{ "log_file": [ "step_108-b_0.log", "step_108-b_1.log", "step_108-b_2.log", "step_108-b_3.log", "step_108-b_4.log", "step_108-b_5.log" ], "query": [ "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:\nDefine the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?\n\n### Response:\n", "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:\nDefine the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?\n\n### Response:\nSEARCH[\"$a \\\\S b = 3a + 5b$\",\"$7 \\\\S 2$\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/197.json\n\n#### Similar Question\nSuppose that $f(x)$ is a linear function satisfying the equation $f(x) = 4f^{-1}(x) + 6$. Given that $f(1) = 4$, find $f(2)$.\n\n#### User Answer\nSince $f(x)$ is a linear function, we can write $f(x) = ax + b$. We want to find the inverse function $g(x)$ defined by $f(g(x))=x$ for every $x$. If we substitute $g(x)$ into the equation for $f$ we get \\[f(g(x))=ag(x)+b.\\]Using that the left side is $f(g(x))=x$ we get \\[x=ag(x)+b.\\]Solving for $g$ we obtain \\[g(x)=\\dfrac{x-b}{a}.\\]Substituting $f(x)$ and $g(x)$ into the given equation, we get \\[ax + b = 4 \\cdot \\frac{x-b}{a} + 6\\]Multiplying both sides by $a$, we get \\[a^2 x + ab = 4x - 4b + 6a.\\]For this equation to hold for $\\emph{all}$ values of $x$, we must have the coefficient of $x$ on both sides equal, and the two constant terms equal. Setting the coefficients of $x$ equal gives $a^2 = 4$, so $a = \\pm2$. Setting constant terms equal gives $ab = -4b + 6a$. If $a = 2$, we have $2b = -4b + 12$, which gives $b = 2$. If $a = -2$, we have $-2b = -4b - 12$, so $b = -6$. Thus we have two possibilities: $f(x) =2x + 2$ or $f(x) = -2x - 6$.\n\nWe're given that $f(1) = 4$, and testing this shows that the first function is the correct choice. So finally, $f(2) = 2(2) + 2 = 6$.\n\n--- RESULTS END ---\n\n", "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:\nDefine the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?\n\n### Response:\n", "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:\nDefine the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?\n\n### Response:\nSEARCH[\"$a \\\\S b = 3a + 5b$\",\"$7 \\\\S 2$\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/1910.json\n\n#### Similar Question\nGiven positive integers $x$ and $y$ such that $\\frac{1}{x} + \\frac{1}{2y} = \\frac{1}{7}$, what is the least possible value of $xy$?\n\n#### User Answer\nMultiplying through by $14xy$, we have $14y + 7x = 2xy$, so $2xy - 7x - 14y = 0$. We then apply Simon's Favorite Factoring Trick by adding $49$ to both sides to get $2xy - 7x - 14y + 49 = 49$. We can then factor this to get $$(x-7)(2y-7) = 49$$Since $49$ factors to $7 \\cdot 7$ and $x$ and $y$ must be positive integers, the only possible solutions $(x,y)$ are $(8, 28), (14,7), \\text{and } (56,4)$. Out of these, $(14,7)$ yields the least possible value $xy$ of $98$.\n\n--- RESULTS END ---\n\n", "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:\nDefine the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?\n\n### Response:\n", "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:\nDefine the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?\n\n### Response:\nSEARCH[\"$a \\\\S b = 3a + 5b$\",\"$7 \\\\S 2$\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/1892.json\n\n#### Similar Question\nA non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to the number of units in its perimeter. What is the number of units in the perimeter of this rectangle?\n\n#### User Answer\nLet the two sides of the rectangle be $a$ and $b$. The problem is now telling us $ab=2a+2b$. Putting everything on one side of the equation, we have $ab-2a-2b=0.$ This looks tricky. However, we can add a number to both sides of the equation to make it factor nicely. 4 works here: $$ab-2a-2b+4=4 \\Rightarrow (a-2)(b-2)=4$$Since we don't have a square, $a$ and $b$ must be different. It doesn't matter which one is which, so we can just say $a=6 $ and $b=3 $. The perimeter is then $2(6+3)=18$\n\n--- RESULTS END ---\n\n" ], "response": [ " SEARCH[\"$a \\\\S b = 3a + 5b$\",\"$7 \\\\S 2$\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe have $7\\,\\S\\, 2 = 3(7) + 5(2) = 21 + 10 = \\boxed{31}$.", " SEARCH[\"$a \\\\S b = 3a + 5b$\",\"$7 \\\\S 2$\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\n$7\\,\\S\\, 2=3(7)+5(2)=21+10=\\boxed{31}$.", " SEARCH[\"$a \\\\S b = 3a + 5b$\",\"$7 \\\\S 2$\"]", " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe have $7\\,\\S\\, 2 = 3(7)+5(2)=21+10=\\boxed{31}$." ], "src_path": [ "train/algebra/1587.json", "train/algebra/1587.json", "train/algebra/1587.json", "train/algebra/1587.json", "train/algebra/1587.json", "train/algebra/1587.json" ], "input": [ "Define the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?", "Define the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?", "Define the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?", "Define the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?", "Define the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?", "Define the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?" ], "output": [ "The problems asks us to substitute $7$ for $a$ and $2$ for $b$ in the expression $3a+5b$. We find that $7\\S 2=3(7)+5(2)=21+10=\\boxed{31}$.", "The problems asks us to substitute $7$ for $a$ and $2$ for $b$ in the expression $3a+5b$. We find that $7\\S 2=3(7)+5(2)=21+10=\\boxed{31}$.", "The problems asks us to substitute $7$ for $a$ and $2$ for $b$ in the expression $3a+5b$. We find that $7\\S 2=3(7)+5(2)=21+10=\\boxed{31}$.", "The problems asks us to substitute $7$ for $a$ and $2$ for $b$ in the expression $3a+5b$. We find that $7\\S 2=3(7)+5(2)=21+10=\\boxed{31}$.", "The problems asks us to substitute $7$ for $a$ and $2$ for $b$ in the expression $3a+5b$. We find that $7\\S 2=3(7)+5(2)=21+10=\\boxed{31}$.", "The problems asks us to substitute $7$ for $a$ and $2$ for $b$ in the expression $3a+5b$. We find that $7\\S 2=3(7)+5(2)=21+10=\\boxed{31}$." ], "rewards": [ 1.0, 1.0, 1.0, 1.0, 1.0, 1.0 ] }