1 + a_2 + \\ldots + a_m, b = b_1 + b_2 + \\ldots + b_n\\}$. Find all elements of the set $M \\oplus N$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\emptyset, \\{(25, 20)\\}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let $M=\{1, 3, 5, 7, 9\}$ and $N=\{2, 4, 6, 8\}$. Define the set $M \oplus N = \{(a, b) | a = a_1 + a_2 + \ldots + a_m, b = b_1 + b_2 + \ldots + b_n\}$. Find all elements of the set $M \oplus N$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $M=\\{1, 3, 5, 7, 9\\}$ and $N=\\{2, 4, 6, 8\\}$. Define the set $M \\oplus N = \\{(a, b) | a = a_1 + a_2 + \\ldots + a_m, b = b_1 + b_2 + \\ldots + b_n\\}$. Find all elements of the set $M \\oplus N$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\emptyset, \{(25, 20)\}
|
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 set $\\{1,2,3,4\\}$, find the probability that the sum of any two distinct numbers chosen from the set is a multiple of $3$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{1}{3}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the set $\{1,2,3,4\}$, find the probability that the sum of any two distinct numbers chosen from the set is a multiple of $3$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the set $\\{1,2,3,4\\}$, find the probability that the sum of any two distinct numbers chosen from the set is a multiple of $3$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{1}{3}
|
numina_synthetic_amc
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Let $y$ be a piecewise linear function defined by:\n\\[\ny = \n\\begin{cases} \n k_1x & \\text{if } x < 0 \\\\\n k_2x & \\text{if } x \\geq 0\n\\end{cases}\n\\]\nGiven $y = 4$ when $x = 2$, and $y = -20$ when $x = -5$, find $k_1$ and $k_2$, and then evaluate $y$ when $x = 10$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "20",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let $y$ be a piecewise linear function defined by:
\[
y =
\begin{cases}
k_1x & \text{if } x < 0 \\
k_2x & \text{if } x \geq 0
\end{cases}
\]
Given $y = 4$ when $x = 2$, and $y = -20$ when $x = -5$, find $k_1$ and $k_2$, and then evaluate $y$ when $x = 10$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $y$ be a piecewise linear function defined by:\n\\[\ny = \n\\begin{cases} \n k_1x & \\text{if } x < 0 \\\\\n k_2x & \\text{if } x \\geq 0\n\\end{cases}\n\\]\nGiven $y = 4$ when $x = 2$, and $y = -20$ when $x = -5$, find $k_1$ and $k_2$, and then evaluate $y$ when $x = 10$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
20
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "If the universal set $U=\\mathbb{R}$, and the set $A=\\{x|x\\geq1\\}\\cup\\{x|x\\leq0\\}$, then $C_UA=\\ $.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(0,1)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
If the universal set $U=\mathbb{R}$, and the set $A=\{x|x\geq1\}\cup\{x|x\leq0\}$, then $C_UA=\ $.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "If the universal set $U=\\mathbb{R}$, and the set $A=\\{x|x\\geq1\\}\\cup\\{x|x\\leq0\\}$, then $C_UA=\\ $. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(0,1)
|
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": "It is known that point P on the number line represents -5. After moving point P by 3 units, the number represented by point P is __ .\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-8, -2",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
It is known that point P on the number line represents -5. After moving point P by 3 units, the number represented by point P is __ .
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "It is known that point P on the number line represents -5. After moving point P by 3 units, the number represented by point P is __ . Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-8, -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 foci of a hyperbola are located on the x-axis and its asymptotes are represented by the equations y = ± 1/2x, determine the eccentricity of the hyperbola.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that the foci of a hyperbola are located on the x-axis and its asymptotes are represented by the equations y = ± 1/2x, determine the eccentricity of the hyperbola.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that the foci of a hyperbola are located on the x-axis and its asymptotes are represented by the equations y = ± 1/2x, determine the eccentricity of the hyperbola. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\sqrt{5}}{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 the derivative of the function $f(x)$ is $f'(x)$, and it satisfies $f'(x) < 2f(x)$, then determine the relationship between $f(\\ln2)$ and $f(\\ln3)$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "e^{2}f(0) > f(1)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the derivative of the function $f(x)$ is $f'(x)$, and it satisfies $f'(x) < 2f(x)$, then determine the relationship between $f(\ln2)$ and $f(\ln3)$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the derivative of the function $f(x)$ is $f'(x)$, and it satisfies $f'(x) < 2f(x)$, then determine the relationship between $f(\\ln2)$ and $f(\\ln3)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
e^{2}f(0) > 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": "Simplify: $-\\left[+\\left(-6\\right)\\right]=\\_\\_\\_\\_\\_\\_.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "6",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Simplify: $-\left[+\left(-6\right)\right]=\_\_\_\_\_\_.$
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Simplify: $-\\left[+\\left(-6\\right)\\right]=\\_\\_\\_\\_\\_\\_.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
6
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that the normal vectors of planes $\\alpha$ and $\\beta$ are $(1, 3, 4)$ and $(x, 1, -2)$ respectively, if $\\alpha \\perp \\beta$, then $x = \\_\\_\\_\\_\\_\\_$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "x = 5",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that the normal vectors of planes $\alpha$ and $\beta$ are $(1, 3, 4)$ and $(x, 1, -2)$ respectively, if $\alpha \perp \beta$, then $x = \_\_\_\_\_\_$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that the normal vectors of planes $\\alpha$ and $\\beta$ are $(1, 3, 4)$ and $(x, 1, -2)$ respectively, if $\\alpha \\perp \\beta$, then $x = \\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
x = 5
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "\\[ \\left[\\frac{\\text{Куб}}{\\text{Куб}}\\right] \\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "6",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
\[ \left[\frac{\text{Куб}}{\text{Куб}}\right] \]
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "\\[ \\left[\\frac{\\text{Куб}}{\\text{Куб}}\\right] \\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
6
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The graph of the equation $2x + 3y + 4 = 0$ is perpendicular to the graph of the equation $bx + 3y + 4 = 0$. What is the value of $b$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-\\frac{9}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The graph of the equation $2x + 3y + 4 = 0$ is perpendicular to the graph of the equation $bx + 3y + 4 = 0$. What is the value of $b$?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The graph of the equation $2x + 3y + 4 = 0$ is perpendicular to the graph of the equation $bx + 3y + 4 = 0$. What is the value of $b$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-\frac{9}{2}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Express $8.\\overline{137}$ as a common fraction in lowest terms.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{2709}{333}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Express $8.\overline{137}$ as a common fraction in lowest terms.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Express $8.\\overline{137}$ as a common fraction in lowest terms. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{2709}{333}
|
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 $\\frac{5}{x} + \\frac{3}{y} = 2 (x > 0, y > 0)$, find the minimum value of $xy$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "15",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $\frac{5}{x} + \frac{3}{y} = 2 (x > 0, y > 0)$, find the minimum value of $xy$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given $\\frac{5}{x} + \\frac{3}{y} = 2 (x > 0, y > 0)$, find the minimum value of $xy$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
15
|
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 system of equations in $x, y$:\n$$\n\\begin{cases}\nax+y-1=0 \\\\\n4x+ay-2=0\n\\end{cases}\n$$\nIf there are infinitely many solutions, then the value of the real number $a$ is $\\boxed{?}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the system of equations in $x, y$:
$$
\begin{cases}
ax+y-1=0 \\
4x+ay-2=0
\end{cases}
$$
If there are infinitely many solutions, then the value of the real number $a$ is $\boxed{?}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the system of equations in $x, y$:\n$$\n\\begin{cases}\nax+y-1=0 \\\\\n4x+ay-2=0\n\\end{cases}\n$$\nIf there are infinitely many solutions, then the value of the real number $a$ is $\\boxed{?}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
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": "In how many ways can 10 people be seated around a round table if one particular person must always sit between two specific other people?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "80,640",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In how many ways can 10 people be seated around a round table if one particular person must always sit between two specific other people?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In how many ways can 10 people be seated around a round table if one particular person must always sit between two specific other people? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
80,640
|
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": "Gretchen now has ten socks, two of each color: magenta, cyan, black, white, and purple. She randomly draws five socks this time. What is the probability that she has exactly one pair of socks with the same color and the rest are all different colors?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{40}{63}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Gretchen now has ten socks, two of each color: magenta, cyan, black, white, and purple. She randomly draws five socks this time. What is the probability that she has exactly one pair of socks with the same color and the rest are all different colors?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Gretchen now has ten socks, two of each color: magenta, cyan, black, white, and purple. She randomly draws five socks this time. What is the probability that she has exactly one pair of socks with the same color and the rest are all different colors? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{40}{63}
|
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 integrals:\na) $\\int \\sin (7 x+3) \\, dx$,\nb) $\\int \\frac{dx}{x+5}$,\nc) $\\int (3x+5)^{99} \\, dx$,\nd) $\\int e^{x^{3}} x^{2} \\, dx$,\ne) $\\int \\sin ^{3} x \\cos x \\, dx$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{\\sin^4(x)}{4} + C",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find the integrals:
a) $\int \sin (7 x+3) \, dx$,
b) $\int \frac{dx}{x+5}$,
c) $\int (3x+5)^{99} \, dx$,
d) $\int e^{x^{3}} x^{2} \, dx$,
e) $\int \sin ^{3} x \cos x \, dx$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find the integrals:\na) $\\int \\sin (7 x+3) \\, dx$,\nb) $\\int \\frac{dx}{x+5}$,\nc) $\\int (3x+5)^{99} \\, dx$,\nd) $\\int e^{x^{3}} x^{2} \\, dx$,\ne) $\\int \\sin ^{3} x \\cos x \\, dx$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{\sin^4(x)}{4} + C
|
numina_aops_forum
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that $ x \\plus{} \\sin y \\equal{} 2008$ and $ x \\plus{} 2008 \\cos y \\equal{} 2007$ , where $ 0 \\leq y \\leq \\pi/2$ , find the value of $ x \\plus{} y$ .\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2007 + \\frac{\\pi}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that $ x \plus{} \sin y \equal{} 2008$ and $ x \plus{} 2008 \cos y \equal{} 2007$ , where $ 0 \leq y \leq \pi/2$ , find the value of $ x \plus{} y$ .
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that $ x \\plus{} \\sin y \\equal{} 2008$ and $ x \\plus{} 2008 \\cos y \\equal{} 2007$ , where $ 0 \\leq y \\leq \\pi/2$ , find the value of $ x \\plus{} y$ . Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
2007 + \frac{\pi}{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 distance from a point $P$ on the parabola $y^2 = 8x$ to the focus $F$ is $4$, determine the area of the triangle $\\Delta PFO$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that the distance from a point $P$ on the parabola $y^2 = 8x$ to the focus $F$ is $4$, determine the area of the triangle $\Delta PFO$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that the distance from a point $P$ on the parabola $y^2 = 8x$ to the focus $F$ is $4$, determine the area of the triangle $\\Delta PFO$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
4
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "A line segment begins at \\((2, 2)\\) and extends 8 units to end at a point \\((x, y)\\) where \\( y > 0 \\). The line segment has a slope of \\( \\frac{3}{4} \\). What are the coordinates of the endpoint \\((x, y)\\)?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(2 - \\frac{4\\sqrt{5475}}{25}, \\frac{3}{4}(2 - \\frac{4\\sqrt{5475}}{25}) + \\frac{1}{2})",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A line segment begins at \((2, 2)\) and extends 8 units to end at a point \((x, y)\) where \( y > 0 \). The line segment has a slope of \( \frac{3}{4} \). What are the coordinates of the endpoint \((x, y)\)?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A line segment begins at \\((2, 2)\\) and extends 8 units to end at a point \\((x, y)\\) where \\( y > 0 \\). The line segment has a slope of \\( \\frac{3}{4} \\). What are the coordinates of the endpoint \\((x, y)\\)? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(2 - \frac{4\sqrt{5475}}{25}, \frac{3}{4}(2 - \frac{4\sqrt{5475}}{25}) + \frac{1}{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": "Tom has a red marble, a green marble, a blue marble, and four identical yellow marbles. How many different groups of three marbles can Tom choose?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "7",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Tom has a red marble, a green marble, a blue marble, and four identical yellow marbles. How many different groups of three marbles can Tom choose?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Tom has a red marble, a green marble, a blue marble, and four identical yellow marbles. How many different groups of three marbles can Tom choose? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
7
|
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": "Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\\circ$ latitude and $110^\\circ \\text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\\circ \\text{ N}$ latitude and $115^\\circ \\text{ W}$ longitude. Assuming the Earth is a perfect sphere with center $C$, determine the degree measure of $\\angle ACB$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "120^\\circ",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assuming the Earth is a perfect sphere with center $C$, determine the degree measure of $\angle ACB$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\\circ$ latitude and $110^\\circ \\text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\\circ \\text{ N}$ latitude and $115^\\circ \\text{ W}$ longitude. Assuming the Earth is a perfect sphere with center $C$, determine the degree measure of $\\angle ACB$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
120^\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": "The sum of two numbers is 60, and their difference is 10. What is their product?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "875",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The sum of two numbers is 60, and their difference is 10. What is their product?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The sum of two numbers is 60, and their difference is 10. What is their product? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
875
|
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": "Piravena must make a trip from city $A$ to city $B$, and then from city $B$ to city $C$. The cities form a right-angled triangle with the right angle at $B$. The distance from $A$ to $B$ is $4000\\text{ km}$ and from $B$ to $C$ is $3000\\text{ km}$. To take a bus, it costs Piravena $\\$0.15$ per kilometer. To take an airplane, it costs her a $\\$120$ booking fee, plus $\\$0.12$ per kilometer. Determine the cost for Piravena to fly from $A$ to $B$ and then take a bus from $B$ to $C$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\$1050",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Piravena must make a trip from city $A$ to city $B$, and then from city $B$ to city $C$. The cities form a right-angled triangle with the right angle at $B$. The distance from $A$ to $B$ is $4000\text{ km}$ and from $B$ to $C$ is $3000\text{ km}$. To take a bus, it costs Piravena $\$0.15$ per kilometer. To take an airplane, it costs her a $\$120$ booking fee, plus $\$0.12$ per kilometer. Determine the cost for Piravena to fly from $A$ to $B$ and then take a bus from $B$ to $C$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Piravena must make a trip from city $A$ to city $B$, and then from city $B$ to city $C$. The cities form a right-angled triangle with the right angle at $B$. The distance from $A$ to $B$ is $4000\\text{ km}$ and from $B$ to $C$ is $3000\\text{ km}$. To take a bus, it costs Piravena $\\$0.15$ per kilometer. To take an airplane, it costs her a $\\$120$ booking fee, plus $\\$0.12$ per kilometer. Determine the cost for Piravena to fly from $A$ to $B$ and then take a bus from $B$ to $C$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\$1050
|
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 middle school environmental knowledge competition, there are a total of 30 questions. For each correct answer, a participant scores 10 points, and for each wrong answer or no answer, 5 points are deducted. To score more than 90 points, how many questions must Xiao Ming answer correctly at the least?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "17",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In a middle school environmental knowledge competition, there are a total of 30 questions. For each correct answer, a participant scores 10 points, and for each wrong answer or no answer, 5 points are deducted. To score more than 90 points, how many questions must Xiao Ming answer correctly at the least?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In a middle school environmental knowledge competition, there are a total of 30 questions. For each correct answer, a participant scores 10 points, and for each wrong answer or no answer, 5 points are deducted. To score more than 90 points, how many questions must Xiao Ming answer correctly at the least? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
17
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The domain of the function $f(x) = \\sqrt{\\tan x - 1} + \\sqrt{1 - x^2}$ is __________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\text{The domain of } f(x) \\text{ is } \\left[\\frac{\\pi}{4}, 1\\right]",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The domain of the function $f(x) = \sqrt{\tan x - 1} + \sqrt{1 - x^2}$ is __________.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The domain of the function $f(x) = \\sqrt{\\tan x - 1} + \\sqrt{1 - x^2}$ is __________. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\text{The domain of } f(x) \text{ is } \left[\frac{\pi}{4}, 1\right]
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "In the Cartesian coordinate system, $O$ is the origin, and points $A(-1,0)$, $B(0, \\sqrt{3})$, $C(3,0)$. A moving point $D$ satisfies $|\\overrightarrow{CD}|=1$, then the maximum value of $|\\overrightarrow{OA}+ \\overrightarrow{OB}+ \\overrightarrow{OD}|$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\sqrt{7}+1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In the Cartesian coordinate system, $O$ is the origin, and points $A(-1,0)$, $B(0, \sqrt{3})$, $C(3,0)$. A moving point $D$ satisfies $|\overrightarrow{CD}|=1$, then the maximum value of $|\overrightarrow{OA}+ \overrightarrow{OB}+ \overrightarrow{OD}|$ is ______.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In the Cartesian coordinate system, $O$ is the origin, and points $A(-1,0)$, $B(0, \\sqrt{3})$, $C(3,0)$. A moving point $D$ satisfies $|\\overrightarrow{CD}|=1$, then the maximum value of $|\\overrightarrow{OA}+ \\overrightarrow{OB}+ \\overrightarrow{OD}|$ is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt{7}+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": "Express $0.00000011$ in scientific notation.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1.1 \\times 10^{-7}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Express $0.00000011$ in scientific notation.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Express $0.00000011$ in scientific notation. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1.1 \times 10^{-7}
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The parabola $y = x^2 + 5$ and the hyperbola $y^2 - nx^2 = 4$ are tangent. Find $n.$\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "10 + \\sqrt{84}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The parabola $y = x^2 + 5$ and the hyperbola $y^2 - nx^2 = 4$ are tangent. Find $n.$
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The parabola $y = x^2 + 5$ and the hyperbola $y^2 - nx^2 = 4$ are tangent. Find $n.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
10 + \sqrt{84}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given an arithmetic sequence $\\{a\\_n\\}$ where all terms are positive, let $S\\_n$ denote the sum of the first $n$ terms. For all $n \\in \\mathbb{N}^*$ and $n \\geq 2$, it is known that $S\\_n = \\frac{n}{n-1}(a\\_n^2 - a\\_1^2)$. Find the value of $S\\_{20} - 2S\\_{10}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "50",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given an arithmetic sequence $\{a\_n\}$ where all terms are positive, let $S\_n$ denote the sum of the first $n$ terms. For all $n \in \mathbb{N}^*$ and $n \geq 2$, it is known that $S\_n = \frac{n}{n-1}(a\_n^2 - a\_1^2)$. Find the value of $S\_{20} - 2S\_{10}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given an arithmetic sequence $\\{a\\_n\\}$ where all terms are positive, let $S\\_n$ denote the sum of the first $n$ terms. For all $n \\in \\mathbb{N}^*$ and $n \\geq 2$, it is known that $S\\_n = \\frac{n}{n-1}(a\\_n^2 - a\\_1^2)$. Find the value of $S\\_{20} - 2S\\_{10}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
50
|
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 $\\sin \\alpha= \\frac {3}{5}$ and $\\alpha \\in (0, \\frac {\\pi}{2})$, then the value of $\\tan 2\\alpha$ is ______.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac {24}{7}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
If $\sin \alpha= \frac {3}{5}$ and $\alpha \in (0, \frac {\pi}{2})$, then the value of $\tan 2\alpha$ is ______.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "If $\\sin \\alpha= \\frac {3}{5}$ and $\\alpha \\in (0, \\frac {\\pi}{2})$, then the value of $\\tan 2\\alpha$ is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {24}{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": "Given 8 volunteer positions to be allocated to 3 schools, with each school receiving at least one position and the allocations being unequal, find the number of ways to distribute the positions.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "12",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given 8 volunteer positions to be allocated to 3 schools, with each school receiving at least one position and the allocations being unequal, find the number of ways to distribute the positions.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given 8 volunteer positions to be allocated to 3 schools, with each school receiving at least one position and the allocations being unequal, find the number of ways to distribute the positions. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
12
|
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 can be the product of several distinct prime numbers if it is divisible by each of them minus 1? Find all possible values of this product.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "6, 42, 1806",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What can be the product of several distinct prime numbers if it is divisible by each of them minus 1? Find all possible values of this product.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What can be the product of several distinct prime numbers if it is divisible by each of them minus 1? Find all possible values of this product. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
6, 42, 1806
|
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 local library is organizing a queue for a book signing event. In how many ways can eight people line up to get their books signed?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "40320",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The local library is organizing a queue for a book signing event. In how many ways can eight people line up to get their books signed?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The local library is organizing a queue for a book signing event. In how many ways can eight people line up to get their books signed? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
40320
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Compute\n\\[\\sin^2 0^\\circ + \\sin^2 2^\\circ + \\sin^2 4^\\circ + \\dots + \\sin^2 90^\\circ.\\]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "23",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Compute
\[\sin^2 0^\circ + \sin^2 2^\circ + \sin^2 4^\circ + \dots + \sin^2 90^\circ.\]
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Compute\n\\[\\sin^2 0^\\circ + \\sin^2 2^\\circ + \\sin^2 4^\\circ + \\dots + \\sin^2 90^\\circ.\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
23
|
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 $\\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.\n\nSuppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then what is the surface area of $DABC$?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "40\\sqrt{1925}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.
Suppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then what is the surface area of $DABC$?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.\n\nSuppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then what is the surface area of $DABC$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
40\sqrt{1925}
|
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 school's science club, there are 8 boys and 10 girls. I need to assemble a team of 7 members to send to a national science competition. The team rules state that I must select 4 boys and 3 girls. In how many ways can I select the team?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "8400",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
In the school's science club, there are 8 boys and 10 girls. I need to assemble a team of 7 members to send to a national science competition. The team rules state that I must select 4 boys and 3 girls. In how many ways can I select the team?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "In the school's science club, there are 8 boys and 10 girls. I need to assemble a team of 7 members to send to a national science competition. The team rules state that I must select 4 boys and 3 girls. In how many ways can I select the team? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
8400
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function $f(x) = \\begin{cases} 1 + \\log_{2}(2 - x), & x < 1 \\\\ 2^{x}, & x \\geq 1 \\end{cases}$, find the value of $f(-2) + f(\\log_{2}6)$.\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 $f(x) = \begin{cases} 1 + \log_{2}(2 - x), & x < 1 \\ 2^{x}, & x \geq 1 \end{cases}$, find the value of $f(-2) + f(\log_{2}6)$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the function $f(x) = \\begin{cases} 1 + \\log_{2}(2 - x), & x < 1 \\\\ 2^{x}, & x \\geq 1 \\end{cases}$, find the value of $f(-2) + f(\\log_{2}6)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
9
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given $a>0$, if for any $x>0$, it holds that $$\\frac {1}{a}- \\frac {1}{x}\\leq2x$$, then the range of values for $a$ is \\_\\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\left[\\frac {\\sqrt {2}}{4}, +\\infty\\right)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given $a>0$, if for any $x>0$, it holds that $$\frac {1}{a}- \frac {1}{x}\leq2x$$, then the range of values for $a$ is \_\_\_\_\_\_.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given $a>0$, if for any $x>0$, it holds that $$\\frac {1}{a}- \\frac {1}{x}\\leq2x$$, then the range of values for $a$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\left[\frac {\sqrt {2}}{4}, +\infty\right)
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that $a > 0$, find the minimum value of $\\frac{(a-1)(4a-1)}{a}$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that $a > 0$, find the minimum value of $\frac{(a-1)(4a-1)}{a}$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that $a > 0$, find the minimum value of $\\frac{(a-1)(4a-1)}{a}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-1
|
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 rectangle \\( R \\) is divided into 2016 smaller rectangles, with each smaller rectangle's sides parallel to the sides of rectangle \\( R \\). The vertices of the smaller rectangles are called \"nodes\". A segment on the side of a smaller rectangle, if both its endpoints are nodes and its interior contains no other nodes, is called a \"basic segment\". Consider all possible ways of dividing the rectangle \\( R \\). Find the maximum and minimum number of basic segments.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "4122 \\text{ and } 6049",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A rectangle \( R \) is divided into 2016 smaller rectangles, with each smaller rectangle's sides parallel to the sides of rectangle \( R \). The vertices of the smaller rectangles are called "nodes". A segment on the side of a smaller rectangle, if both its endpoints are nodes and its interior contains no other nodes, is called a "basic segment". Consider all possible ways of dividing the rectangle \( R \). Find the maximum and minimum number of basic segments.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A rectangle \\( R \\) is divided into 2016 smaller rectangles, with each smaller rectangle's sides parallel to the sides of rectangle \\( R \\). The vertices of the smaller rectangles are called \"nodes\". A segment on the side of a smaller rectangle, if both its endpoints are nodes and its interior contains no other nodes, is called a \"basic segment\". Consider all possible ways of dividing the rectangle \\( R \\). Find the maximum and minimum number of basic segments. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
4122 \text{ and } 6049
|
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 smallest four-digit multiple of 17?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "1013",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
What is the smallest four-digit multiple of 17?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "What is the smallest four-digit multiple of 17? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
1013
|
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 green parabola shown is the graph of the equation $x = py^2 + qy + r$. Find $r$. (Assume that the graph has the standard unit scale.)\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal tickdown=-0.5;\nreal tickdownlength=-0.15inch;\nreal tickdownbase=0.3;\nreal wholetickdown=tickdown;\nvoid rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool\n\nuseticks=false, bool complexplane=false, bool usegrid=true) {\n\nimport graph;\n\nreal i;\n\nif(complexplane) {\n\nlabel(\"$\\textnormal{Re}$\",(xright,0),SE);\n\nlabel(\"$\\textnormal{Im}$\",(0,ytop),NW);\n\n} else {\n\nlabel(\"$x$\",(xright+0.4,-0.5));\n\nlabel(\"$y$\",(-0.5,ytop+0.2));\n\n}\n\nylimits(ybottom,ytop);\n\nxlimits( xleft, xright);\n\nreal[] TicksArrx,TicksArry;\n\nfor(i=xleft+xstep; i<xright; i+=xstep) {\n\nif(abs(i) >0.1) {\n\nTicksArrx.push(i);\n\n}\n\n}\n\nfor(i=ybottom+ystep; i<ytop; i+=ystep) {\n\nif(abs(i) >0.1) {\n\nTicksArry.push(i);\n\n}\n\n}\n\nif(usegrid) {\n\nxaxis(BottomTop(extend=false), Ticks(\"%\", TicksArrx ,pTick=gray\n\n(0.22),extend=true),p=invisible);//,above=true);\n\nyaxis(LeftRight(extend=false),Ticks(\"%\", TicksArry ,pTick=gray(0.22),extend=true),\n\np=invisible);//,Arrows);\n\n}\n\nif(useticks) {\n\nxequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks(\"%\",TicksArry ,\n\npTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));\n\nyequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks(\"%\",TicksArrx ,\n\npTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));\n\n} else {\n\nxequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));\n\nyequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));\n\n}\n};\nreal lowerx, upperx, lowery, uppery;\nreal g(real y) {return -(y-1)*(y-1)/3+4;}\nlowery = -1;\nuppery = 3;\nrr_cartesian_axes(-3,5,lowery,uppery);\ndraw(reflect((0,0),(1,1))*(graph(g,lowery,uppery,operator ..)), green);\n[/asy]\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The green parabola shown is the graph of the equation $x = py^2 + qy + r$. Find $r$. (Assume that the graph has the standard unit scale.)
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real wholetickdown=tickdown;
void rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool
useticks=false, bool complexplane=false, bool usegrid=true) {
import graph;
real i;
if(complexplane) {
label("$\textnormal{Re}$",(xright,0),SE);
label("$\textnormal{Im}$",(0,ytop),NW);
} else {
label("$x$",(xright+0.4,-0.5));
label("$y$",(-0.5,ytop+0.2));
}
ylimits(ybottom,ytop);
xlimits( xleft, xright);
real[] TicksArrx,TicksArry;
for(i=xleft+xstep; i<xright; i+=xstep) {
if(abs(i) >0.1) {
TicksArrx.push(i);
}
}
for(i=ybottom+ystep; i<ytop; i+=ystep) {
if(abs(i) >0.1) {
TicksArry.push(i);
}
}
if(usegrid) {
xaxis(BottomTop(extend=false), Ticks("%", TicksArrx ,pTick=gray
(0.22),extend=true),p=invisible);//,above=true);
yaxis(LeftRight(extend=false),Ticks("%", TicksArry ,pTick=gray(0.22),extend=true),
p=invisible);//,Arrows);
}
if(useticks) {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks("%",TicksArry ,
pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks("%",TicksArrx ,
pTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));
} else {
xequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));
yequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));
}
};
real lowerx, upperx, lowery, uppery;
real g(real y) {return -(y-1)*(y-1)/3+4;}
lowery = -1;
uppery = 3;
rr_cartesian_axes(-3,5,lowery,uppery);
draw(reflect((0,0),(1,1))*(graph(g,lowery,uppery,operator ..)), green);
[/asy]
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The green parabola shown is the graph of the equation $x = py^2 + qy + r$. Find $r$. (Assume that the graph has the standard unit scale.)\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal tickdown=-0.5;\nreal tickdownlength=-0.15inch;\nreal tickdownbase=0.3;\nreal wholetickdown=tickdown;\nvoid rr_cartesian_axes(real xleft, real xright, real ybottom, real ytop, real xstep=1, real ystep=1, bool\n\nuseticks=false, bool complexplane=false, bool usegrid=true) {\n\nimport graph;\n\nreal i;\n\nif(complexplane) {\n\nlabel(\"$\\textnormal{Re}$\",(xright,0),SE);\n\nlabel(\"$\\textnormal{Im}$\",(0,ytop),NW);\n\n} else {\n\nlabel(\"$x$\",(xright+0.4,-0.5));\n\nlabel(\"$y$\",(-0.5,ytop+0.2));\n\n}\n\nylimits(ybottom,ytop);\n\nxlimits( xleft, xright);\n\nreal[] TicksArrx,TicksArry;\n\nfor(i=xleft+xstep; i<xright; i+=xstep) {\n\nif(abs(i) >0.1) {\n\nTicksArrx.push(i);\n\n}\n\n}\n\nfor(i=ybottom+ystep; i<ytop; i+=ystep) {\n\nif(abs(i) >0.1) {\n\nTicksArry.push(i);\n\n}\n\n}\n\nif(usegrid) {\n\nxaxis(BottomTop(extend=false), Ticks(\"%\", TicksArrx ,pTick=gray\n\n(0.22),extend=true),p=invisible);//,above=true);\n\nyaxis(LeftRight(extend=false),Ticks(\"%\", TicksArry ,pTick=gray(0.22),extend=true),\n\np=invisible);//,Arrows);\n\n}\n\nif(useticks) {\n\nxequals(0, ymin=ybottom, ymax=ytop, p=axispen, Ticks(\"%\",TicksArry ,\n\npTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));\n\nyequals(0, xmin=xleft, xmax=xright, p=axispen, Ticks(\"%\",TicksArrx ,\n\npTick=black+0.8bp,Size=ticklength), above=true, Arrows(size=axisarrowsize));\n\n} else {\n\nxequals(0, ymin=ybottom, ymax=ytop, p=axispen, above=true, Arrows(size=axisarrowsize));\n\nyequals(0, xmin=xleft, xmax=xright, p=axispen, above=true, Arrows(size=axisarrowsize));\n\n}\n};\nreal lowerx, upperx, lowery, uppery;\nreal g(real y) {return -(y-1)*(y-1)/3+4;}\nlowery = -1;\nuppery = 3;\nrr_cartesian_axes(-3,5,lowery,uppery);\ndraw(reflect((0,0),(1,1))*(graph(g,lowery,uppery,operator ..)), green);\n[/asy] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
2
|
numina_olympiads
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the function \\( f(x) = 9^x - m \\cdot 3^x + m + 1 \\) is always above the x-axis for \\( x \\in (0, +\\infty) \\), determine the range of values for \\( m \\).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "m < 2 + 2\\sqrt{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the function \( f(x) = 9^x - m \cdot 3^x + m + 1 \) is always above the x-axis for \( x \in (0, +\infty) \), determine the range of values for \( m \).
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the function \\( f(x) = 9^x - m \\cdot 3^x + m + 1 \\) is always above the x-axis for \\( x \\in (0, +\\infty) \\), determine the range of values for \\( m \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
m < 2 + 2\sqrt{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": "Express 8500 billion yuan in scientific notation.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "8.5 \\times 10^{11}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Express 8500 billion yuan in scientific notation.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Express 8500 billion yuan in scientific notation. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
8.5 \times 10^{11}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given the equation $ax^2+bx-4=0$ ($a, b \\in \\mathbb{R}$, and $a>0$) has two real roots, one of which is in the interval $(1, 2)$, \nthen the range of values for $a+b$ is.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "(-\\infty, 4)",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given the equation $ax^2+bx-4=0$ ($a, b \in \mathbb{R}$, and $a>0$) has two real roots, one of which is in the interval $(1, 2)$,
then the range of values for $a+b$ is.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given the equation $ax^2+bx-4=0$ ($a, b \\in \\mathbb{R}$, and $a>0$) has two real roots, one of which is in the interval $(1, 2)$, \nthen the range of values for $a+b$ is. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
(-\infty, 4)
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Determine the greatest common divisor (GCD) of the three numbers 390, 455, and 546.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "13",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Determine the greatest common divisor (GCD) of the three numbers 390, 455, and 546.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Determine the greatest common divisor (GCD) of the three numbers 390, 455, and 546. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
13
|
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": "Four points are chosen uniformly at random on a circle. What is the probability that any three of these points, when combined with the circle's center, always form an acute triangle?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{3}{64}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Four points are chosen uniformly at random on a circle. What is the probability that any three of these points, when combined with the circle's center, always form an acute triangle?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Four points are chosen uniformly at random on a circle. What is the probability that any three of these points, when combined with the circle's center, always form an acute triangle? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{3}{64}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "If the solution set of the inequality $-1 < ax^2 + bx + c < 1$ is $(-1, 3)$, then the range of the real number $a$ is __________.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-\\frac{1}{2} < a < \\frac{1}{2}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
If the solution set of the inequality $-1 < ax^2 + bx + c < 1$ is $(-1, 3)$, then the range of the real number $a$ is __________.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "If the solution set of the inequality $-1 < ax^2 + bx + c < 1$ is $(-1, 3)$, then the range of the real number $a$ is __________. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-\frac{1}{2} < a < \frac{1}{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 graph below shows a red parabola represented by the equation \\(x = ay^2 + by + c\\). Find the value of \\(c\\) if the vertex of the parabola is at \\((2, -3)\\) and it passes through the point \\((7, -1)\\). (Assume the grid lines are spaced one unit apart.)\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac{53}{4}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The graph below shows a red parabola represented by the equation \(x = ay^2 + by + c\). Find the value of \(c\) if the vertex of the parabola is at \((2, -3)\) and it passes through the point \((7, -1)\). (Assume the grid lines are spaced one unit apart.)
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The graph below shows a red parabola represented by the equation \\(x = ay^2 + by + c\\). Find the value of \\(c\\) if the vertex of the parabola is at \\((2, -3)\\) and it passes through the point \\((7, -1)\\). (Assume the grid lines are spaced one unit apart.) Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac{53}{4}
|
numina_synthetic_amc
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Given that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\\pi$, and the arc of the semicircle on segment DF has length $10\\pi$. Determine the radius of the semicircle on segment EF.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\sqrt{136}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\pi$, and the arc of the semicircle on segment DF has length $10\pi$. Determine the radius of the semicircle on segment EF.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\\pi$, and the arc of the semicircle on segment DF has length $10\\pi$. Determine the radius of the semicircle on segment EF. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt{136}
|
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": "It is known that point A on the number line represents the number -2. Point B is to the left of point A, at a distance of 2 units from point A. Point C is 5 units away from point B. Then, the number represented by point C is \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-9 \\text{ or } 1",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
It is known that point A on the number line represents the number -2. Point B is to the left of point A, at a distance of 2 units from point A. Point C is 5 units away from point B. Then, the number represented by point C is \_\_\_\_\_\_\_\_\_\_\_\_.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "It is known that point A on the number line represents the number -2. Point B is to the left of point A, at a distance of 2 units from point A. Point C is 5 units away from point B. Then, the number represented by point C is \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-9 \text{ or } 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": "Simplify and evaluate $\\dfrac{x-3}{x-2}\\div \\left(x+2-\\dfrac{5}{x-2}\\right)$, where $x=\\sqrt {3}-3$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\dfrac{\\sqrt{3}}{3}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Simplify and evaluate $\dfrac{x-3}{x-2}\div \left(x+2-\dfrac{5}{x-2}\right)$, where $x=\sqrt {3}-3$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Simplify and evaluate $\\dfrac{x-3}{x-2}\\div \\left(x+2-\\dfrac{5}{x-2}\\right)$, where $x=\\sqrt {3}-3$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\dfrac{\sqrt{3}}{3}
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The negation of the proposition \"There exists an $x \\in \\mathbb{R}$ such that $x^2 - 2x + 1 \\leq 0$\" is.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\forall x \\in \\mathbb{R},\\, x^2 - 2x + 1 > 0",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The negation of the proposition "There exists an $x \in \mathbb{R}$ such that $x^2 - 2x + 1 \leq 0$" is.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The negation of the proposition \"There exists an $x \\in \\mathbb{R}$ such that $x^2 - 2x + 1 \\leq 0$\" is. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\forall x \in \mathbb{R},\, x^2 - 2x + 1 > 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": "How many zeroes does \\( 12! \\) end with when \\( 12! \\) is written in base 9?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "2",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
How many zeroes does \( 12! \) end with when \( 12! \) is written in base 9?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "How many zeroes does \\( 12! \\) end with when \\( 12! \\) is written in base 9? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
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": "A square has an area of $36.$ A rectangle has the same width as the square. The length of the rectangle is triple its width. What is the area of the rectangle?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "108",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A square has an area of $36.$ A rectangle has the same width as the square. The length of the rectangle is triple its width. What is the area of the rectangle?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A square has an area of $36.$ A rectangle has the same width as the square. The length of the rectangle is triple its width. What is the area of the rectangle? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
108
|
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 an $8 \\times 8$ chessboard, we wrote numbers on each square, placing a 0 in every corner. Each number on the board is not greater than the average of the numbers in the squares directly adjacent to it. What numbers did we write on the chessboard?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "0",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
On an $8 \times 8$ chessboard, we wrote numbers on each square, placing a 0 in every corner. Each number on the board is not greater than the average of the numbers in the squares directly adjacent to it. What numbers did we write on the chessboard?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "On an $8 \\times 8$ chessboard, we wrote numbers on each square, placing a 0 in every corner. Each number on the board is not greater than the average of the numbers in the squares directly adjacent to it. What numbers did we write on the chessboard? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
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 cutting tool is designed in the shape of a notched circle. The radius of the circle is $\\sqrt{80}$ cm, with points $A$, $B$, and $C$ on its boundary such that $AB = 8$ cm and $BC = 3$ cm, forming a right angle at point $B$. Compute the square of the distance from $B$ to the center of the circle.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "80",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A cutting tool is designed in the shape of a notched circle. The radius of the circle is $\sqrt{80}$ cm, with points $A$, $B$, and $C$ on its boundary such that $AB = 8$ cm and $BC = 3$ cm, forming a right angle at point $B$. Compute the square of the distance from $B$ to the center of the circle.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A cutting tool is designed in the shape of a notched circle. The radius of the circle is $\\sqrt{80}$ cm, with points $A$, $B$, and $C$ on its boundary such that $AB = 8$ cm and $BC = 3$ cm, forming a right angle at point $B$. Compute the square of the distance from $B$ to the center of the circle. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
80
|
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 girl has the following coins in her purse: $3$ pennies, $2$ nickels, $2$ dimes, and $2$ quarters. She draws two coins at random, notes their combined value, and then returns them to her purse. She repeats this process multiple times. What is the maximum number of different sums she can record?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "10",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
A girl has the following coins in her purse: $3$ pennies, $2$ nickels, $2$ dimes, and $2$ quarters. She draws two coins at random, notes their combined value, and then returns them to her purse. She repeats this process multiple times. What is the maximum number of different sums she can record?
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "A girl has the following coins in her purse: $3$ pennies, $2$ nickels, $2$ dimes, and $2$ quarters. She draws two coins at random, notes their combined value, and then returns them to her purse. She repeats this process multiple times. What is the maximum number of different sums she can record? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
10
|
numina_synthetic_math
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "Find $x$ such that $\\log_x 256 = \\log_2 64$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\sqrt[3]{16}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Find $x$ such that $\log_x 256 = \log_2 64$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Find $x$ such that $\\log_x 256 = \\log_2 64$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\sqrt[3]{16}
|
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 $\\overrightarrow{a}=(-1,x,3)$, $\\overrightarrow{b}=(2,-4,y)$, and $\\overrightarrow{a}$ is parallel to $\\overrightarrow{b}$, find the value of $x+y$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "-4",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
Given vectors $\overrightarrow{a}=(-1,x,3)$, $\overrightarrow{b}=(2,-4,y)$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $x+y$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "Given vectors $\\overrightarrow{a}=(-1,x,3)$, $\\overrightarrow{b}=(2,-4,y)$, and $\\overrightarrow{a}$ is parallel to $\\overrightarrow{b}$, find the value of $x+y$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
-4
|
numina_cn_k12
|
[
{
"content": "\nWhen tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.\n\n[ASSESS]\n\n[ADVANCE]\n\n[VERIFY]\n\n[SIMPLIFY]\n\n[SYNTHESIZE]\n\n[PIVOT]\n\n[OUTPUT]\n\nYou should strictly follow the format below:\n\n[ACTION NAME]\n\n# Your action step 1\n\n# Your action step 2\n\n# Your action step 3\n\n...\n\nNext action: [NEXT ACTION NAME]\n\n",
"role": "system"
},
{
"content": "The general term formula of the sequence $\\{a_n\\}$ is \n$$\na_n = \\begin{cases} \n\\left( \\frac {1}{3} \\right)^{n}, & 1 \\leq n \\leq 100 \\\\ \n\\frac {2n+1}{5n-1}, & n > 100\n\\end{cases}\n$$\nThen, $\\lim_{n \\to \\infty} a_n = \\_\\_\\_\\_\\_\\_$.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
math
|
{
"ground_truth": "\\frac {2}{5}",
"style": "rule"
}
|
{
"index": 0,
"split": "dummy"
}
|
The general term formula of the sequence $\{a_n\}$ is
$$
a_n = \begin{cases}
\left( \frac {1}{3} \right)^{n}, & 1 \leq n \leq 100 \\
\frac {2n+1}{5n-1}, & n > 100
\end{cases}
$$
Then, $\lim_{n \to \infty} a_n = \_\_\_\_\_\_$.
|
[
{
"content": "You are a mathematical reasoning assistant. For each problem, follow these steps strictly:\n\n1. Solve the problem using step-by-step reasoning and output the final answer within \\boxed{}.\n2. Perform a self-evaluation:\n - You may include reasoning to verify correctness.\n - However, your final self-evaluation **must** be in one of the following formats:\n ```\n [VERIFY] correct.\n ```\n or \n ```\n [VERIFY] wrong.\n ```\n3. If the self-evaluation is `[VERIFY] wrong.`, identify the mistake in your previous reasoning, revise your reasoning path and output a corrected final answer within \\boxed{}.\n\nAlways ensure clarity, correctness, and adherence to the required format.",
"role": "system"
},
{
"content": "The general term formula of the sequence $\\{a_n\\}$ is \n$$\na_n = \\begin{cases} \n\\left( \\frac {1}{3} \\right)^{n}, & 1 \\leq n \\leq 100 \\\\ \n\\frac {2n+1}{5n-1}, & n > 100\n\\end{cases}\n$$\nThen, $\\lim_{n \\to \\infty} a_n = \\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
\frac {2}{5}
|
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