id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-5aaa0ae49c32c50150ba30ad009e21900e4cc50ed6308048aa74730744ea0a5c | Calculate \( A^n \) for all \( n \in \mathbb{N^*} \), where \( A = \begin{pmatrix} 1 & 2 \\ -2 & 1 \end{pmatrix} \). | [
"\\(\\begin{bmatrix} \\mathrm{Re}((1+2i)^n) & \\mathrm{Im}((1+2i)^n) \\\\ -\\mathrm{Im}((1+2i)^n) & \\mathrm{Re}((1+2i)^n) \\end{bmatrix}\\)"
] | [
"Calculate \\( A^n \\) for all \\( n \\in \\mathbb{N^*} \\), where \\( A = \\begin{pmatrix} 1 & 2 \\\\ -2 & 1 \\end{pmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-5ba085c53f895c4b92405259f293a584b1a7393cf1dbf45109995cc8e52831f8 | Calculate \( A^n \) where \( A = \begin{pmatrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{pmatrix} \). | [
"\\( A^{n} = \\begin{pmatrix} \\frac{1}{3} + \\frac{2}{3}(-2)^{n} & \\frac{1}{3} - \\frac{1}{3}(-2)^{n} & \\frac{1}{3} - \\frac{1}{3}(-2)^{n} \\\\ \\frac{1}{3} - \\frac{1}{3}(-2)^{n} & \\frac{1}{3} + \\frac{2}{3}(-2)^{n} & \\frac{1}{3} - \\frac{1}{3}(-2)^{n} \\\\ \\frac{1}{3} - \\frac{1}{3}(-2)^{n} & \\frac{1}{3} -... | [
"Calculate \\( A^n \\) where \\( A = \\begin{pmatrix} -1 & 1 & 1 \\\\ 1 & -1 & 1 \\\\ 1 & 1 & -1 \\end{pmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-2267fd704468b999052e66c09700c85f836cff98f3867484dadcc083888ce135 | Calculate \( A^n \) where \( A = \begin{pmatrix} \cos x & \sin x \\ \sin x & \cos x \end{pmatrix} \). | [
"\\( A^n = \\begin{pmatrix} \\cos(nx) & \\sin(nx) \\\\ \\sin(nx) & \\cos(nx) \\end{pmatrix} \\)"
] | [
"Calculate \\( A^n \\) where \\( A = \\begin{pmatrix} \\cos x & \\sin x \\\\ \\sin x & \\cos x \\end{pmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f9a7fa24414810148bb77f244c37060da5794ecc2c9c9f799d775b81b655162 | Calculate \( A^{-1} \) for the matrix \( A \) defined as: \[ A = \begin{pmatrix} a & b & b & \cdots & b \\ b & a & b & \cdots & b \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ b & b & b & \cdots & a \end{pmatrix} \] | [
"\\( A^{-1} = \\frac{1}{(a - b)(a + (n-2)b)} \\left( (a - b)I - bJ \\right) \\)"
] | [
"Calculate \\( A^{-1} \\) for the matrix \\( A \\) defined as:\n\\[ A = \\begin{pmatrix}\na & b & b & \\cdots & b \\\\\nb & a & b & \\cdots & b \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\nb & b & b & \\cdots & a\n\\end{pmatrix} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 6
} | 17 | 0 | 0 |
omr-ws-56dd6ca94f1608f3ea73e650c04146fcc7e7de6d8095715b2cc34cf86ab820ec | Calculate \( A^{2009} \) where \[ A = \begin{pmatrix} 0 & 0 & 0 & 0 & -1 \\ 0 & -7 & 5 & 3 & 0 \\ 0 & -5 & 4 & 2 & 0 \\ 0 & -9 & 6 & 4 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{pmatrix}. \] | [
"\\begin{pmatrix} 0 & 0 & 0 & 0 & -1 \\\\ 0 & -3 & 3 & 1 & 0 \\\\ 0 & -3 & 3 & 1 & 0 \\\\ 0 & -3 & 3 & 1 & 0 \\\\ 1 & 0 & 0 & 0 & 0 \\end{pmatrix}"
] | [
"Calculate \\( A^{2009} \\) where\n\\[\nA = \\begin{pmatrix}\n0 & 0 & 0 & 0 & -1 \\\\\n0 & -7 & 5 & 3 & 0 \\\\\n0 & -5 & 4 & 2 & 0 \\\\\n0 & -9 & 6 & 4 & 0 \\\\\n1 & 0 & 0 & 0 & 0\n\\end{pmatrix}.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0c626c97e8f8f4df794545e9146a53e006775f53a8b5bdeb0448b76784b7533b | Calculate \( A_n = \int_{a}^{b} (a^n - x^n)^n (b^n - x^n)^n \, dx \). | [
"0"
] | [
"Calculate \\( A_n = \\int_{a}^{b} (a^n - x^n)^n (b^n - x^n)^n \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-75e7df601c701f04d351f209f9095f142a097164b160dc3bf721afb9dd0889ab | Calculate \( B^n \) where \( B = \begin{pmatrix} a & b & b \\ b & a & b \\ b & b & a \end{pmatrix} \). | [
"\\( B^n = \\begin{pmatrix} \\frac{1}{3} \\left( 2(-2)^n + 1 \\right) & \\frac{1}{3} \\left( 1 - (-2)^n \\right) & \\frac{1}{3} \\left( 1 - (-2)^n \\right) \\\\ \\frac{1}{3} \\left( 1 - (-2)^n \\right) & \\frac{1}{3} \\left( 2(-2)^n + 1 \\right) & \\frac{1}{3} \\left( 1 - (-2)^n \\right) \\\\ \\frac{1}{3} \\left( 1... | [
"Calculate \\( B^n \\) where \\( B = \\begin{pmatrix} a & b & b \\\\ b & a & b \\\\ b & b & a \\end{pmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-355de95bf9cc4d81ab5640f078aa7689c1d1d95956dd6c2a60e84195ba7a0bc4 | Calculate \( E((1+\sqrt{2})^{2^{2^{2019}}}) \mod (2^{89}-1) \). | [
"89084454122671745803363121"
] | [
"Calculate \\( E((1+\\sqrt{2})^{2^{2^{2019}}}) \\mod (2^{89}-1) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-143934eeefa0a55fe10548704b637b46634c37e60b63781ec2af64f255063d2f | Calculate \( E\left(\sum_{k=1}^{2^{2^{2020}}} \frac{1}{k}\right) \mod (2^{89} - 1) \), where \( E(x) \) denotes the integer part of \( x \). | [
"395244678588117990215918595"
] | [
"Calculate \\( E\\left(\\sum_{k=1}^{2^{2^{2020}}} \\frac{1}{k}\\right) \\mod (2^{89} - 1) \\), where \\( E(x) \\) denotes the integer part of \\( x \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b5be88d880895bf0f23c3c5abf27cc5e3a62005bb0b74e557e756d91e2305013 | Calculate \( F'(\alpha) \) if \( F(\alpha) = \int\limits_0^\alpha f(x + \alpha, x - \alpha) \, dx \). | [
"\\( F'(\\alpha) = f(2\\alpha, 0) + \\int_0^\\alpha f_1(x + \\alpha, x - \\alpha) \\, dx - \\int_0^\\alpha f_2(x + \\alpha, x - \\alpha) \\, dx \\)"
] | [
"Calculate \\( F'(\\alpha) \\) if \\( F(\\alpha) = \\int\\limits_0^\\alpha f(x + \\alpha, x - \\alpha) \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-2fdb2b5fad6502ff2a78767e7fd36715861c32b3fac32c74b327f51604b7e42f | Calculate \( F'(\alpha) \) if \( F(\alpha) = \int\limits_\alpha^{\alpha^2} e^{-\alpha y^2} \, dy \). | [
"\\( 2\\alpha e^{-\\alpha^5} - e^{-\\alpha^3} - \\int_\\alpha^{\\alpha^2} t^2 e^{-\\alpha t^2} \\, dt \\)"
] | [
"Calculate \\( F'(\\alpha) \\) if \\( F(\\alpha) = \\int\\limits_\\alpha^{\\alpha^2} e^{-\\alpha y^2} \\, dy \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-7f06bb0a685ce197be8a6b0e4eaaf1547cab711fe83d7f8dd3ee45545999e83a | Calculate \( F'(\alpha) \) if \( F(\alpha) = \int\limits_{\sin \alpha}^{\cos \alpha} e^{\alpha \sqrt{1 - x^2}} \, dx \). | [
"\\(-\\cos \\alpha e^{\\alpha |\\cos \\alpha|} - \\sin \\alpha e^{\\alpha |\\sin \\alpha|} + \\int_{\\sin \\alpha}^{\\cos \\alpha} \\sqrt{1 - x^2} e^{\\alpha \\sqrt{1 - x^2}} \\, dx\\)"
] | [
"Calculate \\( F'(\\alpha) \\) if \\( F(\\alpha) = \\int\\limits_{\\sin \\alpha}^{\\cos \\alpha} e^{\\alpha \\sqrt{1 - x^2}} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 7,
"tir": 6
} | 17 | 0 | 0 |
omr-ws-b52f0c99b03816fb78a2d2216c09b598551f788e5b99429c300aded0bb8c2ef7 | Calculate \( F'(\alpha) \) if \( F(\alpha) = \int\limits_{a + \alpha}^{b + \alpha} \frac{\sin \alpha z}{z} \, dz \). | [
"\\(-\\frac{\\sin \\alpha(a+\\alpha)}{a+\\alpha} + \\frac{\\sin \\alpha(b+\\alpha)}{b+\\alpha} + \\int_{a+\\alpha}^{b+\\alpha} \\sin (\\alpha z) \\, dz\\)"
] | [
"Calculate \\( F'(\\alpha) \\) if \\( F(\\alpha) = \\int\\limits_{a + \\alpha}^{b + \\alpha} \\frac{\\sin \\alpha z}{z} \\, dz \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-892421f39a9ec5caea2dc1e07f3bad8a9a3d5321ff82767d7f92136cb4e75dff | Calculate \( I = \int_{0}^{1} \frac{\ln(1+x)}{1+x^{2}}\ dx \), using the substitution \( x = \frac{1-t}{1+t} \). | [
"\\dfrac{\\pi \\ln 2}{8}"
] | [
"Calculate \\( I = \\int_{0}^{1} \\frac{\\ln(1+x)}{1+x^{2}}\\ dx \\), using the substitution \\( x = \\frac{1-t}{1+t} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e9636a409122fc03d25e4b4ecf774ec8b87c712c6340655a363eb53f66ec4e8e | Calculate \( I = \int_{0}^{1} \frac{t-1}{\ln t} \, dt \). | [
"\\ln 2"
] | [
"Calculate \\( I = \\int_{0}^{1} \\frac{t-1}{\\ln t} \\, dt \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cab3feca5e917baa68c8184fa75c6c96bf901b629f2d023bcab1cd42aa02f704 | Calculate \( I \) in terms of the Riemann zeta function \( \zeta \): \[ I = \int_{0}^{1} \int_{0}^{1} \frac{\ln(x) \ln(x) (\ln(x) + \ln(y))^m}{1 - xy} \, dx \, dy \] | [
" (-1)^m \\dfrac{(m + 3)!}{3} \\zeta(m + 4) "
] | [
"Calculate \\( I \\) in terms of the Riemann zeta function \\( \\zeta \\):\n\n\\[ I = \\int_{0}^{1} \\int_{0}^{1} \\frac{\\ln(x) \\ln(x) (\\ln(x) + \\ln(y))^m}{1 - xy} \\, dx \\, dy \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-23a709fe24f6d28741fa4890d39a255128f4db0984fab47f9bf8b251b5484bf7 | Calculate \( I(2) \) where \( I(n) = \int_{0}^{\pi/4}\cdots\int_{0}^{\pi/4}\ln(1+\tan(x_{1})\cdots \tan(x_{n}))dx_{1}\cdots dx_{n} \). | [
"\\dfrac{3\\pi^2}{32} \\ln 2 + \\dfrac{35}{128} \\zeta(3)"
] | [
"Calculate \\( I(2) \\) where \\( I(n) = \\int_{0}^{\\pi/4}\\cdots\\int_{0}^{\\pi/4}\\ln(1+\\tan(x_{1})\\cdots \\tan(x_{n}))dx_{1}\\cdots dx_{n} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37cfc9ce7e73fc407d6a42afeb744ad96f84d90c260c3fa5d23feff4595fc88f | Calculate \( I_{0} \), \( I_{1} \), and \( I_{2} \) where \( I_{n} = \int_{4}^{5} \frac{2x-3}{ x(x-1)(x-2)(x-3) + n} \,dx \). | [
"\\( I_0 = \\log\\frac{\\sqrt{5}}{2} \\), \\( I_1 = \\frac{6}{55} \\), \\( I_2 = \\tan^{-1}\\frac{3}{28} \\)"
] | [
"Calculate \\( I_{0} \\), \\( I_{1} \\), and \\( I_{2} \\) where \\( I_{n} = \\int_{4}^{5} \\frac{2x-3}{ x(x-1)(x-2)(x-3) + n} \\,dx \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-cc884cf3c07ab61f37638315548548d2bfce176e595d92603d08b4988d5913f6 | Calculate \( L = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \frac{f(k)}{k} \), where \( f(k) \) is the greatest odd divisor of \( k \in \mathbb{N} \). | [
"\\( \\frac{2}{3} \\)"
] | [
"Calculate \\( L = \\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{k=1}^{n} \\frac{f(k)}{k} \\), where \\( f(k) \\) is the greatest odd divisor of \\( k \\in \\mathbb{N} \\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-2394fa5644450e48fb55de63bbff3634a1f1adda0c0578d9f5ec00dfe9b03743 | Calculate \( L = \lim_{n \to \infty} \int_0^1 \left( \frac{k}{\sqrt[n]{x} + k - 1} \right)^n \, dx \) for \( k > 1 \). | [
"\\(\\frac{k}{k-1}\\)"
] | [
"Calculate \\( L = \\lim_{n \\to \\infty} \\int_0^1 \\left( \\frac{k}{\\sqrt[n]{x} + k - 1} \\right)^n \\, dx \\) for \\( k > 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 26
} | 43 | 0 | 0 |
omr-ws-a8e09ff81396b39d35fa55a32a680656f09dabaf34819e6d6b67e1569c44dcef | Calculate \( M = 2x^3 + 2x^2 + 1 \) where \( x = \frac{1}{3} \left( \sqrt[3]{\frac{23 + \sqrt{513}}{4}} + \sqrt[3]{\frac{23 - \sqrt{513}}{4}} - 1 \right) \). | [
"2"
] | [
"Calculate \\( M = 2x^3 + 2x^2 + 1 \\) where \\( x = \\frac{1}{3} \\left( \\sqrt[3]{\\frac{23 + \\sqrt{513}}{4}} + \\sqrt[3]{\\frac{23 - \\sqrt{513}}{4}} - 1 \\right) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a3ef1abd683fde93bb6c968502caaa1a4c8ed5a1eedd033f7626b7cc76b2678b | Calculate \( M = F(7, 2) + F(7, 3) + F(7, 5) - F(7, 6) \) where \( F(x, y) = \cos^y\left(\frac{\pi}{x}\right) + \cos^y\left(\frac{3\pi}{x}\right) + \cos^y\left(\frac{5\pi}{x}\right) \). | [
"\\(\\frac{53}{32}\\)"
] | [
"Calculate \\( M = F(7, 2) + F(7, 3) + F(7, 5) - F(7, 6) \\) where \\( F(x, y) = \\cos^y\\left(\\frac{\\pi}{x}\\right) + \\cos^y\\left(\\frac{3\\pi}{x}\\right) + \\cos^y\\left(\\frac{5\\pi}{x}\\right) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 24
} | 43 | 0 | 0 |
omr-ws-0d4b5de3ebe02ded93767bd4f8afb422f1161d0bfd828740070d539ac620a95a | Calculate \( M^{100} \), where \[ M = \begin{pmatrix} 1 & 2 & 0 \\ -3 & -3 & 1 \\ 2 & 2 & -1 \end{pmatrix}. \] | [
"\\( M^{100} = 4950M^2 + 9800M + 4851I_3 \\)"
] | [
"Calculate \\( M^{100} \\), where\n\\[ M = \\begin{pmatrix}\n1 & 2 & 0 \\\\\n-3 & -3 & 1 \\\\\n2 & 2 & -1\n\\end{pmatrix}. \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dbf9c3264530ad7d68e4e2440fc8e8201d7c64ffec9ae3654678b82ee000a123 | Calculate \( P = 16 \cos 20^\circ \cos 40^\circ \cos 80^\circ \) using complex numbers. Given \( z = \cos 20^\circ + i \sin 20^\circ \), express \( P \) in terms of \( z \) and simplify to find the value of \( P \). | [
"2"
] | [
"Calculate \\( P = 16 \\cos 20^\\circ \\cos 40^\\circ \\cos 80^\\circ \\) using complex numbers.\n\nGiven \\( z = \\cos 20^\\circ + i \\sin 20^\\circ \\), express \\( P \\) in terms of \\( z \\) and simplify to find the value of \\( P \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-138d68b9874ad7d5ad300663034063b7f4cb6061db30717a5d6caf562f289a59 | Calculate \( P = \cos \frac{\pi}{12} + \cos \frac{9\pi}{12} + \cos \frac{17\pi}{12} \). | [
"0"
] | [
"Calculate \\( P = \\cos \\frac{\\pi}{12} + \\cos \\frac{9\\pi}{12} + \\cos \\frac{17\\pi}{12} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4ffafb547ce36f20461ba290a039b049cd0d67d0322228256e650a7fed445ff4 | Calculate \( P = \lim_{n \to \infty} \sum_{i=1}^n \frac{1}{n} \left\lfloor \sqrt{\frac{4i}{n}} \right\rfloor \). | [
"\\(\\frac{3}{4}\\)"
] | [
"Calculate \\( P = \\lim_{n \\to \\infty} \\sum_{i=1}^n \\frac{1}{n} \\left\\lfloor \\sqrt{\\frac{4i}{n}} \\right\\rfloor \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-f422522f126492dd07ddd6e209eab97229a8bfcd396bb798f97fa90faa2df137 | Calculate \( Q = \cos \frac{\pi}{12} \cdot \cos \frac{9\pi}{12} + \cos \frac{9\pi}{12} \cdot \cos \frac{17\pi}{12} + \cos \frac{17\pi}{12} \cdot \cos \frac{\pi}{12} \). | [
"-\\dfrac{3}{4}"
] | [
"Calculate \\( Q = \\cos \\frac{\\pi}{12} \\cdot \\cos \\frac{9\\pi}{12} + \\cos \\frac{9\\pi}{12} \\cdot \\cos \\frac{17\\pi}{12} + \\cos \\frac{17\\pi}{12} \\cdot \\cos \\frac{\\pi}{12} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8c4580934208556bd615f0a25af667e581f3bf37c03bfdd07c38f59f4b80dff3 | Calculate \( R = \cos \frac{\pi}{12} \cdot \cos \frac{9\pi}{12} \cdot \cos \frac{17\pi}{12} \). | [
"\\(\\frac{1}{4\\sqrt{2}}\\)"
] | [
"Calculate \\( R = \\cos \\frac{\\pi}{12} \\cdot \\cos \\frac{9\\pi}{12} \\cdot \\cos \\frac{17\\pi}{12} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-d1267c3cc8d5afe680f1b5cf5ddf4e84ccd46b0283f21f6929fc14168f6889af | Calculate \( S = 1 + \frac{\sum_{n=2}^{\infty} \frac{n^2}{n!}}{e} \), where \( e = \sum_{m=0}^{\infty} \frac{1}{m!} \). | [
"\\( 3 - \\frac{1}{e} \\)"
] | [
"Calculate \\( S = 1 + \\frac{\\sum_{n=2}^{\\infty} \\frac{n^2}{n!}}{e} \\), where \\( e = \\sum_{m=0}^{\\infty} \\frac{1}{m!} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-5d07947c3719dfcdc4e2a0c7a84abc10d21a1b4c5585905e02265155c9068d18 | Calculate \( S = 2 \cdot 3^{99} \binom{100}{1} + 2^3 \cdot 3^{97} \binom{100}{3} + \cdots + 2^{99} \cdot 3 \binom{100}{99} \). | [
"\\(\\frac{5^{100} - 1}{2}\\)"
] | [
"Calculate \\( S = 2 \\cdot 3^{99} \\binom{100}{1} + 2^3 \\cdot 3^{97} \\binom{100}{3} + \\cdots + 2^{99} \\cdot 3 \\binom{100}{99} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 10
} | 33 | 0 | 0 |
omr-ws-330c73a8084ffcfa469a6f7dea3fc94f52ea44498215d97ed5148bd39ad75029 | Calculate \( S = \sin \frac{2\pi}{7} + \sin \frac{4\pi}{7} + \sin \frac{6\pi}{7} \). | [
"\\dfrac{\\sqrt{7}}{2}"
] | [
"Calculate \\( S = \\sin \\frac{2\\pi}{7} + \\sin \\frac{4\\pi}{7} + \\sin \\frac{6\\pi}{7} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-a130aea48ee94c336614c08aa6502c5d8be8a7bb0000ff471c957ce7e628bec9 | Calculate \( S = \sum_{k=1}^n \dfrac{k}{(\sqrt{k}+\sqrt{k+1})(\sqrt{k+1}+\sqrt{k+2})(\sqrt{k+2}+\sqrt{k})} \). | [
"\\(\\frac{\\sqrt{2} - \\sqrt{n+2} + n(\\sqrt{2} - 1)}{2}\\)"
] | [
"Calculate \\( S = \\sum_{k=1}^n \\dfrac{k}{(\\sqrt{k}+\\sqrt{k+1})(\\sqrt{k+1}+\\sqrt{k+2})(\\sqrt{k+2}+\\sqrt{k})} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c97f79daaab9d0d6058b52495dfe5bac02e532a4655bd44b4b77e5a6e2001b69 | Calculate \( S(n, n) \) and \( S(p+1, p) \). | [
"\\( S(n, n) = 1 \\) and \\( S(p+1, p) = p \\)"
] | [
"Calculate \\( S(n, n) \\) and \\( S(p+1, p) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b762c2d65e2a48701ccd7c3dbc68c8312b8d5e9f60345c2d72657ab682efd1f4 | Calculate \( S_1 = \sum_{k=0}^{n-1} \sin \frac{2k^2\pi}{n} \). | [
"\\( \\frac{\\sqrt{n}}{2} \\left( \\cos \\frac{n\\pi}{2} - \\sin \\frac{n\\pi}{2} + 1 \\right) \\)"
] | [
"Calculate \\( S_1 = \\sum_{k=0}^{n-1} \\sin \\frac{2k^2\\pi}{n} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-9463788703a45f203b696b9ca3b8ccc00e19c5823893b49d872785ab5067173f | Calculate \( S_2 = \sum_{k=0}^{n-1} \cos \frac{2k^2\pi}{n} \). | [
"\\( \\frac{\\sqrt{n}}{2} \\left( \\cos \\frac{n\\pi}{2} + \\sin \\frac{n\\pi}{2} + 1 \\right) \\)"
] | [
"Calculate \\( S_2 = \\sum_{k=0}^{n-1} \\cos \\frac{2k^2\\pi}{n} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 3,
"tir": 2
} | 20 | 0 | 0 |
omr-ws-110cadd1989bfed572c476502b1b267ae21e66d376f2ba7226d9f1ed75e3e79c | Calculate \( S_k = \sum_{n=1}^{\infty} \frac{x^n}{(3n+k)!} \) for \( k = 0 \). | [
" \\frac{1}{3} \\left( e^{\\sqrt[3]{x}} + 2 e^{-\\frac{\\sqrt[3]{x}}{2}} \\cos\\left( \\frac{\\sqrt{3}}{2} \\sqrt[3]{x} \\right) \\right) - 1 "
] | [
"Calculate \\( S_k = \\sum_{n=1}^{\\infty} \\frac{x^n}{(3n+k)!} \\) for \\( k = 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-b4f67208b997332bf8ed09cc6dfd9bdad1472ff97e9eda1a2dd7e1e5b67d3f32 | Calculate \( S_k = \sum_{n=1}^{\infty} \frac{x^n}{(3n+k)!} \) for \( k = 1 \). | [
" \\frac{e^{\\sqrt[3]{x}} - e^{-\\frac{\\sqrt[3]{x}}{2}} \\left( \\cos\\left( \\frac{\\sqrt{3} \\sqrt[3]{x}}{2} \\right) - \\sqrt{3} \\sin\\left( \\frac{\\sqrt{3} \\sqrt[3]{x}}{2} \\right) \\right)}{3 \\sqrt[3]{x}} - 1 "
] | [
"Calculate \\( S_k = \\sum_{n=1}^{\\infty} \\frac{x^n}{(3n+k)!} \\) for \\( k = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-ea2cf5f2e1912b1a483c967576392caa9cf28605643294ac5cda6e16b8f51f29 | Calculate \( S_k = \sum_{n=1}^{\infty} \frac{x^n}{(3n+k)!} \) for \( k = 2 \). | [
" \\frac{1}{3 x^{2/3}} \\left( e^{x^{1/3}} - e^{-\\frac{x^{1/3}}{2}} \\left( \\cos\\left( \\frac{\\sqrt{3} x^{1/3}}{2} \\right) + \\sqrt{3} \\sin\\left( \\frac{\\sqrt{3} x^{1/3}}{2} \\right) \\right) \\right) - \\frac{1}{2} "
] | [
"Calculate \\( S_k = \\sum_{n=1}^{\\infty} \\frac{x^n}{(3n+k)!} \\) for \\( k = 2 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-e0ac707f0082c4b1b4acb31d13ce2d32be8f31b7d3375dfb79a908f77edd16b3 | Calculate \( S_n = \sum_{k=0}^n \left\lfloor \sqrt{{n \choose k}} \right\rfloor \). | [
"8"
] | [
"Calculate \\( S_n = \\sum_{k=0}^n \\left\\lfloor \\sqrt{{n \\choose k}} \\right\\rfloor \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-1268a80c1c95ca13f939bec31d8bd081f46edbf213d10f8a73ac7c6c8713c614 | Calculate \( S_n = \sum_{k=1}^{2^n} T(k) \), where \( T(k) \) is the number of all pairs "11" in the binary representation of \( k \). | [
"\\( S_n = 2^{n-2}(n-1) \\) for \\( n > 1 \\) and \\( S_1 = 0 \\)."
] | [
"Calculate \\( S_n = \\sum_{k=1}^{2^n} T(k) \\), where \\( T(k) \\) is the number of all pairs \"11\" in the binary representation of \\( k \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect",
"tir"
] | {
"genselect": 8,
"tir": 19
} | 27 | 0 | 0 |
omr-ws-6f21e36feb78d60f6a619351e72fa192faf8a4135e3cd282f3c24828917e5f9e | Calculate \( S_n = \sum_{r=0}^n 2^{r-2n} \binom{2n-r}{n} \). | [
"1"
] | [
"Calculate \\( S_n = \\sum_{r=0}^n 2^{r-2n} \\binom{2n-r}{n} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-61d673793a1969b036c1af069fbfcd7aa146e94778d252dce4bbe493b7a1b206 | Calculate \( S_n = \tan(x) + \frac{1}{2} \tan\left(\frac{x}{2}\right) + \cdots + \frac{1}{2^n} \tan\left(\frac{x}{2^n}\right) \). | [
"\\( S_n = \\frac{1}{2^n} \\cot\\left(\\frac{x}{2^n}\\right) - 2 \\cot(2x) \\)"
] | [
"Calculate \\( S_n = \\tan(x) + \\frac{1}{2} \\tan\\left(\\frac{x}{2}\\right) + \\cdots + \\frac{1}{2^n} \\tan\\left(\\frac{x}{2^n}\\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-b53619f06d1905ce60f72657c0c4e4c0bd834f978d09ee89d90bde33a601a1ff | Calculate \( S_n \) where \[ S_n = \binom{n}{1} - \left(1 + \frac{1}{2}\right) \binom{n}{2} + \left(1 + \frac{1}{2} + \frac{1}{3}\right) \binom{n}{3} - \cdots + (-1)^{n-1} \left(1 + \frac{1}{2} + \cdots + \frac{1}{n}\right) \binom{n}{n} \] | [
"\\(\\frac{1}{n}\\)"
] | [
"Calculate \\( S_n \\) where\n\\[ S_n = \\binom{n}{1} - \\left(1 + \\frac{1}{2}\\right) \\binom{n}{2} + \\left(1 + \\frac{1}{2} + \\frac{1}{3}\\right) \\binom{n}{3} - \\cdots + (-1)^{n-1} \\left(1 + \\frac{1}{2} + \\cdots + \\frac{1}{n}\\right) \\binom{n}{n} \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-69a0d123689c2cbb34eead8b58e5491c99379423d40be4f9b99dce2b9f0a488c | Calculate \( T_0 + T_1 + \cdots + T_G \mod 10000 \), where \( T_n \) is defined as a tower of powers of 3 with \( T_0 = 1 \), \( T_1 = 3 \), \( T_2 = 3^3 \), \( T_3 = 3^{3^3} \), and \( G \) is Graham's number. | [
"2626"
] | [
"Calculate \\( T_0 + T_1 + \\cdots + T_G \\mod 10000 \\), where \\( T_n \\) is defined as a tower of powers of 3 with \\( T_0 = 1 \\), \\( T_1 = 3 \\), \\( T_2 = 3^3 \\), \\( T_3 = 3^{3^3} \\), and \\( G \\) is Graham's number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 4
} | 21 | 0 | 0 |
omr-ws-e8c64cce60d6dba6e097a3cd14bb135f661044e2440a05e33704ead451a22954 | Calculate \( T_1 = \sum \frac{k!}{2010^k (k_1! \cdot k_2! \cdot \cdots \cdot k_n!)} \), where the sum is taken over all n-tuples of the numbers \( k_1, k_2, \ldots, k_n \in \mathbb{N} \cup \{0\} \) satisfying \( k_1 + 2 \cdot k_2 + 3 \cdot k_3 + \ldots + n \cdot k_n = n \) and \( k = k_1 + k_2 + \ldots + k_n \). | [
"\\(\\left(\\frac{2011}{2010}\\right)^{n+1}\\)"
] | [
"Calculate \\( T_1 = \\sum \\frac{k!}{2010^k (k_1! \\cdot k_2! \\cdot \\cdots \\cdot k_n!)} \\), where the sum is taken over all n-tuples of the numbers \\( k_1, k_2, \\ldots, k_n \\in \\mathbb{N} \\cup \\{0\\} \\) satisfying \\( k_1 + 2 \\cdot k_2 + 3 \\cdot k_3 + \\ldots + n \\cdot k_n = n \\) and \\( k = k_1 + k... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f535221afdf41a2eefd28807f122232ab0c4b3f19554884aa16e8869f02534b | Calculate \( T_2 = \sum \frac{(-1)^k k! \binom{\frac{1}{2010}}{1}^{k_1} \binom{\frac{1}{2010}}{2}^{k_2} \cdots \binom{\frac{1}{2010}}{n}^{k_n}}{k_1! \cdot k_2! \cdot \cdots \cdot k_n!} \), where the sum is taken over all n-tuples of the numbers \( k_1, k_2, \ldots, k_n \in \mathbb{N} \cup \{0\} \) satisfying \( k_1 + 2... | [
"\\dfrac{2011}{2 \\times 2010^2}"
] | [
"Calculate \\( T_2 = \\sum \\frac{(-1)^k k! \\binom{\\frac{1}{2010}}{1}^{k_1} \\binom{\\frac{1}{2010}}{2}^{k_2} \\cdots \\binom{\\frac{1}{2010}}{n}^{k_n}}{k_1! \\cdot k_2! \\cdot \\cdots \\cdot k_n!} \\), where the sum is taken over all n-tuples of the numbers \\( k_1, k_2, \\ldots, k_n \\in \\mathbb{N} \\cup \\{0\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-aa6c47eb72a90316040779baeb2fe7bc970ad02f3adc1943aa684c32cd2a55ce | Calculate \( V_n(R_2) = \int \int_{x_1^2 + x_2^2 + \ldots + x_n^2 \leq R^2, \cos(x_1 + x_2 + \ldots + x_n) \geq 0} \ldots \int dx_1 dx_2 \ldots dx_n \). | [
"\\frac{\\pi^{n/2} R^n}{2 \\Gamma\\left(\\frac{n}{2} + 1\\right)}"
] | [
"Calculate \\( V_n(R_2) = \\int \\int_{x_1^2 + x_2^2 + \\ldots + x_n^2 \\leq R^2, \\cos(x_1 + x_2 + \\ldots + x_n) \\geq 0} \\ldots \\int dx_1 dx_2 \\ldots dx_n \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-563c28825417b9b7631ec64f3a04e87ab0e7c238aba3f7d0f251a358a541febc | Calculate \( X = \cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right) \) in two different ways and conclude that \( X_{10}^{1} - X_{10}^{3} + \frac{1}{3} X_{10}^{5} = 2^4 \). | [
"16"
] | [
"Calculate \\( X = \\cos\\left(\\frac{\\pi}{4}\\right) + i \\sin\\left(\\frac{\\pi}{4}\\right) \\) in two different ways and conclude that \\( X_{10}^{1} - X_{10}^{3} + \\frac{1}{3} X_{10}^{5} = 2^4 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e60f7a9e129648b139fc31e74c1760d8bb0bd3756c01d0da6c88e1d70606e812 | Calculate \( \arcsin(i - 1) \). | [
"\\( 2\\pi k - \\tan^{-1} \\varphi^{-1/2} - i \\log (\\varphi - \\varphi^{1/2}) \\) or \\( \\pi(2k+1) + \\tan^{-1} \\varphi^{-1/2} - i \\log (\\varphi + \\varphi^{1/2}) \\)"
] | [
"Calculate \\( \\arcsin(i - 1) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b921bf6bbf6b16646af9187d5024a3e1e1d35ba576d3cd2247e3b82a61d6372a | Calculate \( \arctan \frac{7}{29} \). | [
"0.2368"
] | [
"Calculate \\( \\arctan \\frac{7}{29} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-1a65b12531022a9a576e4ec92d0899cc38319588bd82f715a47745676a0083e2 | Calculate \( \binom{3n+1}{1} + \binom{3n+1}{4} + \cdots + \binom{3n+1}{3n+1} \). | [
"\\(\\frac{2^{3n+1} + 2}{3}\\)"
] | [
"Calculate \\( \\binom{3n+1}{1} + \\binom{3n+1}{4} + \\cdots + \\binom{3n+1}{3n+1} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-83101028c06069675c46086f50e1da912fd2e01d81db7735319feaaa40673e82 | Calculate \( \cos(\cos^{-1}(0.3) + \sin^{-1}(0.4)) \). | [
"\\(-0.10661\\)",
"\\dfrac{3\\sqrt{21} - 2\\sqrt{91}}{50}"
] | [
"Calculate \\( \\cos(\\cos^{-1}(0.3) + \\sin^{-1}(0.4)) \\)."
] | [
"aops_c4_high_school_math",
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 21
} | 60 | 0 | 0 |
omr-ws-27a5dce1e77ffdd201036dcab260ec480132dd36ed5195c5a815c07ffc0bb1e7 | Calculate \( \cot{\frac{\pi}{48}} \). | [
"2 + \\sqrt{2} + \\sqrt{3} + \\sqrt{6} + \\sqrt{16 + 10\\sqrt{2} + 8\\sqrt{3} + 6\\sqrt{6}}"
] | [
"Calculate \\( \\cot{\\frac{\\pi}{48}} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-27c0691df8383ac23976dbf63ff0093942532df0b29d32f0a3db0b586802cccf | Calculate \( \cot{\frac{\pi}{64}} \). | [
"\\sqrt{2 + \\sqrt{2 + \\sqrt{2 + \\sqrt{2 + \\sqrt{2 + \\sqrt{2 + \\sqrt{2 + \\sqrt{2}}}}}}}}"
] | [
"Calculate \\( \\cot{\\frac{\\pi}{64}} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-5b9de179cbc9d61916e2ff45d5f2ba90727f88ef713ca271492a21c65268cec9 | Calculate \( \frac{739\% \text{ of } 383}{628} \). | [
"4.505"
] | [
"Calculate \\( \\frac{739\\% \\text{ of } 383}{628} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 2,
"tir": 8
} | 28 | 0 | 0 |
omr-ws-162f1ff231b660debeffb988b44eb47d96ca6865fd1a98b98a26e075d54ef01e | Calculate \( \frac{d}{d \epsilon } \left( \| \dot{ \mathbf{x} } ^{ \epsilon } \| ^{2} \right) \) where \( \dot{ \mathbf{x} } ^{ \epsilon } = \frac{ \mathbf{x}(t) + \epsilon \mathbf{z} (t) }{ \| \mathbf{x} (t) + \epsilon \mathbf{z} (t) \| } \) at the point where \( \epsilon = 0 \). | [
"0"
] | [
"Calculate \\( \\frac{d}{d \\epsilon } \\left( \\| \\dot{ \\mathbf{x} } ^{ \\epsilon } \\| ^{2} \\right) \\) where \\( \\dot{ \\mathbf{x} } ^{ \\epsilon } = \\frac{ \\mathbf{x}(t) + \\epsilon \\mathbf{z} (t) }{ \\| \\mathbf{x} (t) + \\epsilon \\mathbf{z} (t) \\| } \\) at the point where \\( \\epsilon = 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b582f3629089046de7aab3b0bc0400a5e2f26210db73fdc893e3b641b8f2847b | Calculate \( \iint_{a \le |z| \le b} (1 + f(z)z) \, dx \, dy \), where \( f \) is analytic on the disc \( D(0,R) \) with \( R > b > a > 0 \). | [
"\\(\\pi (b^2 - a^2)\\)"
] | [
"Calculate \\( \\iint_{a \\le |z| \\le b} (1 + f(z)z) \\, dx \\, dy \\), where \\( f \\) is analytic on the disc \\( D(0,R) \\) with \\( R > b > a > 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 16
} | 30 | 0 | 0 |
omr-ws-76c8e4f6f8272815140874e550db418ecb6136cabb3fdd85d47d3c4d324f03ee | Calculate \( \inf \{a + b + c : a, b, c > 0, \sin a \cdot \sin b \cdot \sin c = \frac{3}{\pi} abc \} \). | [
"\\dfrac{\\pi}{6}"
] | [
"Calculate \\( \\inf \\{a + b + c : a, b, c > 0, \\sin a \\cdot \\sin b \\cdot \\sin c = \\frac{3}{\\pi} abc \\} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-da41955b73c9601aadf91db278602ceefd9bff92d5f1fc14be3bbfa32dc376de | Calculate \( \int \int \mathbf{F} \cdot \mathbf{n} \, dS \) where \( \mathbf{F} = \alpha x^3 \mathbf{i} + \beta y^3 \mathbf{j} + \gamma z^3 \mathbf{k} \), \(\alpha, \beta, \gamma\) are constants, over the surface of a sphere of radius \( a \) centered at the origin. | [
"\\(\\frac{4\\pi a^5 (\\alpha + \\beta + \\gamma)}{5}\\)"
] | [
"Calculate \\( \\int \\int \\mathbf{F} \\cdot \\mathbf{n} \\, dS \\) where \\( \\mathbf{F} = \\alpha x^3 \\mathbf{i} + \\beta y^3 \\mathbf{j} + \\gamma z^3 \\mathbf{k} \\), \\(\\alpha, \\beta, \\gamma\\) are constants, over the surface of a sphere of radius \\( a \\) centered at the origin."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-c13d88a633ea64c8dc2935c53c311d6993bf3fa3d45025aaf96f03250d0132dd | Calculate \( \int _0 ^{ \pi } \left( \frac {\sin(nx)}{x} \right)^2 \, dx \) where \( n \in \mathbb{N}^* \). | [
"n \\cdot \\text{Si}(2n\\pi)"
] | [
"Calculate \\( \\int _0 ^{ \\pi } \\left( \\frac {\\sin(nx)}{x} \\right)^2 \\, dx \\) where \\( n \\in \\mathbb{N}^* \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-9d64534de6b4506cc34f5dc62bb6c6da2293fa33c5c77c23aa28dad9e11c3978 | Calculate \( \int _0 ^{ \pi } \left( \frac {\sin(nx)}{x} \right)^3 \, dx \) where \( n \in \mathbb{N}^* \). | [
"\\frac{3n^2}{8} \\left( 3\\,\\text{Si}(3n\\pi) - \\text{Si}(n\\pi) \\right)"
] | [
"Calculate \\( \\int _0 ^{ \\pi } \\left( \\frac {\\sin(nx)}{x} \\right)^3 \\, dx \\) where \\( n \\in \\mathbb{N}^* \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-011acdbbba168e2f2fe0d7653e3095b76b94763122fd1c7573566d847675fe80 | Calculate \( \int _0 ^{ \pi }\frac {\sin(nx)}{x} \, dx \) where \( n \in \mathbb{N}^* \). | [
"\\text{Si}(n\\pi)"
] | [
"Calculate \\( \\int _0 ^{ \\pi }\\frac {\\sin(nx)}{x} \\, dx \\) where \\( n \\in \\mathbb{N}^* \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a91c97704c588ca086a232f8b0af65db5c3c4e609795c024fa9658c7df373ea6 | Calculate \( \int_0^1 \frac{\ln (1-x) + \sum_{i=1}^n \frac{x^i}{i!}}{x^{n+1}} \, dx \), where \( n \) is a natural number. | [
" -\\dfrac{H_n}{n} "
] | [
"Calculate \\( \\int_0^1 \\frac{\\ln (1-x) + \\sum_{i=1}^n \\frac{x^i}{i!}}{x^{n+1}} \\, dx \\), where \\( n \\) is a natural number."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-11b3702d288b1ecdee255b2414c82de706fc7552a7cd24bda33f9c472d8b76e4 | Calculate \( \int_{-1}^{1} f(x) \, dx \) for the function \( f \) defined by \[ f(x) = \begin{cases} x^2 \cdot e^{x^3} & \text{if } x \in [-1,0] \\ x^3 \cdot e^{x^2} & \text{if } x \in (0,1] \end{cases} \] | [
"\\( \\frac{5}{6} - \\frac{1}{3e} \\)"
] | [
"Calculate \\( \\int_{-1}^{1} f(x) \\, dx \\) for the function \\( f \\) defined by \n\\[ f(x) = \\begin{cases} \nx^2 \\cdot e^{x^3} & \\text{if } x \\in [-1,0] \\\\ \nx^3 \\cdot e^{x^2} & \\text{if } x \\in (0,1] \n\\end{cases} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-24fd14a2c69f9dbda9bca02a84a3f337150fb2edde0dd3373cad2b5d15a33ed9 | Calculate \( \int_{-\infty }^{\infty } \frac{x^2\sin^2 x}{x^4+3x^2+2} \, dx \). | [
"\\(\\frac{1}{2} \\left(-1+\\sqrt{2}+\\frac{1}{e^2}-\\sqrt{2} e^{-2 \\sqrt{2}}\\right) \\pi\\)"
] | [
"Calculate \\( \\int_{-\\infty }^{\\infty } \\frac{x^2\\sin^2 x}{x^4+3x^2+2} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-e187ff24a36b08ea009a3bf80fb6e8cb853c87d96cfd45d82a01047484cb8c0d | Calculate \( \int_{-\infty}^{\infty} e^{-x^2} \cos(2x^2) \, dx \). | [
"\\sqrt{\\dfrac{\\pi(\\sqrt{5} + 1)}{10}}"
] | [
"Calculate \\( \\int_{-\\infty}^{\\infty} e^{-x^2} \\cos(2x^2) \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-75bcdd840fed96c9f1d55a927f25b608e8711d74e498b2c979fe2b90e7f0b981 | Calculate \( \int_{0}^{1} \left( \left\lfloor \frac{1}{x} \right\rfloor \right)^{-1} \, dx \). | [
"\\(\\frac{\\pi^2}{6} - 1\\)"
] | [
"Calculate \\( \\int_{0}^{1} \\left( \\left\\lfloor \\frac{1}{x} \\right\\rfloor \\right)^{-1} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-0d6c12be3e2389a64b5cd1c3d73c26e9975f138642dd0ba46d2abbf9f13ae50b | Calculate \( \int_{0}^{2\pi} \cos^n x \, dx \). | [
"\\( 0 \\) if \\( n \\) is odd, \\( 2 B\\left(\\frac{n+1}{2}, \\frac{1}{2}\\right) \\) if \\( n \\) is even."
] | [
"Calculate \\( \\int_{0}^{2\\pi} \\cos^n x \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 4
} | 7 | 0 | 0 |
omr-ws-e69f023963888ecb315565fba02ecce57396b0b57ae8989fbbad27097231d17d | Calculate \( \int_{0}^{2\pi} \sin^n x \, dx \). | [
"\\( 0 \\) if \\( n \\) is odd, \\( 2 B\\left(\\frac{n+1}{2}, \\frac{1}{2}\\right) \\) if \\( n \\) is even."
] | [
"Calculate \\( \\int_{0}^{2\\pi} \\sin^n x \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 6
} | 7 | 0 | 0 |
omr-ws-86253b7ad66df8b85035858f2ecea5464b52ecd6e91b1f98b3e1fde182e769bd | Calculate \( \int_{0}^{2} f(x) \, dx \) where \( f(x) = \begin{cases} \frac{1}{\lfloor \frac{1}{x} \rfloor} & \text{if } 0 < x \le 1 \\ 0 & \text{if } x = 0 \text{ or } x > 1 \end{cases} \). | [
"The integral diverges."
] | [
"Calculate \\( \\int_{0}^{2} f(x) \\, dx \\) where \\( f(x) = \\begin{cases} \\frac{1}{\\lfloor \\frac{1}{x} \\rfloor} & \\text{if } 0 < x \\le 1 \\\\ 0 & \\text{if } x = 0 \\text{ or } x > 1 \\end{cases} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc0e4066a0c8941f42b4668af56cf72e5d8cfe1e7b09e3d92a823a8aae90671a | Calculate \( \int_{0}^{2} f(x) \, dx \) where \( f(x) = \begin{cases} x + \lfloor x \rfloor & \text{if } x \in \mathbb{Q} \\ 0 & \text{if } x \in \mathbb{Q'} \end{cases} \). | [
"0"
] | [
"Calculate \\( \\int_{0}^{2} f(x) \\, dx \\) where \\( f(x) = \\begin{cases} x + \\lfloor x \\rfloor & \\text{if } x \\in \\mathbb{Q} \\\\ 0 & \\text{if } x \\in \\mathbb{Q'} \\end{cases} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2659cfbdfae1fa5700b316d76fccddcfedb342f22f5ae9df69b94fff89dd4735 | Calculate \( \int_{0}^{\frac{1}{2}} f^n(x) \, dx \) where \( f(x) = x(4x^2 - 6x + 3) \) and \( f^n \) denotes the \( n \)-fold composition of \( f \). | [
"\\dfrac{3^n}{4(3^n + 1)}"
] | [
"Calculate \\( \\int_{0}^{\\frac{1}{2}} f^n(x) \\, dx \\) where \\( f(x) = x(4x^2 - 6x + 3) \\) and \\( f^n \\) denotes the \\( n \\)-fold composition of \\( f \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-2fefdc091fff7b201d779833343f1033b05d6fa33269ecfc0fbdae074ec6b5fb | Calculate \( \int_{0}^{\pi} \left( (\cos x)^7 + (\sin \frac{x}{2})^2 + 1 \right) dx \). | [
"\\(\\frac{3\\pi}{2}\\)"
] | [
"Calculate \\( \\int_{0}^{\\pi} \\left( (\\cos x)^7 + (\\sin \\frac{x}{2})^2 + 1 \\right) dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-fa0a8d0168932cca4900dbc433ceba4e842afe8d985a1606f5a0ecbfbab50937 | Calculate \( \int_{\partial \Omega} \phi F \cdot n \, d\sigma \), where \( n \) is the unit normal vector, \( d\sigma \) denotes integration over \( \partial \Omega \), \( F_i = \frac{x_i}{(x_1^2 + x_2^2 + x_3^2)^{\frac{3}{2}}} = \frac{x_i}{r^3} \), and \( \phi(y_1, y_2, y_3) \) is a continuously differentiable functio... | [
"\\( 4\\pi \\)"
] | [
"Calculate \\( \\int_{\\partial \\Omega} \\phi F \\cdot n \\, d\\sigma \\), where \\( n \\) is the unit normal vector, \\( d\\sigma \\) denotes integration over \\( \\partial \\Omega \\), \\( F_i = \\frac{x_i}{(x_1^2 + x_2^2 + x_3^2)^{\\frac{3}{2}}} = \\frac{x_i}{r^3} \\), and \\( \\phi(y_1, y_2, y_3) \\) is a cont... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-30cc8b406fdc1e72f74093384a05509b5319eb61f6fb2d8381ff2e70355acde9 | Calculate \( \left(\tan\frac{\pi}{4} + 0.00023\right)^{10} \). | [
"1.41426511213649"
] | [
"Calculate \\( \\left(\\tan\\frac{\\pi}{4} + 0.00023\\right)^{10} \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23c2cccf4c22fb1e1370166d0caa973f1e3001fffb18540ad9023402618515a3 | Calculate \( \left\lfloor 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{8}} \right\rfloor \), where \( \left\lfloor x \right\rfloor \) denotes the floor function of \( x \). | [
"5"
] | [
"Calculate \\( \\left\\lfloor 1 + \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\cdots + \\frac{1}{\\sqrt{8}} \\right\\rfloor \\), where \\( \\left\\lfloor x \\right\\rfloor \\) denotes the floor function of \\( x \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f5ec7b30e0f1d3075871945beef9f6d776a82cc67d18b2a8b5f3ecab837879a9 | Calculate \( \left\lfloor \frac{(a^2+b^2+c^2)(a+b+c)}{a^3+b^3+c^3} \right\rfloor \), where \( a, b, c \) are the lengths of the sides of a triangle. | [
"2"
] | [
"Calculate \\( \\left\\lfloor \\frac{(a^2+b^2+c^2)(a+b+c)}{a^3+b^3+c^3} \\right\\rfloor \\), where \\( a, b, c \\) are the lengths of the sides of a triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fe6225ca3ea9ecfafbeb59364ea8619cba300391a55f1ad67d26986f2d0111dc | Calculate \( \left\{ \frac{m}{n} \right\} + \left\{ \frac{2m}{n} \right\} + \cdots + \left\{ \frac{(n-1)m}{n} \right\} \) where \( m, n \in \mathbb{N}^* \) and \( (m, n) = 1 \). Here, \( \{ a \} \) denotes the fractional part of \( a \). | [
"\\dfrac{n-1}{2}"
] | [
"Calculate \\( \\left\\{ \\frac{m}{n} \\right\\} + \\left\\{ \\frac{2m}{n} \\right\\} + \\cdots + \\left\\{ \\frac{(n-1)m}{n} \\right\\} \\) where \\( m, n \\in \\mathbb{N}^* \\) and \\( (m, n) = 1 \\). Here, \\( \\{ a \\} \\) denotes the fractional part of \\( a \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9d062e09e04b152e4a8c454795afbcf9b407273ad2b29d567a86ca29f84d85cb | Calculate \( \lim_{N \to \infty} \sum_{n=1}^{N} \left( \sum_{k=1}^{n} \frac{f(k)}{k} - nL \right) \), where \( f(k) \) is the greatest odd divisor of \( k \in \mathbb{N} \) and \( L \) is defined as in Problem 1. | [
"\\(\\infty\\)"
] | [
"Calculate \\( \\lim_{N \\to \\infty} \\sum_{n=1}^{N} \\left( \\sum_{k=1}^{n} \\frac{f(k)}{k} - nL \\right) \\), where \\( f(k) \\) is the greatest odd divisor of \\( k \\in \\mathbb{N} \\) and \\( L \\) is defined as in Problem 1."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-389ea178fc187cd8039a53699e4ecabded1773466fa741e8609dd29cee8cf4a0 | Calculate \( \lim_{\stackrel{h\to 0}{h>0}} \sum_{n=1}^{\infty } \frac{h}{1+n^2h^2} \). | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"Calculate \\( \\lim_{\\stackrel{h\\to 0}{h>0}} \\sum_{n=1}^{\\infty } \\frac{h}{1+n^2h^2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-25687bf151ec3b20e81e3dfac7677e8e9e5928bb921003b6691dd2c251cd367a | Calculate \( \lim_{m \to \infty} m \cdot I_m \). | [
"\\(\\ln(1 + \\sqrt{2})\\)"
] | [
"Calculate \\( \\lim_{m \\to \\infty} m \\cdot I_m \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3572bda560564e138ee10e51db1fe3333f12e21e5f2814b62d35e7987119f23b | Calculate \( \lim_{n \rightarrow \infty} a_n \int_{0}^{1}x^{2n}\sin\frac {\pi x}{2}dx \), where \( a_n = \sum_{k = 1}^{n}\sin \frac {\pi k}{2n} \). | [
"\\( \\frac{1}{\\pi} \\)"
] | [
"Calculate \\( \\lim_{n \\rightarrow \\infty} a_n \\int_{0}^{1}x^{2n}\\sin\\frac {\\pi x}{2}dx \\), where \\( a_n = \\sum_{k = 1}^{n}\\sin \\frac {\\pi k}{2n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-733e9487d28ecdd7ff3fcd04ee25998c92f74ed5459e3fb98883cce85fac4383 | Calculate \( \lim_{n \to \infty } n^{2}(a_{n}-b_{n}) \) where \( a_{n}=\sqrt{n+\sqrt{n-1+\cdots+\sqrt{2+\sqrt{1}}}} \) and \( b_{n}=\sqrt{1+\sqrt{2+\cdots+\sqrt{n-1+\sqrt{n}}}} \). | [
"\\dfrac{1}{8}"
] | [
"Calculate \\( \\lim_{n \\to \\infty } n^{2}(a_{n}-b_{n}) \\) where \\( a_{n}=\\sqrt{n+\\sqrt{n-1+\\cdots+\\sqrt{2+\\sqrt{1}}}} \\) and \\( b_{n}=\\sqrt{1+\\sqrt{2+\\cdots+\\sqrt{n-1+\\sqrt{n}}}} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-32afec813ae97bd98607d28cff1885153f327de3396feb977b99e03e8e1eb131 | Calculate \( \lim_{n \to \infty } n^{2}(e^{x_{n+1}} - e^{x_{n}}) \) where \( x_{n} = \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{3n} \). | [
"1"
] | [
"Calculate \\( \\lim_{n \\to \\infty } n^{2}(e^{x_{n+1}} - e^{x_{n}}) \\) where \\( x_{n} = \\frac{1}{n+1} + \\frac{1}{n+2} + \\cdots + \\frac{1}{3n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fc662736eae1eb27013102c4e24e7a5d7c526efdb09faa73f91629b1da7901f5 | Calculate \( \lim_{n \to \infty} \frac{n}{\ln n} (2 - x_n) \). | [
"\\(\\frac{\\pi^2}{6}\\)"
] | [
"Calculate \\( \\lim_{n \\to \\infty} \\frac{n}{\\ln n} (2 - x_n) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08a363ee129e5e8f13cb34d607467fed552ddef76db6512f322ca98c822f5900 | Calculate \( \lim_{n \to \infty} \int_{0}^{\pi} \frac{\sin x}{1 + \cos^2(nx)} \, dx \). | [
"\\(\\sqrt{2}\\)"
] | [
"Calculate \\( \\lim_{n \\to \\infty} \\int_{0}^{\\pi} \\frac{\\sin x}{1 + \\cos^2(nx)} \\, dx \\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 29
} | 52 | 0 | 0 |
omr-ws-fc9f813c88f77537302d44e27e5fedede2111b6008d1bd73d741b0fd81898a9e | Calculate \( \lim_{n \to \infty} \int_{a}^{b} f(x) \cos(nx) \, dx \) for a continuous function \( f: [a, b] \rightarrow \mathbb{R} \). | [
"0"
] | [
"Calculate \\( \\lim_{n \\to \\infty} \\int_{a}^{b} f(x) \\cos(nx) \\, dx \\) for a continuous function \\( f: [a, b] \\rightarrow \\mathbb{R} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-dd130eba4fc106c880c4e084c6b2a147f10a70f064e807116993a08a5841859c | Calculate \( \lim_{n \to \infty} \left( t(n) \frac{\ln n}{n} \right) \). | [
"1"
] | [
"Calculate \\( \\lim_{n \\to \\infty} \\left( t(n) \\frac{\\ln n}{n} \\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-230cb400a7d39454ae738b1ca40a57a069fa46f7103fc8b0f2d627a415c65df2 | Calculate \( \lim_{n \to \infty} \sin^{2008}(\pi \sqrt{n^2 - n}) \). | [
"1"
] | [
"Calculate \\( \\lim_{n \\to \\infty} \\sin^{2008}(\\pi \\sqrt{n^2 - n}) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-c766d571bacb1564a76f338ac75b32044c2e9d0cc009619e5b66f264b708dd39 | Calculate \( \lim_{n \to \infty} a_{n} \) where \( a_{n} = 2n \left( \frac{1}{n^{2}+2n} + \frac{1}{n^{2}+4n} + \ldots + \frac{1}{n^{2}+2n^{2}} \right) \). | [
"\\(\\ln 3\\)"
] | [
"Calculate \\( \\lim_{n \\to \\infty} a_{n} \\) where \\( a_{n} = 2n \\left( \\frac{1}{n^{2}+2n} + \\frac{1}{n^{2}+4n} + \\ldots + \\frac{1}{n^{2}+2n^{2}} \\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-5f893f3054d0c235206f225a83328b3ef02c13c42376aa8210e5a595f203ac8a | Calculate \( \lim_{n \to \infty} n I_n \) where \( I_n = \int_0^1 x^n \cdot \sqrt{x^2 + 3} \, \mathrm{dx} \). | [
"2"
] | [
"Calculate \\( \\lim_{n \\to \\infty} n I_n \\) where \\( I_n = \\int_0^1 x^n \\cdot \\sqrt{x^2 + 3} \\, \\mathrm{dx} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0c93191d9c9263049e08011eef81239a6dd6370bbbe301125d770f6c55505386 | Calculate \( \lim_{n \to \infty} n I_n \), where \( I_n = \int_0^{\pi/4} e^{\sin x \cos x} (\cos x - \sin x)^{2n} (\cos x + \sin x) \, dx \). | [
"\\dfrac{1}{2}"
] | [
"Calculate \\( \\lim_{n \\to \\infty} n I_n \\), where \\( I_n = \\int_0^{\\pi/4} e^{\\sin x \\cos x} (\\cos x - \\sin x)^{2n} (\\cos x + \\sin x) \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf8b93c6b7cfbedbaafc710295358d216c1e3f9011ffd301080e69bb82400a9d | Calculate \( \lim_{n \to \infty} n \left[ L - \int_0^1 \left( \frac{k}{\sqrt[n]{x} + k - 1} \right)^n \, dx \right] \) for \( k > 1 \). | [
"\\(\\dfrac{k}{(k-1)^2}\\)"
] | [
"Calculate \\( \\lim_{n \\to \\infty} n \\left[ L - \\int_0^1 \\left( \\frac{k}{\\sqrt[n]{x} + k - 1} \\right)^n \\, dx \\right] \\) for \\( k > 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 6
} | 19 | 0 | 0 |
omr-ws-425bb9bc868ec5e91d121da543e1a8bf1ebc18df209aae30a7045abf17839635 | Calculate \( \lim_{n\rightarrow \infty} \left( \frac{\sqrt{n}+\sqrt{n+2}}{2\sqrt{n+1}} \right)^n \). | [
"1"
] | [
"Calculate \\( \\lim_{n\\rightarrow \\infty} \\left( \\frac{\\sqrt{n}+\\sqrt{n+2}}{2\\sqrt{n+1}} \\right)^n \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-eba33afd8a1669ef00d736997ede333585127ed15e1e0a3210c8c26e4ac5eb20 | Calculate \( \lim_{n\rightarrow \infty} \sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2+\sqrt{3}}}}}, \) where there are \( n \) radicals. | [
"0"
] | [
"Calculate \\( \\lim_{n\\rightarrow \\infty} \\sqrt{2-\\sqrt{2+\\sqrt{2+\\cdots+\\sqrt{2+\\sqrt{3}}}}}, \\) where there are \\( n \\) radicals."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-d0c759e0c7ef03715d7a1bb0ea6e828de9db3c17072c0316bac9a661bfd0c5a3 | Calculate \( \lim_{n\rightarrow\infty}\sum_{k=1}^{n} \frac{1}{k}\left(\frac{1}{2^k}-\frac{1}{5^k}\right) \). | [
"\\ln \\frac{8}{5}"
] | [
"Calculate \\( \\lim_{n\\rightarrow\\infty}\\sum_{k=1}^{n} \\frac{1}{k}\\left(\\frac{1}{2^k}-\\frac{1}{5^k}\\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d88f99807a0a784174cc3707d05ecc7d4f61aabc0703fd6940b42138bb8cd5a | Calculate \( \lim_{n\to\infty } \frac{\ln a_n}{n} \). | [
"\\(\\sqrt{2}\\)"
] | [
"Calculate \\( \\lim_{n\\to\\infty } \\frac{\\ln a_n}{n} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-382b0a7e86a0e0de7471309ab7a2f1c9d0e6e21fdbac0fe0b4d95b52d7dd146d | Calculate \( \lim_{n\to\infty } \frac{f(1)+(f(2))^2+\cdots +(f(n))^n}{(f(n))^n} \), where \( f:\mathbb{R}\longrightarrow\mathbb{R}_{>0} \) is an unbounded and nondecreasing function. | [
"1"
] | [
"Calculate \\( \\lim_{n\\to\\infty } \\frac{f(1)+(f(2))^2+\\cdots +(f(n))^n}{(f(n))^n} \\), where \\( f:\\mathbb{R}\\longrightarrow\\mathbb{R}_{>0} \\) is an unbounded and nondecreasing function."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9c952358a9475ca16cdef941611d4bc2ed45c95ae433c5827ae1495f6a3db5c9 | Calculate \( \lim_{n\to\infty }\left(\lim_{x\to 0} \left( -\frac{n}{x}+1+\frac{1}{x}\sum_{r=2}^{n+1}\sqrt[r!]{1+\sin rx}\right)\right) . \) | [
"e"
] | [
"Calculate \\( \\lim_{n\\to\\infty }\\left(\\lim_{x\\to 0} \\left( -\\frac{n}{x}+1+\\frac{1}{x}\\sum_{r=2}^{n+1}\\sqrt[r!]{1+\\sin rx}\\right)\\right) . \\)"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2b5e105d57fb5b4931d4d50199c4a42f61fe4af71ad287e6800538e4a960389a | Calculate \( \lim_{n\to\infty }\sqrt[n]{\binom{kn}{n}} \), where \( k \) is an integer greater than 1. | [
"\\(\\frac{k^k}{(k-1)^{k-1}}\\)"
] | [
"Calculate \\( \\lim_{n\\to\\infty }\\sqrt[n]{\\binom{kn}{n}} \\), where \\( k \\) is an integer greater than 1."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
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