| { | |
| "log_file": [ | |
| "step_0-b_0.log", | |
| "step_0-b_1.log", | |
| "step_0-b_2.log", | |
| "step_0-b_3.log", | |
| "step_0-b_4.log", | |
| "step_0-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:\nA function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n\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:\nA function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n\n\n### Response:\nSEARCH[\"f(x) = 2004.\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/1030.json\n\n#### Similar Question\nIf $f(3)=1$ and $f(2x)=2f(x)$ for all $x$, find $f^{-1}(64)$.\n\n#### User Answer\nWe are looking for some $x$ such that $f(x)=64$. We notice that by doubling $x$ we can double $f(x)$ as well and also that $f(3)=1$.\n\nApplying $f(2x)=2f(x)$ repeatedly, we have: \\begin{align*}\nf(3)&=1,\\\\\nf(6)&=2,\\\\\nf(12)&=4,\\\\\nf(24)&=8,\\\\\nf(48)&=16,\\\\\nf(96)&=32,\\\\\nf(192)&=64.\n\\end{align*}So $f^{-1}(64)=192$.\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:\nA function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n\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:\nA function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n\n\n### Response:\nSEARCH[\"f(x) = 2004.\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/intermediate_algebra/1879.json\n\n#### Similar Question\nA certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x$, and that $f(x) = 1 - |x - 2|$ for $1\\leq x \\leq 3$. Find the smallest $x$ for which $f(x) = f(2001)$.\n\n#### User Answer\nUsing the given $f(3x) = 3f(x)$ repeatedly, we have that \\[f(2001) = 3f\\left(\\frac{2001}{3}\\right) = 3^2f\\left(\\frac{2001}{3^2}\\right) = \\dots = 3^6f\\left(\\frac{2001}{3^6}\\right).\\]Since $1 \\le 2001/3^6 \\le 3,$ we can apply the second part of the definition of $f$ to get \\[f(2001) = 3^6\\left(1 - \\left|\\frac{2001}{3^6} - 2\\right|\\right) = 3 \\cdot 3^6 - 2001 = 186.\\]Therefore, we want the smallest $x$ for which $f(x) = 186.$ Note that the range of $f(x) $ in the interval $x \\in [1, 3]$ is $[0, 1].$ Since $f(3x) = 3f(x)$ for all $x,$ it follows that the range of $f(x)$ in the interval $x \\in [3, 9]$ is $[0,3].$ Similarly, for each $k,$ the range of $f(x)$ in the interval $x \\in [3^k, 3^{k+1}]$ is $[0, 3^k].$ Therefore, if $f(x) = 186,$ then $3^k \\ge 186,$ so $k \\ge 5.$\n\nWe search the interval $x \\in [3^5, 3^6] = [243, 729].$ We want $f(x) = 186,$ and for any $x$ in this interval, we have $f(x) = 3^5f\\left(\\frac{x}{3^5}\\right).$ Therefore, letting $y = \\frac{x}{3^5},$ we want $f(y) = \\frac{186}{3^5} = \\frac{186}{243},$ where $y \\in [1, 3].$ That is, \\[1 - |y-2| = \\frac{186}{243} \\implies |y-2| = \\frac{57}{243}.\\]The smaller of the two solutions to this equation is $y = 2 - \\frac{57}{243} = \\frac{429}{243}.$ Thus, $x = 3^5y = 429.$\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:\nA function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n\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:\nA function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n\n\n### Response:\nSEARCH[\"f(x) = 2004.\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/1947.json\n\n#### Similar Question\nLet \\[f(x) = \\left\\{\n\\begin{array}{cl}\n-x + 3 & \\text{if } x \\le 0, \\\\\n2x - 5 & \\text{if } x > 0.\n\\end{array}\n\\right.\\]How many solutions does the equation $f(f(x)) = 4$ have?\n\n#### User Answer\nTo solve the equation $f(f(x)) = 4,$ we first find the values $x$ such that $f(x) = 4.$\n\nEither $f(x) = -x + 3$ (for $x \\le 0$) or $f(x) = 2x - 5$ (for $x > 0$). If $-x + 3 = 4,$ then $x = -1.$ Note that this value satisfies $x \\le 0.$ If $2x - 5 = 4,$ then $x = 9/2.$ Note that this value satisfies $x > 0.$ Therefore, the solutions to $f(x) = 4$ are $x = -1$ and $x = 9/2.$\n\nNext, we solve for the values $x$ such that $f(x) = -1.$ If $-x + 3 = -1,$ then $x = 4.$ This value does not satisfy $x \\le 0.$ If $2x - 5 = -1,$ then $x = 2.$ This value satisfies $x > 0.$\n\nFinally, we solve for the values $x$ such that $f(x) = 9/2.$ If $-x + 3 = 9/2,$ then $x = -3/2.$ This value satisfies $x \\le 0.$ If $2x - 5 = 9/2,$ then $x = 19/4.$ This value satisfies $x > 0.$\n\nTherefore, the equation $f(f(x)) = 4$ has the solutions $x = 2,$ $-3/2,$ and $19/4,$ for a total of $3$ solutions.\n\n--- RESULTS END ---\n\n" | |
| ], | |
| "response": [ | |
| " SEARCH[\"f(x) = 2004.\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nLet $x_1$ and $x_2$ be two nonzero solutions to the equation $2f(x) + f\\left(\\frac{1}{x}\\right) = 5x + 4$. Substituting $\\frac{1}{x_1}$ and $\\frac{1}{x_2}$ into the equation, we have\n\\[2f\\left(\\frac{1}{x_1}\\right) + f\\left(\\frac{1}{x_2}\\right) = 5\\frac{1}{x_1} + 4\\]and\n\\[2f\\left(\\frac{1}{x_2}\\right) + f\\left(\\frac{1}{x_1}\\right) = 5\\frac{1}{x_2} + 4.\\]Adding the two equations, we get\n\\[4f\\left(\\frac{1}{x_1}\\right) + 4f\\left(\\frac{1}{x_2}\\right) = 10\\left(\\frac{1}{x_1} + \\frac{1}{x_2}\\right) + 8 = 10\\cdot\\frac{x_1+x_2}{x_1\\cdot x_2} + 8,\\]which simplifies to\n\\[4\\left(f\\left(\\frac{1}{x_1}\\right) + f\\left(\\frac{1}{x_2}\\right)\\right) = 10\\cdot\\frac{x_1+x_2}{x_1\\cdot x_2} + 8.\\]Since $x_1$ and $x_2$ are nonzero, we can divide both sides by $4$ to get\n\\[f\\left(\\frac{1}{x_1}\\right) + f\\left(\\frac{1}{x_2}\\right) = \\frac{5}{2} + \\frac{1}{2} = \\frac{5}{2} = 2.5\\]The only value of $f(x)$ that satisfies this equation is $f(x) = 2$. Thus, the only nonzero solutions to the original equation are $x = \\frac{1}{2}$ and $x = 4$.\n\nWe have that $x = 4$ is a solution to the equation $2f(x) + f\\left(\\frac{1}{x}\\right) = 5x + 4$ if and only if $f(4) = 2$. Similarly, $x = \\frac{1}{2}$ is a solution to the equation $2f(x) + f\\left(\\frac{1}{x}\\right) = 5x + 4$ if and only if $f\\left(\\frac{1}{2}\\right) = 2$. Thus, all nonzero solutions to the equation $2f(x) + f\\left(\\frac{1}{x}\\right) = 5x + 4$ are $x = 4$ and $x = \\frac{1}{2}$.\n\nTherefore, the sum of all of the values of $x$ for which $f(x) = 2004$ is $4 + \\frac{1}{2} = \\boxed{4.5}$.</s>", | |
| " SEARCH[\"f(x) = 2004.\"]</s>", | |
| "The result might be helpful, I will try using it to answer the question only if it is useful.\n\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n\nWe can rewrite the equation as $2f(x) + f\\left(\\frac{1}{x}\\right) = 5x + 4$. Let $y = \\frac{1}{x}$. Then, we have $2f(y) + f(y) = 5y + 4$. We can solve for $f(y)$ as follows:\n\n$f(y) = \\frac{5y + 4 - 2f(y)}{1 - 2} = \\frac{5y + 4 - 2f(y)}{-1} = -5y - 4 + 2f(y)$.\n\nNow, we can substitute $y = \\frac{1}{x}$ back into the equation to get $f\\left(\\frac{1}{x}\\right) = -5\\left(\\frac{1}{x}\\right) - 4 + 2f(x)$. We can solve this equation for $f(x)$ as follows:\n\n$f(x) = \\frac{-5\\left(\\frac{1}{x}\\right) - 4 + 2f(x)}{1 - 2} = \\frac{-5\\left(\\frac{1}{x}\\right) - 4 + 2f(x)}{-1} = -5\\left(\\frac{1}{x}\\right) - 4 + 2f(x) = -5 + 2f(x) - 4\\left(\\frac{1}{x}\\right)$.\n\nNow, we can substitute $y = \\frac{1}{x}$ back into the equation to get $f\\left(\\frac{1}{x}\\right) = -5 + 2f(y) - 4\\left(\\frac{1}{y}\\right) = -5 + 2f(y) - 4\\left(\\frac{1}{y}\\right) = -5 + 2f(x) - 4\\left(\\frac{1}{x}\\right)$. We can solve this equation for $f(y)$ as follows:\n\n$f(y)", | |
| " SEARCH[\"f(x) = 2004.\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe can substitute $x= \\frac{1}{u}$ into the given equation to get\n\\[2f\\left(\\frac{1}{u}\\right) + f\\left(u\\right) = 5\\left(\\frac{1}{u}\\right) + 4 = \\frac{5}{u} + 4.\\]Then\n\\[2f\\left(\\frac{1}{u}\\right) + f\\left(u\\right) = \\frac{5}{u} + 4.\\]Multiplying both sides by $2u$ (or cross-multiplying) gives\n\\[2f\\left(\\frac{1}{u}\\right) + 2f\\left(u\\right) = 5u + 8.\\]Then\n\\[2f\\left(\\frac{1}{u}\\right) + 2f\\left(u\\right) = 5u + 8.\\]Dividing both sides by 2 gives\n\\[f\\left(\\frac{1}{u}\\right) + f\\left(u\\right) = \\frac{5u + 8}{2} = \\frac{5}{2}u + \\frac{4}{2}.\\]Then\n\\[f\\left(\\frac{1}{u}\\right) = \\frac{5}{2}u + \\frac{2}{2} - \\frac{3}{2}u = \\frac{1}{2}u + \\frac{1}{2}.\\]Since $u$ can be any nonzero real number, $f\\left(\\frac{1}{u}\\right)$ can take on any real number. In particular, it can take on 2004. Therefore, the sum of all values of $x$ for which $f(x) = 2004$ is $\\boxed{2005}.$</s>" | |
| ], | |
| "src_path": [ | |
| "train/intermediate_algebra/9009.json", | |
| "train/intermediate_algebra/9009.json", | |
| "train/intermediate_algebra/9009.json", | |
| "train/intermediate_algebra/9009.json", | |
| "train/intermediate_algebra/9009.json", | |
| "train/intermediate_algebra/9009.json" | |
| ], | |
| "input": [ | |
| "A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n", | |
| "A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n", | |
| "A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n", | |
| "A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n", | |
| "A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n", | |
| "A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have\n\\[2f\\left(x\\right) + f\\left(\\frac{1}{x}\\right) = 5x + 4\\]\nLet $S$ denote the sum of all of the values of $x$ for which $f(x) = 2004$. Compute the integer nearest to $S$.\n" | |
| ], | |
| "output": [ | |
| "Substituting $\\frac{1}{x}$, we have\n\\[2f\\left(\\frac 1x\\right) + f\\left(x\\right) = \\frac{5}{x} + 4\\]\nThis gives us two equations, which we can eliminate $f\\left(\\frac 1x\\right)$ from (the first equation multiplied by two, subtracting the second):\n\\begin{align*} 3f(x) &= 10x + 4 - \\frac 5x \\\\ 0 &= x^2 - \\frac{3 \\times 2004 - 4}{10}x + \\frac 52\\end{align*}\nClearly, the discriminant of the quadratic equation $\\Delta > 0$, so both roots are real. By Vieta's formulas, the sum of the roots is the coefficient of the $x$ term, so our answer is $\\left[\\frac{3 \\times 2004 - 4}{10}\\right] = \\boxed{601}$.", | |
| "Substituting $\\frac{1}{x}$, we have\n\\[2f\\left(\\frac 1x\\right) + f\\left(x\\right) = \\frac{5}{x} + 4\\]\nThis gives us two equations, which we can eliminate $f\\left(\\frac 1x\\right)$ from (the first equation multiplied by two, subtracting the second):\n\\begin{align*} 3f(x) &= 10x + 4 - \\frac 5x \\\\ 0 &= x^2 - \\frac{3 \\times 2004 - 4}{10}x + \\frac 52\\end{align*}\nClearly, the discriminant of the quadratic equation $\\Delta > 0$, so both roots are real. By Vieta's formulas, the sum of the roots is the coefficient of the $x$ term, so our answer is $\\left[\\frac{3 \\times 2004 - 4}{10}\\right] = \\boxed{601}$.", | |
| "Substituting $\\frac{1}{x}$, we have\n\\[2f\\left(\\frac 1x\\right) + f\\left(x\\right) = \\frac{5}{x} + 4\\]\nThis gives us two equations, which we can eliminate $f\\left(\\frac 1x\\right)$ from (the first equation multiplied by two, subtracting the second):\n\\begin{align*} 3f(x) &= 10x + 4 - \\frac 5x \\\\ 0 &= x^2 - \\frac{3 \\times 2004 - 4}{10}x + \\frac 52\\end{align*}\nClearly, the discriminant of the quadratic equation $\\Delta > 0$, so both roots are real. By Vieta's formulas, the sum of the roots is the coefficient of the $x$ term, so our answer is $\\left[\\frac{3 \\times 2004 - 4}{10}\\right] = \\boxed{601}$.", | |
| "Substituting $\\frac{1}{x}$, we have\n\\[2f\\left(\\frac 1x\\right) + f\\left(x\\right) = \\frac{5}{x} + 4\\]\nThis gives us two equations, which we can eliminate $f\\left(\\frac 1x\\right)$ from (the first equation multiplied by two, subtracting the second):\n\\begin{align*} 3f(x) &= 10x + 4 - \\frac 5x \\\\ 0 &= x^2 - \\frac{3 \\times 2004 - 4}{10}x + \\frac 52\\end{align*}\nClearly, the discriminant of the quadratic equation $\\Delta > 0$, so both roots are real. By Vieta's formulas, the sum of the roots is the coefficient of the $x$ term, so our answer is $\\left[\\frac{3 \\times 2004 - 4}{10}\\right] = \\boxed{601}$.", | |
| "Substituting $\\frac{1}{x}$, we have\n\\[2f\\left(\\frac 1x\\right) + f\\left(x\\right) = \\frac{5}{x} + 4\\]\nThis gives us two equations, which we can eliminate $f\\left(\\frac 1x\\right)$ from (the first equation multiplied by two, subtracting the second):\n\\begin{align*} 3f(x) &= 10x + 4 - \\frac 5x \\\\ 0 &= x^2 - \\frac{3 \\times 2004 - 4}{10}x + \\frac 52\\end{align*}\nClearly, the discriminant of the quadratic equation $\\Delta > 0$, so both roots are real. By Vieta's formulas, the sum of the roots is the coefficient of the $x$ term, so our answer is $\\left[\\frac{3 \\times 2004 - 4}{10}\\right] = \\boxed{601}$.", | |
| "Substituting $\\frac{1}{x}$, we have\n\\[2f\\left(\\frac 1x\\right) + f\\left(x\\right) = \\frac{5}{x} + 4\\]\nThis gives us two equations, which we can eliminate $f\\left(\\frac 1x\\right)$ from (the first equation multiplied by two, subtracting the second):\n\\begin{align*} 3f(x) &= 10x + 4 - \\frac 5x \\\\ 0 &= x^2 - \\frac{3 \\times 2004 - 4}{10}x + \\frac 52\\end{align*}\nClearly, the discriminant of the quadratic equation $\\Delta > 0$, so both roots are real. By Vieta's formulas, the sum of the roots is the coefficient of the $x$ term, so our answer is $\\left[\\frac{3 \\times 2004 - 4}{10}\\right] = \\boxed{601}$." | |
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