| { | |
| "log_file": [ | |
| "step_129-b_0.log", | |
| "step_129-b_1.log", | |
| "step_129-b_2.log", | |
| "step_129-b_3.log", | |
| "step_129-b_4.log", | |
| "step_129-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:\nIf the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?\n\n### Response:\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$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:\nIf the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?\n\n### Response:\nSEARCH[\"maximum \\\\cos(33)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/399.json\n\n#### Similar Question\nGiven that $x$ is real and $x^3+\\frac{1}{x^3}=52$, find $x+\\frac{1}{x}$.\n\n#### User Answer\nWe know that $$\\left(x+\\frac{1}{x}\\right)^3=x^3+3(x^2)\\left(\\frac{1}{x}\\right)+3(x)\\left(\\frac{1}{x}\\right)^2+\\left(\\frac{1}{x}\\right)^3=x^3+\\frac{1}{x^3}+3\\left(x+\\frac{1}{x}\\right).$$Let $x+\\frac{1}{x}=a$. Then our equation is $a^3=x^3+\\frac{1}{x^3}+3a$. We know $x^3+\\frac{1}{x^3}=52$, so we have $a^3=52+3a$ or $a^3-3a-52=0$. By the rational root theorem, the possible roots of this polynomial equation are the divisors of 52 as well as their negatives: $\\pm1, \\pm 2, \\pm 4, \\pm 13, \\pm 26, \\pm 52$. Both $\\pm1$ and $\\pm2$ are easy to check by substitution. For $\\pm 4$ we can use synthetic division (or substitution), and we find that that $a=4$ is a root. (We could also see this by inspection by writing $a^3-3a=52$ and noting that $4$ works.)\n\nAre there other solutions? Use synthetic division to divide:\n\n\\begin{tabular}{c|cccc}\n$4$&$1$&$0$&$-3$&$-52$\\\\\n$$&$\\downarrow$&$4$&$16$&$52$\\\\ \\hline\n$$&$1$&$4$&13$$&$0$\n\\end{tabular}\nThe quotient is $a^2+4a+13$, so $a^3-3a-52 = (a-4)(a^2+4a+13)$. The discriminant of $a^2+4a+13$ is $4^2-4(1)(13)=16-52=-36$, which is negative, so there are no other real solutions for $a$. If $x$ is real, $a$ must be real, so we conclude that there are no other values of $x+\\frac{1}{x}$. Thus $x+\\frac{1}{x}=a=4$.\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:\nIf the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?\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:\nIf the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?\n\n### Response:\nSEARCH[\"maximum \\\\cos(33)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/precalculus/1116.json\n\n#### Similar Question\nTwo rays with common endpoint $O$ form a $30^\\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=1$. What is the maximum possible length of $\\overline{OB}$?\n\n#### User Answer\nBy the Law of Sines applied to triangle $OAB$, $$\\frac{OB}{\\sin\\angle\nOAB}=\\frac{AB}{\\sin\\angle AOB}.$$With $AB = 1$ and $\\angle AOB = 30^\\circ$, we have \\[\\frac{OB}{\\sin \\angle OAB} = \\frac{1}{\\sin 30^\\circ} = 2,\\]so so $OB=2\\sin\\angle OAB$. Thus, $OB \\le 2$, with equality if and only if $\\angle OAB=90^\\circ$.\n\n[asy]\nunitsize(1.5 cm);\n\npair O, A, B;\n\nO = (0,0);\nA = sqrt(3)*dir(30);\nB = (2,0);\n\ndraw((0,0)--3*dir(30),Arrow(6));\ndraw((0,0)--(3,0),Arrow(6));\ndraw(A--B);\ndraw(rightanglemark(O,A,B,4));\n\nlabel(\"$A$\", A, NW);\nlabel(\"$B$\", B, S);\nlabel(\"$O$\", O, W);\nlabel(\"$1$\", (A + B)/2, NE, red);\n[/asy]\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:\nIf the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?\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:\nIf the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?\n\n### Response:\nSEARCH[\"maximum \\\\cos(33)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/2609.json\n\n#### Similar Question\nThe new Perry Hotter book will have a cover price of $\\$25$. The local bookstore is offering two discounts $\\$4.00$ off and $20\\%$ off. A clever shopper realizes that the prices will be different depending on the order in which she claims her discounts. How much more money will she save by taking the better valued approach rather than the other approach? Express your answer in cents.\n\n#### User Answer\nIf the clever shopper takes $4 off followed by $20\\%$ off, the book will cost $0.8 \\times (\\$25 - \\$4) = 0.8 \\times \\$21 = \\$16.80$. If she takes $20\\%$ off followed by $4 off, it will cost $(0.8 \\times \\$25) - \\$4 = \\$20 - \\$4 = \\$16.00$. She will save $\\$16.80 - 16.00 = \\$0.80 = 80\\text{ cents}$ by taking the better-valued approach.\n\n--- RESULTS END ---\n\n" | |
| ], | |
| "response": [ | |
| " SEARCH[\"maximum \\\\cos(33)\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\n$\\$32$ is equal to 3200 cents. A stamp costs 33 cents, so we can buy $\\boxed{96}$ stamps.</s>", | |
| " SEARCH[\"maximum \\\\cos(33)\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\n$\\$32$ is equal to 3200 cents. Therefore, the maximum number of stamps that could be purchased is $\\boxed{3200/33}$.</s>", | |
| " SEARCH[\"maximum \\\\cos(33)\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\n$\\$32$ is equal to 3200 cents. Since $n$ stamps cost 33 cents each, the maximum number of stamps that can be purchased with $\\$32$ is $\\frac{3200}{33} = \\boxed{96}$.</s>" | |
| ], | |
| "src_path": [ | |
| "train/prealgebra/640.json", | |
| "train/prealgebra/640.json", | |
| "train/prealgebra/640.json", | |
| "train/prealgebra/640.json", | |
| "train/prealgebra/640.json", | |
| "train/prealgebra/640.json" | |
| ], | |
| "input": [ | |
| "If the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?", | |
| "If the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?", | |
| "If the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?", | |
| "If the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?", | |
| "If the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?", | |
| "If the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\\$32$?" | |
| ], | |
| "output": [ | |
| "$\\$32$ is equal to 3200 cents. Since $n$ stamps cost $33n$ cents, we can buy $n$ stamps only if $33n \\le 3200$. Dividing both sides of this inequality by $33$, we get $$n\\le \\frac{3200}{33}.$$We want to know the largest integer $n$ which satisfies this inequality (since we can only buy an integer number of stamps). We note that \\begin{align*}\n\\frac{3200}{33} &= \\frac{3300}{33} - \\frac{100}{33} \\\\\n&= 100 - 3\\frac{1}{33} \\\\\n&= 96\\frac{32}{33},\n\\end{align*}so the largest number of stamps we can buy is $\\boxed{96}$.", | |
| "$\\$32$ is equal to 3200 cents. Since $n$ stamps cost $33n$ cents, we can buy $n$ stamps only if $33n \\le 3200$. Dividing both sides of this inequality by $33$, we get $$n\\le \\frac{3200}{33}.$$We want to know the largest integer $n$ which satisfies this inequality (since we can only buy an integer number of stamps). We note that \\begin{align*}\n\\frac{3200}{33} &= \\frac{3300}{33} - \\frac{100}{33} \\\\\n&= 100 - 3\\frac{1}{33} \\\\\n&= 96\\frac{32}{33},\n\\end{align*}so the largest number of stamps we can buy is $\\boxed{96}$.", | |
| "$\\$32$ is equal to 3200 cents. Since $n$ stamps cost $33n$ cents, we can buy $n$ stamps only if $33n \\le 3200$. Dividing both sides of this inequality by $33$, we get $$n\\le \\frac{3200}{33}.$$We want to know the largest integer $n$ which satisfies this inequality (since we can only buy an integer number of stamps). We note that \\begin{align*}\n\\frac{3200}{33} &= \\frac{3300}{33} - \\frac{100}{33} \\\\\n&= 100 - 3\\frac{1}{33} \\\\\n&= 96\\frac{32}{33},\n\\end{align*}so the largest number of stamps we can buy is $\\boxed{96}$.", | |
| "$\\$32$ is equal to 3200 cents. Since $n$ stamps cost $33n$ cents, we can buy $n$ stamps only if $33n \\le 3200$. Dividing both sides of this inequality by $33$, we get $$n\\le \\frac{3200}{33}.$$We want to know the largest integer $n$ which satisfies this inequality (since we can only buy an integer number of stamps). We note that \\begin{align*}\n\\frac{3200}{33} &= \\frac{3300}{33} - \\frac{100}{33} \\\\\n&= 100 - 3\\frac{1}{33} \\\\\n&= 96\\frac{32}{33},\n\\end{align*}so the largest number of stamps we can buy is $\\boxed{96}$.", | |
| "$\\$32$ is equal to 3200 cents. Since $n$ stamps cost $33n$ cents, we can buy $n$ stamps only if $33n \\le 3200$. Dividing both sides of this inequality by $33$, we get $$n\\le \\frac{3200}{33}.$$We want to know the largest integer $n$ which satisfies this inequality (since we can only buy an integer number of stamps). We note that \\begin{align*}\n\\frac{3200}{33} &= \\frac{3300}{33} - \\frac{100}{33} \\\\\n&= 100 - 3\\frac{1}{33} \\\\\n&= 96\\frac{32}{33},\n\\end{align*}so the largest number of stamps we can buy is $\\boxed{96}$.", | |
| "$\\$32$ is equal to 3200 cents. Since $n$ stamps cost $33n$ cents, we can buy $n$ stamps only if $33n \\le 3200$. Dividing both sides of this inequality by $33$, we get $$n\\le \\frac{3200}{33}.$$We want to know the largest integer $n$ which satisfies this inequality (since we can only buy an integer number of stamps). We note that \\begin{align*}\n\\frac{3200}{33} &= \\frac{3300}{33} - \\frac{100}{33} \\\\\n&= 100 - 3\\frac{1}{33} \\\\\n&= 96\\frac{32}{33},\n\\end{align*}so the largest number of stamps we can buy is $\\boxed{96}$." | |
| ], | |
| "rewards": [ | |
| 1.0, | |
| 1.0, | |
| 0.0, | |
| 0.0, | |
| 1.0, | |
| 1.0 | |
| ] | |
| } |