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-05794cfbc8b27c1d8970ee02a426fbc3db2e2c4db1d497b697cbfdb6f2a811fd | Calculate \( \lim_{n\to\infty }\sum_{t=1}^n\frac{1}{n+t+\sqrt{n^2+nt}} \). | [
"\\(2\\ln \\left(\\frac{\\sqrt{2}+1}{2}\\right)\\)"
] | [
"Calculate \\( \\lim_{n\\to\\infty }\\sum_{t=1}^n\\frac{1}{n+t+\\sqrt{n^2+nt}} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ecf22cc155f749a43f1fd28963022d73b71ecd3d22975acba4f293ff96c73464 | Calculate \( \lim_{n\to\infty} J_n \sqrt{n} \), where \( J_n = \int_{(1+n)^2}^{1+(1+n)^2} \sqrt{\frac{x-1-n-n^2}{x-1}} \, dx \). | [
"1"
] | [
"Calculate \\( \\lim_{n\\to\\infty} J_n \\sqrt{n} \\), where \\( J_n = \\int_{(1+n)^2}^{1+(1+n)^2} \\sqrt{\\frac{x-1-n-n^2}{x-1}} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5291a6a07fccfd995e3dd37a3e10ce89089076d586dd8f1e6e1770a207617b7 | Calculate \( \lim_{n\to\infty} \frac{a^n}{(1+a)(1+a^2)\cdots (1+a^n)} \) where \( a \) is a positive real number. | [
"0"
] | [
"Calculate \\( \\lim_{n\\to\\infty} \\frac{a^n}{(1+a)(1+a^2)\\cdots (1+a^n)} \\) where \\( a \\) is a positive real number."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c5f3dd0047c8ade77b3752b5293d576d01c4474fe2b78a398593e477a0b321b3 | Calculate \( \lim_{n\to\infty} \int_0^{1} \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx \). | [
"1"
] | [
"Calculate \\( \\lim_{n\\to\\infty} \\int_0^{1} \\ln \\left( 1+x+x^2+\\cdots +x^{n-1} \\right) dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a482ea5366bb5198e282c9f0cc04d8ec9f2ac6bed8586b29257a18b9dd0c95e8 | Calculate \( \lim_{n\to\infty} \int_0^{1} x^n \cdot \cos x \, \mathrm{dx} \). | [
"0"
] | [
"Calculate \\( \\lim_{n\\to\\infty} \\int_0^{1} x^n \\cdot \\cos x \\, \\mathrm{dx} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-24032286611603e2c32984ca15de411b41790cc3640c20db100e5e7ba8937427 | Calculate \( \lim_{n\to\infty} \int_0^{\alpha} \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx \) for all \( \alpha \in (0,1) \). | [
"\\( a + (1-a) \\ln(1-a) \\)"
] | [
"Calculate \\( \\lim_{n\\to\\infty} \\int_0^{\\alpha} \\ln \\left( 1+x+x^2+\\cdots +x^{n-1} \\right) dx \\) for all \\( \\alpha \\in (0,1) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-95835f17cf4b9d2859b381266fd9eb6cea40380ed6d384e22c44f0ad42fcea81 | Calculate \( \lim_{n\to\infty} \left( \frac{1 + \tan\frac{\pi}{2n}}{1 + \sin\frac{\pi}{3n}} \right)^n \). | [
"\\( e^{\\pi/6} \\)"
] | [
"Calculate \\( \\lim_{n\\to\\infty} \\left( \\frac{1 + \\tan\\frac{\\pi}{2n}}{1 + \\sin\\frac{\\pi}{3n}} \\right)^n \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-278b4a3e16ed233a3ebdc100d151e70ed3c9617a35d011cedff69f428a94ca63 | Calculate \( \lim_{n\to\infty}\frac{x_n}{x_{n+1}} \) for the sequence \( \{x_n\} \) defined by: \[ x_1 = 2 \] \[ x_{n+1} = x_n^2 - x_n + 1 \] | [
"0"
] | [
"Calculate \\( \\lim_{n\\to\\infty}\\frac{x_n}{x_{n+1}} \\) for the sequence \\( \\{x_n\\} \\) defined by:\n\\[ x_1 = 2 \\]\n\\[ x_{n+1} = x_n^2 - x_n + 1 \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-fab2a9a16445a5e055a1e1e87cbebf09340aa64e2ab6dd0280c47415f4807aec | Calculate \( \lim_{n\to\infty}\sum_{k=1}^{n}{\sum_{p=1}^{n}\frac{k^2+p\cdot{k}+p}{k\cdot{(k+1)}\cdot{(k+p)!}}} \). | [
"\\( e \\)"
] | [
"Calculate \\( \\lim_{n\\to\\infty}\\sum_{k=1}^{n}{\\sum_{p=1}^{n}\\frac{k^2+p\\cdot{k}+p}{k\\cdot{(k+1)}\\cdot{(k+p)!}}} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a3f07fad0b94fedba52296c91329302ac097403e7ef8f4e67c8c116e9443bcf3 | Calculate \( \lim_{p \to +\infty} S_p \) where \( S_p = \lim_{n \to +\infty} \sum_{k=1}^{n} \left( \sqrt{1 + \frac{k}{n^2}} + \sqrt[3]{1 + \frac{k^2}{n^3}} + \cdots + \sqrt[p]{1 + \frac{k^{p-1}}{n^p}} + \sqrt[p+1]{1 + \frac{k^p}{n^{p+1}}} - p \right) \). | [
"\\(\\frac{\\pi^2}{6} - 1\\)"
] | [
"Calculate \\( \\lim_{p \\to +\\infty} S_p \\) where \\( S_p = \\lim_{n \\to +\\infty} \\sum_{k=1}^{n} \\left( \\sqrt{1 + \\frac{k}{n^2}} + \\sqrt[3]{1 + \\frac{k^2}{n^3}} + \\cdots + \\sqrt[p]{1 + \\frac{k^{p-1}}{n^p}} + \\sqrt[p+1]{1 + \\frac{k^p}{n^{p+1}}} - p \\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-1decdf53bde5fe99070cc4d4d3ce1a30a3a53f08a888904a7c43cf1ae443cf4d | Calculate \( \lim_{t \to \infty} t \int_1^a \frac{dx}{1 + x^t} \). | [
"\\(\\ln 2\\)."
] | [
"Calculate \\( \\lim_{t \\to \\infty} t \\int_1^a \\frac{dx}{1 + x^t} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-47c55f57549b3ef88c6b51fcf3297f8b533fb8e573e44b735db8169197b2e359 | Calculate \( \lim_{x \rightarrow 0-} (1 - 2^x)^{\sin x} \). | [
"1"
] | [
"Calculate \\( \\lim_{x \\rightarrow 0-} (1 - 2^x)^{\\sin x} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0e7747b1a860036d0f161e0cef64f16132b96526db0af0ee99f8c0b1ee9d0a73 | Calculate \( \lim_{x \rightarrow 0} \left( \frac{1}{\log (x + \sqrt{1 + x^2})} - \frac{1}{\log (1 + x)} \right) \). | [
"\\(-\\frac{1}{2}\\)"
] | [
"Calculate \\( \\lim_{x \\rightarrow 0} \\left( \\frac{1}{\\log (x + \\sqrt{1 + x^2})} - \\frac{1}{\\log (1 + x)} \\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-79ab14fdc821e7a6d72761a9cd5a606b2782a95990d1cc99da8553374509e97d | Calculate \( \lim_{x \rightarrow \dfrac{\pi}{4}} f(x) \) when \( f(x) = \dfrac{\tan\left(\dfrac{\pi}{3}-x\right) \cdot \tan\left(\dfrac{\pi}{3}+x\right)}{\tan\left(\dfrac{\pi}{2}-x\right)} \). | [
"-1"
] | [
"Calculate \\( \\lim_{x \\rightarrow \\dfrac{\\pi}{4}} f(x) \\) when \\( f(x) = \\dfrac{\\tan\\left(\\dfrac{\\pi}{3}-x\\right) \\cdot \\tan\\left(\\dfrac{\\pi}{3}+x\\right)}{\\tan\\left(\\dfrac{\\pi}{2}-x\\right)} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-79821e73d584a815b47caf61bf809b1b1df50a951f88088b3010b3d95311f793 | Calculate \( \lim_{x \rightarrow \infty} \frac{f^{-1}(x)}{x} \) for the function \( f(x) = 2^x + x - 1 \). | [
"0"
] | [
"Calculate \\( \\lim_{x \\rightarrow \\infty} \\frac{f^{-1}(x)}{x} \\) for the function \\( f(x) = 2^x + x - 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f8174f0bbabceca0ced0d2a2d8af5bd45c0e279c40ab14860e601b8bc7cee457 | Calculate \( \lim_{x \to 0^+} (x^{x^x} - x^x) \) | [
"-1"
] | [
"Calculate \\( \\lim_{x \\to 0^+} (x^{x^x} - x^x) \\)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-aaa78849f01d899a174fd104019a96f46ed7666b7b115ee9612e95617aec761d | Calculate \( \lim_{x \to 0} \frac{\sin^3(x) + \sin^3(2x) + \sin^3(3x)}{\sin^4(x) + \sin^4(2x) + \sin^4(3x)} \). | [
"\\infty"
] | [
"Calculate \\( \\lim_{x \\to 0} \\frac{\\sin^3(x) + \\sin^3(2x) + \\sin^3(3x)}{\\sin^4(x) + \\sin^4(2x) + \\sin^4(3x)} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a8cd3a770827c919c05dbc273ce0bbbd0cb0eaa1b413cf88e742b39b34f7925 | Calculate \( \lim_{x \to 1^{-}} \frac{f(x)}{\ln^2(x)} \) where \( f(x) = 1 - x \). | [
"\\( +\\infty \\)"
] | [
"Calculate \\( \\lim_{x \\to 1^{-}} \\frac{f(x)}{\\ln^2(x)} \\) where \\( f(x) = 1 - x \\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-1f9ab16b77f67eff5b9a1a9f6b6f818b5aaa456a587b6f062b7d2fbb97c2a1bd | Calculate \( \lim_{x \to \infty} \left( (x + 3)^{1 + \frac{1}{x}} - x^{1 + \frac{1}{x + 3}} \right) \). | [
"3"
] | [
"Calculate \\( \\lim_{x \\to \\infty} \\left( (x + 3)^{1 + \\frac{1}{x}} - x^{1 + \\frac{1}{x + 3}} \\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50fb091f8e788cd69e32ab16d137e986012d57b1e474a5df3c8af057f641a186 | Calculate \( \lim_{x \to \infty} \left( \sqrt[3]{x^3 + x^2 + x + 1} - \sqrt{x^2 + x + 1} \frac{\ln(e^x + x)}{x} \right) \). | [
"\\(-\\frac{1}{6}\\)"
] | [
"Calculate \\( \\lim_{x \\to \\infty} \\left( \\sqrt[3]{x^3 + x^2 + x + 1} - \\sqrt{x^2 + x + 1} \\frac{\\ln(e^x + x)}{x} \\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-7d21218c2446d8459d569c241fff4a9086e574046ae4985aadfc7d4ecf896e41 | Calculate \( \lim_{x\to0}\frac{\int^{x^{2}}_{0}\ln(1+t+t^{2})\,dt}{\int^{x}_{0}\ln(1+t^{3})\,dt} \). | [
"2"
] | [
"Calculate \\( \\lim_{x\\to0}\\frac{\\int^{x^{2}}_{0}\\ln(1+t+t^{2})\\,dt}{\\int^{x}_{0}\\ln(1+t^{3})\\,dt} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f4f07b97b1b36c46fd66b14a4396715b2d11ff638e3c8f5eb87529ab1ef0927 | Calculate \( \mbox{Hom}(\mathbb{Z}_n, \mathbb{Z}) \) for any \( n \). | [
"\\( \\mbox{Hom}(\\mathbb{Z}_n, \\mathbb{Z}) = 0 \\)"
] | [
"Calculate \\( \\mbox{Hom}(\\mathbb{Z}_n, \\mathbb{Z}) \\) for any \\( n \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-ccc31b4062ab905071b0d4b7bb7dff465e96c277405d87a517ca1b7fe8ae0a2c | Calculate \( \sin 20^\circ \cdot \sin 40^\circ \cdot \sin 60^\circ \cdot \sin 80^\circ \). | [
"\\(\\frac{3}{16}\\)"
] | [
"Calculate \\( \\sin 20^\\circ \\cdot \\sin 40^\\circ \\cdot \\sin 60^\\circ \\cdot \\sin 80^\\circ \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-4d3d15c103b1dac7e5044c341cf8340df0dd3c95be6a518729c0a839e2d0d286 | Calculate \( \sum_{\sigma \in \sigma_n} \frac{(-1)^{I(\sigma)}}{(v(\sigma) + 1)(I(\sigma) + \pi(\sigma) + 1)} \). Here, \( \sigma_n \) denotes the set of all permutations of the numbers \( 1, 2, 3, \ldots, n \). For a permutation \( \sigma \in \sigma_n \): - \( v(\sigma) \) denotes the number of fixed points of \( \sig... | [
" \\dfrac{ (-1)^{n+1} \\cdot 2n }{ (n + 1)(n^2 - n + 2) } "
] | [
"Calculate \\( \\sum_{\\sigma \\in \\sigma_n} \\frac{(-1)^{I(\\sigma)}}{(v(\\sigma) + 1)(I(\\sigma) + \\pi(\\sigma) + 1)} \\).\n\nHere, \\( \\sigma_n \\) denotes the set of all permutations of the numbers \\( 1, 2, 3, \\ldots, n \\). For a permutation \\( \\sigma \\in \\sigma_n \\):\n- \\( v(\\sigma) \\) denotes th... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-2493b7e049ebcd8994dd5e3ef538abef412f94e7abd1aeb88139c6ec0388a680 | Calculate \( \sum_{\sigma \in \sigma_n} \frac{(-1)^{I(\sigma)}}{I(\sigma) + \pi(\sigma) + 1} \). | [
"\\(\\frac{2}{n + 1}\\)"
] | [
"Calculate \\( \\sum_{\\sigma \\in \\sigma_n} \\frac{(-1)^{I(\\sigma)}}{I(\\sigma) + \\pi(\\sigma) + 1} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc169843f84231fdc243efbf120ba8dc88a32d128c6eb1dd14bf7cb0fc716d4c | Calculate \( \sum_{\sigma\in S_n }\sum_{i=1}^n |i-\sigma (i)| \), where \( n \ge 3 \) is a natural number. | [
"\\(\\frac{n! (n+1)}{3}\\)"
] | [
"Calculate \\( \\sum_{\\sigma\\in S_n }\\sum_{i=1}^n |i-\\sigma (i)| \\), where \\( n \\ge 3 \\) is a natural number."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-93ca4ef6e0407a73a217205e5fc2a721420d64e916be192a73f319a9bed60788 | Calculate \( \sum_{j=0}^{\infty} \frac{4^j}{\binom{2j}{j}} \) | [
"\\infty"
] | [
"Calculate \\( \\sum_{j=0}^{\\infty} \\frac{4^j}{\\binom{2j}{j}} \\)"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-98b743a127b2cc6bb61cb906c00b2f6d19b0328914ec2774057941ce12d86e0e | Calculate \( \sum_{j=1}^n (-1)^{j-1} j^2 \). | [
"\\( (-1)^{n-1}\\frac{n(n+1)}{2} \\)"
] | [
"Calculate \\( \\sum_{j=1}^n (-1)^{j-1} j^2 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-1fea744ee847b0f97f73e1b8f0265cc832733ba4a436b8f70a091c9ab1fedf1a | Calculate \( \sum_{k=0}^m \binom{s-1}{m-k} \binom{n-s}{k-1} \), where \( n > s > m \). | [
"\\( \\binom{n}{m} \\)"
] | [
"Calculate \\( \\sum_{k=0}^m \\binom{s-1}{m-k} \\binom{n-s}{k-1} \\), where \\( n > s > m \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb659acc58aff58cb38bb27b1291521b5955238e970ed9c15298a6d7d875cfdc | Calculate \( \sum_{k=0}^n (-1)^k \binom{n}{k} (n - 2k)^m \) for \( m = n \). | [
"\\( 2^n n! \\)"
] | [
"Calculate \\( \\sum_{k=0}^n (-1)^k \\binom{n}{k} (n - 2k)^m \\) for \\( m = n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-47e85377c6478f78aa9ebfd9c236ac3fff7ca4ede836b39ab205b4393a687a57 | Calculate \( \sum_{k=0}^n 2^k \cdot \binom{n}{2n-k} \). | [
"\\( 2^{2n} \\)"
] | [
"Calculate \\( \\sum_{k=0}^n 2^k \\cdot \\binom{n}{2n-k} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a2ec4ab0e1ae624222cbded41c34e68a95febddbda4784a539ba57b68ebfecf | Calculate \( \sum_{k=0}^{n-1} {2n \choose 2k+1} {k \choose j} \). | [
"\\( 2^{n - 2j - 1} \\binom{n - j - 1}{j} \\)"
] | [
"Calculate \\( \\sum_{k=0}^{n-1} {2n \\choose 2k+1} {k \\choose j} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-638771b431474e4a1c18a4dffdbe4aa54201cc9611c9f61f152c45354a54b243 | Calculate \( \sum_{k=1}^{2n-1} \frac{(-1)^{k-1}}{\binom{2n}{k}} \). | [
"\\dfrac{1}{n+1}"
] | [
"Calculate \\( \\sum_{k=1}^{2n-1} \\frac{(-1)^{k-1}}{\\binom{2n}{k}} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-eaaf8f3d993a00c6a0dda72347498033e0af93fc8020b48f5e78c5b287e5890d | Calculate \( \sum_{k=1}^{n} \binom{n}{k}^2 \cos(kx) \) where \( x \in [0, \pi] \). | [
"\\binom{2n}{n} \\cos^n\\left(\\frac{x}{2}\\right) P_n\\left(\\cos\\left(\\frac{x}{2}\\right)\\right) - 1"
] | [
"Calculate \\( \\sum_{k=1}^{n} \\binom{n}{k}^2 \\cos(kx) \\) where \\( x \\in [0, \\pi] \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-611246e9fc4fcf06b1a12c1c748231406ad76e97dc6e4da8266d4a37c44d9d3a | Calculate \( \sum_{n = 1}^{\infty}\frac{\cos n}{n^2 + 1} \). | [
"\\(\\frac{\\pi}{2}\\left[\\coth(\\pi) \\cosh(1) - \\sinh(1)\\right] - \\frac{1}{2}\\)"
] | [
"Calculate \\( \\sum_{n = 1}^{\\infty}\\frac{\\cos n}{n^2 + 1} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-577c718bc053f67097eacaf276af77a437f87f90b7498c7dfece217b54a146e6 | Calculate \( \sum_{n,k=0}^{\infty} \frac{1}{n!k!(n+k+1)} \). | [
"\\(\\frac{e^2 - 1}{2}\\)"
] | [
"Calculate \\( \\sum_{n,k=0}^{\\infty} \\frac{1}{n!k!(n+k+1)} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-fd9452c659b7b3427effe7ada97628cac7a87cb94250a078062f909363328924 | Calculate \( \sum_{n,k=0}^{\infty} \frac{n!k!}{(n+k+2)!} \). | [
"\\(\\frac{\\pi^2}{6}\\)"
] | [
"Calculate \\( \\sum_{n,k=0}^{\\infty} \\frac{n!k!}{(n+k+2)!} \\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 8
} | 18 | 0 | 0 |
omr-ws-2a42102682b9c0e0f30091d9c23b68b17bbc57465ab79840d4421d71f9971378 | Calculate \( \sum_{n=1, \ n \ \text{odd}}^{\infty} \frac{n^2}{(n^2-4)^2} \) | [
"\\( \\frac{\\pi^2}{16} \\)"
] | [
"Calculate \\( \\sum_{n=1, \\ n \\ \\text{odd}}^{\\infty} \\frac{n^2}{(n^2-4)^2} \\)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 19
} | 33 | 0 | 0 |
omr-ws-5edcac2b4741acdf3c693c6bccc2c36baa65f4e58368205e9c73f0f50e5e581a | Calculate \( \sum_{n=1}^{+\infty} \arctan{\frac{8n}{n^4 - 2n^2 + 5}} \). | [
"\\( 2\\arctan{2} + \\arctan{\\frac{1}{2}} \\)"
] | [
"Calculate \\( \\sum_{n=1}^{+\\infty} \\arctan{\\frac{8n}{n^4 - 2n^2 + 5}} \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-6ad4f391d70db04a690c85e8b9e21892004567c5e24b1d55b0c38f8abc8eb65a | Calculate \( \sum_{n=1}^{\infty} \frac{1}{n!} \cdot \sum_{k=0}^{\infty} \frac{{(-1)}^k}{k!} \). | [
"\\( 1 - \\frac{1}{e} \\)"
] | [
"Calculate \\( \\sum_{n=1}^{\\infty} \\frac{1}{n!} \\cdot \\sum_{k=0}^{\\infty} \\frac{{(-1)}^k}{k!} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-e4ed411dc0c06d8ea1825b8c154b626d5304f93f0a8c0e1d66af1cbde6e15d78 | Calculate \( \sum_{n=4}^{\infty} I_n \) where \( I_n = \sum_{k=2}^{n-2} \frac{1}{k \binom{n}{k}} \). | [
"\\(\\frac{1}{2}\\)"
] | [
"Calculate \\( \\sum_{n=4}^{\\infty} I_n \\) where \\( I_n = \\sum_{k=2}^{n-2} \\frac{1}{k \\binom{n}{k}} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 5,
"tir": 8
} | 21 | 0 | 0 |
omr-ws-b6fc086cb2bd0f383513fac69effb8b11b77037a3cccf19bd97799cdb1cdcf5b | Calculate \( \sum_{r=0}^{m} \left( \frac{r}{2} \right) \binom{m}{r} \) in terms of \( m \). | [
"\\( m(m-1) \\cdot 2^{m-3} \\)"
] | [
"Calculate \\( \\sum_{r=0}^{m} \\left( \\frac{r}{2} \\right) \\binom{m}{r} \\) in terms of \\( m \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-229c8acfc218d21d139e106de8c05bb1aef2e5571474a8b4ed42b410d5a30851 | Calculate \( \sum_{x=0}^{p-1} \left( \frac{ax^3 + bx^2 + cx + d}{p} \right) \), where \( \left( \frac{\cdot}{p} \right) \) denotes the Legendre symbol. | [
"0"
] | [
"Calculate \\( \\sum_{x=0}^{p-1} \\left( \\frac{ax^3 + bx^2 + cx + d}{p} \\right) \\), where \\( \\left( \\frac{\\cdot}{p} \\right) \\) denotes the Legendre symbol."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-e3742f51305e7aa1c05004d7413dd1cc55ade009307f0feebc37792eb052f70b | Calculate \( \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 60^\circ \cdot \tan 80^\circ \). | [
"3"
] | [
"Calculate \\( \\tan 20^\\circ \\cdot \\tan 40^\\circ \\cdot \\tan 60^\\circ \\cdot \\tan 80^\\circ \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a49ac9c402998b0be19234c7dacf74755f74128c3c0b517a477047ef03aa60d | Calculate \( \tan \left(\frac{3 \pi}{11}\right) + 4 \sin \left(\frac{2 \pi}{11}\right) \). | [
"\\sqrt{11}"
] | [
"Calculate \\( \\tan \\left(\\frac{3 \\pi}{11}\\right) + 4 \\sin \\left(\\frac{2 \\pi}{11}\\right) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e7ba39c09d7f1730313249c83265a445bb03c5f25a7d54260acb726d289768a8 | Calculate \( \tan\left(\arctan \frac{1}{2} + \arctan \frac{1}{8} + \cdots + \arctan \frac{1}{2n^2}\right) \). | [
"\\(\\frac{n}{n+1}\\)"
] | [
"Calculate \\( \\tan\\left(\\arctan \\frac{1}{2} + \\arctan \\frac{1}{8} + \\cdots + \\arctan \\frac{1}{2n^2}\\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-2e5a02b85dd892d3ebe27d0280597a615f0a7324c7f5090ab3df775f8eff8e83 | Calculate \( \tan\left(\frac{3\pi}{11}\right) + 4 \sin\left(\frac{2\pi}{11}\right) \). | [
"\\(\\sqrt{11}\\)"
] | [
"Calculate \\( \\tan\\left(\\frac{3\\pi}{11}\\right) + 4 \\sin\\left(\\frac{2\\pi}{11}\\right) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-612b9123fec8cf6da78cf9e2b146e1066f9ca2cfc333913298af6c0107f3cd07 | Calculate \( \text{Var}(X) \) given that \( \text{Var}(X) = \text{Cov}(X,X) \). | [
"\\(\\text{Cov}(X,X)\\)"
] | [
"Calculate \\( \\text{Var}(X) \\) given that \\( \\text{Var}(X) = \\text{Cov}(X,X) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-86ee47956517b63fb183bdebc82e3019fc13a3dcca320e542f22f5cc8eb8f041 | Calculate \( \| \phi(x) \|_{X'} \) for \( x = (\alpha_j) \in l^{\infty} \) and hence find a norm on \( l^{\infty} \) such that \( \phi \) becomes an isometry. | [
"\\|x\\|_{\\infty}"
] | [
"Calculate \\( \\| \\phi(x) \\|_{X'} \\) for \\( x = (\\alpha_j) \\in l^{\\infty} \\) and hence find a norm on \\( l^{\\infty} \\) such that \\( \\phi \\) becomes an isometry."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a98769a911415128a09e4bb7f76e8ab528846c1f4d8a789ab889b4b47d54da6e | Calculate \( a \cdot I(a) \cdot I(a-1) \) where \( I(a) = \int_{0}^{\frac{\pi}{2}} \sin^a x \, dx \) and \( a \in \mathbb{N^*} \). | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"Calculate \\( a \\cdot I(a) \\cdot I(a-1) \\) where \\( I(a) = \\int_{0}^{\\frac{\\pi}{2}} \\sin^a x \\, dx \\) and \\( a \\in \\mathbb{N^*} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-af8d09ddd3cc0af4f9c8da2c02d7fb94e648ce01b42858953e3f5cafaa9df8b9 | Calculate \( a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \) if you know that \( a_{4} = 2 \), \( S_{2} = 23 \), and \( S_{4} = 71 \). | [
"\\( a_1 = -\\frac{5}{2}, a_2 = -1, a_3 = \\frac{1}{2}, a_4 = 2, a_5 = \\frac{7}{2} \\)"
] | [
"Calculate \\( a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \\) if you know that \\( a_{4} = 2 \\), \\( S_{2} = 23 \\), and \\( S_{4} = 71 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-980a88b53900f29c1048e59c826f455795a240d869284510895c190f2c8786e2 | Calculate \( b \) given the following equations: \[ \lim_{n\to\infty}\sum_{t=0}^n \log_{p}5^{t+1} = 4 \] \[ p = b^{(t+1) \cdot 2^t} \] | [
"\\( b = \\sqrt{5} \\)"
] | [
"Calculate \\( b \\) given the following equations:\n\\[\n\\lim_{n\\to\\infty}\\sum_{t=0}^n \\log_{p}5^{t+1} = 4\n\\]\n\\[\np = b^{(t+1) \\cdot 2^t}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-b8f393abad833216391882691797debb1a804d98b9ef1d219e97f1468183446c | Calculate \( c(n,k) + 4c(n,k-1) + 6c(n,k-2) + 4c(n,k-3) + c(n,k-4) \) where \( 4 \leq k \leq n \). | [
"\\(\\binom{n+4}{k}\\)"
] | [
"Calculate \\( c(n,k) + 4c(n,k-1) + 6c(n,k-2) + 4c(n,k-3) + c(n,k-4) \\) where \\( 4 \\leq k \\leq n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-d16a8c7481597ee9258186a66280725620b320d173beb35a8b5c9734052e97b2 | Calculate \( d(1) + d(2) + d(3) + \cdots + d(2^{2016}) \), where \( d(n) \) is the greatest divisor of the positive integer \( n \). | [
"2^{2015}(2^{2016} + 1)"
] | [
"Calculate \\( d(1) + d(2) + d(3) + \\cdots + d(2^{2016}) \\), where \\( d(n) \\) is the greatest divisor of the positive integer \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-778d940fde8af9e4eaba773972b00cd467078ae9b74423e8f3edf846c07dfb42 | Calculate \( e^{xA} \) for the square matrix \( A \) using its eigenvectors, where \[ A = \begin{pmatrix} -1 & 0 & -1 \\ 1 & -2 & -1 \\ 1 & 0 & -3 \end{pmatrix} \] Note: The eigenvalue is \(-2\) with a multiplicity of 3, and the matrix has 2 eigenvectors: \(\{1,0,1\}\) and \(\{0,1,0\}\). | [
"e^{-2x} \\begin{pmatrix} 1 + x & 0 & -x \\\\ x & 1 & -x \\\\ x & 0 & 1 - x \\end{pmatrix}"
] | [
"Calculate \\( e^{xA} \\) for the square matrix \\( A \\) using its eigenvectors, where\n\n\\[ A = \\begin{pmatrix}\n-1 & 0 & -1 \\\\\n1 & -2 & -1 \\\\\n1 & 0 & -3\n\\end{pmatrix} \\]\n\nNote: The eigenvalue is \\(-2\\) with a multiplicity of 3, and the matrix has 2 eigenvectors: \\(\\{1,0,1\\}\\) and \\(\\{0,1,0\\... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-de679c6ab86bd1077e7b76f293b38815cfd11be6fd5348777673eb3e926d6996 | Calculate \( f'(0) \), \( f'(2) \), and \( f'(-3) \) for the function \( f(x) = (|x+1|)(|x-1|) - |x| \). | [
"\\( f'(0) \\) does not exist, \\( f'(2) = 3 \\), \\( f'(-3) = -5 \\)"
] | [
"Calculate \\( f'(0) \\), \\( f'(2) \\), and \\( f'(-3) \\) for the function \\( f(x) = (|x+1|)(|x-1|) - |x| \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-bedf280657bb1d269a0ebe59bdb3fa6e39d0a9031002b6498299757cb12438ff | Calculate \( f'(1) \) and \( f'(-1) \) for the function \( f(x) = \frac{|x| - 2}{x - 2} \). | [
"\\( f'(1) = 0 \\) and \\( f'(-1) = \\frac{4}{9} \\)"
] | [
"Calculate \\( f'(1) \\) and \\( f'(-1) \\) for the function \\( f(x) = \\frac{|x| - 2}{x - 2} \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-44521ae5e8f000f77a6acd7e89a4a03483581dc47e9c1e2e447e83497e6e62c2 | Calculate \( f(1.05) \) and \( p_3(1.05) \), and find the error in percent when using the Taylor polynomial instead of the real function value. | [
"0.000023\\%"
] | [
"Calculate \\( f(1.05) \\) and \\( p_3(1.05) \\), and find the error in percent when using the Taylor polynomial instead of the real function value."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-bd046f351bbc7d8ff9a818d488da0c89b44704091c24360dc8624d73b2a2f5d0 | Calculate \( f(\sqrt[3]{2} - 1) \), where \[ f(x) = x^{1999} + 3x^{1998} + 4x^{1997} + 2x^{1996} + 4x^{1995} + 2x^{1994} + \cdots + 4x^3 + 2x^2 + 3x + 1. \] | [
"2"
] | [
"Calculate \\( f(\\sqrt[3]{2} - 1) \\), where\n\\[ f(x) = x^{1999} + 3x^{1998} + 4x^{1997} + 2x^{1996} + 4x^{1995} + 2x^{1994} + \\cdots + 4x^3 + 2x^2 + 3x + 1. \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-5e0cc530771b8db8f7f017358aa326f900a00f1f951edae71168ff81f7ae78a6 | Calculate \( f^{-1}(2012) \). | [
"1006"
] | [
"Calculate \\( f^{-1}(2012) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-9e32b66ac68c11c2662137f762cffb964dae439d2e14d606988f2ca13b15b4b0 | Calculate \( p(11) \) for the polynomial \( p(x) = x^{17} - 12x^{16} + 12x^{15} - 12x^{14} + \cdots - 12x^2 + 12x + 1 \). | [
"12"
] | [
"Calculate \\( p(11) \\) for the polynomial \\( p(x) = x^{17} - 12x^{16} + 12x^{15} - 12x^{14} + \\cdots - 12x^2 + 12x + 1 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-03fd2c416d46ee9ca32825fafddab48dd5696e32780d6c4f5e8b786b9ab274dd | Calculate \( x \) if \[ x = \frac{(11 + 6 \cdot \sqrt{2}) \cdot \sqrt{11 - 6 \cdot \sqrt{2}} - (11 - 6 \cdot \sqrt{2}) \cdot \sqrt{11 + 6 \cdot \sqrt{2}}}{(\sqrt{\sqrt{5} + 2} + \sqrt{\sqrt{5} - 2}) - (\sqrt{\sqrt{5}+1})} \] | [
"14",
"\\( 7(2+\\sqrt{2})\\sqrt{\\sqrt{5} - 1} \\)"
] | [
"Calculate \\( x \\) if \n\\[ x = \\frac{(11 + 6 \\cdot \\sqrt{2}) \\cdot \\sqrt{11 - 6 \\cdot \\sqrt{2}} - (11 - 6 \\cdot \\sqrt{2}) \\cdot \\sqrt{11 + 6 \\cdot \\sqrt{2}}}{(\\sqrt{\\sqrt{5} + 2} + \\sqrt{\\sqrt{5} - 2}) - (\\sqrt{\\sqrt{5}+1})} \\]"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 7
} | 20 | 0 | 0 |
omr-ws-27446ee90ff029d13a762a3a3ac2a3a6b143c5bc673cbf0bbef9a56fdd08247d | Calculate \((1 - \cos a + i\sin a)^6\) for \(a \in \mathbb{R}\). | [
"\\(-64 \\sin^{6} \\left(\\frac{a}{2}\\right) \\left( \\cos(3a) + i \\sin(3a) \\right)\\)"
] | [
"Calculate \\((1 - \\cos a + i\\sin a)^6\\) for \\(a \\in \\mathbb{R}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 7
} | 16 | 0 | 0 |
omr-ws-a27b3feb0ed661affc5392e7c8b5f6f8ca2e916bf773a8706d2b555a115c120d | Calculate \(1 + 4 - 7 - 10 + 13 + 16 - 19 - 22 + \ldots + 301\). | [
"1"
] | [
"Calculate \\(1 + 4 - 7 - 10 + 13 + 16 - 19 - 22 + \\ldots + 301\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-323d76412acdac958102dfb03cdc428117b0036ff4f0e41c5f905e22191d798b | Calculate \(11!\). | [
"39916800"
] | [
"Calculate \\(11!\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ebb4b983ccad6bb675528fa4f89e75094cf6310d529f62fded1bd4de6f2af391 | Calculate \(2019^{2019^{2019}} \mod 100\). | [
"79"
] | [
"Calculate \\(2019^{2019^{2019}} \\mod 100\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 18
} | 40 | 0 | 0 |
omr-ws-0c640010b7d5a0911ed6b55cc58157aacb9474ab522d4696f41d221634c3eba7 | Calculate \(2^{37 \times 73 - 1} \mod (37 \times 73)\). | [
"1"
] | [
"Calculate \\(2^{37 \\times 73 - 1} \\mod (37 \\times 73)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0825cd827684d88445f71ae988d8c5980b219188aa5b581b690a0192cf32d0c4 | Calculate \(2^{64} \mod 360\). | [
"16"
] | [
"Calculate \\(2^{64} \\mod 360\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-c09c8e6bef9756c64b0f653c65124d565258574d28a1542b02dbb0503a7c687a | Calculate \(7^{1016} \mod 31\). | [
"20"
] | [
"Calculate \\(7^{1016} \\mod 31\\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-983c9b06636e43cf8ab59dc01288eb2b03abdf7669858fe9ae337f161189d327 | Calculate \(7^{99}\) without using a calculator. | [
"3"
] | [
"Calculate \\(7^{99}\\) without using a calculator."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 31,
"genselect": 2
} | 33 | 0 | 0 |
omr-ws-eda3f2e82c6b8847e43077606979cdf6d6d1ca315b1832705f3c24d7eca7ad46 | Calculate \(A^n\), \(B^n\), and \(C^n\) for \(n \geq 1\), where \(C = 3A + 7B\) and \(A\) and \(B\) are the given matrices. | [
"\\(A^n = 3^{n-1}A\\), \\(B^n = 3^{n-1}B\\), \\(C^n = 3^{2n-1}A + 7\\cdot21^{n-1}B\\)"
] | [
"Calculate \\(A^n\\), \\(B^n\\), and \\(C^n\\) for \\(n \\geq 1\\), where \\(C = 3A + 7B\\) and \\(A\\) and \\(B\\) are the given matrices."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04aeade55d792552a117088f571d0e8032ccd727ebe61c5ac49b6d7b3476cbcd | Calculate \(A_n\), the number of subsets of \(\{1, \ldots, n\}\) that do not contain two consecutive integers. | [
"\\(A_n = F_{n+2}\\)"
] | [
"Calculate \\(A_n\\), the number of subsets of \\(\\{1, \\ldots, n\\}\\) that do not contain two consecutive integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 15
} | 63 | 0 | 0 |
omr-ws-82d348cec662b0ce456630c65283cd45b1ca346bcdbc676884fad83451ee2775 | Calculate \(\cos 15^\circ 12' 51''\) correct to five decimal places. | [
"0.96495"
] | [
"Calculate \\(\\cos 15^\\circ 12' 51''\\) correct to five decimal places."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0a00f20c5401010c18c6d87ac2d0185dab37febb9c926c34cd569c17f085a626 | Calculate \(\cos(M)\) where \(M\) is the matrix \[ M = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \] using the power series definition of the cosine function for matrices. | [
"\\[\n\\begin{pmatrix}\n\\cos 2 & -\\sin 2 & 0 \\\\\n0 & \\cos 2 & 0 \\\\\n0 & 0 & \\cos 1\n\\end{pmatrix}\n\\]"
] | [
"Calculate \\(\\cos(M)\\) where \\(M\\) is the matrix\n\n\\[\nM = \\begin{pmatrix}\n2 & 1 & 0 \\\\\n0 & 2 & 0 \\\\\n0 & 0 & 1\n\\end{pmatrix}\n\\]\n\nusing the power series definition of the cosine function for matrices."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-c7eb62fffbad81b2a855622f53f1cd9c3caf42622ceabefd4b6f7785ff9951e1 | Calculate \(\dfrac{\sum_{k=1}^{2499} \sqrt{10+\sqrt{50+\sqrt{k}}}}{\sum_{k=1}^{2499} \sqrt{10-\sqrt{50+\sqrt{k}}}}\). | [
"\\(1 + \\sqrt{2} + \\sqrt{4 + 2\\sqrt{2}}\\)"
] | [
"Calculate \\(\\dfrac{\\sum_{k=1}^{2499} \\sqrt{10+\\sqrt{50+\\sqrt{k}}}}{\\sum_{k=1}^{2499} \\sqrt{10-\\sqrt{50+\\sqrt{k}}}}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e363503ae3bc3e9077785f3b3fd275b5037c81d9e42780096b8a32cadff6841 | Calculate \(\dim(\alpha + \beta)\) for the planes \(\alpha\) and \(\beta\) given by the equations: \[ \alpha: \left\{ \begin{array}{l} 2x_1 + x_2 + x_3 + x_4 - 1 = 0 \\ -x_1 + x_2 - x_4 + 2 = 0 \end{array} \right. \] \[ \beta: \left\{ \begin{array}{l} x_1 - x_2 - x_4 + 1 = 0 \\ -2x_1 + x_2 + 2x_3 + x_4 + 3 = 0 \end{arr... | [
"4"
] | [
"Calculate \\(\\dim(\\alpha + \\beta)\\) for the planes \\(\\alpha\\) and \\(\\beta\\) given by the equations:\n\\[\n\\alpha: \\left\\{\n\\begin{array}{l}\n2x_1 + x_2 + x_3 + x_4 - 1 = 0 \\\\\n-x_1 + x_2 - x_4 + 2 = 0\n\\end{array}\n\\right.\n\\]\n\\[\n\\beta: \\left\\{\n\\begin{array}{l}\nx_1 - x_2 - x_4 + 1 = 0 \... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89915a5f95290886d977340074bfc9bbabd9685b4ebf8ce921d434e9f1dcabc6 | Calculate \(\frac{(1 + x)^{\frac{1}{x}} - e}{x}\) without using derivatives. | [
"-\\dfrac{e}{2}"
] | [
"Calculate \\(\\frac{(1 + x)^{\\frac{1}{x}} - e}{x}\\) without using derivatives."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9b9bb468aec5f165d51ba454f529c476ec699c61b160741f590df491f4ef5911 | Calculate \(\frac{16^{1007} - 1}{5} \mod 210\). | [
"51"
] | [
"Calculate \\(\\frac{16^{1007} - 1}{5} \\mod 210\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 11
} | 29 | 0 | 0 |
omr-ws-fec65274af817bbded53c384fb0359ba9fce0600f9f1ab1228e38895f93809ee | Calculate \(\gcd(17^{2007} - 1, 17^{1115} - 1)\). | [
"\\(17^{223} - 1\\)"
] | [
"Calculate \\(\\gcd(17^{2007} - 1, 17^{1115} - 1)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-7adcf2d84a9cde0a8e2d8b33478131bde6d0cb36815a31b324058409126da43f | Calculate \(\int_{0}^{\frac{\pi}{4}} \arctan \sqrt{\frac{\cos 2x}{2 \cos^2 x}} \, dx\). | [
"\\dfrac{\\pi^2}{32}"
] | [
"Calculate \\(\\int_{0}^{\\frac{\\pi}{4}} \\arctan \\sqrt{\\frac{\\cos 2x}{2 \\cos^2 x}} \\, dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-8961dd533f00eae43562f5be3dcb7d3f0a3b9f8a326dde7111a309b7d18d456a | Calculate \(\left(\frac{x^2 + y^2 + z^2 - \overline{x^2 + y^2 + z^2}}{4i}\right)^2\). | [
"\\left( \\frac{\\operatorname{Im}(x^2 + y^2 + z^2)}{2} \\right)^2"
] | [
"Calculate \\(\\left(\\frac{x^2 + y^2 + z^2 - \\overline{x^2 + y^2 + z^2}}{4i}\\right)^2\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-e2392ff93317a3753eef80b1be5ad0879c6cdbb92c51ae1095ce1c589aca5953 | Calculate \(\lim \frac{3^n - 2.5^n}{7 + 3.5^n}\). | [
"0"
] | [
"Calculate \\(\\lim \\frac{3^n - 2.5^n}{7 + 3.5^n}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-707dda2305f82f81f2ebdfa8f805f0bc7ca7c006ca2466b946511693cc70e02f | Calculate \(\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+\frac{a_1x+a_3x^3+\cdots+a_{2n-1}x^{2n-1}}{n} \right)^ndx}\) where \(\left(a_n\right)_{n\geq1}\) is a sequence of nonnegative real numbers converging to \(a \in \mathbb{R}\). | [
"1 + a"
] | [
"Calculate \\(\\lim \\limits_{n\\to \\infty}\\sqrt[n]{\\int \\limits_0^1 \\left(1+\\frac{a_1x+a_3x^3+\\cdots+a_{2n-1}x^{2n-1}}{n} \\right)^ndx}\\) where \\(\\left(a_n\\right)_{n\\geq1}\\) is a sequence of nonnegative real numbers converging to \\(a \\in \\mathbb{R}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-b4b7a59bd3e76682773c82d13b70be2db9981beab331577e995fe3ccb027a245 | Calculate \(\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+a_nx^n \right)^ndx}\) where \(\left(a_n\right)_{n\geq1}\) is a sequence of nonnegative real numbers converging to \(a \in \mathbb{R}\). | [
"1 + a"
] | [
"Calculate \\(\\lim \\limits_{n\\to \\infty}\\sqrt[n]{\\int \\limits_0^1 \\left(1+a_nx^n \\right)^ndx}\\) where \\(\\left(a_n\\right)_{n\\geq1}\\) is a sequence of nonnegative real numbers converging to \\(a \\in \\mathbb{R}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d15fee3fe9ea5b8204aed8910f18ad7ef515f2afd643f17b82410128c4dfff46 | Calculate \(\lim \limits_{x \rightarrow \infty} \left( a \sqrt{x+1} + b\sqrt{4x+1} + c\sqrt{9x+1} \right)\), where \(a, b, c \in \mathbb{R}\). | [
"\\( \\lim \\limits_{x \\rightarrow \\infty} \\left( a \\sqrt{x+1} + b\\sqrt{4x+1} + c\\sqrt{9x+1} \\right) = \\begin{cases} \n+\\infty & \\text{if } a + 2b + 3c > 0 \\\\\n-\\infty & \\text{if } a + 2b + 3c < 0 \\\\\n0 & \\text{if } a + 2b + 3c = 0 \n\\end{cases} \\)"
] | [
"Calculate \\(\\lim \\limits_{x \\rightarrow \\infty} \\left( a \\sqrt{x+1} + b\\sqrt{4x+1} + c\\sqrt{9x+1} \\right)\\), where \\(a, b, c \\in \\mathbb{R}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 23
} | 40 | 0 | 0 |
omr-ws-bebab0de26f69dfd3922db94d6bf9f24a7ccbd70c773696e83b414fcbcffc4b2 | Calculate \(\lim_{n \rightarrow \infty} n \left( \sqrt[3]{n^3 + 3n^2 + 2n + 1} + \sqrt{n^2 - 2n + 3} - 2n \right)\). | [
"\\(\\frac{2}{3}\\)"
] | [
"Calculate \\(\\lim_{n \\rightarrow \\infty} n \\left( \\sqrt[3]{n^3 + 3n^2 + 2n + 1} + \\sqrt{n^2 - 2n + 3} - 2n \\right)\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-5fb0c8617b1235e301b26276b8b844c233d7c37a73065edbbaea0fdde25accb4 | Calculate \(\lim_{n \to +\infty} \cos^n(2\pi n! x)\) where \(x \notin \mathbb{Q}\). | [
"0"
] | [
"Calculate \\(\\lim_{n \\to +\\infty} \\cos^n(2\\pi n! x)\\) where \\(x \\notin \\mathbb{Q}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ab7ac29c1c321ff95027e6ddff8de484dfefa17d0cf6106a7e97196afbc77ce9 | Calculate \(\lim_{n \to +\infty} \left(1 + n + \frac{n^2}{2!} + \cdots + \frac{n^n}{n!}\right) e^{-n}\). | [
"\\dfrac{1}{2}"
] | [
"Calculate \\(\\lim_{n \\to +\\infty} \\left(1 + n + \\frac{n^2}{2!} + \\cdots + \\frac{n^n}{n!}\\right) e^{-n}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-daca1fe1645f0f24b78cdf4373f4dbe40bf0a7fefc46bd0da659a8d120dd3d2e | Calculate \(\lim_{n \to +\infty} \sqrt{n} \int_{-\infty}^{+\infty} \frac{\cos x + \sin x}{(1 + x^2)^n} \, dx\). | [
"\\sqrt{\\pi}"
] | [
"Calculate \\(\\lim_{n \\to +\\infty} \\sqrt{n} \\int_{-\\infty}^{+\\infty} \\frac{\\cos x + \\sin x}{(1 + x^2)^n} \\, dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8aa5b0dbbff7e08802f8ef3bdd371beefe26356b4d52ee812efdb5deda2c4ebd | Calculate \(\lim_{n \to \infty} \frac{1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n - 1)}{3^n \cdot n!}\). | [
"0"
] | [
"Calculate \\(\\lim_{n \\to \\infty} \\frac{1 \\cdot 3 \\cdot 5 \\cdot \\ldots \\cdot (2n - 1)}{3^n \\cdot n!}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2f38740b3cf30cbdb041b558070755092e096cbc5547286c61483e6977f1f925 | Calculate \(\lim_{n \to \infty} \left( \frac{1^2}{n^3+n^2+1} + \frac{2^2}{n^3+n^2+1} + \cdots + \frac{n^2}{n^3+n^2+1} \right)\). | [
"\\(\\frac{1}{3}\\)"
] | [
"Calculate \\(\\lim_{n \\to \\infty} \\left( \\frac{1^2}{n^3+n^2+1} + \\frac{2^2}{n^3+n^2+1} + \\cdots + \\frac{n^2}{n^3+n^2+1} \\right)\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-df5f02226d570afddb5224fc15da60d4f8e78907f14bde202e58223f8d834dd9 | Calculate \(\lim_{n \to \infty} \left( \frac{1}{n^2+1} + \frac{2}{n^2+2} + \cdots + \frac{n}{n^2+n} \right)\). | [
"\\(\\frac{1}{2}\\)"
] | [
"Calculate \\(\\lim_{n \\to \\infty} \\left( \\frac{1}{n^2+1} + \\frac{2}{n^2+2} + \\cdots + \\frac{n}{n^2+n} \\right)\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-39cb765a4249f2dd2af24e1759af4316e5e74c0f131019b4eb23ed6f0fdc7dc2 | Calculate \(\lim_{n \to \infty} \left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{6}\right) \cdots \left(1 - \frac{1}{\binom{n+1}{2}}\right)\). | [
"\\(\\frac{1}{3}\\)"
] | [
"Calculate \\(\\lim_{n \\to \\infty} \\left(1 - \\frac{1}{3}\\right)\\left(1 - \\frac{1}{6}\\right) \\cdots \\left(1 - \\frac{1}{\\binom{n+1}{2}}\\right)\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-bf25b234c018a4b85d9495003fe4df1d7d50981c507f1a3cdcddb89cc01c0760 | Calculate \(\lim_{n \to \infty} \prod_{k=1}^{n} (n+k)^{1/p_{n+k}}\), where \(p_n\) denotes the \(n\)-th prime number. | [
"\\(\\sqrt{e}\\)"
] | [
"Calculate \\(\\lim_{n \\to \\infty} \\prod_{k=1}^{n} (n+k)^{1/p_{n+k}}\\), where \\(p_n\\) denotes the \\(n\\)-th prime number."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-192e4167eed33ebc7cd285a3133bcf173e3c36662e94b1ef704ef0b8a0384ff8 | Calculate \(\lim_{n \to \infty} \sqrt{n} \cdot I_n\). | [
"1"
] | [
"Calculate \\(\\lim_{n \\to \\infty} \\sqrt{n} \\cdot I_n\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8201b787e676d7a2f52dfd582c1e94af7c5abcddbac33b73d799207eaa2e9e46 | Calculate \(\lim_{n \to \infty} \sum_{k=0}^{n} \frac{\ln(n+k)}{\sqrt{n+k}}\). | [
"\\(+\\infty\\)"
] | [
"Calculate \\(\\lim_{n \\to \\infty} \\sum_{k=0}^{n} \\frac{\\ln(n+k)}{\\sqrt{n+k}}\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-755dcc25011bd3b81a2b00728146d0d926ca366a1b98dfd143dd8111aeff6970 | Calculate \(\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{\binom{2k}{k}}\). | [
"\\(\\frac{1}{3} + \\frac{2\\pi}{9\\sqrt{3}}\\)"
] | [
"Calculate \\(\\lim_{n \\to \\infty} \\sum_{k=1}^{n} \\frac{1}{\\binom{2k}{k}}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-ae4c33d3e4ead89dd182b576d10d11182feb8220c465b2dbe4d6bc821d8ba879 | Calculate \(\lim_{n \to \infty} n \left( n^{1/n} - 1 \right)\). | [
"\\(\\ln(n)\\)"
] | [
"Calculate \\(\\lim_{n \\to \\infty} n \\left( n^{1/n} - 1 \\right)\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 5
} | 6 | 0 | 0 |
omr-ws-57b56f6750748da24de04b0e880418c0215d168c0989a6f876f3caf408827995 | Calculate \(\lim_{n\rightarrow \infty}(\sqrt{4n^{2}+1}+\sqrt{n^{2}+3n+1}-3n)\). | [
"\\(\\frac{3}{2}\\)"
] | [
"Calculate \\(\\lim_{n\\rightarrow \\infty}(\\sqrt{4n^{2}+1}+\\sqrt{n^{2}+3n+1}-3n)\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-3c0bb182bfa43053397e3f5ca2d87fe1a421ed8fee3de5a57806ecfa618c8731 | Calculate \(\lim_{n\rightarrow \infty}(\sqrt{x^{2}+x+1}-\sqrt{x^{2}-2x+4})\). | [
"\\dfrac{3}{2}"
] | [
"Calculate \\(\\lim_{n\\rightarrow \\infty}(\\sqrt{x^{2}+x+1}-\\sqrt{x^{2}-2x+4})\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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