| { | |
| "log_file": [ | |
| "step_109-b_0.log", | |
| "step_109-b_1.log", | |
| "step_109-b_2.log", | |
| "step_109-b_3.log", | |
| "step_109-b_4.log", | |
| "step_109-b_5.log" | |
| ], | |
| "query": [ | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?\n\n### Response:\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?\n\n### Response:\nSEARCH[\"greatest positive integer\", \"sum of the first ten terms\", \"arithmetic sequence\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/2600.json\n\n#### Similar Question\nThe sum of four positive integers that form an arithmetic sequence is 46. Of all such possible sequences, what is the greatest possible third term?\n\n#### User Answer\nLet the first term be $a$, and let the common difference be $d$. Then the four positive integers are $a$, $a + d$, $a + 2d$, and $a + 3d$. The sum of these four positive integers is $4a + 6d = 46$, so $2a + 3d = 23$. Solving for $d$, we find $d = (23 - 2a)/3$.\n\nThe third term is \\[a + 2d = a + 2 \\cdot \\frac{23 - 2a}{3} = \\frac{46 - a}{3}.\\] Thus, to maximize this expression, we should minimize $a$. Since $a$ is a positive integer, the smallest possible value of $a$ is 1. Furthermore, when $a = 1$, $d = (23 - 2)/3 = 7$, which gives us the arithmetic sequence 1, 8, 15, 22. Therefore, the greatest possible third term is $15$.\n\n--- RESULTS END ---\n\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?\n\n### Response:\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?\n\n### Response:\nSEARCH[\"greatest positive integer\", \"sum of the first ten terms\", \"arithmetic sequence\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/number_theory/1263.json\n\n#### Similar Question\nWhat is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?\n\n#### User Answer\nThe first 10 terms of any arithmetic sequence can be represented as $x$, $x+c$, $x+2c$, $\\ldots x+9c$, where $x$ is the first term and $c$ is the constant difference between each consecutive term. So, the sum of all of these terms will include $10x$ and $(1+2+\\ldots+9)c$, which equals $45c$. As a result, the sum of all the terms is $10x+45c$ and the greatest number we can factor out is $5$, where we end up with $5(2x+9c)$.\n\n--- RESULTS END ---\n\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?\n\n### Response:\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\nWhat is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?\n\n### Response:\nSEARCH[\"greatest positive integer\", \"sum of the first ten terms\", \"arithmetic sequence\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/6132.json\n\n#### Similar Question\nThe degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.\n\n#### User Answer\nThe average angle in an 18-gon is $160^\\circ$. In an arithmetic sequence the average is the same as the median, so the middle two terms of the sequence average to $160^\\circ$. Thus for some positive (the sequence is increasing and thus non-constant) integer $d$, the middle two terms are $(160-d)^\\circ$ and $(160+d)^\\circ$. Since the step is $2d$ the last term of the sequence is $(160 + 17d)^\\circ$, which must be less than $180^\\circ$, since the polygon is convex. This gives $17d < 20$, so the only suitable positive integer $d$ is 1. The first term is then $(160-17)^\\circ = 143.$\n\n--- RESULTS END ---\n\n" | |
| ], | |
| "response": [ | |
| " SEARCH[\"greatest positive integer\", \"sum of the first ten terms\", \"arithmetic sequence\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so the sum of the first 10 terms of an arithmetic sequence whose terms are positive integers is a multiple of $10 \\cdot 5 = 50$. The greatest multiple of 50 that is less than 100 is $\\boxed{50}$.</s>", | |
| " SEARCH[\"greatest positive integer\", \"sum of the first ten terms\", \"arithmetic sequence\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so the sum of the first ten terms of any arithmetic sequence whose terms are positive integers is $10 \\cdot 20 = 200$. The greatest positive integer that must divide this sum is $\\boxed{200}$.</s>", | |
| " SEARCH[\"greatest positive integer\", \"sum of the first ten terms\", \"arithmetic sequence\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so the sum of the first ten terms of any arithmetic sequence whose terms are positive integers is $\\frac{1 + 100}{2} \\cdot 10 = 55 \\cdot 10 = \\boxed{550}$.</s>" | |
| ], | |
| "src_path": [ | |
| "train/number_theory/1263.json", | |
| "train/number_theory/1263.json", | |
| "train/number_theory/1263.json", | |
| "train/number_theory/1263.json", | |
| "train/number_theory/1263.json", | |
| "train/number_theory/1263.json" | |
| ], | |
| "input": [ | |
| "What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?", | |
| "What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?", | |
| "What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?", | |
| "What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?", | |
| "What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?", | |
| "What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?" | |
| ], | |
| "output": [ | |
| "The first 10 terms of any arithmetic sequence can be represented as $x$, $x+c$, $x+2c$, $\\ldots x+9c$, where $x$ is the first term and $c$ is the constant difference between each consecutive term. So, the sum of all of these terms will include $10x$ and $(1+2+\\ldots+9)c$, which equals $45c$. As a result, the sum of all the terms is $10x+45c$ and the greatest number we can factor out is $\\boxed{5}$, where we end up with $5(2x+9c)$.", | |
| "The first 10 terms of any arithmetic sequence can be represented as $x$, $x+c$, $x+2c$, $\\ldots x+9c$, where $x$ is the first term and $c$ is the constant difference between each consecutive term. So, the sum of all of these terms will include $10x$ and $(1+2+\\ldots+9)c$, which equals $45c$. As a result, the sum of all the terms is $10x+45c$ and the greatest number we can factor out is $\\boxed{5}$, where we end up with $5(2x+9c)$.", | |
| "The first 10 terms of any arithmetic sequence can be represented as $x$, $x+c$, $x+2c$, $\\ldots x+9c$, where $x$ is the first term and $c$ is the constant difference between each consecutive term. So, the sum of all of these terms will include $10x$ and $(1+2+\\ldots+9)c$, which equals $45c$. As a result, the sum of all the terms is $10x+45c$ and the greatest number we can factor out is $\\boxed{5}$, where we end up with $5(2x+9c)$.", | |
| "The first 10 terms of any arithmetic sequence can be represented as $x$, $x+c$, $x+2c$, $\\ldots x+9c$, where $x$ is the first term and $c$ is the constant difference between each consecutive term. So, the sum of all of these terms will include $10x$ and $(1+2+\\ldots+9)c$, which equals $45c$. As a result, the sum of all the terms is $10x+45c$ and the greatest number we can factor out is $\\boxed{5}$, where we end up with $5(2x+9c)$.", | |
| "The first 10 terms of any arithmetic sequence can be represented as $x$, $x+c$, $x+2c$, $\\ldots x+9c$, where $x$ is the first term and $c$ is the constant difference between each consecutive term. So, the sum of all of these terms will include $10x$ and $(1+2+\\ldots+9)c$, which equals $45c$. As a result, the sum of all the terms is $10x+45c$ and the greatest number we can factor out is $\\boxed{5}$, where we end up with $5(2x+9c)$.", | |
| "The first 10 terms of any arithmetic sequence can be represented as $x$, $x+c$, $x+2c$, $\\ldots x+9c$, where $x$ is the first term and $c$ is the constant difference between each consecutive term. So, the sum of all of these terms will include $10x$ and $(1+2+\\ldots+9)c$, which equals $45c$. As a result, the sum of all the terms is $10x+45c$ and the greatest number we can factor out is $\\boxed{5}$, where we end up with $5(2x+9c)$." | |
| ], | |
| "rewards": [ | |
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