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-908e44b8bbe08886bb3f7eaa0fdaf6671943996c9b226c1ea9886823ba8b317d | Calculate the following in significant figures: 1. \( 2.0 \times 3.625 \) 2. \( 0.0000003 \div 1.0 \) | [
"0.0000003"
] | [
"Calculate the following in significant figures:\n1. \\( 2.0 \\times 3.625 \\)\n2. \\( 0.0000003 \\div 1.0 \\)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-571ec3f23d6d73f723308b485887367433b2357d931738aebe1ed931d4da6804 | Calculate the following integral for given positive integers $d, n_0, \ldots, n_d$: \[ \int_{0}^{1}\int_{0}^{1-x_1}\int_{0}^{1-x_1-x_2}\cdots\int_{0}^{1-x_1-\cdots-x_d}(1-x_1-x_2-\cdots-x_d)^{n_0}x_1^{n_{1}}x_2^{n_{2}}\cdots x_d^{n_{d}}(x_1+x_2+\cdots+x_d)(1-x_1)(1-x_2)\cdots(1-x_d)\, \mathrm{d}x_d\, \mathrm{d}x_{d-1}\... | [
"\\dfrac{\\Gamma(n_0 + 1) \\prod_{i=1}^d \\Gamma(n_i + 1)}{\\Gamma(n_0 + \\sum_{i=1}^d n_i + d + 2)} \\left( \\sum_{i=1}^d \\frac{n_i + 1}{n_0 + \\sum_{j=1}^d n_j + d + 2} \\right)"
] | [
"Calculate the following integral for given positive integers $d, n_0, \\ldots, n_d$:\n\\[\n\\int_{0}^{1}\\int_{0}^{1-x_1}\\int_{0}^{1-x_1-x_2}\\cdots\\int_{0}^{1-x_1-\\cdots-x_d}(1-x_1-x_2-\\cdots-x_d)^{n_0}x_1^{n_{1}}x_2^{n_{2}}\\cdots x_d^{n_{d}}(x_1+x_2+\\cdots+x_d)(1-x_1)(1-x_2)\\cdots(1-x_d)\\, \\mathrm{d}x_d... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2bd2662e4c4444385d4cfc2c0589f43cf3ab721d82290f6b7181d1c48822cedf | Calculate the following integral: $$\int\limits_{0\leq p\leq 1, \sum p\leq k}\sum\limits_{1\leq i_{1}< i_{2}< \ldots i_{l}\leq N}\prod\limits_{j= 1}^{l}p_{i_{j}}\prod\limits_{i\neq i_{j}, \forall_{j}\; 1\leq j\leq l}\left ( 1- p_{i} \right ){\rm d}p_{1}{\rm d}p_{2}\ldots{\rm d}p_{N}$$ with the given numbers $p_{1}, p_{... | [
"\\binom{N}{l} \\frac{k^{N}}{N!}"
] | [
"Calculate the following integral:\n$$\\int\\limits_{0\\leq p\\leq 1, \\sum p\\leq k}\\sum\\limits_{1\\leq i_{1}< i_{2}< \\ldots i_{l}\\leq N}\\prod\\limits_{j= 1}^{l}p_{i_{j}}\\prod\\limits_{i\\neq i_{j}, \\forall_{j}\\; 1\\leq j\\leq l}\\left ( 1- p_{i} \\right ){\\rm d}p_{1}{\\rm d}p_{2}\\ldots{\\rm d}p_{N}$$\nw... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-d1844e44f50124af8fce6f0aa457b9fbb72c8a4f3584e107c02a40c03b566f99 | Calculate the following integral: $$\int\limits_{0}^{1}\int\limits_{0}^{1}\int\limits_{0}^{1}\sqrt{x^2+y^2+z^2}dxdydz$$ | [
"\\(\\frac{\\sqrt{3}}{4}-\\frac{\\pi}{24}+\\operatorname{arcth}\\left ( \\frac{1}{\\sqrt{3}} \\right )\\)"
] | [
"Calculate the following integral:\n$$\\int\\limits_{0}^{1}\\int\\limits_{0}^{1}\\int\\limits_{0}^{1}\\sqrt{x^2+y^2+z^2}dxdydz$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-68f81e78e930f3fd964d2de577d4776954d9560374ea4974906558383701c031 | Calculate the following integral: $$\int_0^{\pi/2}\,dx\int_0^{\pi/2}\,dy\ e^{\sin{x}\sin{y}} \frac{\cos^2{x}\sin^2{y}}{\cos^2{x} + \cos^2{y}}$$ | [
"\\(\\frac{(e-1)\\pi}{4}\\)"
] | [
"Calculate the following integral:\n$$\\int_0^{\\pi/2}\\,dx\\int_0^{\\pi/2}\\,dy\\ e^{\\sin{x}\\sin{y}} \\frac{\\cos^2{x}\\sin^2{y}}{\\cos^2{x} + \\cos^2{y}}$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-7dcc650bd331917377fb38e5386f9a0219aed5d8e4f72b74b32acd217f855eea | Calculate the following integral: $$\mathcal{M} = \int_0^{\infty} \int_0^\infty \cdots \int_0^\infty \frac{\prod \limits_{m=1}^{n} \cos (x_m)}{\sum \limits_{m=1}^{n} x_m} \, {\rm d}(x_1, x_2, \dots, x_n)$$ | [
"\\(\\frac{1}{2}\\frac{\\Gamma \\left( \\frac{n + 1}{2} \\right)\\Gamma \\left( \\frac{n - 1}{2} \\right)}{\\Gamma \\left( n \\right)}\\)"
] | [
"Calculate the following integral:\n\n$$\\mathcal{M} = \\int_0^{\\infty} \\int_0^\\infty \\cdots \\int_0^\\infty \\frac{\\prod \\limits_{m=1}^{n} \\cos (x_m)}{\\sum \\limits_{m=1}^{n} x_m} \\, {\\rm d}(x_1, x_2, \\dots, x_n)$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 2
} | 7 | 0 | 0 |
omr-ws-98531944c4c89c81d8509f7a33a602425cb79d02e824335a6b1f517b508294b7 | Calculate the following integral: \[ \iiint\limits_{\begin{subarray}{c} 0 < x,y < 1 \\ 0 < z \end{subarray} } \frac{dx \, dy \, dz}{\left( 1 + x^2 z^2 \right)\left( 1 + y^2 z^2 \right)}. \] | [
"\\( \\pi \\ln 2 \\)"
] | [
"Calculate the following integral:\n\\[ \\iiint\\limits_{\\begin{subarray}{c} 0 < x,y < 1 \\\\ 0 < z \\end{subarray} } \\frac{dx \\, dy \\, dz}{\\left( 1 + x^2 z^2 \\right)\\left( 1 + y^2 z^2 \\right)}. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-f640b4f6b8c1b0c2524d810d624841191e52f6e60a3523bb1e2f341711c8aa0d | Calculate the following integral: \[ \int \sqrt{x^3 \sqrt{x^4 \sqrt{x^5 \sqrt{\cdots}}}} \, dx \] | [
"\\(\\frac{x^5}{5} + C\\)"
] | [
"Calculate the following integral:\n\\[\n\\int \\sqrt{x^3 \\sqrt{x^4 \\sqrt{x^5 \\sqrt{\\cdots}}}} \\, dx\n\\]"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-ae87bc4f06331834ac58f2c0fc955255e68879426a0f22a29cd771050ae52ab6 | Calculate the following integral: \[ \int_0^1 \sqrt{1-x^2} \log \left|1-\frac{1}{x^2}\right| \, dx. \] | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"Calculate the following integral:\n\\[\n\\int_0^1 \\sqrt{1-x^2} \\log \\left|1-\\frac{1}{x^2}\\right| \\, dx.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 20
} | 40 | 0 | 0 |
omr-ws-6839c06209aea44806fc229fb6895292421c0675ff9cf3a7b5a793a07dee224c | Calculate the following integral: \[ \int_0^{\pi/2}\frac{x\sin(2x)}{1+\cos(2x)^2}\,dx \] | [
"\\(\\frac{\\pi^2}{16}\\)"
] | [
"Calculate the following integral:\n\\[\n\\int_0^{\\pi/2}\\frac{x\\sin(2x)}{1+\\cos(2x)^2}\\,dx\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-4bd94db9ebdc44b237a83c1014cf782c2c36c759c35571a1c184ffd10ad85fa9 | Calculate the following integral: \[ \int_{-\frac{ab^{2}}{a+b}}^{\frac{a^{2}b}{a+b}} \left(\log\left(\frac{ab}{a+b}-\frac{x}{a}\right) \cdot \log\left(\frac{ab}{a+b}+\frac{x}{b}\right)\right) \, dx, \quad a, b > 0. \] | [
"\\( ab\\left(\\ln a \\ln b - \\ln(ab) + 2 - \\frac{\\pi^2}{6}\\right) \\)"
] | [
"Calculate the following integral:\n\n\\[\n\\int_{-\\frac{ab^{2}}{a+b}}^{\\frac{a^{2}b}{a+b}} \\left(\\log\\left(\\frac{ab}{a+b}-\\frac{x}{a}\\right) \\cdot \\log\\left(\\frac{ab}{a+b}+\\frac{x}{b}\\right)\\right) \\, dx, \\quad a, b > 0.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 8
} | 29 | 0 | 0 |
omr-ws-cd83cda338dcf4de27a49e84029f5976181c6cb7b3d35fc349804c4f0dcb4ad0 | Calculate the following integral: \[ \int_{0}^{1}\ln(\sqrt{1+x}+\sqrt{1-x})\ln(\sqrt{1+x}-\sqrt{1-x})dx. \] | [
"-0.64243"
] | [
"Calculate the following integral:\n\n\\[\n\\int_{0}^{1}\\ln(\\sqrt{1+x}+\\sqrt{1-x})\\ln(\\sqrt{1+x}-\\sqrt{1-x})dx.\n\\]"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-9ccef0e65cd05bdf9a8052979c982dfa84a1b083a22f9e1db7dfa3ee9e7666e3 | Calculate the following integrals. $\int _0^3 \left(\frac{1}{\sqrt {x^2+4x+4}}+\sqrt {x^2-4x+4}\right)dx$. | [
"$\\ln 2.5+2.5$"
] | [
"Calculate the following integrals. $\\int _0^3 \\left(\\frac{1}{\\sqrt {x^2+4x+4}}+\\sqrt {x^2-4x+4}\\right)dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fff2ab52e49229600114800ed09df6e59a7c108bd9004c71fee605fbd519a2cc | Calculate the following integrals. $\int _0^\pi \cos ^2 \left(\frac{3}{8}\pi -\frac{x}{4}\right) -\cos ^2 \left(\frac{11}{8}\pi +\frac{x}{4}\right)dx$. | [
"$\\sqrt 2$"
] | [
"Calculate the following integrals. $\\int _0^\\pi \\cos ^2 \\left(\\frac{3}{8}\\pi -\\frac{x}{4}\\right) -\\cos ^2 \\left(\\frac{11}{8}\\pi +\\frac{x}{4}\\right)dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e1301dac071456eecae4b405f3471ebd5e2c7ca11a4ed4e6f2649ff40f97db12 | Calculate the following integrals. $\int _0^\pi \cos x\sin 3xdx$. | [
"0"
] | [
"Calculate the following integrals. $\\int _0^\\pi \\cos x\\sin 3xdx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe63df4ea5e01419e8cbfc1b4df963a31bffdf5c31415c224151b03f74ac0ef7 | Calculate the following integrals. $\int _0^{7/3}\frac{x+1}{\sqrt [3]{3x+1}}dx$. | [
"$\\frac{46}{15}$"
] | [
"Calculate the following integrals. $\\int _0^{7/3}\\frac{x+1}{\\sqrt [3]{3x+1}}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-710d5e0e315fe809dc6e7fb458f5bbbc45953dd509e3a89b7a4ebff4ba0b2d66 | Calculate the following integrals. $\int _0^{\pi /4}(\sin 2t-\cos 2t)^2 dt$. | [
"$\\frac{\\pi -2}{4}$"
] | [
"Calculate the following integrals. $\\int _0^{\\pi /4}(\\sin 2t-\\cos 2t)^2 dt$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9cb20625bfb4b3ef7470ecae48b848c3e1fe1b69eed7b9b006383c4f31cd14a | Calculate the following integrals. $\int _1^2 \frac{dx}{\sqrt {x-1}+\sqrt {1+x}}$. | [
"$\\frac{3^{3/2}-2^{3/2}-1}{3}$"
] | [
"Calculate the following integrals. $\\int _1^2 \\frac{dx}{\\sqrt {x-1}+\\sqrt {1+x}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b9477993f14beeedc17682bb8dc9e4e026448597250e8c68089b24015ee4119 | Calculate the following integrals. $\int _3^5 \left(\sqrt {x+2\sqrt {2x-3}}+\sqrt {x-2\sqrt {2x-4}}\right)dx$. | [
"$2\\sqrt 2+\\frac{4}{3}(3^{3/2}-2^{3/2})$"
] | [
"Calculate the following integrals. $\\int _3^5 \\left(\\sqrt {x+2\\sqrt {2x-3}}+\\sqrt {x-2\\sqrt {2x-4}}\\right)dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c928be61462d5f757b8b92b7dcc41dd7549fa413a7a2473dc95123e53353c12b | Calculate the following integrals. $\int _{-1/2}^{1/2}\left[\left(\frac{x+1}{x-1}\right)^2 +\left(\frac{x-1}{x+1}\right)^2-2\right]^{1/2}dx$. | [
"$4\\ln \\frac{4}{3}$"
] | [
"Calculate the following integrals. $\\int _{-1/2}^{1/2}\\left[\\left(\\frac{x+1}{x-1}\\right)^2 +\\left(\\frac{x-1}{x+1}\\right)^2-2\\right]^{1/2}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1d7e03e7090c71369b622125f2b64d7dcf7a27a4c32f3aafee98426953d4237 | Calculate the following integrals. $\int _{-4}^2 \frac{xdx}{\sqrt {2-x/2}}$. | [
"$-2\\frac{2}{3}$"
] | [
"Calculate the following integrals. $\\int _{-4}^2 \\frac{xdx}{\\sqrt {2-x/2}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b28d4709448445b5d66264d65373019cbf10200be8da046922745b8d32c2db53 | Calculate the following integrals. $\int _{-\pi /4}^{\pi /2}\sqrt {1-\cos ^2 x}dx$. | [
"$2-\\frac{\\sqrt 2}{2}$"
] | [
"Calculate the following integrals. $\\int _{-\\pi /4}^{\\pi /2}\\sqrt {1-\\cos ^2 x}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3985104bf59cc18069f45faa3aa2b01082d5040e30db34651b569bc8953c7ae1 | Calculate the following integrals. $\int _{1/3}^{5/3}(x-2)\sqrt {3x-1}dx$. | [
"$-1\\frac{73}{135}$"
] | [
"Calculate the following integrals. $\\int _{1/3}^{5/3}(x-2)\\sqrt {3x-1}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f961dd10e1f2c769ec38968f943e52db5b3d09bbdd474b01dfd6ed6e180a9018 | Calculate the following integrals. $\int _{\pi /2}^{3\pi /2}\sqrt {1-\cos 2x}dx$. | [
"$2\\sqrt 2$"
] | [
"Calculate the following integrals. $\\int _{\\pi /2}^{3\\pi /2}\\sqrt {1-\\cos 2x}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f01e677723ce2deff2fad5cbf70f5e579174063a508d3f34de61d5987e08cbed | Calculate the following integrals. $\int _{\pi /4}^{3\pi /2}\sqrt {1+\cos 2x}dx$. | [
"$2\\sqrt 2-1$"
] | [
"Calculate the following integrals. $\\int _{\\pi /4}^{3\\pi /2}\\sqrt {1+\\cos 2x}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d7e28608cfd9c6374ca20d6d5886028bbb07e39e8128fa39256cc79ef16a42f | Calculate the following integrals. $\int _{\pi /6}^{\pi /3}\left[1-\frac{1}{1-\sin ^{-1}(2x+3\pi /2)} \right]dx$. | [
"$\\frac{1}{2}\\left(\\tan \\frac{\\pi }{3}-\\tan\\frac{\\pi }{6}\\right)$"
] | [
"Calculate the following integrals. $\\int _{\\pi /6}^{\\pi /3}\\left[1-\\frac{1}{1-\\sin ^{-1}(2x+3\\pi /2)} \\right]dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c4034d9d080366b81e275d1e98fdc52da82b7a0e9e894c8d0ae533f9ba5aede8 | Calculate the following limit without using Scholze's theorem: \[ \lim_{n\to \infty}{\frac{s_n(a+1)}{ns_n(a)}} \] where \( s_n(a) = \sum_{k=1}^n k^a \). Distinguish cases for \( a > 0 \), \( a = 0 \), and \( a < 0 \). | [
"\\[\n\\begin{cases} \n\\frac{a+1}{a+2} & \\text{if } a > 0, \\\\\n\\frac{1}{2} & \\text{if } a = 0, \\\\\n0 & \\text{if } a < 0.\n\\end{cases}\n\\]"
] | [
"Calculate the following limit without using Scholze's theorem:\n\n\\[\n\\lim_{n\\to \\infty}{\\frac{s_n(a+1)}{ns_n(a)}}\n\\]\n\nwhere \\( s_n(a) = \\sum_{k=1}^n k^a \\). Distinguish cases for \\( a > 0 \\), \\( a = 0 \\), and \\( a < 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-a556167d89e577792353e6ef060dc4c94bf85bb017394cd540545a20d4b94fc2 | Calculate the following limit: $$ \lim_{n\rightarrow\infty} \int_{-1}^{1} \left(x^{2n+1} + \frac{1}{x^{2n+1}}\right) \ln (1+e^{xn}) \, dx.$$ | [
"\\infty"
] | [
"Calculate the following limit:\n\n$$ \\lim_{n\\rightarrow\\infty} \\int_{-1}^{1} \\left(x^{2n+1} + \\frac{1}{x^{2n+1}}\\right) \\ln (1+e^{xn}) \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-25960067251bca83b822b74237a0c212de0780a024664c0c15795180f6c503fc | Calculate the following limit: $$ \lim_{n\rightarrow\infty} n\left(\left(1+\frac{x}{n}\right)^x - e^x\right) $$ | [
"-\\dfrac{x^2 e^x}{2}"
] | [
"Calculate the following limit:\n$$\n\\lim_{n\\rightarrow\\infty} n\\left(\\left(1+\\frac{x}{n}\\right)^x - e^x\\right)\n$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-02508abfcb2549ca6441085c38fdffc33b007bdadca6980280b94b9033647b11 | Calculate the following limit: $$\displaystyle{\ell=\lim_{n \rightarrow +\infty} \left[ \sum_{i = 1}^n \sum_{j = 1}^n \frac{1}{i^2+j^2}-\frac{\pi \log n}{2} \right]}$$ | [
"\\frac{\\pi}{2} \\left( \\gamma + \\log \\frac{4}{\\pi} \\right)"
] | [
"Calculate the following limit:\n\n$$\\displaystyle{\\ell=\\lim_{n \\rightarrow +\\infty} \\left[ \\sum_{i = 1}^n \\sum_{j = 1}^n \\frac{1}{i^2+j^2}-\\frac{\\pi \\log n}{2} \\right]}$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-1ed340db8c8da4a132118d702ce1c0bc9a4790a762a83968fd18467075e5984c | Calculate the following limit: $$\lim_{n \to \infty} \int_{0}^{1/2} \sqrt[n]{\left(\frac{1}{2}+x\right)^n + \left(\frac{1}{2}-x\right)^n +\left(\frac12\right)^n}dx$$ | [
"\\(\\frac{3}{8}\\)"
] | [
"Calculate the following limit:\n$$\\lim_{n \\to \\infty} \\int_{0}^{1/2} \\sqrt[n]{\\left(\\frac{1}{2}+x\\right)^n + \\left(\\frac{1}{2}-x\\right)^n +\\left(\\frac12\\right)^n}dx$$"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-3a1685f8f6ce65a9918e8a109e457ae2a2adc4b52e7798c07ac5e2f5335e3e6c | Calculate the following limit: $$\lim_{n\to+\infty}\int_0^{\frac{\pi}{2}}\frac{n}{(\sqrt2 \cos x)^n+(\sqrt2 \sin x)^n}{\rm d}x$$ | [
"\\dfrac{\\pi}{2}"
] | [
"Calculate the following limit:\n$$\\lim_{n\\to+\\infty}\\int_0^{\\frac{\\pi}{2}}\\frac{n}{(\\sqrt2 \\cos x)^n+(\\sqrt2 \\sin x)^n}{\\rm d}x$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d7f8d49d50b95218ea77b252d367b8a09351aea949e79316b00afe10cc30a90d | Calculate the following limit: $$\lim_{n\to\infty}\left(\frac{\sqrt[n(n+1)]{1!.2!\dots n!}}{\sqrt{n}}\right)$$ | [
"\\( e^{-\\frac{3}{4}} \\)"
] | [
"Calculate the following limit:\n$$\\lim_{n\\to\\infty}\\left(\\frac{\\sqrt[n(n+1)]{1!.2!\\dots n!}}{\\sqrt{n}}\\right)$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 26
} | 49 | 0 | 0 |
omr-ws-50c3d3cddcf8caecb68b9f761ee7ade39c8de3981ef0649b9228281a50f87497 | Calculate the following limit: $$\lim_{x \to 0} \frac{(\cos x + 1)(\cos x \cos^2(2x) + 1) \cdots (\cos x \cos^2(2x) \cdots \cos^n(nx) + 1) - 2^n}{x^2}$$ | [
"\\(-\\frac{2^{n-2}n(n+1)(3n^{3}+12n^{2}+13n+2)}{60}\\)"
] | [
"Calculate the following limit:\n$$\\lim_{x \\to 0} \\frac{(\\cos x + 1)(\\cos x \\cos^2(2x) + 1) \\cdots (\\cos x \\cos^2(2x) \\cdots \\cos^n(nx) + 1) - 2^n}{x^2}$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5e6bf12e7b70df5768f45a26160f17ff852e5ce667456d1831db7a79468fc656 | Calculate the following limit: \[ \lim_{n \to +\infty} \frac{1}{n} \sum_{k=0}^{n-1} \begin{bmatrix} \cos\left(\frac{m\pi}{n}\right) & 0 & \cdots & 0 & -\sin\left(\frac{m\pi}{n}\right) \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & \cos\left(\frac{2\pi}{n}\right) & -\sin\left(\frac{2\pi}{n}\right) \\ ... | [
"\\frac{1 - (-1)^m}{m\\pi} \\begin{bmatrix} 0 & -1 \\\\ 1 & 0 \\end{bmatrix}"
] | [
"Calculate the following limit:\n\\[\n\\lim_{n \\to +\\infty} \\frac{1}{n} \\sum_{k=0}^{n-1} \\begin{bmatrix}\n\\cos\\left(\\frac{m\\pi}{n}\\right) & 0 & \\cdots & 0 & -\\sin\\left(\\frac{m\\pi}{n}\\right) \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & \\cos\\left(\\frac{2\\pi}{n}\\ri... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-eb7a062829f02e0fed6fca32101bdc1153ee94114d52ce35b649767d2b9fca72 | Calculate the following limit: \[ \lim_{n \to \infty} \int_{0}^{\pi} \cos \left(\frac{x}{x + n^2}\right) \, dx \] | [
"\\(\\pi\\)"
] | [
"Calculate the following limit:\n\\[\n\\lim_{n \\to \\infty} \\int_{0}^{\\pi} \\cos \\left(\\frac{x}{x + n^2}\\right) \\, dx\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-8d0b6e8da402b5db811e5c3482d3d654a3f11c4d46e19fafa185c5a8851f209e | Calculate the following limit: \[ \lim_{n \to \infty} n^2 \left( \left(1 + \frac{1}{n+1}\right)^{n+1} - \left(1 + \frac{1}{n}\right)^n \right). \] | [
"\\(\\frac{e}{2}\\)"
] | [
"Calculate the following limit:\n\\[\n\\lim_{n \\to \\infty} n^2 \\left( \\left(1 + \\frac{1}{n+1}\\right)^{n+1} - \\left(1 + \\frac{1}{n}\\right)^n \\right).\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-857f23b424cc281a477c965d389709f2e200596af1d5c66608a14d2362573c03 | Calculate the following limit: \[ \lim_{n\to\infty}\int_{0}^{\infty}e^{-x}\arctan(nx)(\cos(x)+\sin(x))\,dx \] | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"Calculate the following limit:\n\n\\[\n\\lim_{n\\to\\infty}\\int_{0}^{\\infty}e^{-x}\\arctan(nx)(\\cos(x)+\\sin(x))\\,dx\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-7dfdc000e4997042efeabff6ed532c8b8a92e13bf53dbf86ba2ef6fa16e06f26 | Calculate the following limit: \[ \lim_{x \to 2} \frac{\sqrt{x + 2} + \sqrt{x^2 + 5} - 5}{\sqrt{3x - 2} + \sqrt{4x^2 + 5x + 23} - 9} \] | [
"\\(\\frac{11}{27}\\)"
] | [
"Calculate the following limit:\n\n\\[\n\\lim_{x \\to 2} \\frac{\\sqrt{x + 2} + \\sqrt{x^2 + 5} - 5}{\\sqrt{3x - 2} + \\sqrt{4x^2 + 5x + 23} - 9}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-d785cd89eec31a29715b6a2ca857594e96ba64dd9ab24486ba945afc79b2fad7 | Calculate the following limits by means of a division of the numerator and the denominator of the fraction by leading power of $n$: $\lim\limits_{n\to \infty }\frac{\sqrt [4]{n^5+2}-\sqrt [3]{n^2+1}} {\sqrt [5]{n^4+2}+\sqrt {n^3+1}}$. | [
"0"
] | [
"Calculate the following limits by means of a division of the numerator and the denominator of the fraction by leading power of $n$: $\\lim\\limits_{n\\to \\infty }\\frac{\\sqrt [4]{n^5+2}-\\sqrt [3]{n^2+1}} {\\sqrt [5]{n^4+2}+\\sqrt {n^3+1}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30010cf876836aefdcd6d4d5112707c51bb0702086c94a58c84c9742f4a24c2f | Calculate the following limits. $\lim\limits_{n\to \infty }\left(\frac{3n+1}{4n+5}\right)^n$. | [
"0"
] | [
"Calculate the following limits. $\\lim\\limits_{n\\to \\infty }\\left(\\frac{3n+1}{4n+5}\\right)^n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-284faff8eb1c17f210e8ad38a68399e3f9ec7ba27b0d9b406758cd3e0785fb90 | Calculate the following limits. $\lim\limits_{n\to \infty }\left[\frac{n^2-1}{(2n+1)(n+2)}\right]^n$. | [
"0"
] | [
"Calculate the following limits. $\\lim\\limits_{n\\to \\infty }\\left[\\frac{n^2-1}{(2n+1)(n+2)}\\right]^n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e530f99f3668a2053f2204b217d7a9f2cca53b0e95312446a46838265299fca1 | Calculate the following limits. $\lim\limits_{x\to -\infty }x\left(\sqrt {4x^2+7}+2x\right)$. | [
"$-1.75$"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to -\\infty }x\\left(\\sqrt {4x^2+7}+2x\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3efc36463715a82c7db7ce267db6e814edf24cf6cdc4fa9da8c7857e43d7cf46 | Calculate the following limits. $\lim\limits_{x\to 0}\frac{1-\cos x}{\sin ^3 x}$. | [
"$\\infty $"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 0}\\frac{1-\\cos x}{\\sin ^3 x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ed9a65763678d3c42ba42a04dcbd60880225ade4c951c6a4a5c44e7b1076966c | Calculate the following limits. $\lim\limits_{x\to 0}\frac{\cos x\sin x-\tan x}{x^2\sin x}$. | [
"$-1$"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 0}\\frac{\\cos x\\sin x-\\tan x}{x^2\\sin x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-345cbbbfc25894ace3694caed1ac0797f298651127849eb20339efc5784fcf45 | Calculate the following limits. $\lim\limits_{x\to 0}\frac{\sin (2x-\pi )}{\cos 3(x+\pi /2)}$. | [
"2/3"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 0}\\frac{\\sin (2x-\\pi )}{\\cos 3(x+\\pi /2)}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23c641bdda9ef56f57e8a986f720ece098231db261a6e12b466d25def2a37807 | Calculate the following limits. $\lim\limits_{x\to 0}\frac{\sqrt {x^2+4}-2}{\sqrt {x^2+9}-3}$. | [
"3/2"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 0}\\frac{\\sqrt {x^2+4}-2}{\\sqrt {x^2+9}-3}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b681598a67b495e4d8d58f1e0537b3593ae2373030730b6e9a856aeb2f04720b | Calculate the following limits. $\lim\limits_{x\to 0}\frac{\tan x-\sin x}{\sin ^3 x}$. | [
"1/2"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 0}\\frac{\\tan x-\\sin x}{\\sin ^3 x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43493d01ab7766378f6e4b49e64d99cb3039cf587a33e157aba37bd11b1e323f | Calculate the following limits. $\lim\limits_{x\to 0}\frac{\tan x}{1-\cos x}$. | [
"$\\infty $"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 0}\\frac{\\tan x}{1-\\cos x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5593372ca56e20751852781ca0910a745c3c1ba323c9821dcea1947aa5e0cd85 | Calculate the following limits. $\lim\limits_{x\to 1}\frac{\sqrt x+\sqrt {x-1}-1}{\sqrt {x^2-1}}$. | [
"$\\sqrt 2/2$"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 1}\\frac{\\sqrt x+\\sqrt {x-1}-1}{\\sqrt {x^2-1}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a4adac3caf6d5fa945172ea1a78a3fceec71b4c32876158322615ebd7d39d5a | Calculate the following limits. $\lim\limits_{x\to 2}\left(\frac{1}{2-x}-\frac{3}{8-x^3}\right)$. | [
"$\\infty $"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 2}\\left(\\frac{1}{2-x}-\\frac{3}{8-x^3}\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5ea530e0c0ebcec0e75558525f5e0ff34fd75489235e5af951ae1c45cf277a7 | Calculate the following limits. $\lim\limits_{x\to 3\pi /2}\frac{\sin 2x}{1+\sin x}$. | [
"$\\infty $"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 3\\pi /2}\\frac{\\sin 2x}{1+\\sin x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41bbfba148e5764ec2c35b7fec0c4da1696c390e2912bd116d34c7a681819ee0 | Calculate the following limits. $\lim\limits_{x\to 3}\frac{x-3}{\sqrt [3]{x^2-1}-2}$. | [
"2"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 3}\\frac{x-3}{\\sqrt [3]{x^2-1}-2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-210f758bdd6f517089f53b3fd6cf4485999392cda598e87af0e98524c8160eda | Calculate the following limits. $\lim\limits_{x\to 4}\frac{\sqrt x-2}{\sqrt {x+5}-3}$. | [
"1.5"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to 4}\\frac{\\sqrt x-2}{\\sqrt {x+5}-3}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c7de883230f40c47a8edd9b69556d808da367dc7aea56fe3ac34dd32b6bcb4c | Calculate the following limits. $\lim\limits_{x\to \pi /2}[(1-\sin x)\tan^2 x]$. | [
"0"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to \\pi /2}[(1-\\sin x)\\tan^2 x]$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fcf21267092b3e52339fe1f76556407d98f36ee82a403105222a00f0c5109ed2 | Calculate the following limits. $\lim\limits_{x\to \pi /2}\frac{\sec 2x+1}{\cos x}$. | [
"0"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to \\pi /2}\\frac{\\sec 2x+1}{\\cos x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c98cea0d71d26da5e3d45116ebb5cf6dd830bc258ec0ac45251cdf49b8f0c6a | Calculate the following limits. $\lim\limits_{x\to \pi /4}\frac{1-\cot^3 x}{2-\cot x-\cot^3 x}$. | [
"3/4"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to \\pi /4}\\frac{1-\\cot^3 x}{2-\\cot x-\\cot^3 x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-205f5ee518542a22a500d79927058cd9a7e9b0fba15395d94e7161b8f0fea5e2 | Calculate the following limits. $\lim\limits_{x\to \pi /4}\frac{\sin (\pi /4-x)}{\cos ^2 x-1/2}$. | [
"1"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to \\pi /4}\\frac{\\sin (\\pi /4-x)}{\\cos ^2 x-1/2}$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2f036fb492f1a4cdb435a47ff666f9fe6654bf0406172bb4adece5f3a6cb7958 | Calculate the following limits. $\lim\limits_{x\to \pi /4}\frac{\tan^2 x+\tan x-2}{\sin x-\cos x}$. | [
"$3\\sqrt 2$"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to \\pi /4}\\frac{\\tan^2 x+\\tan x-2}{\\sin x-\\cos x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe76772c23eb63d50e4f2bf1a062b646729a1417eec73964e6451675cdaead93 | Calculate the following limits. $\lim\limits_{x\to \pi }\frac{1-\sin (x/2)}{\pi -x}$. | [
"0"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to \\pi }\\frac{1-\\sin (x/2)}{\\pi -x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4cdc501d8c03d8acd49c890b17eb75ba86497999656c5f6cc0ce23940e5d5219 | Calculate the following limits. $\lim\limits_{x\to \pi }\frac{\sin 2x}{1+\cos ^3 x}: \lim\limits_{x\to 0}\frac{\sqrt {2x+1}-1}{\sqrt {3x+4}-2}$. | [
"$\\infty $"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to \\pi }\\frac{\\sin 2x}{1+\\cos ^3 x}: \\lim\\limits_{x\\to 0}\\frac{\\sqrt {2x+1}-1}{\\sqrt {3x+4}-2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d196bd5598941a43c58befeb06d768ed6610c4480c1f593267b67909fdddd3e | Calculate the following limits. $\lim\limits_{x\to a}\frac{\sin ^2 x-\sin ^2 a}{\sin (x-a)}$. | [
"$\\sin 2a$"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to a}\\frac{\\sin ^2 x-\\sin ^2 a}{\\sin (x-a)}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2d9b369236e0ded45c4ec09be0c574d27df0cde9882d4502cbce4e0cc2b30eee | Calculate the following limits. $\lim\limits_{x\to a}\frac{\tan^2 x-\tan^2 a}{\tan (x-a)}$. | [
"$\\sin 2a/\\cos ^4 a$"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to a}\\frac{\\tan^2 x-\\tan^2 a}{\\tan (x-a)}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-687bde9214a5b5149815d1df2cfec03fd0c5fde52b00e31679cec7f9293bb2bd | Calculate the following limits. $\lim\limits_{x\to a}\left(\sin \frac{a-x}{2}\tan\frac{\pi x}{2a}\right)$. | [
"$a/\\pi $"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to a}\\left(\\sin \\frac{a-x}{2}\\tan\\frac{\\pi x}{2a}\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a7a4143efbb0911611883efd11584a20b4c63ed89c943d485b963e5b6a4adf9 | Calculate the following limits. $\lim\limits_{x\to a}\left[(a-x)\sec\frac{\pi x}{2a}\right]$. | [
"$2a/\\pi $"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to a}\\left[(a-x)\\sec\\frac{\\pi x}{2a}\\right]$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad50b7a73262314f3708a72d4178a9128555ae6d58109f628130792de6857a9a | Calculate the following limits. $\lim\limits_{x\to a}\left[\frac{(x^3-2ax^2-a^2x+2a^3)(x-2a)^{-1}+2a(x+a)}{x^3-a^2x} -\frac{2}{x-a}\right]$. | [
"$\\frac{1}{a}$ for $a\\ne 0$, for $a=0$ there is no limit"
] | [
"Calculate the following limits. $\\lim\\limits_{x\\to a}\\left[\\frac{(x^3-2ax^2-a^2x+2a^3)(x-2a)^{-1}+2a(x+a)}{x^3-a^2x} -\\frac{2}{x-a}\\right]$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b157d1a8347a330cc8603fbd1f48c04d1fda42a0f4abbd6210cbe41adcadb77 | Calculate the following limits. (a) $\lim\limits_{x\to 1/2}f(x)$, (b) $\lim\limits_{x\to 3/2}f(x)$, where $$f(x)=\frac{\frac{|x-1|}{2}+x|x-1|+2-\frac{2}{x}}{\sqrt {x-2+1/x}}.$$ | [
"(a) $-\\frac{3}{4}\\sqrt 2$; (b) $\\frac{21}{4}\\sqrt {\\frac{2}{3}}$"
] | [
"Calculate the following limits. (a) $\\lim\\limits_{x\\to 1/2}f(x)$, (b) $\\lim\\limits_{x\\to 3/2}f(x)$, where $$f(x)=\\frac{\\frac{|x-1|}{2}+x|x-1|+2-\\frac{2}{x}}{\\sqrt {x-2+1/x}}.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f69d6d9ee4172b7ea5932bd9d334305a76addc3565f6e0c99130558c818038e | Calculate the following limits. (a) $\lim\limits_{x\to 1}f(x)$, (b) $\lim\limits_{x\to -1}f(x)$, where $$f(x)=\frac{x|x-3|}{(x^2-x-6)|x|}.$$ | [
"(a) $-1/3$; (b) 1"
] | [
"Calculate the following limits. (a) $\\lim\\limits_{x\\to 1}f(x)$, (b) $\\lim\\limits_{x\\to -1}f(x)$, where $$f(x)=\\frac{x|x-3|}{(x^2-x-6)|x|}.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d7e0d43f088311be59d07c032826e410fcd0768792d8a7a26625eb810e08c1ea | Calculate the following limits: - \(\lim_{x \to 0} xE(f(x))\) - \(\lim_{x \to +\infty} \frac{E(f(x))}{x^2 + E(x)}\) | [
"0, 0"
] | [
"Calculate the following limits:\n - \\(\\lim_{x \\to 0} xE(f(x))\\)\n - \\(\\lim_{x \\to +\\infty} \\frac{E(f(x))}{x^2 + E(x)}\\)"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-0d78c9e341365acfa8fb6606c2f407372222d2dca27684f9e994b7bd02b10d5d | Calculate the following sum for positive integers \( n \) and \( p \): \[ \sum_{x=1}^n \sum_{y=1}^n \left\lfloor \frac{px}{y} \right\rfloor \] | [
"\\frac{p n(n+1)}{2} H_n"
] | [
"Calculate the following sum for positive integers \\( n \\) and \\( p \\):\n\\[ \\sum_{x=1}^n \\sum_{y=1}^n \\left\\lfloor \\frac{px}{y} \\right\\rfloor \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-7db3c9a1b4f4c4884607333409732caedf3da08e02a665662db5842fd13d6219 | Calculate the following sum where \( n \) is an integer between 1 and 10: \[ \sum_{a_1+a_2+\cdots+a_n=10 \ \text{and} \ a_i \neq 0, \ \forall i} \binom{10}{a_1,a_2,\ldots,a_n} \] | [
"\\(\\sum_{k=0}^{n} (-1)^k \\binom{n}{k} (n-k)^{10}\\)"
] | [
"Calculate the following sum where \\( n \\) is an integer between 1 and 10:\n\n\\[\n\\sum_{a_1+a_2+\\cdots+a_n=10 \\ \\text{and} \\ a_i \\neq 0, \\ \\forall i} \\binom{10}{a_1,a_2,\\ldots,a_n}\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 26,
"genselect": 4
} | 30 | 0 | 0 |
omr-ws-6ede57e589ecd511b345669ab53bb3221a27b9f725599841d49e2f81c4f30a17 | Calculate the following sum: $$\sum_{m=1}^{p+1}(-1)^{m}\frac{m+p}{p+1}\binom{p+1}{m-1}2^{p-m+1}$$ | [
"$(-1)^{p+1}$"
] | [
"Calculate the following sum:\n$$\\sum_{m=1}^{p+1}(-1)^{m}\\frac{m+p}{p+1}\\binom{p+1}{m-1}2^{p-m+1}$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-9009d3a572b2831f2a05814166c79fe50d85428d46fdb1d902a8b7a38895ec9b | Calculate the following sum: $\sum_{s=2}^{\infty}\sum_{n=2}^{\infty}\frac{1}{n^{s}}$. | [
"1"
] | [
"Calculate the following sum: $\\sum_{s=2}^{\\infty}\\sum_{n=2}^{\\infty}\\frac{1}{n^{s}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-483c270325b950402f34ba62035ed79ffa46cc01766c4d02ea226e049e965704 | Calculate the following sum: \[ \sum_{k=1}^m \frac{(-1)^k (4m - 2k)!}{(k + 1)!(2m - k)!(2m - 2k)!} \] | [
"-1"
] | [
"Calculate the following sum:\n\\[ \\sum_{k=1}^m \\frac{(-1)^k (4m - 2k)!}{(k + 1)!(2m - k)!(2m - 2k)!} \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-b094ca999a8adcaac9edcaada87d8af6862879e165abf4401dfcf8315b3a5053 | Calculate the following sum: \[ \sum_{m=1}^{n} (-1)^m \sum_{1 \leq k_1 < k_2 < \cdots < k_m \leq n} \binom{k_1^3 + k_2^3 + \cdots + k_m^3 + 2^n}{n} \] | [
"(-1)^n (n!)^3 - \\dbinom{2^n}{n}"
] | [
"Calculate the following sum:\n\\[ \\sum_{m=1}^{n} (-1)^m \\sum_{1 \\leq k_1 < k_2 < \\cdots < k_m \\leq n} \\binom{k_1^3 + k_2^3 + \\cdots + k_m^3 + 2^n}{n} \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3aa5fa0664a1c02149d2fd1dc4355bc31ca96b12be49be1ef057963d472ced01 | Calculate the following trigonometric values by hand: \[ \cos 36^\circ \] \[ \cos 72^\circ \] \[ \cos 54^\circ \] \[ \cos 18^\circ \] | [
"\\[\n\\cos 36^\\circ = \\frac{1 + \\sqrt{5}}{4}, \\quad \\cos 72^\\circ = \\frac{\\sqrt{5} - 1}{4}, \\quad \\cos 54^\\circ = \\sqrt{\\frac{5 - \\sqrt{5}}{8}}, \\quad \\cos 18^\\circ = \\sqrt{\\frac{5 + \\sqrt{5}}{8}}\n\\]"
] | [
"Calculate the following trigonometric values by hand:\n\\[ \\cos 36^\\circ \\]\n\\[ \\cos 72^\\circ \\]\n\\[ \\cos 54^\\circ \\]\n\\[ \\cos 18^\\circ \\]"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-311afe0d0c59d2d2676340e5b5bbcb427a4b135d3e50a985e2c7622f4dcd1b8b | Calculate the following triple integral using Cylindrical Coordinates: $$\int_0^1 \int_0^{\sqrt{1-x^2}} \int_{\sqrt{x^2+y^2}}^{\sqrt{2-x^2-y^2}} z^2 \, dz \, dy \, dx.$$ | [
"\\(\\frac{\\pi}{15}\\left(2\\sqrt{2}-1\\right)\\)"
] | [
"Calculate the following triple integral using Cylindrical Coordinates:\n$$\\int_0^1 \\int_0^{\\sqrt{1-x^2}} \\int_{\\sqrt{x^2+y^2}}^{\\sqrt{2-x^2-y^2}} z^2 \\, dz \\, dy \\, dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-7cb93fd52e5ff905750af9db028e81c8f5cc6bda479a9bca27dc7d5c8c73c2d2 | Calculate the force that affects $Q_{3}$ when electrically charged particles $Q_{1}=30 \times 10^{-10} \, \text{C}$, $Q_{2}=-45 \times 10^{-10} \, \text{C}$, and $Q_{3}=60 \times 10^{-10} \, \text{C}$ are placed at the vertices of an equilateral triangle with side length $a=5 \, \text{cm}$. Perform the calculation and ... | [
"\\( 8.566 \\times 10^{-10} \\, \\text{N} \\)"
] | [
"Calculate the force that affects $Q_{3}$ when electrically charged particles $Q_{1}=30 \\times 10^{-10} \\, \\text{C}$, $Q_{2}=-45 \\times 10^{-10} \\, \\text{C}$, and $Q_{3}=60 \\times 10^{-10} \\, \\text{C}$ are placed at the vertices of an equilateral triangle with side length $a=5 \\, \\text{cm}$. Perform the ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ddd2090eb55423b561312dae8b258c851857425597df3a1cd36b39e89cccfc82 | Calculate the full asymptotic expansions of the integral $$\int\limits_{-\infty }^{\infty }\exp \left( -k|x|-\frac{x^2}{a^2} \right)dx$$ at the limits $ka \ll 1$ and $ka \gg 1$. | [
"\\frac{2}{k} \\left( 1 - \\frac{2}{(ka)^2} + \\frac{12}{(ka)^4} - \\frac{120}{(ka)^6} + \\cdots \\right)"
] | [
"Calculate the full asymptotic expansions of the integral\n$$\\int\\limits_{-\\infty }^{\\infty }\\exp \\left( -k|x|-\\frac{x^2}{a^2} \\right)dx$$\nat the limits $ka \\ll 1$ and $ka \\gg 1$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-f53873e7a02fbfd380fc2c636d8a77b7a3c881658228574ca008184f3e080ccb | Calculate the function \[ f(x) = \frac{\sqrt{2} \sin x + \cos x}{\sin x + \sqrt{1 - \sin x}} \] for \( 0 \le x \le \pi \). | [
"\\( f(x) = \\sqrt{2} \\cdot \\begin{cases} \n1 & \\text{if } 0 \\le x \\le \\frac{\\pi}{2} \\\\\n\\tan\\left(\\frac{3\\pi}{4} - x\\right) & \\text{if } \\frac{\\pi}{2} < x \\le \\pi \n\\end{cases} \\)"
] | [
"Calculate the function \n\\[ f(x) = \\frac{\\sqrt{2} \\sin x + \\cos x}{\\sin x + \\sqrt{1 - \\sin x}} \\]\nfor \\( 0 \\le x \\le \\pi \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-d6e3f3d2e77113259a239890dd51cd934ce9c859b4d1bc3dd977595eadc3b1f9 | Calculate the fundamental group $\pi_1(X, x_{0})$ and the first homology group $H_{1}(X)$ for the quotient space $X$ obtained from an 8-sided polygonal region $P$ by the labelling scheme $abcdad^{-1}cb^{-1}$. | [
"$\\pi_1(X, x_{0}) = \\langle a,c,d \\rangle / \\langle acdad^{-1}c \\rangle$, $H_{1}(X) = \\mathbb{Z}/2 \\oplus \\mathbb{Z}^2$"
] | [
"Calculate the fundamental group $\\pi_1(X, x_{0})$ and the first homology group $H_{1}(X)$ for the quotient space $X$ obtained from an 8-sided polygonal region $P$ by the labelling scheme $abcdad^{-1}cb^{-1}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f669f631bf99ac302fda5e78d0c637ccda89dad3a595a841de9bacd17f736f92 | Calculate the general sum \(1 + \sqrt[k]{2} + \cdots + \sqrt[k]{n}\). | [
"\\sum_{m=1}^{n} m^{1/k}"
] | [
"Calculate the general sum \\(1 + \\sqrt[k]{2} + \\cdots + \\sqrt[k]{n}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-a422c6bd72636f0648ebb74a23285342df2cc9c9c1d416989e58d4925e050576 | Calculate the generalized integral \(\int_{0}^{+\infty} \frac{x^2}{x^4 + 1} \, dx\). | [
"\\(\\frac{\\pi}{2\\sqrt{2}}\\)"
] | [
"Calculate the generalized integral \\(\\int_{0}^{+\\infty} \\frac{x^2}{x^4 + 1} \\, dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-f23b8093f652e4b364d791b42362c7a6f48cfac17ad4140f7909a9a3672adca2 | Calculate the given expression $$\sum_{k=0}^{n} \frac{2^k}{3^{2^k}+1}$$ | [
"\\(\\frac12-\\frac{2^{n+1}}{3^{2^{n+1}}-1}\\)"
] | [
"Calculate the given expression $$\\sum_{k=0}^{n} \\frac{2^k}{3^{2^k}+1}$$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-6a2cd5aa459fefc3eb81d485db30daf6476a940c41d3b90689be817c32a3b06c | Calculate the gradient $\bigtriangledown f$ of the function $f(x_1, ..., x_m) = f(x) = \Vert b - Ax \Vert_2^2$. | [
"\\( 2(A^TAx - A^Tb) \\)"
] | [
"Calculate the gradient $\\bigtriangledown f$ of the function $f(x_1, ..., x_m) = f(x) = \\Vert b - Ax \\Vert_2^2$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-fb058d6604ea7d37a295cbc3c832debc00faf40f1e1c784553357a2bc259e66e | Calculate the greatest common divisor (gcd) of \(5^m + 7^m\) and \(5^n + 7^n\) given that \(m\) and \(n\) are natural numbers and \(\gcd(m, n) = 1\). | [
"12"
] | [
"Calculate the greatest common divisor (gcd) of \\(5^m + 7^m\\) and \\(5^n + 7^n\\) given that \\(m\\) and \\(n\\) are natural numbers and \\(\\gcd(m, n) = 1\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-7fccba10653cec7beec78b75ed6c57227033e5a06cb38016dbcecccd5639a06f | Calculate the greatest common divisor (gcd) of the sequence $2002+2, 2002^2+2, 2002^3+2, \ldots$. | [
"6"
] | [
"Calculate the greatest common divisor (gcd) of the sequence $2002+2, 2002^2+2, 2002^3+2, \\ldots$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-4de6b4dc822700e488a5e7381102908b9eb252394b18d324fa28dcf8079a2fe8 | Calculate the growth rate of the function $$ f(n)=\sum_{i=1}^n\sum_{j=i}^n\sum_{k=j}^n [\mathrm{lcm}(i,j)\le n][\mathrm{lcm}(j,k)\le n][\mathrm{lcm}(i,k)\le n] $$ as \( n \rightarrow +\infty \), where \(\mathrm{lcm}\) denotes the least common multiple and \([ \cdot ]\) is the Iverson bracket. | [
"\\( \\Theta(n^{1.5}) \\)"
] | [
"Calculate the growth rate of the function\n\n$$\nf(n)=\\sum_{i=1}^n\\sum_{j=i}^n\\sum_{k=j}^n [\\mathrm{lcm}(i,j)\\le n][\\mathrm{lcm}(j,k)\\le n][\\mathrm{lcm}(i,k)\\le n]\n$$\n\nas \\( n \\rightarrow +\\infty \\), where \\(\\mathrm{lcm}\\) denotes the least common multiple and \\([ \\cdot ]\\) is the Iverson bra... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-8a608f1d721c6c0c1d45bade542bb3e09ef972245863885708579eff9ccb3be0 | Calculate the height of a trapezoid with bases measuring 9 cm and 23 cm, and the other two sides measuring 13 cm and 15 cm. | [
"12"
] | [
"Calculate the height of a trapezoid with bases measuring 9 cm and 23 cm, and the other two sides measuring 13 cm and 15 cm."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-5d1dae3e567e95537ef8772feb441a6d7053d44d87a923c3693b27e65f4c9eb5 | Calculate the homology group $H_{1201}(\mathbb{R}P^{2019} \setminus \mathbb{R}P^{2018})$. | [
"0"
] | [
"Calculate the homology group $H_{1201}(\\mathbb{R}P^{2019} \\setminus \\mathbb{R}P^{2018})$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a87ad0fd55795bf91c677934851aed243f9fd03be2f920b7b7d3b5a2f1051dc0 | Calculate the homology group $H_{9}(\mathbb{R}P^9 \setminus \mathbb{R}P^8)$. | [
"0"
] | [
"Calculate the homology group $H_{9}(\\mathbb{R}P^9 \\setminus \\mathbb{R}P^8)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-c3bf63028fcbe1905812af676d4ab502cced6d90d494476b295b7c32cd666b30 | Calculate the homology of the Klein Bottle with coefficients in $\mathbb{F}_2$. | [
"\\mathbb{F}_2"
] | [
"Calculate the homology of the Klein Bottle with coefficients in $\\mathbb{F}_2$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-c369c392054b0f42e3cf6ab2e9a0ecaf9a36dbead84109392d0691c48c5d27e6 | Calculate the homology of the complex $C(3)_{*}\otimes C(4)_{*}$ with the usual differential on the product complex, where $C(k)_*$ is the chain complex $0\rightarrow\mathbb{Z}\stackrel{2}{\rightarrow}\mathbb{Z}\stackrel{0}{\rightarrow}\cdots\rightarrow\mathbb{Z}\to 0$ (with a total of $k$ copies of $\mathbb{Z}$) and t... | [
"H_n = \\begin{cases} \n\\mathbb{Z}_2 & \\text{if } n = 0, 1, 3, 4, \\\\\n\\mathbb{Z}_2 \\oplus \\mathbb{Z}_2 & \\text{if } n = 2, \\\\\n0 & \\text{otherwise.}\n\\end{cases}"
] | [
"Calculate the homology of the complex $C(3)_{*}\\otimes C(4)_{*}$ with the usual differential on the product complex, where $C(k)_*$ is the chain complex $0\\rightarrow\\mathbb{Z}\\stackrel{2}{\\rightarrow}\\mathbb{Z}\\stackrel{0}{\\rightarrow}\\cdots\\rightarrow\\mathbb{Z}\\to 0$ (with a total of $k$ copies of $\... | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-c4615d0e5902afa8b81d25538bad7106c7043fca63d2186359052c281a42425e | Calculate the improper integral $\displaystyle \int _{1}^{+\infty }\frac{\sin x}{x\left( e^{\sin x} -1\right)} dx$. | [
"\\text{The integral diverges}"
] | [
"Calculate the improper integral $\\displaystyle \\int _{1}^{+\\infty }\\frac{\\sin x}{x\\left( e^{\\sin x} -1\\right)} dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-2298fba64c499c563deee75aa0e5d51b192901f93454004ad89756ef4c28b457 | Calculate the improper integral $\iint_{D}\ln(y^2-x^2)dxdy$ where $D$ is the square with vertices $(2,2), (3,3), (2,4), (1,3)$. | [
"\\( 2(3 \\log 3 - 2) \\)"
] | [
"Calculate the improper integral $\\iint_{D}\\ln(y^2-x^2)dxdy$ where $D$ is the square with vertices $(2,2), (3,3), (2,4), (1,3)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-bb7ad4918db8e51d6133c2e712dc27950862a504bd232d64dab5e3abf60e5e46 | Calculate the improper integral $\int_0^\infty e^{-\frac {(lnx - lna)^2} {2\sigma^2}}dx$, where $a$ and $\sigma$ are positive parameters. | [
"\\(\\sigma\\sqrt{2\\pi}\\)"
] | [
"Calculate the improper integral $\\int_0^\\infty e^{-\\frac {(lnx - lna)^2} {2\\sigma^2}}dx$, where $a$ and $\\sigma$ are positive parameters."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-294818f374a351725641a5051ff74a8d1eb7827b2c05ce8128747b24c5e19697 | Calculate the improper integral \( \int_{-\infty}^\infty f(x) \, dx \) for the function \( f(z) = \frac{e^{(1+i)z}}{(e^z+1)^2} \). | [
"\\(\\frac{2e^{\\pi}\\pi}{e^{2\\pi}-1}\\)"
] | [
"Calculate the improper integral \\( \\int_{-\\infty}^\\infty f(x) \\, dx \\) for the function \\( f(z) = \\frac{e^{(1+i)z}}{(e^z+1)^2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-a41c5cdf0c9890fa2c0664ee67b9b7bf54e5f366171ffc4cd7fead56796ce2af | Calculate the indefinite integral $$ \int \sqrt{x^3 + x^4} \, dx. $$ | [
"\\frac{2}{231} (8 - 22x + 33x^2) \\sqrt{x^4 + x^3} + \\frac{11}{16} \\ln\\left| \\frac{\\sqrt{1+x} - \\sqrt{x}}{\\sqrt{1+x} + \\sqrt{x}} \\right| + C"
] | [
"Calculate the indefinite integral $$ \\int \\sqrt{x^3 + x^4} \\, dx. $$"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-d67433ddbe9fda3c3e4e911a448875032391a28007d71817d4e817e42a4903cf | Calculate the indefinite integral $\int (2x+1)\sqrt{x+2}\ dx$. | [
"\\(\\frac{4}{5}(x+2)^{\\frac{5}{2}} - 2(x+2)^{\\frac{3}{2}} + C\\) or \\(\\frac{2}{5}(x+2)(2x-1)\\sqrt{x+2} + C\\)"
] | [
"Calculate the indefinite integral $\\int (2x+1)\\sqrt{x+2}\\ dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-e7e0a79b913e31f3fc3c48b58ef35a74b6e805551536077909586432df5d6760 | Calculate the indefinite integral $\int (3^x-2)^2 \, dx$. | [
"\\(\\frac{3^{2x}}{2\\ln{3}} - \\frac{4 \\cdot 3^x}{\\ln{3}} + 4x\\)"
] | [
"Calculate the indefinite integral $\\int (3^x-2)^2 \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 3
} | 9 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.