logs/step_127-b_0.log

{
  "log_file": [
    "step_127-b_0.log",
    "step_127-b_1.log",
    "step_127-b_2.log",
    "step_127-b_3.log",
    "step_127-b_4.log",
    "step_127-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:\nThe polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$\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:\nThe polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$\n\n### Response:\nSEARCH[\"zube(2\\\\cos)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/number_theory/608.json\n\n#### Similar Question\nFind the smallest three-digit palindrome whose product with 101 is not a five-digit palindrome.\n\n#### User Answer\nWe can use the distributive property of multiplication to multiply a three-digit palindrome $aba$ (where $a$ and $b$ are digits) with 101: $$ 101 \\cdot aba = (100 + 1) \\cdot aba = aba00 + aba = ab(2a)ba. $$Here, the digits of the product are $a$, $b$, $2a$, $b$, and $a$, unless carrying occurs.  In fact, this product is a palindrome unless carrying occurs, and that could only happen when $2a \\ge 10$.  Since we want the smallest such palindrome in which carrying occurs, we want the smallest possible value of $a$ such that $2a \\ge 10$ and the smallest possible value of $b$.  This gives us $505$ as our answer and we see that $101 \\cdot 505 = 51005$ is not a palindrome.\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:\nThe polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$\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:\nThe polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$\n\n### Response:\nSEARCH[\"zube(2\\\\cos)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/counting_and_probability/5117.json\n\n#### Similar Question\nWhat is the value of the sum $S=\\sum_{k=0}^{49}(-1)^k\\binom{99}{2k}=\\binom{99}{0}-\\binom{99}{2}+\\binom{99}{4}-\\cdots -\\binom{99}{98}?$\n(A) $-2^{50}$ (B) $-2^{49}$ (C) 0 (D) $2^{49}$ (E) $2^{50}$\n\n#### User Answer\nBy the Binomial Theorem, $(1+i)^{99}=\\sum_{n=0}^{99}\\binom{99}{j}i^n =$ $\\binom{99}{0}i^0+\\binom{99}{1}i^1+\\binom{99}{2}i^2+\\binom{99}{3}i^3+\\binom{99}{4}i^4+\\cdots +\\binom{99}{98}i^{98}$.\nUsing the fact that $i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$, and $i^{n+4}=i^n$, the sum becomes:\n$(1+i)^{99}=\\binom{99}{0}+\\binom{99}{1}i-\\binom{99}{2}-\\binom{99}{3}i+\\binom{99}{4}+\\cdots -\\binom{99}{98}$.\nSo, $Re[(1+i)^{99}]=\\binom{99}{0}-\\binom{99}{2}+\\binom{99}{4}-\\cdots -\\binom{99}{98} = S$.\nUsing De Moivre's Theorem, $(1+i)^{99}=[\\sqrt{2}cis(45^\\circ)]^{99}=\\sqrt{2^{99}}\\cdot cis(99\\cdot45^\\circ)=2^{49}\\sqrt{2}\\cdot cis(135^\\circ) = -2^{49}+2^{49}i$.\nAnd finally, $S=Re[-2^{49}+2^{49}i] = -2^{49}$.\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:\nThe polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$\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:\nThe polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$\n\n### Response:\nSEARCH[\"zube(2\\\\cos)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/precalculus/824.json\n\n#### Similar Question\nCompute the least positive value of $t$ such that\n\\[\\arcsin (\\sin \\alpha), \\ \\arcsin (\\sin 2 \\alpha), \\ \\arcsin (\\sin 7 \\alpha), \\ \\arcsin (\\sin t \\alpha)\\]is a geometric progression for some $\\alpha$ with $0 < \\alpha < \\frac{\\pi}{2}.$\n\n#### User Answer\nLet $r$ be the common ratio.  Since $0 < \\alpha < \\frac{\\pi}{2},$ both $\\arcsin (\\sin \\alpha)$ and $\\arcsin (\\sin 2 \\alpha)$ are positive, so $r$ is positive.  The positive portions of the graphs of $y = \\arcsin (\\sin x),$ $y = \\arcsin (2 \\sin x),$ and $y = \\arcsin (7 \\sin x)$ are shown below.  (Note that each graph is piece-wise linear.)\n\n[asy]\nunitsize(4 cm);\n\ndraw((0,0)--(pi/2,pi/2),red);\ndraw((0,0)--(pi/4,pi/2)--(pi/2,0),green);\ndraw((0,0)--(pi/14,pi/2)--(pi/7,0),blue);\ndraw((2*pi/7,0)--(5/14*pi,pi/2)--(3*pi/7,0),blue);\ndraw((0,0)--(pi/2,0));\ndraw((0,0)--(0,pi/2));\n\ndraw((1.8,1.2)--(2.2,1.2),red);\ndraw((1.8,1.0)--(2.2,1.0),green);\ndraw((1.8,0.8)--(2.2,0.8),blue);\n\nlabel(\"$0$\", (0,0), S);\nlabel(\"$\\frac{\\pi}{2}$\", (pi/2,0), S);\nlabel(\"$\\frac{\\pi}{7}$\", (pi/7,0), S);\nlabel(\"$\\frac{2 \\pi}{7}$\", (2*pi/7,0), S);\nlabel(\"$\\frac{3 \\pi}{7}$\", (3*pi/7,0), S);\n\nlabel(\"$0$\", (0,0), W);\nlabel(\"$\\frac{\\pi}{2}$\", (0,pi/2), W);\n\nlabel(\"$y = \\arcsin (\\sin x)$\", (2.2,1.2), E);\nlabel(\"$y = \\arcsin (\\sin 2x)$\", (2.2,1.0), E);\nlabel(\"$y = \\arcsin (\\sin 7x)$\", (2.2,0.8), E);\n[/asy]\n\nNote that $\\arcsin (\\sin x) = x.$  If $0 < x \\le \\frac{\\pi}{4},$ then\n\\[\\arcsin (\\sin 2x) = 2x,\\]and if $\\frac{\\pi}{4} \\le x < \\frac{\\pi}{2},$ then\n\\[\\arcsin (\\sin 2x) = \\pi - 2x.\\]If $0 < x \\le \\frac{\\pi}{14},$ then\n\\[\\arcsin (\\sin 7x) = 7x.\\]The first three terms become $x,$ $2x,$ $7x,$ which cannot form a geometric progression.\n\nIf $\\frac{\\pi}{14} \\le x \\le \\frac{\\pi}{7},$ then\n\\[\\arcsin (\\sin 7x) = \\pi - 7x.\\]The first three terms become $x,$ $2x,$ $\\pi - 7x.$  If these form a geometric progression, then\n\\[(2x)^2 = x(\\pi - 7x).\\]Solving, we find $x = \\frac{\\pi}{11}.$  The common ratio $r$ is then 2, and the fourth term is\n\\[2^3 \\cdot \\frac{\\pi}{11} = \\frac{8 \\pi}{11}.\\]But this is greater than $\\frac{\\pi}{...\n\n--- RESULTS END ---\n\n"
  ],
  "response": [
    " SEARCH[\"zube(2\\\\cos)\"]</s>",
    " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nLet $z_k = r_k \\left[\\cos (2 \\pi \\alpha_k) + i \\sin (2 \\pi \\alpha_k) \\right],$ where $r_k > 0$ and $0 < \\alpha_k < 1.$  Then\n\\[|z_k|^2 = r_k^2 [\\cos^2 (2 \\pi \\alpha_k) + 2 \\cos (2 \\pi \\alpha_k) \\sin (2 \\pi \\alpha_k) + \\sin^2 (2 \\pi \\alpha_k)].\\]Since $z_k$ is a zero of $P(x),$\n\\[r_k^2 [\\cos^2 (2 \\pi \\alpha_k) + 2 \\cos (2 \\pi \\alpha_k) \\sin (2 \\pi \\alpha_k) + \\sin^2 (2 \\pi \\alpha_k)] = x^{17} P(x) = 0.\\]Thus, $r_k^2 [\\cos^2 (2 \\pi \\alpha_k) + \\sin^2 (2 \\pi \\alpha_k)] = 0,$ so $r_k = 0$ or $\\cos^2 (2 \\pi \\alpha_k) = \\sin^2 (2 \\pi \\alpha_k) = 0.$  Since $0 < \\alpha_k < 1,$ $r_k = 0.$  Hence, $z_k = 0.$  In other words, $0$ is a zero of $P(x),$ so\n\\[\\cos (2 \\pi \\alpha_1) + i \\sin (2 \\pi \\alpha_1) = 0.\\]Similarly, $\\cos (2 \\pi \\alpha_2) + i \\sin (2 \\pi \\alpha_2) = 0,$ $\\cos (2 \\pi \\alpha_3) + i \\sin (2 \\pi \\alpha_3) = 0,$ $\\cos (2 \\pi \\alpha_4) + i \\sin (2 \\pi \\alpha_4) = 0,$ and $\\cos (2 \\pi \\alpha_5) + i \\sin (2 \\pi \\alpha_5) = 0.$  Thus,\n\\[\\alpha_1 + \\alpha_2 + \\alpha_3 + \\alpha_4 + \\alpha_5 = \\frac{5 \\pi}{2}.\\]Therefore, $\\alpha_1 + \\alpha_2 + \\alpha_3 + \\alpha_4 + \\alpha_5 = \\boxed{\\frac{5 \\pi}{2} - \\frac{\\pi}{2}} = \\frac{4 \\pi}{2} = \\pi.}</s>",
    " SEARCH[\"zube(2\\\\cos)\"]</s>",
    "The result might be helpful, I will try using it to answer the question only if it is useful.\n\nLet $x_1, x_2, \\ldots, x_{34}$ be the zeros of $P(x)$. Then, we have the following equation:\n\n$$(1+x_1+x_1^2+\\ldots+x_1^{17})^2-x_1^{17}=0$$\n\nSimilarly, we have the following equation:\n\n$$(1+x_2+x_2^2+\\ldots+x_2^{17})^2-x_2^{17}=0$$\n\nAnd so on. We can write these equations as:\n\n$$(1+x_1+x_1^2+\\ldots+x_1^{17})^2-x_1^{17}=x_1^{34}$$\n$$(1+x_2+x_2^2+\\ldots+x_2^{17})^2-x_2^{17}=x_2^{34}$$\n$$(1+x_3+x_3^2+\\ldots+x_3^{17})^2-x_3^{17}=x_3^{34}$$\n$$\\cdots$$\n$$(1+x_{34}+x_{34}^2+\\ldots+x_{34}^{17})^2-x_{34}^{17}=x_{34}^{34}$$\n\nAdding all of these equations together, we get:\n\n$$\\sum_{k=1}^{34}(1+x_k+x_k^2+\\ldots+x_k^{17})^2-x_k^{17}=\\sum_{k=1}^{34}x_k^{34}$$\n\nSimplifying the left-hand side, we get:\n\n$$\\sum_{k=1}^{34}(1+x_k+x_k^2+\\ldots+x_k^{17})^2-x_k^{17}=\\sum_{k=1}^{34}(x_k+x_k^2+\\ldots+x_k^{17})^2$$\n\nSince each term in the sum has a power of $x_k$ that is greater than or equal to $2$, we can use the AM-GM inequality to find an upper bound for the sum. The AM-GM inequality states that:\n\n$$\\sum_{k=1}^{n}a_k^p\\leq\\left(\\sum_{k=1}^{n}a_k^q\\right)^p$$\n\nwhere $p$ is the number of terms, $q$ is the average power of $a_k$, and $a_k$ are non-negative real numbers. In this case, we have $p=2$ and $q=2$. Thus, we have:\n\n$$\\sum_{k=1}^{34}(x_k+x_k^2+\\ldots+x_k^{17})^2\\leq\\left(\\sum_{k=1}^{3",
    " SEARCH[\"zube(2\\\\cos)\"]</s>",
    "The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nLet $z_k = r_k \\left[\\cos \\left( \\frac{2 \\pi \\alpha_k}{34} \\right) + i \\sin \\left( \\frac{2 \\pi \\alpha_k}{34} \\right) \\right],$ where $r_k > 0$ and $0 < \\alpha_k < 1.$  Then the polynomial can be written as\n\\begin{align*}\n& (1 + x + x^2 + \\dots + x^{17})^2 - x^{17} \\\\\n&= \\left[ (1 + x + x^2 + \\dots + x^{17}) - x^{17} \\right]^2 \\\\\n&= \\left[ (1 + x + x^2 + \\dots + x^{17}) - x^{17} - \\left( x^{17} - x^{17} + x^{"
  ],
  "src_path": [
    "train/precalculus/463.json",
    "train/precalculus/463.json",
    "train/precalculus/463.json",
    "train/precalculus/463.json",
    "train/precalculus/463.json",
    "train/precalculus/463.json"
  ],
  "input": [
    "The polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$",
    "The polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$",
    "The polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$",
    "The polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$",
    "The polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$",
    "The polynomial $$P(x)=(1+x+x^2+\\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\\left[\\cos(2\\pi\\alpha_k)\n+i\\sin(2\\pi\\alpha_k)\\right]$, $k=1,2,3,\\ldots,34$, with $0<\\alpha_1\\le\\alpha_2\\le\\alpha_3\\le\\dots\\le\\alpha_{34}<1$ and $r_k>0$. Find $\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5.$"
  ],
  "output": [
    "Note that for $x\\ne1$, \\begin{align*}\nP(x)&=\\left(\\frac{x^{18}-1}{x-1}\\right)^2-x^{17} \\end{align*}so \\begin{align*}\n\\cr (x-1)^2P(x)&=(x^{18}-1)^2-x^{17}(x-1)^2\\cr\n&=x^{36}-2x^{18}+1-x^{19}+2x^{18}-x^{17}\\cr\n&=x^{36}-x^{19}-x^{17}+1\\cr &=x^{19}(x^{17}-1)-(x^{17}-1)\\cr\n&=(x^{19}-1)(x^{17}-1). \\end{align*}Then\n\\[P(x)=\\frac{(x^{19}-1)(x^{17}-1)}{(x-1)^2}.\\]Thus the zeros of $P(x)$ are the 34 complex numbers other than 1 which satisfy $x^{17}=1$ or $x^{19}=1$. It follows that $\\alpha_1= \\frac{1}{19},$ $\\alpha_2= \\frac{1}{17},$ $\\alpha_3= \\frac{2}{19},$ $\\alpha_4= \\frac{2}{17},$ and $\\alpha_5= \\frac{3}{19},$ so\n\\[\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5= \\boxed{\\frac{159}{323}}.\\]",
    "Note that for $x\\ne1$, \\begin{align*}\nP(x)&=\\left(\\frac{x^{18}-1}{x-1}\\right)^2-x^{17} \\end{align*}so \\begin{align*}\n\\cr (x-1)^2P(x)&=(x^{18}-1)^2-x^{17}(x-1)^2\\cr\n&=x^{36}-2x^{18}+1-x^{19}+2x^{18}-x^{17}\\cr\n&=x^{36}-x^{19}-x^{17}+1\\cr &=x^{19}(x^{17}-1)-(x^{17}-1)\\cr\n&=(x^{19}-1)(x^{17}-1). \\end{align*}Then\n\\[P(x)=\\frac{(x^{19}-1)(x^{17}-1)}{(x-1)^2}.\\]Thus the zeros of $P(x)$ are the 34 complex numbers other than 1 which satisfy $x^{17}=1$ or $x^{19}=1$. It follows that $\\alpha_1= \\frac{1}{19},$ $\\alpha_2= \\frac{1}{17},$ $\\alpha_3= \\frac{2}{19},$ $\\alpha_4= \\frac{2}{17},$ and $\\alpha_5= \\frac{3}{19},$ so\n\\[\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5= \\boxed{\\frac{159}{323}}.\\]",
    "Note that for $x\\ne1$, \\begin{align*}\nP(x)&=\\left(\\frac{x^{18}-1}{x-1}\\right)^2-x^{17} \\end{align*}so \\begin{align*}\n\\cr (x-1)^2P(x)&=(x^{18}-1)^2-x^{17}(x-1)^2\\cr\n&=x^{36}-2x^{18}+1-x^{19}+2x^{18}-x^{17}\\cr\n&=x^{36}-x^{19}-x^{17}+1\\cr &=x^{19}(x^{17}-1)-(x^{17}-1)\\cr\n&=(x^{19}-1)(x^{17}-1). \\end{align*}Then\n\\[P(x)=\\frac{(x^{19}-1)(x^{17}-1)}{(x-1)^2}.\\]Thus the zeros of $P(x)$ are the 34 complex numbers other than 1 which satisfy $x^{17}=1$ or $x^{19}=1$. It follows that $\\alpha_1= \\frac{1}{19},$ $\\alpha_2= \\frac{1}{17},$ $\\alpha_3= \\frac{2}{19},$ $\\alpha_4= \\frac{2}{17},$ and $\\alpha_5= \\frac{3}{19},$ so\n\\[\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5= \\boxed{\\frac{159}{323}}.\\]",
    "Note that for $x\\ne1$, \\begin{align*}\nP(x)&=\\left(\\frac{x^{18}-1}{x-1}\\right)^2-x^{17} \\end{align*}so \\begin{align*}\n\\cr (x-1)^2P(x)&=(x^{18}-1)^2-x^{17}(x-1)^2\\cr\n&=x^{36}-2x^{18}+1-x^{19}+2x^{18}-x^{17}\\cr\n&=x^{36}-x^{19}-x^{17}+1\\cr &=x^{19}(x^{17}-1)-(x^{17}-1)\\cr\n&=(x^{19}-1)(x^{17}-1). \\end{align*}Then\n\\[P(x)=\\frac{(x^{19}-1)(x^{17}-1)}{(x-1)^2}.\\]Thus the zeros of $P(x)$ are the 34 complex numbers other than 1 which satisfy $x^{17}=1$ or $x^{19}=1$. It follows that $\\alpha_1= \\frac{1}{19},$ $\\alpha_2= \\frac{1}{17},$ $\\alpha_3= \\frac{2}{19},$ $\\alpha_4= \\frac{2}{17},$ and $\\alpha_5= \\frac{3}{19},$ so\n\\[\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5= \\boxed{\\frac{159}{323}}.\\]",
    "Note that for $x\\ne1$, \\begin{align*}\nP(x)&=\\left(\\frac{x^{18}-1}{x-1}\\right)^2-x^{17} \\end{align*}so \\begin{align*}\n\\cr (x-1)^2P(x)&=(x^{18}-1)^2-x^{17}(x-1)^2\\cr\n&=x^{36}-2x^{18}+1-x^{19}+2x^{18}-x^{17}\\cr\n&=x^{36}-x^{19}-x^{17}+1\\cr &=x^{19}(x^{17}-1)-(x^{17}-1)\\cr\n&=(x^{19}-1)(x^{17}-1). \\end{align*}Then\n\\[P(x)=\\frac{(x^{19}-1)(x^{17}-1)}{(x-1)^2}.\\]Thus the zeros of $P(x)$ are the 34 complex numbers other than 1 which satisfy $x^{17}=1$ or $x^{19}=1$. It follows that $\\alpha_1= \\frac{1}{19},$ $\\alpha_2= \\frac{1}{17},$ $\\alpha_3= \\frac{2}{19},$ $\\alpha_4= \\frac{2}{17},$ and $\\alpha_5= \\frac{3}{19},$ so\n\\[\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5= \\boxed{\\frac{159}{323}}.\\]",
    "Note that for $x\\ne1$, \\begin{align*}\nP(x)&=\\left(\\frac{x^{18}-1}{x-1}\\right)^2-x^{17} \\end{align*}so \\begin{align*}\n\\cr (x-1)^2P(x)&=(x^{18}-1)^2-x^{17}(x-1)^2\\cr\n&=x^{36}-2x^{18}+1-x^{19}+2x^{18}-x^{17}\\cr\n&=x^{36}-x^{19}-x^{17}+1\\cr &=x^{19}(x^{17}-1)-(x^{17}-1)\\cr\n&=(x^{19}-1)(x^{17}-1). \\end{align*}Then\n\\[P(x)=\\frac{(x^{19}-1)(x^{17}-1)}{(x-1)^2}.\\]Thus the zeros of $P(x)$ are the 34 complex numbers other than 1 which satisfy $x^{17}=1$ or $x^{19}=1$. It follows that $\\alpha_1= \\frac{1}{19},$ $\\alpha_2= \\frac{1}{17},$ $\\alpha_3= \\frac{2}{19},$ $\\alpha_4= \\frac{2}{17},$ and $\\alpha_5= \\frac{3}{19},$ so\n\\[\\alpha_1+\\alpha_2+\\alpha_3+\\alpha_4+\\alpha_5= \\boxed{\\frac{159}{323}}.\\]"
  ],
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