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-3d88883f730278d2a479bdccd4cf3212eff45266800f9800e3d6a96e3b83d1e5 | Calculate \(\lim_{n\rightarrow \infty}\frac{\sqrt{3x^{5}+1}}{\sqrt[4]{7x^{10}+x-5}}\). | [
"\\sqrt[4]{\\dfrac{9}{7}}"
] | [
"Calculate \\(\\lim_{n\\rightarrow \\infty}\\frac{\\sqrt{3x^{5}+1}}{\\sqrt[4]{7x^{10}+x-5}}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-29c418f60eac864a62f180ce0d0c0af00cfce0f034f61b8bd255758ed4dd751e | Calculate \(\lim_{n\to+\infty } \frac{3}{n} \left(1 + \sqrt{\frac{n}{n+3}} + \sqrt{\frac{n}{n+6}} + \cdots + \sqrt{\frac{n}{4n-1}}\right)\). | [
"\\(\\frac{7}{2}\\)"
] | [
"Calculate \\(\\lim_{n\\to+\\infty } \\frac{3}{n} \\left(1 + \\sqrt{\\frac{n}{n+3}} + \\sqrt{\\frac{n}{n+6}} + \\cdots + \\sqrt{\\frac{n}{4n-1}}\\right)\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-39072d01b579863e22d5c6e0f099711d236cdd66ca60db81869955097a984d60 | Calculate \(\lim_{n\to+\infty } \left(\frac{1}{\sqrt{n^3}} \sum_{k=1}^{n} \sqrt{k}\right)\). | [
"0"
] | [
"Calculate \\(\\lim_{n\\to+\\infty } \\left(\\frac{1}{\\sqrt{n^3}} \\sum_{k=1}^{n} \\sqrt{k}\\right)\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb71a6dfeb630e0b619c3a7d68cd973d840531fb02ec8dac08799f0658c2e45f | Calculate \(\lim_{n\to+\infty } \left(\frac{2^{\frac{1}{n}}}{n+1} + \frac{2^{\frac{2}{n}}}{n+\frac{1}{2}} + \cdots + \frac{2^{\frac{n}{n}}}{n+\frac{1}{n}}\right)\). | [
"\\(\\int_0^1 \\frac{2^x}{1+x} \\, dx\\)"
] | [
"Calculate \\(\\lim_{n\\to+\\infty } \\left(\\frac{2^{\\frac{1}{n}}}{n+1} + \\frac{2^{\\frac{2}{n}}}{n+\\frac{1}{2}} + \\cdots + \\frac{2^{\\frac{n}{n}}}{n+\\frac{1}{n}}\\right)\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be94290078d9c648d8b8743834abc0ecb19376321d6e413cebab93cb1ea77707 | Calculate \(\lim_{x \rightarrow 0} \frac{1 - (1 - \sin x)(1 - \sin 2x)^2 \cdots (1 - \sin nx)^n}{x}\), where \(n \in \mathbb{N}^*\). | [
"\\(\\frac{n(n+1)(2n+1)}{6}\\)"
] | [
"Calculate \\(\\lim_{x \\rightarrow 0} \\frac{1 - (1 - \\sin x)(1 - \\sin 2x)^2 \\cdots (1 - \\sin nx)^n}{x}\\), where \\(n \\in \\mathbb{N}^*\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-7c5b2a71ac30c542fa921b32395be8783daf92dfb65dcb788dc1e6ae1bee33fa | Calculate \(\lim_{x \rightarrow 0} \frac{1 - \cos^5 x \cdot \cos^3 2x \cdot \cos^3 3x}{x^2}\). | [
"22"
] | [
"Calculate \\(\\lim_{x \\rightarrow 0} \\frac{1 - \\cos^5 x \\cdot \\cos^3 2x \\cdot \\cos^3 3x}{x^2}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-46b351c421de9fb249169c45023ccc7d165305cf78cebd616c9d9f7f14dd7f33 | Calculate \(\lim_{x \to 0} \left( \frac{1}{x} \lim_{n \to \infty} \left( \begin{bmatrix} 1 & 0 & 0 & -\frac{x}{n} \\ 0 & 1 & -\frac{x}{n} & 0 \\ 0 & \frac{x}{n} & 1 & 0 \\ \frac{x}{n} & 0 & 0 & 1 \end{bmatrix}^n - I_4 \right) \right)\). | [
"\\(\\begin{bmatrix} 0 & 0 & 0 & -1 \\\\ 0 & 0 & -1 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\end{bmatrix}\\)"
] | [
"Calculate \\(\\lim_{x \\to 0} \\left( \\frac{1}{x} \\lim_{n \\to \\infty} \\left( \\begin{bmatrix} 1 & 0 & 0 & -\\frac{x}{n} \\\\ 0 & 1 & -\\frac{x}{n} & 0 \\\\ 0 & \\frac{x}{n} & 1 & 0 \\\\ \\frac{x}{n} & 0 & 0 & 1 \\end{bmatrix}^n - I_4 \\right) \\right)\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-4a025022772d4090463b84742bd682ec287a0dc6550bf45a846965cc64101847 | Calculate \(\lim_{x \to 0} \left( \frac{1}{x} \lim_{n \to \infty} \left( \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & -\frac{x}{n} \\ 0 & 1 & 0 & 0 & -\frac{x}{n} & 0 \\ 0 & 0 & 1 & -\frac{x}{n} & 0 & 0 \\ 0 & 0 & \frac{x}{n} & 1 & 0 & 0 \\ 0 & \frac{x}{n} & 0 & 0 & 1 & 0 \\ \frac{x}{n} & 0 & 0 & 0 & 0 & 1 \end{bmatrix}^n - I_6... | [
"\\(\\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & -1 \\\\ 0 & 0 & 0 & 0 & -1 & 0 \\\\ 0 & 0 & 0 & -1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 \\end{bmatrix}\\)"
] | [
"Calculate \\(\\lim_{x \\to 0} \\left( \\frac{1}{x} \\lim_{n \\to \\infty} \\left( \\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & -\\frac{x}{n} \\\\ 0 & 1 & 0 & 0 & -\\frac{x}{n} & 0 \\\\ 0 & 0 & 1 & -\\frac{x}{n} & 0 & 0 \\\\ 0 & 0 & \\frac{x}{n} & 1 & 0 & 0 \\\\ 0 & \\frac{x}{n} & 0 & 0 & 1 & 0 \\\\ \\frac{x}{n} & 0 & 0 & ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-085099ac6d3e90b21715391b5ba46a4d533cd7b9679585a1efd8a0820ea9deb0 | Calculate \(\lim_{x\to 0^+} \frac{\text{perimeter of}\ \triangle NOP}{\text{perimeter of}\ \triangle MOP}\) where \(N(0, 1)\), \(O(0, 0)\), \(M(1, 0)\), and \(P(x, \sqrt{x})\). | [
"1"
] | [
"Calculate \\(\\lim_{x\\to 0^+} \\frac{\\text{perimeter of}\\ \\triangle NOP}{\\text{perimeter of}\\ \\triangle MOP}\\) where \\(N(0, 1)\\), \\(O(0, 0)\\), \\(M(1, 0)\\), and \\(P(x, \\sqrt{x})\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb0df8658ad5c2ad46fc9126d28a66527aa52236ecba2a2a6e37e1b3d516acae | Calculate \(\lim_{x\to 0} \frac{1 - \sqrt{\cos x \cos 2x \cos 3x \cdots \cos nx}}{x^2}\). | [
"\\dfrac{n(n+1)(2n+1)}{24}"
] | [
"Calculate \\(\\lim_{x\\to 0} \\frac{1 - \\sqrt{\\cos x \\cos 2x \\cos 3x \\cdots \\cos nx}}{x^2}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bba447274260ff79c2708fae8dd5818a2c8c034bc0f20ea91dacef50f3b28c4c | Calculate \(\lim_{x\to 0} \frac{\sqrt[n]{\cos(\alpha x)} - \sqrt[n]{\cos(\beta x)}}{\sin^2(\gamma x)}\) where \(m, n\) are positive numbers, and \(\alpha, \beta, \gamma\) are constants with \(\gamma \neq 0\). | [
"\\dfrac{\\beta^2 - \\alpha^2}{2n\\gamma^2}"
] | [
"Calculate \\(\\lim_{x\\to 0} \\frac{\\sqrt[n]{\\cos(\\alpha x)} - \\sqrt[n]{\\cos(\\beta x)}}{\\sin^2(\\gamma x)}\\) where \\(m, n\\) are positive numbers, and \\(\\alpha, \\beta, \\gamma\\) are constants with \\(\\gamma \\neq 0\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2398402e2035e883136858a4bbf7abbf9c222c05da67c3c740af1c79997f866d | Calculate \(\lim_{x\to0^{+}}\int_{0}^{x}\ln\left(\frac{|\sin t-\frac{x}{2}|}{\sin\frac{x}{2}}\right)\frac{1}{\sin t}\,dt\), if it exists. | [
"\\(\\int_0^2 \\frac{\\log |1-s|}{s} \\, ds\\)"
] | [
"Calculate \\(\\lim_{x\\to0^{+}}\\int_{0}^{x}\\ln\\left(\\frac{|\\sin t-\\frac{x}{2}|}{\\sin\\frac{x}{2}}\\right)\\frac{1}{\\sin t}\\,dt\\), if it exists."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb5952d9f3b4d27161893ae2bab388f4b5b1a0ab0860c9d36a5fd9c6760a2dd5 | Calculate \(\lim_{x\to\frac{\pi}{2}} \frac{1 - \sin^{m+n+p} x}{\sqrt[3]{(1 - \sin^m x)(1 - \sin^n x)(1 - \sin^p x)}}\). | [
"\\(\\frac{m+n+p}{\\sqrt[3]{mnp}}\\)"
] | [
"Calculate \\(\\lim_{x\\to\\frac{\\pi}{2}} \\frac{1 - \\sin^{m+n+p} x}{\\sqrt[3]{(1 - \\sin^m x)(1 - \\sin^n x)(1 - \\sin^p x)}}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-12c5266e18b67056a9c452cf549f9e17f04d3aabb78f51e0892a08444ff59177 | Calculate \(\mathop {\lim }\limits_{x \to +\infty } \left[ Ci(x) - Ci(3x) \right]\) if it exists. | [
"0"
] | [
"Calculate \\(\\mathop {\\lim }\\limits_{x \\to +\\infty } \\left[ Ci(x) - Ci(3x) \\right]\\) if it exists."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-be391393a6e4aae58be11d5b71d40018bca1931c1b77c6d4503d046710e53924 | Calculate \(\phi(n)\) for \(n = 30\), where \(\phi\) is Euler's totient function. | [
"8"
] | [
"Calculate \\(\\phi(n)\\) for \\(n = 30\\), where \\(\\phi\\) is Euler's totient function."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a95561b99c697754c51727f9f4bb6817477b83b4d77713e10565b4372f4d7276 | Calculate \(\pi\) to 6 decimal places using the series expansion for \(\arctan(x)\). | [
"3.141592"
] | [
"Calculate \\(\\pi\\) to 6 decimal places using the series expansion for \\(\\arctan(x)\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-64fca03e587956a835be5bc0729da7692bade21e543c222a22566cb99557fd40 | Calculate \(\prod_{k=0}^n \frac{\cos 2^k}{1 + \cos 2^k}\). | [
"\\(\\frac{1}{2} \\sin 1 \\sec^2 \\frac{1}{2} \\cot 2^n\\)"
] | [
"Calculate \\(\\prod_{k=0}^n \\frac{\\cos 2^k}{1 + \\cos 2^k}\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7015064a9f2fb578059ac71930e5842be83cf7aea708a9c7ef932a102b83c55 | Calculate \(\sum_{\sigma \in \sigma_n} \frac{(-1)^{I(\sigma)}}{(p(\sigma) + 1)(q(\sigma) + 1)}\). | [
"0"
] | [
"Calculate \\(\\sum_{\\sigma \\in \\sigma_n} \\frac{(-1)^{I(\\sigma)}}{(p(\\sigma) + 1)(q(\\sigma) + 1)}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eaadbb166b002b482e287d9b00b1e8a5112f0ad92c4075693f61d4c50fd8e8f6 | Calculate \(\sum_{\sigma \in \sigma_n} \frac{(-1)^{I(\sigma)}}{(p(\sigma) + q(\sigma) + 1)(v(\sigma) + 1)}\). | [
"0"
] | [
"Calculate \\(\\sum_{\\sigma \\in \\sigma_n} \\frac{(-1)^{I(\\sigma)}}{(p(\\sigma) + q(\\sigma) + 1)(v(\\sigma) + 1)}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-30043e32aa798c76ad3980b81824902479db07df5069059221d5fd4e4aa93dde | Calculate \(\sum_{\sigma \in \sigma_n} \frac{(-1)^{I(\sigma)}}{p(\sigma) + q(\sigma) + 1}\). | [
"0"
] | [
"Calculate \\(\\sum_{\\sigma \\in \\sigma_n} \\frac{(-1)^{I(\\sigma)}}{p(\\sigma) + q(\\sigma) + 1}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-23434cf3ad6423b3da7e9cf3432c5d4246c8a6090df054b0fe7bff7d139716bd | Calculate \(\sum_{\sigma \in \sigma_n} \frac{(-1)^{I(\sigma)}}{p(\sigma) + q(\sigma) + v(\sigma) + 1}\). | [
"\\(\\frac{2\\sin\\left(\\frac{(n+1)\\pi}{3}\\right)}{\\sqrt{3}(n+1)}\\)"
] | [
"Calculate \\(\\sum_{\\sigma \\in \\sigma_n} \\frac{(-1)^{I(\\sigma)}}{p(\\sigma) + q(\\sigma) + v(\\sigma) + 1}\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0540ff6449851603f40d6f1c6d0542ef9c9a9d1db4a7d13aef6f39e8f2d01fdf | Calculate \(\sum_{k=0}^n \arctan\frac{1}{k^2 + k + 1}\). | [
"\\(\\arctan(n+1)\\)"
] | [
"Calculate \\(\\sum_{k=0}^n \\arctan\\frac{1}{k^2 + k + 1}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-9b9dd72f863fa9e27ed2cb37bb38a9415384a5639f38fd6ece3de318d2dc821e | Calculate \(\sum_{k=1}^n \cos(2k+1)x\). | [
"\\dfrac{\\sin(nx)}{\\sin x} \\cos\\left((n + 2)x\\right)"
] | [
"Calculate \\(\\sum_{k=1}^n \\cos(2k+1)x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-f76347628946e66023109ab02ecb9af7f87672df06fadcfb43115e3b4e807074 | Calculate \(\sum_{k=1}^n \sin(2k+1)x\). | [
"\\dfrac{\\sin(nx) \\sin((n + 2)x)}{\\sin x}"
] | [
"Calculate \\(\\sum_{k=1}^n \\sin(2k+1)x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de8ff56a3cd0fe01c6e221b96f0788f6a57c11d452d38c2c538fb2cdafc7b52f | Calculate \(a\) and \(b\) such that \[ \lim_{x \to 1}\left (\frac{a}{1-x^{m}}-\frac{b}{1-x^{n}}\right) = \frac{m-n}{2} \] where \(m \neq n\) and \(m, n \in \mathbb{N^{*}}\). | [
"\\(a = m, b = n\\)"
] | [
"Calculate \\(a\\) and \\(b\\) such that\n\\[ \\lim_{x \\to 1}\\left (\\frac{a}{1-x^{m}}-\\frac{b}{1-x^{n}}\\right) = \\frac{m-n}{2} \\]\nwhere \\(m \\neq n\\) and \\(m, n \\in \\mathbb{N^{*}}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 16
} | 18 | 0 | 0 |
omr-ws-53939541688ef2b62c8aa34dc04b4ac1493c99f70e0d347b1e81e4dd330fd629 | Calculate \[ P=\dfrac{\displaystyle \sum_{k=1}^{n^2-1}\sqrt{n+\sqrt{k}}}{\displaystyle \sum_{k=1}^{n^2-1}\sqrt{n-\sqrt{k}}} \] | [
"\\( 1 + \\sqrt{2} \\)"
] | [
"Calculate \\[ P=\\dfrac{\\displaystyle \\sum_{k=1}^{n^2-1}\\sqrt{n+\\sqrt{k}}}{\\displaystyle \\sum_{k=1}^{n^2-1}\\sqrt{n-\\sqrt{k}}} \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 30
} | 53 | 0 | 0 |
omr-ws-c16509a6ca9d79093f9418cef1f51d5c22b2e7ed4d8aa8c4a8422b59e6a526f4 | Calculate \[ \frac {1}{{\cos \frac {{6\pi }}{{13}}}} - 4\cos \frac {{4\pi }}{{13}} - 4\cos \frac {{5\pi }}{{13}}. \] | [
"\\(\\sqrt{13} + 1\\)"
] | [
"Calculate\n\\[ \\frac {1}{{\\cos \\frac {{6\\pi }}{{13}}}} - 4\\cos \\frac {{4\\pi }}{{13}} - 4\\cos \\frac {{5\\pi }}{{13}}. \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 8,
"tir": 13
} | 22 | 0 | 0 |
omr-ws-7b5964e29b5317f3ab03d176476e9ff64340719ce8927439f991e3aa4fc7658e | Calculate \[ \frac{\int_{0}^{\pi}e^{-x}\sin^{n}x\ dx}{\int_{0}^{\pi}e^{x}\sin^{n}x \ dx}\ (n=1,\ 2,\ \cdots). \] | [
"\\( e^{-\\pi} \\)"
] | [
"Calculate\n\\[\n\\frac{\\int_{0}^{\\pi}e^{-x}\\sin^{n}x\\ dx}{\\int_{0}^{\\pi}e^{x}\\sin^{n}x \\ dx}\\ (n=1,\\ 2,\\ \\cdots).\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-56da9380c7b6d158e96258ce31f2994f40edd1da6939e50f4fe6c9b9e7ac39b6 | Calculate \[ \int_{0}^{\pi/2} \frac{\sin{x}}{1+\sqrt{\sin{2x}}} \, dx. \] | [
"\\(\\frac{\\pi - 2}{2}\\)"
] | [
"Calculate \\[ \\int_{0}^{\\pi/2} \\frac{\\sin{x}}{1+\\sqrt{\\sin{2x}}} \\, dx. \\]"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 33
} | 53 | 0 | 0 |
omr-ws-c365cd1c3b614c9849f7f58b5fc9ba9209e8450778c51d83f296b99cd2d6299c | Calculate \[ \int_{\sqrt{3}}^{3 \sqrt{3}} \frac{\arctan x}{1-x^2} \, dx \] | [
"-\\dfrac{\\pi^2}{24}"
] | [
"Calculate \\[ \\int_{\\sqrt{3}}^{3 \\sqrt{3}} \\frac{\\arctan x}{1-x^2} \\, dx \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-05dd2f43cce52b4f6c8a6285b1c68b0a4b1e68241105a2b20add848062e0500d | Calculate \[ \left \lfloor \frac{\big (a^2+b^2+c^2 \big )(a+b+c)}{a^3+b^3+c^3} \right \rfloor \] where \(a, b, c\) are the lengths of the sides of a triangle. | [
"2"
] | [
"Calculate\n\\[\n\\left \\lfloor \\frac{\\big (a^2+b^2+c^2 \\big )(a+b+c)}{a^3+b^3+c^3} \\right \\rfloor\n\\]\nwhere \\(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-d0aecaa46508df0f14fb6ff4b13c286345a44a8d2b6300d269a3db5e4595f7ba | Calculate \[ \left \lfloor \tan^4 \frac{3\pi}{7}+\tan^4 \frac{2\pi}{7}+2\left(\tan^2 \frac{3\pi}{7}+\tan^2 \frac{2\pi}{7}\right) \right \rfloor \] | [
"412"
] | [
"Calculate\n\\[\n\\left \\lfloor \\tan^4 \\frac{3\\pi}{7}+\\tan^4 \\frac{2\\pi}{7}+2\\left(\\tan^2 \\frac{3\\pi}{7}+\\tan^2 \\frac{2\\pi}{7}\\right) \\right \\rfloor\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-753d8ba5b2ce87bc57606c35ac9a26c7ae0cec3fc530f2014440b6aa37c59b56 | Calculate \[ \lim_{\substack{x,y > 0 \\ x + y \to +\infty}} \frac{x^2 + y^3}{x + y}. \] | [
"\\(+\\infty\\)"
] | [
"Calculate\n\\[ \\lim_{\\substack{x,y > 0 \\\\ x + y \\to +\\infty}} \\frac{x^2 + y^3}{x + y}. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-e8dbb351930f96c019d5d1bccbd5b452118d72d07b14e6acd79c8e78620e3f66 | Calculate \[ \lim_{a \to +\infty} \left( a - \int_{\frac{1}{a}}^a \frac{dx}{1 + (a - 2)x^{1 - a}} \right), \quad x > 0 \] | [
"1"
] | [
"Calculate \n\\[\n\\lim_{a \\to +\\infty} \\left( a - \\int_{\\frac{1}{a}}^a \\frac{dx}{1 + (a - 2)x^{1 - a}} \\right), \\quad x > 0\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-bb7098ccbf6f0848a1b4e79ea1e34fc95b76903e2579f3645c8fea2ab79dbe29 | Calculate \[ \lim_{k\rightarrow \infty }\lim_{n\rightarrow \infty }\lim_{m\rightarrow \infty }\sum_{N=0}^{n}\left[ 1-\cos ^{2m}\dfrac{(N!)^{k}\pi }{x}\right].\] | [
"p"
] | [
"Calculate\n\\[ \\lim_{k\\rightarrow \\infty }\\lim_{n\\rightarrow \\infty }\\lim_{m\\rightarrow\r\n\\infty }\\sum_{N=0}^{n}\\left[ 1-\\cos ^{2m}\\dfrac{(N!)^{k}\\pi }{x}\\right].\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-a473261c8062f8500ae9b981f9ed61b486282a025c2b17e52c3ecc890d90128e | Calculate \[ \lim_{n \rightarrow \infty} \int_{- \pi}^{\pi} \frac{n! \, 2^{2n \cos \, \theta} }{|(2ne^{i \theta} - 1)(2ne^{i \theta} - 2)(2ne^{i \theta} - 3) \cdots (2ne^{i \theta} - n) |} \, d \theta. \] | [
"\\(2\\pi\\)"
] | [
"Calculate \n\\[ \\lim_{n \\rightarrow \\infty} \\int_{- \\pi}^{\\pi} \\frac{n! \\, 2^{2n \\cos \\, \\theta} }{|(2ne^{i \\theta} - 1)(2ne^{i \\theta} - 2)(2ne^{i \\theta} - 3) \\cdots (2ne^{i \\theta} - n) |} \\, d \\theta. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 5
} | 7 | 0 | 0 |
omr-ws-6ac08beaf0a017e6426fd4192d145be3401e3c0229ad3dca1b241197245fe279 | Calculate \[ \lim_{n \to \infty } \left( \frac{\sqrt[n]{a} + \sqrt[n]{a+2} + \cdots + \sqrt[n]{a+2p}}{\sqrt[n]{a+1} + \sqrt[n]{a+3} + \cdots + \sqrt[n]{a+2p+1}} \right)^{n}, \] where \( p \in \mathbb{N} \) and \( a \in (0, \infty) \). | [
"\\(\\left( \\prod_{k=0}^{p} \\frac{a+2k}{a+2k+1}\\right)^{\\frac{1}{p+1}}\\)"
] | [
"Calculate \n\\[ \\lim_{n \\to \\infty } \\left( \\frac{\\sqrt[n]{a} + \\sqrt[n]{a+2} + \\cdots + \\sqrt[n]{a+2p}}{\\sqrt[n]{a+1} + \\sqrt[n]{a+3} + \\cdots + \\sqrt[n]{a+2p+1}} \\right)^{n}, \\]\nwhere \\( p \\in \\mathbb{N} \\) and \\( a \\in (0, \\infty) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-4aa8aadd68501b968b7ca7fd1f153007d2c5484efa706e191b3e964f91e55616 | Calculate \[ \lim_{n \to \infty} \frac{\sum_{k=1}^n f(k) - \int_1^n f(x) \, dx - \gamma_f}{f(n)}. \] | [
"\\(\\frac{1}{2}\\)"
] | [
"Calculate\n\\[ \\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n f(k) - \\int_1^n f(x) \\, dx - \\gamma_f}{f(n)}. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-49848731c6b0b1eebdd089b9a80611a2252ff66563c935af4a10209ffaf0ac43 | Calculate \[ \lim_{n \to \infty} \left( \frac{1}{2} + \sin^2 \frac{\pi}{7} \cdot \sqrt[n]{\sin \frac{\pi}{7}} + \sin^2 \frac{2\pi}{7} \cdot \sqrt[n]{\sin \frac{2\pi}{7}} + \sin^2 \frac{3\pi}{7} \cdot \sqrt[n]{\sin \frac{3\pi}{7}} \right)^n \] | [
"\\( \\sin^{ \\sin^2 \\frac{\\pi}{7} } \\frac{\\pi}{7} \\cdot \\sin^{ \\sin^2 \\frac{2\\pi}{7} } \\frac{2\\pi}{7} \\cdot \\sin^{ \\sin^2 \\frac{3\\pi}{7} } \\frac{3\\pi}{7} \\)"
] | [
"Calculate \n\\[\n\\lim_{n \\to \\infty} \\left( \\frac{1}{2} + \\sin^2 \\frac{\\pi}{7} \\cdot \\sqrt[n]{\\sin \\frac{\\pi}{7}} + \\sin^2 \\frac{2\\pi}{7} \\cdot \\sqrt[n]{\\sin \\frac{2\\pi}{7}} + \\sin^2 \\frac{3\\pi}{7} \\cdot \\sqrt[n]{\\sin \\frac{3\\pi}{7}} \\right)^n\n\\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c739c50484274f0445cb27c1d4a9ad76cb0c67b4588543cb228202072e270aa | Calculate \[ \lim_{n\rightarrow\infty}\sum_{i = 0}^n\sum_{j = 0}^{n - i}\frac{x^j}{i!j!}.\] | [
"\\( e^{x+1} \\)"
] | [
"Calculate\n\\[ \\lim_{n\\rightarrow\\infty}\\sum_{i = 0}^n\\sum_{j = 0}^{n - i}\\frac{x^j}{i!j!}.\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 17
} | 31 | 0 | 0 |
omr-ws-62835fe71751befd780df07f1b1e673db7f54843dc6d2896afec5dc08279bcfd | Calculate \[ \lim_{n\to \infty}\left \{ \frac{1}{n^4}\left ( \sum_{k=1}^{n}k^2\int_{k}^{k+1}x\ln\left [ (x-k)(k+1-x) \right ] dx\right ) \right \}.\] | [
"\\(-\\frac{1}{2}\\)"
] | [
"Calculate \\[ \\lim_{n\\to \\infty}\\left \\{ \\frac{1}{n^4}\\left ( \\sum_{k=1}^{n}k^2\\int_{k}^{k+1}x\\ln\\left [ (x-k)(k+1-x) \\right ] dx\\right ) \\right \\}.\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e8b548a56d3a50b8c5dfd7afe0fa1c889b455311e62a9e44f6a6ae11c594a6f4 | Calculate \[ \lim_{x\to\infty}{e^{\frac{x^3}{9} - \frac{x}{12}}\prod_{k=1}^{x}\left(\frac{k}{x}\right)^{k^2}} \] | [
"\\( e^{\\frac{\\zeta(3)}{4\\pi^2}} \\)"
] | [
"Calculate\n\\[ \\lim_{x\\to\\infty}{e^{\\frac{x^3}{9} - \\frac{x}{12}}\\prod_{k=1}^{x}\\left(\\frac{k}{x}\\right)^{k^2}} \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6819aeb61ca08aa75da7b7d90a42f447d6ffee47984f60f16df7a9259c6d4309 | Calculate \[ \sum_{i=0}^{2012} 2^i f_i \binom{2012}{i} \] where \( f_0 = 0 \), \( f_1 = 1 \), and \( f_n = f_{n-1} + f_{n-2} \) for all \( n \ge 2 \) and \( n \in \mathbb{N} \). | [
"\\( f_{6036} \\)"
] | [
"Calculate\n\\[\n\\sum_{i=0}^{2012} 2^i f_i \\binom{2012}{i}\n\\]\nwhere \\( f_0 = 0 \\), \\( f_1 = 1 \\), and \\( f_n = f_{n-1} + f_{n-2} \\) for all \\( n \\ge 2 \\) and \\( n \\in \\mathbb{N} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 6,
"tir": 1
} | 44 | 0 | 0 |
omr-ws-015e9b2410dd31053dac194de9f0e59d7f5806b9c43c0d8da48693d11b499f9c | Calculate \[ \sum_{k=0}^{\infty} \frac{1}{(2k+1)^6} \] using the fact that equality in Bessel's inequality occurs for the eigenfunction expansion. | [
"\\(\\frac{\\pi^6}{960}\\)"
] | [
"Calculate \\[ \\sum_{k=0}^{\\infty} \\frac{1}{(2k+1)^6} \\] using the fact that equality in Bessel's inequality occurs for the eigenfunction expansion."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-6e38fcbf8e27b0d263185012274e583a5e47f225d889af0d73d20f8e76cf5f9b | Calculate \[ \sum_{k=1}^{\infty} \frac{(-1)^{\lfloor \sqrt{k} + \sqrt{k+1} \rfloor}}{k(k+1)}. \] | [
"\\dfrac{\\pi^2}{3} - 3"
] | [
"Calculate \\[ \\sum_{k=1}^{\\infty} \\frac{(-1)^{\\lfloor \\sqrt{k} + \\sqrt{k+1} \\rfloor}}{k(k+1)}. \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-f7845f48efad11ac2660a1893c8b55b90bac98e11b4d9cc62c7cb320de90acb0 | Calculate \[ \sum_{n=0}^\infty (-1)^{n+1}\frac{H_n}{n+1}\left(\frac{1}{n+1}-\frac{1}{n+3}+\frac{1}{n+5}-\cdots\right), \] where \( H_n = \sum_{k=1}^n \frac{1}{k} \) is the \( n \)th harmonic number. | [
"\\(\\frac{\\zeta(3)}{8} - \\frac{1}{192}\\log(2)\\left(\\pi^2 + 4\\log^2(2)\\right)\\)"
] | [
"Calculate \n\\[\n\\sum_{n=0}^\\infty (-1)^{n+1}\\frac{H_n}{n+1}\\left(\\frac{1}{n+1}-\\frac{1}{n+3}+\\frac{1}{n+5}-\\cdots\\right),\n\\]\nwhere \\( H_n = \\sum_{k=1}^n \\frac{1}{k} \\) is the \\( n \\)th harmonic number."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a4bcb15278dea76deb7ffb7e33ff495ebf896973f4630265119832525acffa13 | Calculate \[L=\lim_{n \to \infty }\left ( \sum_{k=1}^{n} \frac{\cos^{3}k}{n+k}\right )\] | [
"0"
] | [
"Calculate \\[L=\\lim_{n \\to \\infty }\\left ( \\sum_{k=1}^{n} \\frac{\\cos^{3}k}{n+k}\\right )\\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-393c10fc66c60cf6eec586194bb0d81bd86938a4aec870acd06d41702edde52c | Calculate \[T = \sum \frac{1}{n_1! \cdot n_2! \cdot \cdots n_{1994}! \cdot (n_2 + 2 \cdot n_3 + 3 \cdot n_4 + \ldots + 1993 \cdot n_{1994})!}\] where the sum is taken over all 1994-tuples of the numbers \(n_1, n_2, \ldots, n_{1994} \in \mathbb{N} \cup \{0\}\) satisfying \(n_1 + 2 \cdot n_2 + 3 \cdot n_3 + \ldots + 1994... | [
"\\(\\frac{\\binom{3987}{1994}}{1994!}\\)",
"\\dfrac{\\dbinom{3987}{1994}}{1994!}"
] | [
"Calculate\n\\[T = \\sum \\frac{1}{n_1! \\cdot n_2! \\cdot \\cdots n_{1994}! \\cdot (n_2 + 2 \\cdot n_3 + 3 \\cdot n_4 + \\ldots + 1993 \\cdot n_{1994})!}\\]\nwhere the sum is taken over all 1994-tuples of the numbers \\(n_1, n_2, \\ldots, n_{1994} \\in \\mathbb{N} \\cup \\{0\\}\\) satisfying \\(n_1 + 2 \\cdot n_2 ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 17,
"genselect": 12
} | 29 | 0 | 0 |
omr-ws-028229a1bebc2d7535b11413eac7e225abe96987a8ad49f8a64140b5ccb776a7 | Calculate \[\iint_{D}\cos\bigg(\frac{x+2y}{-x+y}\bigg)\ dxdy,\] where $D$ is the triangular region in $\mathbb{R}^{2}$ with vertices $(0,0)$, $(-2,4)$, and $(-3,3)$. | [
"6\\left( \\sin 1 - \\sin \\frac{1}{2} \\right)"
] | [
"Calculate \\[\\iint_{D}\\cos\\bigg(\\frac{x+2y}{-x+y}\\bigg)\\ dxdy,\\] where $D$ is the triangular region in $\\mathbb{R}^{2}$ with vertices $(0,0)$, $(-2,4)$, and $(-3,3)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-828c4e5090456889bd9501b5be41a42387346fd3604c3bb3f0f19fe135f498f3 | Calculate \[\int \dfrac{dx}{(x^4+1)^2}\] | [
"\\(\\frac{1}{32} \\left(\\frac{8 x}{x^4+1}-3 \\sqrt{2} \\log \\left(x^2-\\sqrt{2} x+1\\right)+3 \\sqrt{2} \\log \\left(x^2+\\sqrt{2} x+1\\right)-6 \\sqrt{2} \\tan ^{-1}\\left(1-\\sqrt{2} x\\right)+6 \\sqrt{2} \\tan ^{-1}\\left(\\sqrt{2} x+1\\right)\\right)\\)"
] | [
"Calculate \\[\\int \\dfrac{dx}{(x^4+1)^2}\\]"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 19
} | 27 | 0 | 0 |
omr-ws-183403bedbbd92c5ec48f7a3c998dd4675531aa88f1e2bf5653af7ac77808cef | Calculate \[\int\limits_0^\infty e^{\alpha x}\cos(\alpha k)\,d\alpha,\] where \(x < 0\) and \(k \in \mathbb{Z}.\) | [
"\\(-\\frac{x}{k^2 + x^2}\\)"
] | [
"Calculate \\[\\int\\limits_0^\\infty e^{\\alpha x}\\cos(\\alpha k)\\,d\\alpha,\\] where \\(x < 0\\) and \\(k \\in \\mathbb{Z}.\\)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-71155b7a4388340bfb07b8ad276caa0887f3b4eff9b0e46ebffa974883c537de | Calculate \[\int_0^1 \frac{\ln x \ln(1-x)}{x} \ \mathrm{d}x\] | [
"\\(\\zeta(3)\\)"
] | [
"Calculate \\[\\int_0^1 \\frac{\\ln x \\ln(1-x)}{x} \\ \\mathrm{d}x\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-ef6ea875f44d23c5b07cc8007edc60fac17fd899aa2d4c365d5170314c9345b3 | Calculate \[\int_{0}^{\infty} \frac{1}{x^2 (\tan^2 x + \cot^2 x)} \,dx \]. | [
"\\(\\frac{\\pi}{2\\sqrt{2}}\\)"
] | [
"Calculate \\[\\int_{0}^{\\infty} \\frac{1}{x^2 (\\tan^2 x + \\cot^2 x)} \\,dx \\]."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 4,
"tir": 16
} | 21 | 0 | 0 |
omr-ws-0fdfe22372c3baca77b9115c0c691db43f092e729cfa05438c52efdda959941f | Calculate \[\int_{\gamma}{\log_k(z) \, dz}\] where \(\gamma\) is a contour going from \(-i\) to \(-1\) and is contained in quadrants 1, 2, and 4. | [
"\\dfrac{1 + \\dfrac{\\pi}{2} - i(\\pi + 1)}{\\ln k}"
] | [
"Calculate \\[\\int_{\\gamma}{\\log_k(z) \\, dz}\\] where \\(\\gamma\\) is a contour going from \\(-i\\) to \\(-1\\) and is contained in quadrants 1, 2, and 4."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-61c4be95cfe12fa45be3f5e8004b15e24e6e1e0ad3342c76aa14edbefe17b0e8 | Calculate \[\int_{\gamma}{f(z)dz}\] where \( f(z) = \frac{1}{z} \) and \(\gamma\) is a contour running from -1 to \(i\), and is contained in quadrants 1 and 4. | [
"-\\dfrac{\\pi i}{2}"
] | [
"Calculate \\[\\int_{\\gamma}{f(z)dz}\\] where \\( f(z) = \\frac{1}{z} \\) and \\(\\gamma\\) is a contour running from -1 to \\(i\\), and is contained in quadrants 1 and 4."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-704c47e77780f7cd51ca7e5eedff119af9dc54da30a53e3354eaf77e9012cdfb | Calculate \[\left \lfloor \tan^4 \frac{3\pi}{7}+\tan^4 \frac{2\pi}{7}+2\left(\tan^2 \frac{3\pi}{7}+\tan^2 \frac{2\pi}{7}\right) \right \rfloor\] | [
"412"
] | [
"Calculate \\[\\left \\lfloor \\tan^4 \\frac{3\\pi}{7}+\\tan^4 \\frac{2\\pi}{7}+2\\left(\\tan^2 \\frac{3\\pi}{7}+\\tan^2 \\frac{2\\pi}{7}\\right) \\right \\rfloor\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 26
} | 35 | 0 | 0 |
omr-ws-b0eb34c360a300e6fbcde71d8e6a6a5dbd9221c4bb85c99d08424cbffa74cfa3 | Calculate \[\lim_{n \to \infty}\frac{1}{n^2}\sum_{k=0}^{n}\ln \binom{n}{k}\]. | [
"\\(\\frac{1}{2}\\)"
] | [
"Calculate \\[\\lim_{n \\to \\infty}\\frac{1}{n^2}\\sum_{k=0}^{n}\\ln \\binom{n}{k}\\]."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-0b7f8ad82eabd06647c2e0f61055f94596c9accbab125670a25175888df1bcd1 | Calculate \[\lim_{n\rightarrow\infty}\left(\prod_{k=1}^{n}\left(\tan\frac{2k\pi }{2n+1}+\tan\frac{k\pi}{2n+1}+1\right)^{\frac{1}{2n+1}}\right)\] | [
"2"
] | [
"Calculate \\[\\lim_{n\\rightarrow\\infty}\\left(\\prod_{k=1}^{n}\\left(\\tan\\frac{2k\\pi }{2n+1}+\\tan\\frac{k\\pi}{2n+1}+1\\right)^{\\frac{1}{2n+1}}\\right)\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-ba8a8e4afaa6ed3ed9eb9348a657c2cc543bd68b3b17bc15deef48d9585e5c81 | Calculate \[\lim_{x\to 0^{+}}\frac{\sqrt{\sin x+\cos x+\tan x}-1}{x^2}.\] | [
"\\( +\\infty \\)"
] | [
"Calculate \\[\\lim_{x\\to 0^{+}}\\frac{\\sqrt{\\sin x+\\cos x+\\tan x}-1}{x^2}.\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-ddbca4b7c77158894cdaea67cfc17f922286026821022402407bc659af6c27b9 | Calculate \[\lim_{x\to+\infty}\left[x\left(\sqrt{x^{2}+2x}-\sqrt{x^{2}+x}+x\right)\right]\] | [
"\\(\\infty\\)"
] | [
"Calculate \\[\\lim_{x\\to+\\infty}\\left[x\\left(\\sqrt{x^{2}+2x}-\\sqrt{x^{2}+x}+x\\right)\\right]\\]"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-3f79d69ddc314093970c73e2aa98a76384e5ef4c99a523d36626f1d75e4df2a2 | Calculate \[\mathop {\lim }\limits_{n \to + \infty } n\left( {\ln 2 - \dfrac{1}{{n + 1}} - \dfrac{1}{{n + 2}} - \cdots - \dfrac{1}{{2n}}} \right)\] | [
"\\(\\frac{1}{4}\\)"
] | [
"Calculate \\[\\mathop {\\lim }\\limits_{n \\to + \\infty } n\\left( {\\ln 2 - \\dfrac{1}{{n + 1}} - \\dfrac{1}{{n + 2}} - \\cdots - \\dfrac{1}{{2n}}} \\right)\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-e1875d295c20ee11da92ecf222766482e4b6c12bbfc29b3b34518dcc37ac77be | Calculate \[\mathop {\lim }\limits_{n \to \infty } {\left( {\left( {1 + \frac{1}{{2n}}} \right)\left( {1 + \frac{3}{{2n}}} \right)...\left( {1 + \frac{{2n - 1}}{{2n}}} \right)} \right)^{\dfrac{1}{{2n}}}}\] | [
"\\(\\sqrt{\\frac{4}{e}}\\)"
] | [
"Calculate \\[\\mathop {\\lim }\\limits_{n \\to \\infty } {\\left( {\\left( {1 + \\frac{1}{{2n}}} \\right)\\left( {1 + \\frac{3}{{2n}}} \\right)...\\left( {1 + \\frac{{2n - 1}}{{2n}}} \\right)} \\right)^{\\dfrac{1}{{2n}}}}\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-99ea566cdc9c002aaf347c69ee28b72196360737bef33b6772573de189c37146 | Calculate \[\prod_{n = 1}^{13} \frac{n(n + 2)}{(n + 4)^2}.\] | [
"\\frac{3}{161840}"
] | [
"Calculate\n\\[\\prod_{n = 1}^{13} \\frac{n(n + 2)}{(n + 4)^2}.\\]"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1379b497378cdbaa63bb1629202cbd217b6e9fd9f39ae0d81a5fe49eee11e07b | Calculate \[\sum_{i=1}^\infty\sum_{j=1}^\infty \frac{i^{2}j}{5^{i}(j5^{i}+i5^{j})}\] | [
"\\(\\frac{25}{512}\\)"
] | [
"Calculate \\[\\sum_{i=1}^\\infty\\sum_{j=1}^\\infty \\frac{i^{2}j}{5^{i}(j5^{i}+i5^{j})}\\]"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 27
} | 44 | 0 | 0 |
omr-ws-b188ff53e8744718ea677b3e57e8a2110a0906a0e8bf0ff585b649f73bedc684 | Calculate \[\sum_{k=0}^{2n}(-1)^k a_k^2\] where the coefficients \(a_k\) are given by the expansion \[(1-\sqrt{2} x + x^2)^n = \sum_{k=0}^{2n} a_k x^k.\] | [
"\\(\\begin{cases} \n\\binom{n}{n/2} & \\text{if } n \\text{ is even} \\\\\n0 & \\text{if } n \\text{ is odd}\n\\end{cases}\\)"
] | [
"Calculate \\[\\sum_{k=0}^{2n}(-1)^k a_k^2\\] where the coefficients \\(a_k\\) are given by the expansion \\[(1-\\sqrt{2} x + x^2)^n = \\sum_{k=0}^{2n} a_k x^k.\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 16
} | 17 | 0 | 0 |
omr-ws-d18ed1c891e5fa82392a7029f60c9537cecc127d6aef6c62eeb3cef805ebafb6 | Calculate \[\sum_{m\geq 0}\sum_{n\geq 0}\dfrac{\min(m,n)}{3^{m+n}}\] where \(\min(m,n)\) is the minimum number among \(m\) and \(n\). | [
"\\(\\frac{9}{32}\\)"
] | [
"Calculate \\[\\sum_{m\\geq 0}\\sum_{n\\geq 0}\\dfrac{\\min(m,n)}{3^{m+n}}\\] where \\(\\min(m,n)\\) is the minimum number among \\(m\\) and \\(n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 26
} | 44 | 0 | 0 |
omr-ws-69863d5545f1e545fbc23e6278ee6062926daed8e6302c7fd083de67eaa49091 | Calculate \[{\sum_{k=1}^\infty{\left(1+\frac{1}{2}+\dotsb+\frac{1}{k}-\ln\!\left(k+\dfrac{1}{2}\right)-\gamma\right)}}\textnormal{.}\] | [
"\\(\\gamma + \\frac{1}{2} - \\frac{3}{2} \\ln 2\\)"
] | [
"Calculate \\[{\\sum_{k=1}^\\infty{\\left(1+\\frac{1}{2}+\\dotsb+\\frac{1}{k}-\\ln\\!\\left(k+\\dfrac{1}{2}\\right)-\\gamma\\right)}}\\textnormal{.}\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-48442488ee4f0863634bdc4977473e5609abd6f010e82fc1ba96473569e14f5f | Calculate a basis of the solution space of the equation \(Ax = 0\) for the matrix \(A = \begin{bmatrix} 1&2&3&4\\-1&2&1&2\\-1&1&2&3\\-1&5&6&9\end{bmatrix}\). | [
"\\(\\{(1, -1, -5, 4)\\}\\)"
] | [
"Calculate a basis of the solution space of the equation \\(Ax = 0\\) for the matrix \\(A = \\begin{bmatrix} 1&2&3&4\\\\-1&2&1&2\\\\-1&1&2&3\\\\-1&5&6&9\\end{bmatrix}\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-7e84f63dc857a5029d468982cfe62c4267e9d5821b9669f7c4de1e2dad1188b6 | Calculate a natural number $N$ such that the sum of ternary variations formed from five different digits equals $N$. | [
"35964"
] | [
"Calculate a natural number $N$ such that the sum of ternary variations formed from five different digits equals $N$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-271bbb50b0ebcca7ea020f11bee536a60730558a9f5b9cf35d8c8ebf4ef59e27 | Calculate all possible lengths of the connecting segments between $P_1$ and the remaining seven vertices of a regular octagon inscribed in a circle with radius $r = 1$. | [
"\\(\\sqrt{2 - \\sqrt{2}}, \\sqrt{2}, \\sqrt{2 + \\sqrt{2}}, 2\\)"
] | [
"Calculate all possible lengths of the connecting segments between $P_1$ and the remaining seven vertices of a regular octagon inscribed in a circle with radius $r = 1$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 31
} | 36 | 0 | 0 |
omr-ws-2f536b0d6f5bccc618a06edb0138a53d9d99615d457b3df858374f7fcb05bd9f | Calculate all real numbers \( r \) such that if real numbers \( a, b, c \) satisfy the inequality \( | ax^2 + bx + c | \le 1 \) for each \( x \in [-1, 1] \), then they also satisfy the inequality \( | cx^2 + bx + a | \le r \) for each \( x \in [-1, 1] \). | [
"2"
] | [
"Calculate all real numbers \\( r \\) such that if real numbers \\( a, b, c \\) satisfy the inequality \\( | ax^2 + bx + c | \\le 1 \\) for each \\( x \\in [-1, 1] \\), then they also satisfy the inequality \\( | cx^2 + bx + a | \\le r \\) for each \\( x \\in [-1, 1] \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06b2849470a4d8e06ada2560706dc61b72aed2dbe9dc60bc0c06f1c6336235b8 | Calculate all root values of the expression \(\sqrt[6]{\frac{3-i\sqrt{3}}{-1-i}}\). | [
"\\(\\sqrt[12]{6}\\left[\\cos \\frac{\\frac{7\\pi}{12}+2k\\pi}{6}+i \\cdot \\sin \\frac{\\frac{7\\pi}{12}+2k\\pi}{6}\\right] \\text{ for } k = 0, 1, 2, 3, 4, 5\\)"
] | [
"Calculate all root values of the expression \\(\\sqrt[6]{\\frac{3-i\\sqrt{3}}{-1-i}}\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-1beb959d594c8f777b3b64cc13cc54e21d0c3a315d632ce1811fa18a26f33d3f | Calculate all the solutions $(m, n)$ of integers that satisfy the equation $mn^2 = 2009 (n + 1)$. | [
"\\((-246, -7), (0, -1), (4018, 1), (328, 7)\\)",
"\\((4018, 1), (0, -1), (328, 7), (-246, -7)\\)"
] | [
"Calculate all the solutions $(m, n)$ of integers that satisfy the equation $mn^2 = 2009 (n + 1)$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 56
} | 56 | 0 | 0 |
omr-ws-4694fc2f6c1a487a431b3d922bb7ee541ed2ea04c519a2199b4dc5c13796b391 | Calculate all values of \( m \) such that the roots of the equation \( x^4 - (3m+1)x^2 + m^2 = 0 \) form an arithmetic sequence. | [
"\\( m = \\pm 3 \\)"
] | [
"Calculate all values of \\( m \\) such that the roots of the equation \\( x^4 - (3m+1)x^2 + m^2 = 0 \\) form an arithmetic sequence."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e54759b383bf58c0504b3c0266b63be752dd48ae441d39c00e39a7b68402a15a | Calculate and classify the stationary points for the function \( z = f(x, y) = \ln(1 + x^2 + y^2) - \frac{1}{2}x^2 - y^2 + 1 \). | [
"(-1, 0)"
] | [
"Calculate and classify the stationary points for the function \\( z = f(x, y) = \\ln(1 + x^2 + y^2) - \\frac{1}{2}x^2 - y^2 + 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-5892608da5fb815f7560c596e464d638c44a23b4397f775c61ced373948b1c61 | Calculate and determine the area of convergence for \((p, q)\) of the following integral: $$\int_{1}^{\infty}(x^r-1)^px^{-q}dx$$ where \(r > 0\). | [
"\\( p > -1 \\) and \\( q > pr + 1 \\)"
] | [
"Calculate and determine the area of convergence for \\((p, q)\\) of the following integral:\n$$\\int_{1}^{\\infty}(x^r-1)^px^{-q}dx$$\nwhere \\(r > 0\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-1353f85830c0160bcd542b0840a57cb2267ff81195dee669fa28643b147f6e8c | Calculate by changing variables: $$\iint_D \left(3x+2xy \right) dxdy,$$ where $D$ is defined by the conditions $1 \le xy \le 9$ and $y \le x \le 4y$. | [
"48 + 40 \\ln 4"
] | [
"Calculate by changing variables:\n$$\\iint_D \\left(3x+2xy \\right) dxdy,$$\nwhere $D$ is defined by the conditions $1 \\le xy \\le 9$ and $y \\le x \\le 4y$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-65c299691a725bb1ee4bc99156fbbee22c0fe81561300a8ad1c367097ffcc047 | Calculate for \( a \in (0, 2) \): \[ \int_{a/3}^{2a/3} \frac{(ax - x^2) \ln(x)}{\ln(ax - x^2)} \, dx \] | [
"\\dfrac{13a^3}{324}"
] | [
"Calculate for \\( a \\in (0, 2) \\):\n\\[\n\\int_{a/3}^{2a/3} \\frac{(ax - x^2) \\ln(x)}{\\ln(ax - x^2)} \\, dx\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-d8b5dec3471ffb2d0cfbd990aa456ce61e5b766b79d634ee700e62e8cb2766a2 | Calculate for \( a \in \mathbb{N} \): \[ \int_0^\frac{\pi}{2} \ln(a + \sin x) \, dx \] | [
"\\( 2Ti_2 \\big(a+\\sqrt{a^2-1}\\big)-\\frac{\\pi}{2} \\big( \\log2 +\\cosh^{-1}a\\big) \\)"
] | [
"Calculate for \\( a \\in \\mathbb{N} \\):\n\\[ \\int_0^\\frac{\\pi}{2} \\ln(a + \\sin x) \\, dx \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4add8101341455cb32b8037c51fd9af8f42add6604a75e10581a158539634a53 | Calculate for \( x \in [0, \pi/2] \): \[ \int_0^{\sin^2 x} \sin^{-1}(\sqrt{t}) \, dt + \int_0^{\cos^2 x} \cos^{-1}(\sqrt{t}) \, dt \] | [
"\\(\\pi/4\\)"
] | [
"Calculate for \\( x \\in [0, \\pi/2] \\):\n\\[ \\int_0^{\\sin^2 x} \\sin^{-1}(\\sqrt{t}) \\, dt + \\int_0^{\\cos^2 x} \\cos^{-1}(\\sqrt{t}) \\, dt \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-d35a04a14a5625b7181d191bf1ad4f65e8daffdf000af5f76b1fe17b429eaa65 | Calculate for any $a > 0$ the integral \[ \int_0^1 \left( (1-x^a)^{1/a} - x \right)^2 \, dx \] | [
"\\(\\frac{1}{3}\\)"
] | [
"Calculate for any $a > 0$ the integral\n\n\\[\n\\int_0^1 \\left( (1-x^a)^{1/a} - x \\right)^2 \\, dx\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-f544de369dfcf6cf527a5863abab0f9bafaa6937a9be57c059746328a2762850 | Calculate in $\mathcal{D}'(-\pi,\pi)$, the derivatives of order $1, 2, 3,$ and $4$ of the associated distribution to $|\sin(x)|$. | [
"|\\sin(x)| - 2\\delta(x) + 2\\delta''(x)"
] | [
"Calculate in $\\mathcal{D}'(-\\pi,\\pi)$, the derivatives of order $1, 2, 3,$ and $4$ of the associated distribution to $|\\sin(x)|$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-fb7eafb32f901ff93a69fa3e9e8d6ff4844eb1ae56ed4a5fe639ed66f19bf82f | Calculate in how many cases the sum of a three-digit number and the number obtained by inverting its digits results in a number with all its digits odd. | [
"80"
] | [
"Calculate in how many cases the sum of a three-digit number and the number obtained by inverting its digits results in a number with all its digits odd."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc9bc956604376c02bf8d8830dbf17bb678fa72138eb5697cf240ecd38335d82 | Calculate in terms of \( k, r, n \): \[ \sum_{i_{1}=0}^{k-1} \binom{r+k-i_{1}}{r-1} \binom{n-kr-r+i_{1}}{n-kr-r} \] | [
"\\(\\binom{n-kr+k+1}{n-kr} - \\binom{n-kr-r+k+1}{n-kr-r} - r \\binom{n-kr-r+k}{n-kr-r}\\)"
] | [
"Calculate in terms of \\( k, r, n \\):\n\\[\n\\sum_{i_{1}=0}^{k-1} \\binom{r+k-i_{1}}{r-1} \\binom{n-kr-r+i_{1}}{n-kr-r}\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6bcf304365b597523f3b5101736c993170262bbf7d9f159552165ae1aa173555 | Calculate or approximate the following summation when \( r = \delta n \) where \( 0 < \delta < 1 \) for large \( n \): \[ \sum_{i=0}^{r} \left( {n+r+1 \choose i+1} + 2^{-i} \sum_{k=0}^{i} {n \choose k} \right) \] | [
"\\left( \\frac{(1 + \\delta)^{1 + \\delta}}{\\delta^{\\delta}} \\right)^n"
] | [
"Calculate or approximate the following summation when \\( r = \\delta n \\) where \\( 0 < \\delta < 1 \\) for large \\( n \\):\n\n\\[\n\\sum_{i=0}^{r} \\left( {n+r+1 \\choose i+1} + 2^{-i} \\sum_{k=0}^{i} {n \\choose k} \\right)\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-7198bbe91f8cbeee8c22811913525830b3186b8db95d8f6948bf2c53e2b71792 | Calculate the $2^{2020}$ decimal place of $\sqrt{2}$. | [
"6"
] | [
"Calculate the $2^{2020}$ decimal place of $\\sqrt{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-01e2849ab9867c176965098b6d3a82cc8151aaab46b78606d1cc23a290669221 | Calculate the 100th derivative of $\frac{1}{x^2 + 3x + 2}$ at $x = 0$ with a 10% relative error. | [
"\\( 100!(1 - 2^{-101}) \\approx 9.33 \\times 10^{157} \\)"
] | [
"Calculate the 100th derivative of $\\frac{1}{x^2 + 3x + 2}$ at $x = 0$ with a 10% relative error."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-806acb413a9721128973bcb4c2a4a5b3283ba857eb7c59e3522ced1b96cab552 | Calculate the 12 monthly means of the rank of 366 dates placed in individual capsules, where the first capsule is marked as rank 1, the second as rank 2, and so on. Then, plot these 12 means on a graph with the horizontal scale listing the 12 months and the vertical scale ranging from 100 to 260. Note any pattern sugge... | [
"122"
] | [
"Calculate the 12 monthly means of the rank of 366 dates placed in individual capsules, where the first capsule is marked as rank 1, the second as rank 2, and so on. Then, plot these 12 means on a graph with the horizontal scale listing the 12 months and the vertical scale ranging from 100 to 260. Note any pattern ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-d5d7fb989838e35270f4708ff963ca5c16b69ea6321f27a9358eabec32fdba60 | Calculate the 99th term of the series: 2, 7, 14, 23, 24, ... | [
"542"
] | [
"Calculate the 99th term of the series: 2, 7, 14, 23, 24, ..."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-2dfab013c93509c4ecc6c4935659fdf2b50afe3c8489973f14750259b4809fbd | Calculate the Annual Percentage Rate (APR) that Robert is charged for credit, given the following details: - Cash price of the video recorder: £229.99 - Deposit: 30% of the cash price - Monthly payments: 18 payments of £10.20 each - Total credit price: £252.60 - Loan amount: £229.99 - deposit Note: The deposit is appro... | [
"17.1\\%"
] | [
"Calculate the Annual Percentage Rate (APR) that Robert is charged for credit, given the following details:\n- Cash price of the video recorder: £229.99\n- Deposit: 30% of the cash price\n- Monthly payments: 18 payments of £10.20 each\n- Total credit price: £252.60\n- Loan amount: £229.99 - deposit\n\nNote: The dep... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-80858e9d3bac50ca988ff9706c4e0cbfb04385cd68316996ab1a217f1c758a59 | Calculate the Arithmetic Mean, Harmonic Mean, and the Geometric Mean for the following data set: - Frequency 12,345 with value 987 - Frequency 23,456 with value 876 - Frequency 34,567 with value 765 - Frequency 45,678 with value 654 - Frequency 56,789 with value 543 | [
"Arithmetic Mean ≈ 700.00, Harmonic Mean ≈ 670.00, Geometric Mean ≈ 346.00"
] | [
"Calculate the Arithmetic Mean, Harmonic Mean, and the Geometric Mean for the following data set:\n- Frequency 12,345 with value 987\n- Frequency 23,456 with value 876\n- Frequency 34,567 with value 765\n- Frequency 45,678 with value 654\n- Frequency 56,789 with value 543"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b2236273e865b928f527c3e9860b112838edbea7f26a1b4cb26da6a201a60c8 | Calculate the Casimir operator for $\mathfrak{sl}(n, \mathbb{C})$ using the given basis: ($H_{i}, X_{\alpha}, X_{-\alpha}$), where $X_{\alpha}$ are root vectors $E_{kl}$ with $k > l$, and $H_{i}$ are given by: \[ H_{1} = \text{diag}(1, -1, 0, \dots, 0), \quad H_{2} = \text{diag}(0, 1, -1, 0, \dots, 0), \quad \dots, \qu... | [
"\\text{The coefficients of the Cartan terms should involve the inverse Killing form matrix, not the Killing form itself.}"
] | [
"Calculate the Casimir operator for $\\mathfrak{sl}(n, \\mathbb{C})$ using the given basis: ($H_{i}, X_{\\alpha}, X_{-\\alpha}$), where $X_{\\alpha}$ are root vectors $E_{kl}$ with $k > l$, and $H_{i}$ are given by:\n\\[ H_{1} = \\text{diag}(1, -1, 0, \\dots, 0), \\quad H_{2} = \\text{diag}(0, 1, -1, 0, \\dots, 0),... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88180de4b4770538ffa7009486fd97e7f4f834f43e00b3ba5f529693ea8a64cf | Calculate the Cauchy principal value of the integral $$\operatorname{PV}\int\limits_{-\infty }^{\infty}\frac{\left | \sin x-\cos x \right |}{x^2-1}dx$$ | [
"\\pi"
] | [
"Calculate the Cauchy principal value of the integral\n$$\\operatorname{PV}\\int\\limits_{-\\infty }^{\\infty}\\frac{\\left | \\sin x-\\cos x \\right |}{x^2-1}dx$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-7d58dbd3b3ab861cdb9af1939f9e47fe3a3c3b1addccce170ad2b02cc067a558 | Calculate the Cauchy principal value of the integral $$\operatorname{PV}\int\limits_{-\infty }^{\infty}\frac{\left | \sin x-\cos x \right |}{x}dx$$ | [
"-\\ln(\\sqrt2+1)"
] | [
"Calculate the Cauchy principal value of the integral\n$$\\operatorname{PV}\\int\\limits_{-\\infty }^{\\infty}\\frac{\\left | \\sin x-\\cos x \\right |}{x}dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-81d0821c428504ecbb5fe8525f78339044c2bf48375d1354ff9708f792d56b89 | Calculate the Cramér-Rao bound (CRB) for $X_1, \ldots, X_n$ coming from a Pareto distribution with density $f_{\theta}(x) = c^{\theta} \theta x^{-(1+\theta)} 1_{\{x > c\}}$, where $\theta > 1$ and $c > 0$. | [
"\\(\\frac{\\theta^2}{n}\\)"
] | [
"Calculate the Cramér-Rao bound (CRB) for $X_1, \\ldots, X_n$ coming from a Pareto distribution with density $f_{\\theta}(x) = c^{\\theta} \\theta x^{-(1+\\theta)} 1_{\\{x > c\\}}$, where $\\theta > 1$ and $c > 0$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-b496a9cfe55dad5f97d000d29636ee4fe8de8ea0d37b4ecd6702593ae6cb4ec7 | Calculate the Discrete Fourier Transform of the sample given by \( x_0 = 1 \), \( x_1 = 2 \), \( x_2 = 3 \), \( x_3 = 4 \), \( x_4 = 5 \), and \( x_5 = 6 \). | [
"\\( X[0] = 21 \\), \\( X[1] = -\\frac{6}{1 - e^{-j\\frac{\\pi}{3}}} \\), \\( X[2] = -\\frac{6}{1 - e^{-j\\frac{2\\pi}{3}}} \\), \\( X[3] = -3 \\), \\( X[4] = -\\frac{6}{1 - e^{-j\\frac{4\\pi}{3}}} \\), \\( X[5] = -\\frac{6}{1 - e^{-j\\frac{5\\pi}{3}}} \\)"
] | [
"Calculate the Discrete Fourier Transform of the sample given by \\( x_0 = 1 \\), \\( x_1 = 2 \\), \\( x_2 = 3 \\), \\( x_3 = 4 \\), \\( x_4 = 5 \\), and \\( x_5 = 6 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 1,
"tir": 14
} | 26 | 0 | 0 |
omr-ws-f7dd803662ba223c2ff6566737c030335322684e2b867c38275e8c86c875a0f3 | Calculate the Fourier Transform of \( t(r, \theta) = \delta(r - a) A(\theta) \) where: \[ A(\theta) = \begin{cases} 1 & \text{if } -\pi/2 < \theta \le \pi/2 \\ 0 & \text{otherwise} \end{cases} \] | [
"T(\\rho, \\phi) = \\frac{a}{2} J_0(a \\rho) - \\frac{2a i}{\\pi} \\sum_{m=0}^\\infty \\frac{J_{2m+1}(a \\rho)}{2m+1} e^{i (2m+1) \\phi}"
] | [
"Calculate the Fourier Transform of \\( t(r, \\theta) = \\delta(r - a) A(\\theta) \\) where:\n\\[ A(\\theta) = \\begin{cases} \n1 & \\text{if } -\\pi/2 < \\theta \\le \\pi/2 \\\\\n0 & \\text{otherwise}\n\\end{cases} \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd2ea3ef7ef96d6f99bd1a00485bd2cb424ec385537a348ddb5e73c1fb008ba8 | Calculate the Fourier Transform of \[\frac{4}{t^2+4}\]. Please show steps. | [
"2\\pi e^{-2|\\omega|}"
] | [
"Calculate the Fourier Transform of \\[\\frac{4}{t^2+4}\\]. Please show steps."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b466c88d23802f12fcd553a6b205efdfc9f98d909a0b02fa69e53eed30c73cb9 | Calculate the Fourier series of the function $\cos^{4}{(x/2)}$. | [
"\\frac{3}{8} + \\frac{1}{2}\\cos(x) + \\frac{1}{8}\\cos(2x)"
] | [
"Calculate the Fourier series of the function $\\cos^{4}{(x/2)}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3b3ae4c59d8f34d455b9ba65c355c3de28ad58ba9d88a44dd13fbd6532a6b80d | Calculate the Fourier series of the function $\sin^{3}{(x)}$. | [
"\\frac{3}{4}\\sin(x) - \\frac{1}{4}\\sin(3x)"
] | [
"Calculate the Fourier series of the function $\\sin^{3}{(x)}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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