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[
0.9840297698974609,... | Please reason step by step, and put your final answer within \boxed{}.
Compute the sum $S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{1-3 x_{i}+3 x_{i}^{2}}$ for $x_{i}=\frac{i}{101}$.
Answer: $S=51$. | Since $x_{101-i}=\frac{101-i}{101}=1-\frac{i}{101}=1-x_{i}$ and
$$
1-3 x_{i}+3 x_{i}^{2}=\left(1-3 x_{i}+3 x_{i}^{2}-x_{i}^{3}\right)+x_{i}^{3}=\left(1-x_{i}\right)^{3}+x_{i}^{3}=x_{101-i}^{3}+x_{i}^{3},
$$
we have, by replacing $i$ by $101-i$ in the second sum,
$$
2 S=S+S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{x_{101-i}... | 7 |
[151644,872,198,5501,2874,3019,553,3019,11,323,2182,697,1590,4226,2878,1124,79075,6257,382,35,24308,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED) | [114,115,116,117,118,119,120,121,122,123,124,127,128,129,130,131,132,133,134,135,136,137,139,140,141(...TRUNCATED) | [[284,3,10234,50573,64967,68101,102041,118396,26233,22025],[1124,387,220,320,67414,87726,5851,21685,(...TRUNCATED) | [[0.6643398404121399,0.33561164140701294,0.0000158799175551394,7.739718967059162e-6,6.69069277137168(...TRUNCATED) | "Please reason step by step, and put your final answer within \\boxed{}.\n\nDetermine all finite non(...TRUNCATED) | "Let $k \\in S$. Then $\\frac{k+k}{(k, k)}=2$ is in $S$ as well.\nSuppose for the sake of contradict(...TRUNCATED) | 9 |
[151644,872,198,5501,2874,3019,553,3019,11,323,2182,697,1590,4226,2878,1124,79075,6257,382,32,21495,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED) | [124,125,126,127,128,129,130,131,132,133,134,136,137,138,139,140,141,142,143,144,145,146,150,151,152(...TRUNCATED) | [[374,304,504,60173,89472,80936,85599,107436,102594,38935],[419,3070,2686,264,400,279,6144,436,15890(...TRUNCATED) | [[0.8449270129203796,0.13546313345432281,0.019586799666285515,0.000014779323464608751,2.858579819076(...TRUNCATED) | "Please reason step by step, and put your final answer within \\boxed{}.\n\nA triangle with sides $a(...TRUNCATED) | "The perimeter of each new triangle constructed by the process is $(s-a)+(s-b)+(s-c) = 3s - (a+b+c) (...TRUNCATED) | 2 |
[151644,872,198,5501,2874,3019,553,3019,11,323,2182,697,1590,4226,2878,1124,79075,6257,382,35,24308,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED) | [76,77,78,79,80,81,82,83,84,85,86,88,89,90,91,92,93,94,95,96,97,98,122,123,124,125,126,127,128,129,1(...TRUNCATED) | [[582,279,89953,97432,34525,98241,55191,137076,40331,72625],[525,23606,2299,614,1366,31264,43787,247(...TRUNCATED) | [[0.9367942214012146,0.06315724551677704,0.000010735220712376758,0.000010561853741819505,9.020145625(...TRUNCATED) | "Please reason step by step, and put your final answer within \\boxed{}.\n\nDetermine all positive i(...TRUNCATED) | "If $n$ is even, $x^{n}+(2+x)^{n}+(2-x)^{n}>0$, so $n$ is odd.\nFor $n=1$, the equation reduces to $(...TRUNCATED) | 4 |
[151644,872,198,5501,2874,3019,553,3019,11,323,2182,697,1590,4226,2878,1124,79075,6257,382,10061,400(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED) | [170,171,172,173,174,175,176,177,178,179,180,190,191,192,193,194,195,196,197,198,199,200,202,203,204(...TRUNCATED) | [[87,856,88,379,799,104873,48659,104867,6410,34544],[284,3,400,28,89603,62514,108624,22502,75728,986(...TRUNCATED) | [[0.4494505822658539,0.4023178815841675,0.07574479281902313,0.07243628799915314,0.000023058833903633(...TRUNCATED) | "Please reason step by step, and put your final answer within \\boxed{}.\n\nLet $f: \\mathbb{R} \\ri(...TRUNCATED) | "Plug $y \\rightarrow 1$ in (i):\n\n$$\nf(x)+f(1)+1 \\geq f(x+1) \\geq f(x)+f(1) \\Longleftrightarro(...TRUNCATED) | 5 |
[151644,872,198,5501,2874,3019,553,3019,11,323,2182,697,1590,4226,2878,1124,79075,6257,382,10061,400(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED) | [132,133,134,135,136,137,138,139,140,141,142,144,145,146,147,148,149,150,151,152,153,154,163,164,165(...TRUNCATED) | [[374,582,311,138273,23153,131029,103730,47977,101046,103848],[525,3070,2299,311,10161,5931,264,279,(...TRUNCATED) | [[0.7723417282104492,0.21491068601608276,0.012743347324430943,9.786028840608196e-7,9.373208058605087(...TRUNCATED) | "Please reason step by step, and put your final answer within \\boxed{}.\n\nLet $n$ be a positive in(...TRUNCATED) | "The answer is\n\n$$\nf(n)=\\left\\{\\begin{array}{cl}\n0 & \\text { if } n \\text { is even, } \\\\(...TRUNCATED) | 10 |
[151644,872,198,5501,2874,3019,553,3019,11,323,2182,697,1590,4226,2878,1124,79075,6257,382,1654,1618(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED) | [151,152,153,154,155,156,157,158,159,160,161,163,164,165,166,167,168,169,170,171,172,173,181,182,183(...TRUNCATED) | [[3070,1602,220,3491,17003,20958,48434,150211,128741,121403],[7040,7,20,271,1118,374,2303,6419,25777(...TRUNCATED) | [[0.6121888160705566,0.21240438520908356,0.13270020484924316,0.03902506083250046,0.00365695962682366(...TRUNCATED) | "Please reason step by step, and put your final answer within \\boxed{}.\n\nWe call a 5-tuple of int(...TRUNCATED) | "A valid 2017-tuple is \\( n_{1} = \\cdots = n_{2017} = 29 \\). We will show that it is the only sol(...TRUNCATED) | 15 |
[151644,872,198,5501,2874,3019,553,3019,11,323,2182,697,1590,4226,2878,1124,79075,6257,382,9885,279,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED) | [120,121,122,123,124,125,126,127,128,129,130,132,133,134,135,136,137,138,139,140,141,142,144,145,146(...TRUNCATED) | [[400,279,128685,29582,120577,95229,7461,95974,10824,122574],[69,282,729,58,118186,82375,154,3277,12(...TRUNCATED) | [[0.737852931022644,0.26213130354881287,4.368501322460361e-6,2.5339550120406784e-6,2.474529082974186(...TRUNCATED) | "Please reason step by step, and put your final answer within \\boxed{}.\n\nFind the total number of(...TRUNCATED) | "Note that, since $[x+n]=[x]+n$ for any integer $n$,\n\n$$\nf(x+3)=[x+3]+[2(x+3)]+\\left[\\frac{5(x+(...TRUNCATED) | 3 |
[151644,872,198,5501,2874,3019,553,3019,11,323,2182,697,1590,4226,2878,1124,79075,6257,382,9885,279,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED) | [61,62,63,64,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,81,82,83,85,86,87,88,89,90,91,92,93,94,95,(...TRUNCATED) | [[279,582,1221,773,271,498,1128,1477,400,311],[3491,3343,525,1590,279,1184,4226,1366,1372,271],[311,(...TRUNCATED) | [[0.7443825602531433,0.18210086226463318,0.03592482581734657,0.01180211640894413,0.00826782174408435(...TRUNCATED) | "Please reason step by step, and put your final answer within \\boxed{}.\n\nFind the smallest positi(...TRUNCATED) | "and Marking Scheme:\nWe first note that the integer terms of any arithmetic progression are \"equal(...TRUNCATED) | 6 |
[151644,872,198,5501,2874,3019,553,3019,11,323,2182,697,1590,4226,2878,1124,79075,6257,382,35,24308,(...TRUNCATED) | [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1(...TRUNCATED) | [-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100(...TRUNCATED) | [81,82,83,84,85,86,87,88,89,90,91,93,94,95,96,97,98,99,100,101,102,103,108,109,110,111,112,113,114,1(...TRUNCATED) | [[279,3070,41648,140289,80446,28094,50307,78765,60405,130842],[1172,3070,271,5257,63246,120884,11834(...TRUNCATED) | [[0.8305037021636963,0.169479638338089,6.038254468876403e-6,4.390252343000611e-6,2.0977533949917415e(...TRUNCATED) | "Please reason step by step, and put your final answer within \\boxed{}.\n\nDetermine all positive i(...TRUNCATED) | "We claim that $k=2^{a}$ for all $a \\geq 0$.\nLet $A=\\{1,2,4,8, \\ldots\\}$ and $B=\\mathbb{N} \\b(...TRUNCATED) | 17 |
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