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-b28f783a3e5248fde3ab512fd0a95813c3581fe6d2b565eac28b15c67b873e9f | Calculate $\frac{A}{B}$, where \[ A = \frac{1}{1 \cdot 2} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{21 \cdot 22} \] \[ B = \frac{1}{12 \cdot 22} + \frac{1}{13 \cdot 21} + \dots + \frac{1}{22 \cdot 12} \] | [
"17"
] | [
"Calculate $\\frac{A}{B}$, where\n\n\\[ A = \\frac{1}{1 \\cdot 2} + \\frac{1}{3 \\cdot 4} + \\dots + \\frac{1}{21 \\cdot 22} \\]\n\n\\[ B = \\frac{1}{12 \\cdot 22} + \\frac{1}{13 \\cdot 21} + \\dots + \\frac{1}{22 \\cdot 12} \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-403193fa7d77ae3b8f894cb3d2784abc9078883d68cd3505cd325e8c9b31120f | Calculate $\frac{\log _3 135}{\log _{15}3}-\frac{\log _3 5}{\log _{405}3}$ without resort to tables. | [
"3"
] | [
"Calculate $\\frac{\\log _3 135}{\\log _{15}3}-\\frac{\\log _3 5}{\\log _{405}3}$ without resort to tables."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e9f428b590812fcc5846bec856fd028bf98711d13a0b68fab0d1eef826a39d13 | Calculate $\frac{\sqrt[4]{7}+\sqrt[8]{10}}{\sqrt[3]{5}+\sqrt[4]{2}+\sqrt[7]{8}}$. | [
"0.697"
] | [
"Calculate $\\frac{\\sqrt[4]{7}+\\sqrt[8]{10}}{\\sqrt[3]{5}+\\sqrt[4]{2}+\\sqrt[7]{8}}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f7b0aa05fdcac12859c7ba8ed6fe72700fda22588eb101f852320b3d89d53107 | Calculate $\frac{\text{d}^{100}}{\text{d}x^{100}}\frac{1}{x^2+4}$ without using complex numbers. | [
"\\dfrac{(100!) \\cdot 2^{100}}{(x^{2} + 4)^{101}}"
] | [
"Calculate $\\frac{\\text{d}^{100}}{\\text{d}x^{100}}\\frac{1}{x^2+4}$ without using complex numbers."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f388cc5a7edc0fb97700b90e676c099f0627c06385474d382dd3138fd797268e | Calculate $\gcd(n! + 1, (n + 1)!)$ for a positive integer $n$. | [
"\\(\\gcd(n! + 1, (n + 1)!) = \\begin{cases} \nn+1 & \\text{if } n+1 \\text{ is prime} \\\\\n1 & \\text{otherwise}\n\\end{cases}\\)"
] | [
"Calculate $\\gcd(n! + 1, (n + 1)!)$ for a positive integer $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 17
} | 24 | 0 | 0 |
omr-ws-04fe923d19a75498cfde78be55550dfa578ddcf4ef960ce7ed0c622c44cb489e | Calculate $\gcd(n! + 1, (n + 1)!)$ where $n$ is an integer. | [
"\\(\\gcd(n! + 1, (n + 1)!) = \\begin{cases} \n1 & \\text{if } n+1 \\text{ is not prime} \\\\\nn+1 & \\text{if } n+1 \\text{ is prime}\n\\end{cases}\\)"
] | [
"Calculate $\\gcd(n! + 1, (n + 1)!)$ where $n$ is an integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 19
} | 29 | 0 | 0 |
omr-ws-eaac1ef2b552e9a45c335d15ce37c72dfff79f290f43a6f512ce246d8bca1896 | Calculate $\gcd(n!+1, (n+1)!)$ for a positive integer $n$. | [
"\\(\\gcd(n!+1, (n+1)!) = \\begin{cases} \n1 & \\text{if } n+1 \\text{ is composite} \\\\\nn+1 & \\text{if } n+1 \\text{ is prime}\n\\end{cases}\\)"
] | [
"Calculate $\\gcd(n!+1, (n+1)!)$ for a positive integer $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 16
} | 29 | 0 | 0 |
omr-ws-9a72f49668746f49dbf4b238acb58029f313210396487d21bee78b3924850991 | Calculate $\iint_\Sigma (\bar{\nabla} \times \bar{A}) \cdot dS$ where $\bar{A} = -y\hat{x} + x\hat{y} + f(z)\hat{z}$, $\Sigma$ is the surface of the hemisphere, and $\gamma = \partial \Sigma$ is the large circle (base) of the hemisphere lying in the $xy$-plane. $f(z)$ is an arbitrary nice function. | [
"\\(2\\pi R^2\\)"
] | [
"Calculate $\\iint_\\Sigma (\\bar{\\nabla} \\times \\bar{A}) \\cdot dS$ where $\\bar{A} = -y\\hat{x} + x\\hat{y} + f(z)\\hat{z}$, $\\Sigma$ is the surface of the hemisphere, and $\\gamma = \\partial \\Sigma$ is the large circle (base) of the hemisphere lying in the $xy$-plane. $f(z)$ is an arbitrary nice function."... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 11
} | 22 | 0 | 0 |
omr-ws-130d1cc4450c3af73374d0dc4ae99feab27b565f8d685e4109aafa9a0e0ebe08 | Calculate $\int \frac{1}{\sqrt{a + \frac{1}{x}}} \, dx$. | [
"\\frac{\\sqrt{x(a x + 1)}}{a} - \\frac{1}{a^{3/2}} \\ln\\left( \\sqrt{a x} + \\sqrt{a x + 1} \\right) + C"
] | [
"Calculate $\\int \\frac{1}{\\sqrt{a + \\frac{1}{x}}} \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c6a0321637231ea069cc1b8d2fd768ea792582e2e53af8e96c67a8f1d2eff359 | Calculate $\int \frac{e^{2x}\cdot\arcsin e^{x}}{\sqrt{1-e^{2x}}} dx.$ | [
"\\( e^x - \\sqrt{1 - e^{2x}} \\arcsin(e^x) + C \\)"
] | [
"Calculate $\\int \\frac{e^{2x}\\cdot\\arcsin e^{x}}{\\sqrt{1-e^{2x}}} dx.$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 32
} | 46 | 0 | 0 |
omr-ws-65c3a84079fcdba7ad9e4e7ae4cb49e8701743a3932408de5c42b4e5c59560b6 | Calculate $\int \frac{xe^{\arctan x}}{(1 + x^2)^{\frac{3}{2}}} \, dx$. | [
"\\dfrac{(x - 1)e^{\\arctan x}}{2\\sqrt{1 + x^2}} + C"
] | [
"Calculate $\\int \\frac{xe^{\\arctan x}}{(1 + x^2)^{\\frac{3}{2}}} \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4c7658caf574fe49b4af57e659773a624b126077ca95d9dd32589d2f8547bbee | Calculate $\int \sqrt{\cos x} \, dx$. | [
"\\( 2\\,{\\frac {\\sqrt { \\left( \\sin \\left( x/2 \\right) \\right) ^{2}}\\sqrt {-2\\, \\left( \\cos \\left( x/2 \\right) \\right) ^{2}+1}{\\it EllipticE} \\left( \\cos \\left( x/2 \\right) ,\\sqrt {2} \\right) }{\\sin \\left( x/2 \\right) \\sqrt {2\\, \\left( \\cos \\left( x/2 \\right) \\right) ^{2}-1}}} \\)... | [
"Calculate $\\int \\sqrt{\\cos x} \\, dx$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-6b87f44866530b794da5f6bd623a00af96f129917e849b1ccf12b6bcbea52621 | Calculate $\int \sqrt{\cosh x} \, dx$. | [
"\\(-2\\,{\\frac {\\sqrt {- \\left( \\sinh \\left( x/2 \\right) \\right) ^{2}}\n\\sqrt {-2\\, \\left( \\cosh \\left( x/2 \\right) \\right) ^{2}+1}{\\it \nEllipticE} \\left( \\cosh \\left( x/2 \\right) ,\\sqrt {2} \\right) }{\\sinh\n \\left( x/2 \\right) \\sqrt {2\\, \\left( \\cosh \\left( x/2 \\right) \n\\right) ... | [
"Calculate $\\int \\sqrt{\\cosh x} \\, dx$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f50e6d99bc69cc60f733f7bad06d07384092a8138e76fb39ed421fd748e0d1f | Calculate $\int \sqrt{a + \frac{1}{x}} \, dx$. | [
"\\sqrt{x(ax + 1)} + \\frac{1}{\\sqrt{a}} \\ln\\left( \\sqrt{ax} + \\sqrt{ax + 1} \\right) + C"
] | [
"Calculate $\\int \\sqrt{a + \\frac{1}{x}} \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-9458fbcddace427f4c9fbc68527f55329437dd0d2099497a8b75527d063e7cd3 | Calculate $\int _{C} y \, dx + z \, dy + x \, dz$ where $C$ is the intersection between $z=xy$ and $x^{2}+y^{2}=1$, and the curve has a counter-clockwise direction when viewed from the $xy$-plane. | [
"-\\pi"
] | [
"Calculate $\\int _{C} y \\, dx + z \\, dy + x \\, dz$ where $C$ is the intersection between $z=xy$ and $x^{2}+y^{2}=1$, and the curve has a counter-clockwise direction when viewed from the $xy$-plane."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b3b1cf97719f6555d142a9332bf8a56d5e04d7cef6b5c28fd0cc4e8ec4901d15 | Calculate $\int _{C} y \, dx + z \, dy + x \, dz$ where $C$ is the intersection curve between $x+y=2$ and $x^{2}+y^{2}+z^{2} = 2(x+y)$, and the curve goes in a clockwise direction when viewed from $(0,0,0)$. | [
"$-2\\sqrt{2}\\pi$"
] | [
"Calculate $\\int _{C} y \\, dx + z \\, dy + x \\, dz$ where $C$ is the intersection curve between $x+y=2$ and $x^{2}+y^{2}+z^{2} = 2(x+y)$, and the curve goes in a clockwise direction when viewed from $(0,0,0)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 8
} | 12 | 0 | 0 |
omr-ws-7c6c7b4beecbba3a37e74167591a3db14349e54b2c7cb8364b3996e217ba9600 | Calculate $\int e^{-2x^2}\ \text{d}x$. | [
"\\(\\frac{1}{4}\\sqrt{2}\\sqrt{\\pi} \\text{erf}(\\sqrt{2}x) + C\\)"
] | [
"Calculate $\\int e^{-2x^2}\\ \\text{d}x$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-35484751df227f0000a83e0687ccb2309808423a3cbb0afd02d852a728582328 | Calculate $\int e^{\theta}\csc\theta \, d\theta$. | [
"-e^{\\theta} \\ln|\\csc\\theta + \\cot\\theta| + C"
] | [
"Calculate $\\int e^{\\theta}\\csc\\theta \\, d\\theta$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-bf8026550e96f0a67a949015c355fc3ffd28134fe7d8320f50272c7ce084e2a0 | Calculate $\int e^{\theta}\sec\theta \, d\theta$. | [
"e^\\theta \\sec \\theta - \\int e^\\theta \\sec \\theta \\tan \\theta \\, d\\theta"
] | [
"Calculate $\\int e^{\\theta}\\sec\\theta \\, d\\theta$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-68fa1b335af0008ad4449af945303f3c7b4ce5e706b39d47da48f16f6c260345 | Calculate $\int e^{t^2} dt$. | [
"\\frac{\\sqrt{\\pi}}{2} \\text{erfi}(t) + C"
] | [
"Calculate $\\int e^{t^2} dt$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-78cd290406f881e6aa3a278559047cec67b3d50a41a97594ff01ecedf708c95e | Calculate $\int e^{x+e^x}\ \text{d}x$. | [
"\\( e^{e^x} + C \\)"
] | [
"Calculate $\\int e^{x+e^x}\\ \\text{d}x$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-8e78e3f8ddf5ee044e081b008c6b089ec5d28fdc0b9b16323b2626eaf7eb28fc | Calculate $\int\int_{D}(x-y)dxdy,$ where $D={(x,y):0 \le x \le 2, 3 \le y \le x^2}.$ | [
"\\(\\frac{19}{5}\\)"
] | [
"Calculate $\\int\\int_{D}(x-y)dxdy,$ where $D={(x,y):0 \\le x \\le 2, 3 \\le y \\le x^2}.$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-43ba809c386393488509dd06d1ae713bf295ba761d134a886cfef880cb1d7851 | Calculate $\int\int_{D}(yx^2+y^2 )dxdy,$ where $D={(x,y):1 \le x \le 3, x \le y \le 2x}.$ | [
"\\(\\frac{1789}{15}\\)"
] | [
"Calculate $\\int\\int_{D}(yx^2+y^2 )dxdy,$ where $D={(x,y):1 \\le x \\le 3, x \\le y \\le 2x}.$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-7449e09db8ecc4af759416ad08790fd41278d2cc63b06e0d03782990270a2d7c | Calculate $\int\int_{D}yx \, dxdy,$ where $D$ is the region in the plane $Oxy$ between the parabola $y=x^2$ and the lines $y=0$ and $x=2.$ | [
"\\dfrac{16}{3}"
] | [
"Calculate $\\int\\int_{D}yx \\, dxdy,$ where $D$ is the region in the plane $Oxy$ between the parabola $y=x^2$ and the lines $y=0$ and $x=2.$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2527203661de9b719638db01556d66eeec66075dac350a723e0bc2053c9145ef | Calculate $\int\limits_{0}^{\frac{1}{\sqrt{2}}}{\frac{{{x}^{8}}}{{{\left( {{x}^{4}}-1 \right)}^{2}}}}dx$ | [
"\\( \\frac{2}{3}\\sqrt{2} - \\frac{5}{8}\\arctan\\left(\\frac{\\sqrt{2}}{2}\\right) - \\frac{5}{16}\\ln(2+\\sqrt{2}) + \\frac{5}{16}\\ln(2-\\sqrt{2}) \\)"
] | [
"Calculate $\\int\\limits_{0}^{\\frac{1}{\\sqrt{2}}}{\\frac{{{x}^{8}}}{{{\\left( {{x}^{4}}-1 \\right)}^{2}}}}dx$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-20feb6490abdbb2fcb3059cc69c27e39c62a782027429bf8f726716cd948500f | Calculate $\int\limits_{0}^{\frac{\pi }{4}}{\frac{x}{{{\cos x}^{4}}}}dx$ | [
"\\(\\frac{\\pi}{3} - \\frac{\\ln(2)}{3} - \\frac{1}{6}\\)"
] | [
"Calculate $\\int\\limits_{0}^{\\frac{\\pi }{4}}{\\frac{x}{{{\\cos x}^{4}}}}dx$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-69ed82219a74b797e1aecda39c77dfa1238c3f4810f2c0d8ca3686651e56402a | Calculate $\int\limits_{1}^{2\sqrt{2}}{\frac{\sqrt[3]{x-{{x}^{3}}}}{{{x}^{4}}}}dx$ | [
"\\(-\\frac{3}{8} \\left( -\\frac{7}{8} \\right)^{\\frac{4}{3}}\\)"
] | [
"Calculate $\\int\\limits_{1}^{2\\sqrt{2}}{\\frac{\\sqrt[3]{x-{{x}^{3}}}}{{{x}^{4}}}}dx$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 11
} | 12 | 0 | 0 |
omr-ws-7e02e41b234e7958fdb38184c3db4c8fcf1d24126ed070f6a20e29cd5d8f95cb | Calculate $\int\limits_{1}^{2\sqrt{2}}{\frac{\sqrt[3]{x-{{x}^{3}}}}{{{x}^{5}}}}dx$ | [
"-\\dfrac{3}{16}"
] | [
"Calculate $\\int\\limits_{1}^{2\\sqrt{2}}{\\frac{\\sqrt[3]{x-{{x}^{3}}}}{{{x}^{5}}}}dx$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-13a1ac23e97e309e1668469eddd925016855ace3a5fd6a1826efdc6e8415bd8e | Calculate $\int_L \text{Re}z \, dz$, where $L$ is the lower semicircle $|z|=4$ starting at $-4$. | [
"\\( 8\\pi i \\)"
] | [
"Calculate $\\int_L \\text{Re}z \\, dz$, where $L$ is the lower semicircle $|z|=4$ starting at $-4$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 4,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-cdb33bc12c24aba581694275dfd908520bf15462f97f40f2c7cbda24b681a3a7 | Calculate $\int_\varphi z^m (1-z)^n dz$ for the path $\varphi(t)=2e^{2\pi it}$, where $t\in[0,1]$ and $m,n\in \mathbb{Z}$. | [
"2\\pi i \\left( (-1)^{m+1} \\dbinom{n}{-m-1} + (-1)^n \\dbinom{m}{-n-1} \\right)"
] | [
"Calculate $\\int_\\varphi z^m (1-z)^n dz$ for the path $\\varphi(t)=2e^{2\\pi it}$, where $t\\in[0,1]$ and $m,n\\in \\mathbb{Z}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-f6d2cb8170692732643c427cd2a701cdf1ff1d4737ab6180c9ab450e00a55dc5 | Calculate $\int_{ - \pi }^\pi {\left( {\cos \theta } \right)} ^m \cos n\theta d\theta$, where $m, n \in \mathbb{Z}$. | [
"\\(\\begin{cases} \n\\frac{\\pi}{2^{m-1}} \\binom{m}{\\frac{m-|n|}{2}} & \\text{if } m \\equiv n \\ (\\mathrm{mod} \\ 2) \\ \\text{and} \\ 0 \\leq |n| \\leq m \\\\\n0 & \\text{otherwise}\n\\end{cases}\\)"
] | [
"Calculate $\\int_{ - \\pi }^\\pi {\\left( {\\cos \\theta } \\right)} ^m \\cos n\\theta d\\theta$, where $m, n \\in \\mathbb{Z}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 6
} | 24 | 0 | 0 |
omr-ws-ab3d7f86ba87cf1bc1f6bbcec62555582b38d098a2ce75149bcf42febe9c3c29 | Calculate $\int_{ - \pi }^\pi {e^{\cos \theta } \cos \left( {\sin \theta - n\theta } \right)} d\theta$. | [
"\\(\\frac{2\\pi}{n!}\\)"
] | [
"Calculate $\\int_{ - \\pi }^\\pi {e^{\\cos \\theta } \\cos \\left( {\\sin \\theta - n\\theta } \\right)} d\\theta$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-18a87a744e025860449250ed0e0a21fa203e4a612900dbd751a7e9123b6b59f4 | Calculate $\int_{-2}^{2} x^2 \, d\alpha(x)$, where $\alpha(x) = \begin{cases} x+2 & \text{if } -2 \le x \le -1 \\ 2 & \text{if } -1 < x < 0 \\ x^2 + 3 & \text{if } 0 \le x \le 2 \end{cases}$. | [
"$\\frac{34}{3}$"
] | [
"Calculate $\\int_{-2}^{2} x^2 \\, d\\alpha(x)$, where $\\alpha(x) = \\begin{cases} \nx+2 & \\text{if } -2 \\le x \\le -1 \\\\\n2 & \\text{if } -1 < x < 0 \\\\\nx^2 + 3 & \\text{if } 0 \\le x \\le 2 \n\\end{cases}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e7a0416b82f1bb17fb92cd52ac893a69a5a056ca4a0608b1de52f26eaefbbc16 | Calculate $\int_{0}^{1}\ln\{ x \}\ln\{ 2x \}...\ln\{ nx \}dx$, where $n \in \mathbb{N}$. | [
"(-1)^n n!"
] | [
"Calculate $\\int_{0}^{1}\\ln\\{ x \\}\\ln\\{ 2x \\}...\\ln\\{ nx \\}dx$, where $n \\in \\mathbb{N}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-130b5ccc5cbe06d86d0614d9e71b7ad636d8b9f9618ccff84fb78bedf3ef0107 | Calculate $\int_{0}^{1}\ln\{ x \}\ln\{ 2x \}dx$. | [
"\\(2 - \\frac{\\pi^2}{24}\\)"
] | [
"Calculate $\\int_{0}^{1}\\ln\\{ x \\}\\ln\\{ 2x \\}dx$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-39b5ffc8d37300d31526746eeea5662b2079f6267b4b2539be5a4f034a7f6e8d | Calculate $\int_{0}^{2\pi} p(\cos(x)) \, dx$ where $p(x)$ is a polynomial with real coefficients. | [
"\\(2\\pi \\left(a_0 + \\frac{1}{2} a_2 + \\frac{1 \\cdot 3}{2 \\cdot 4} a_4 + \\frac{1 \\cdot 3 \\cdot 5}{2 \\cdot 4 \\cdot 6} a_6 + \\cdots \\right)\\)"
] | [
"Calculate $\\int_{0}^{2\\pi} p(\\cos(x)) \\, dx$ where $p(x)$ is a polynomial with real coefficients."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 24
} | 46 | 0 | 0 |
omr-ws-c259f90d4a5f43c1d0414b4ecf082d18037d25848e8c013ca3f2b7fe90363364 | Calculate $\int_{0}^{2\pi}r(\theta )^{2}\ d\theta$, where $r(\theta)$ is the length of the perpendicular from the origin to the tangent line of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\ (a>0,\ b>0)$, and $\theta$ is the angle between the perpendicular and the positive direction of the $x$-axis. | [
"$\\pi(a^{2}+b^{2})$"
] | [
"Calculate $\\int_{0}^{2\\pi}r(\\theta )^{2}\\ d\\theta$, where $r(\\theta)$ is the length of the perpendicular from the origin to the tangent line of the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1\\ (a>0,\\ b>0)$, and $\\theta$ is the angle between the perpendicular and the positive direction of the $x$-... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 24
} | 44 | 0 | 0 |
omr-ws-8ff6c10c5effd2686ba2fa3e608c651d200a8c5e015cbe6dd212c2e0fab06909 | Calculate $\int_{0}^{\infty} \frac{dx}{x^{n}+x^{\frac{1}{n}}}$ for $n \neq 1$ and $n \in \mathbb{N}$. | [
"\\(\\frac{\\pi}{(n+1)\\sin(\\frac{\\pi}{n+1})}\\)"
] | [
"Calculate $\\int_{0}^{\\infty} \\frac{dx}{x^{n}+x^{\\frac{1}{n}}}$ for $n \\neq 1$ and $n \\in \\mathbb{N}$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-2cb50b6f96cf07ed0c3f8c50a1e7feffbf259eea20eea9ec5f7a2271e51654f0 | Calculate $\int_{0}^{\pi/4} \arctan(\cot^2(x)) \, dx$. | [
"\\dfrac{\\pi^2}{16}"
] | [
"Calculate $\\int_{0}^{\\pi/4} \\arctan(\\cot^2(x)) \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-1cdcf8e9584fe0bd751deb3e46543854b45059852ebe3e667a5daa469b5dea44 | Calculate $\int_{2}^{3} \frac{\ln x}{x-2} \, dx$. | [
"\\text{The integral diverges}"
] | [
"Calculate $\\int_{2}^{3} \\frac{\\ln x}{x-2} \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5c81fb940fccc196c2af228169435cb4d8787c2a80d239f1c263aab545436c0 | Calculate $\int_{C}ydx +zdy +xdz$, where C is the curve of intersection of the surfaces $x+y=2$ and $x^2+y^2+z^2=2(x+y)$. The curve is to be traversed once in a direction that appears clockwise when viewed from the origin. | [
"-2\\pi\\sqrt{2}"
] | [
"Calculate $\\int_{C}ydx +zdy +xdz$, where C is the curve of intersection of the surfaces $x+y=2$ and $x^2+y^2+z^2=2(x+y)$. The curve is to be traversed once in a direction that appears clockwise when viewed from the origin."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 10
} | 17 | 0 | 0 |
omr-ws-3ac2f000084a39df35d8d089dca7a9a68a94a2c55a4b43d0f3a0ec20d249b421 | Calculate $\int_{C}ydx +zdy +xdz$, where C is the intersection of the surfaces $z=xy$ and $x^2+y^2=1$, traversed once in a direction that appears counterclockwise when viewed from high above the $xy-$plane. | [
"\\(\\pi\\)"
] | [
"Calculate $\\int_{C}ydx +zdy +xdz$, where C is the intersection of the surfaces $z=xy$ and $x^2+y^2=1$, traversed once in a direction that appears counterclockwise when viewed from high above the $xy-$plane."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d853aa26c3d498e16e351a7d3848b25ef23ce4012caab82a01afa2227761d65 | Calculate $\int{\frac{1}{\sqrt{\tan{x}}}}dx$. | [
"\\(-\\frac{\\tan ^{-1}\\left(1-\\sqrt{2} \\sqrt{\\tan (x)}\\right)}{\\sqrt{2}}+\\frac{\\tan ^{-1}\\left(1+\\sqrt{2} \\sqrt{\\tan (x)}\\right)}{\\sqrt{2}}-\\frac{\\log \\left(1-\\sqrt{2} \\sqrt{\\tan (x)}+\\tan (x)\\right)}{2 \\sqrt{2}}+\\frac{\\log \\left(1+\\sqrt{2} \\sqrt{\\tan (x)}+\\tan (x)\\right)}{2 \\sqrt{2... | [
"Calculate $\\int{\\frac{1}{\\sqrt{\\tan{x}}}}dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 16
} | 29 | 0 | 0 |
omr-ws-69cbb35c3d238e806c494cd48d1763a3753485108c550b270cf17599e7bed40d | Calculate $\left(\frac{1}{2} + \cos \frac{\pi}{20}\right)\left(\frac{1}{2} + \cos \frac{3\pi}{20}\right)\left(\frac{1}{2} + \cos \frac{9\pi}{20}\right)\left(\frac{1}{2} + \cos \frac{27\pi}{20}\right)$. | [
"\\dfrac{1}{16}"
] | [
"Calculate $\\left(\\frac{1}{2} + \\cos \\frac{\\pi}{20}\\right)\\left(\\frac{1}{2} + \\cos \\frac{3\\pi}{20}\\right)\\left(\\frac{1}{2} + \\cos \\frac{9\\pi}{20}\\right)\\left(\\frac{1}{2} + \\cos \\frac{27\\pi}{20}\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-2888d5b2316a28677fe8a610991925ce6ead2b1df2113e070d172be5415fd5fe | Calculate $\left(\sin\frac{\pi}{5}\right)\cdot\left(\sin\frac{2\pi}{5}\right)$ without directly calculating the values of $\sin\frac{\pi}{5}$ or $\sin\frac{2\pi}{5}$. | [
"$\\frac{\\sqrt{5}}{4}$"
] | [
"Calculate $\\left(\\sin\\frac{\\pi}{5}\\right)\\cdot\\left(\\sin\\frac{2\\pi}{5}\\right)$ without directly calculating the values of $\\sin\\frac{\\pi}{5}$ or $\\sin\\frac{2\\pi}{5}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-5970503308ee6c1d4207c7a257ef34c5ab62812f9a83c8361a7ffe7bdae68d64 | Calculate $\left[\sum_{n=1}^{10^{9}}\frac{1}{\sqrt[3]{n^{2}}}\right]$. | [
"2997"
] | [
"Calculate $\\left[\\sum_{n=1}^{10^{9}}\\frac{1}{\\sqrt[3]{n^{2}}}\\right]$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 8
} | 29 | 0 | 0 |
omr-ws-b4e50ec59a38619c710fbfa428f009ceb1caab7be202c51193f231343369ad28 | Calculate $\left\{\left[i^{100}-\left(\frac{1-i}{1+i}\right)^5\right]^2+\left(\frac{1+i}{\sqrt 2}\right)^{20}\right\}(1+2i)$. | [
"-5"
] | [
"Calculate $\\left\\{\\left[i^{100}-\\left(\\frac{1-i}{1+i}\\right)^5\\right]^2+\\left(\\frac{1+i}{\\sqrt 2}\\right)^{20}\\right\\}(1+2i)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-5affb9caaf68e1b0efd4c2da0f249f9c365e1df8e783271f671f30305cf64fca | Calculate $\lfloor (\sqrt{2}+\sqrt{3})^6\rfloor$. | [
"969"
] | [
"Calculate $\\lfloor (\\sqrt{2}+\\sqrt{3})^6\\rfloor$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-0de27a4ecb8825707cff0a88d7e23563011b830929c131efef314793a31cefb3 | Calculate $\lim\limits_{n \to \infty}\int_{0}^{1} x\cdot\sin\left(\frac{\pi}{2}\lfloor2nx\rfloor\right) dx$. | [
"0"
] | [
"Calculate $\\lim\\limits_{n \\to \\infty}\\int_{0}^{1} x\\cdot\\sin\\left(\\frac{\\pi}{2}\\lfloor2nx\\rfloor\\right) dx$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a3da6bdce8c5e0cd5f21573ae06d2e5d0485c2836148c6249e3853d381305eda | Calculate $\lim\limits_{x \to 0} (\cot^2 x - \frac{1}{x^2})$ without using L'Hôpital's rule or Taylor series. | [
"\\(-\\frac{2}{3}\\)"
] | [
"Calculate $\\lim\\limits_{x \\to 0} (\\cot^2 x - \\frac{1}{x^2})$ without using L'Hôpital's rule or Taylor series."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-413c9b371a0258afe83c5bb6f9d4bc6faf270111b3d998fbcf599cdfb86bccc9 | Calculate $\lim_{(x,y) \rightarrow (2,0)} \frac{xy^2}{y^2+(x-2)^2}$. | [
"The limit does not exist."
] | [
"Calculate $\\lim_{(x,y) \\rightarrow (2,0)} \\frac{xy^2}{y^2+(x-2)^2}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-d62d96b5edd0f0b964e6c41c611e1f2b84901af6619fa8adad14b6206fdb545d | Calculate $\lim_{(x,y) \rightarrow (2,0)} \frac{xy}{2(x-2)}$. | [
"\\text{DNE}"
] | [
"Calculate $\\lim_{(x,y) \\rightarrow (2,0)} \\frac{xy}{2(x-2)}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a45f620c6d8bda6cfbcfd2f517d3f6026473b0db3b714ab010e9809cc0368a69 | Calculate $\lim_{\lambda\rightarrow0^+}\lambda F(\lambda)$ where $F(\lambda)=\int_{\mathbb R_+}e^{-\lambda x}\frac{x^2}{\sqrt{1+x^4}}\ell(dx)$ and $\ell$ is the Lebesgue measure on $\mathbb R$. | [
"1"
] | [
"Calculate $\\lim_{\\lambda\\rightarrow0^+}\\lambda F(\\lambda)$ where $F(\\lambda)=\\int_{\\mathbb R_+}e^{-\\lambda x}\\frac{x^2}{\\sqrt{1+x^4}}\\ell(dx)$ and $\\ell$ is the Lebesgue measure on $\\mathbb R$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2fa4d2457b4213ce6fe6a37dc55f78c07583e104fd36aa27ecb70ec8798b3e29 | Calculate $\lim_{m\to\infty}\lim_{n\to\infty}\cos^{2n}(m!\pi x)$. | [
"\\(\\begin{cases} \n1 & \\text{if } x \\text{ is rational} \\\\\n0 & \\text{if } x \\text{ is irrational}\n\\end{cases}\\)"
] | [
"Calculate $\\lim_{m\\to\\infty}\\lim_{n\\to\\infty}\\cos^{2n}(m!\\pi x)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-58e787092651dd829a178fe65b6b4057c6127f994f1fce5949f1aa63125bacbb | Calculate $\lim_{n \to \infty} I_n$, where $I_{n}= n \int_{0}^{1} \dfrac{\cos(x)}{1+e^{nx}} dx.$ | [
"\\(\\ln 2\\)"
] | [
"Calculate $\\lim_{n \\to \\infty} I_n$, where $I_{n}= n \\int_{0}^{1} \\dfrac{\\cos(x)}{1+e^{nx}} dx.$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7e92b23313d301ea00fd08882cb0944bd54ad1da74496b5f3ac23049ff865d54 | Calculate $\lim_{n \to \infty} \left(1-\frac{1}{a}\right)\left(1-\frac{1}{a^2}\right) \cdots \left(1-\frac{1}{a^n}\right)$, where $a > 1$. | [
"\\prod_{k=1}^{\\infty} \\left(1 - \\frac{1}{a^k}\\right)"
] | [
"Calculate $\\lim_{n \\to \\infty} \\left(1-\\frac{1}{a}\\right)\\left(1-\\frac{1}{a^2}\\right) \\cdots \\left(1-\\frac{1}{a^n}\\right)$, where $a > 1$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5af8956c0d6ebac731d7d8fe61a0beb7f953eeb170bb27086d03cc2dd822bab | Calculate $\lim_{n \to \infty}\frac{1}{n^4}\sum_{k=0}^{n-1}\frac{1}{\sin^4(\frac{x+k\pi}{n})}$, where $x \in \mathbb{R} \setminus \{p \pi | p \in \mathbb{Z} \}$. | [
"\\dfrac{1}{45}"
] | [
"Calculate $\\lim_{n \\to \\infty}\\frac{1}{n^4}\\sum_{k=0}^{n-1}\\frac{1}{\\sin^4(\\frac{x+k\\pi}{n})}$, where $x \\in \\mathbb{R} \\setminus \\{p \\pi | p \\in \\mathbb{Z} \\}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-8394d298ed86b839b7639a25834c87053ca121ed0e2f01cfd7ddf085cd09565f | Calculate $\lim_{n\rightarrow \infty} \int_{0}^{1} nx(1-x^2)^n \, dx$. | [
"$\\frac{1}{2}$"
] | [
"Calculate $\\lim_{n\\rightarrow \\infty} \\int_{0}^{1} nx(1-x^2)^n \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2c60ebecff388fe511d1468ee272fecf0857726fb77895296dcfc8111d0a71aa | Calculate $\lim_{n\rightarrow \infty}\int_{-1}^1 (1-x^2)^n dx$. | [
"0"
] | [
"Calculate $\\lim_{n\\rightarrow \\infty}\\int_{-1}^1 (1-x^2)^n dx$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-99e4e9d8e452bcc32003e4e90ed329fdd0b0c2ebb8e08805119858a256523f61 | Calculate $\lim_{n\rightarrow \infty}\int_{1}^e (\ln x)^n dx$. | [
"0"
] | [
"Calculate $\\lim_{n\\rightarrow \\infty}\\int_{1}^e (\\ln x)^n dx$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-263efde2e7e7b1b6deefd5ca9f19f9a905d1fbe293abfd20f97b3f013858c312 | Calculate $\lim_{n\to \infty} \cos \left(\frac{\pi}{4}\right) \cdot \cos \left(\frac{\pi}{8}\right) \cdot \ldots \cdot \cos \left(\frac{\pi}{2^n}\right)$. | [
"\\(\\frac{2}{\\pi}\\)"
] | [
"Calculate $\\lim_{n\\to \\infty} \\cos \\left(\\frac{\\pi}{4}\\right) \\cdot \\cos \\left(\\frac{\\pi}{8}\\right) \\cdot \\ldots \\cdot \\cos \\left(\\frac{\\pi}{2^n}\\right)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-b237af8bb7f2af47fab4627c76621960bd1c0527ab123146e1e96d732bedf926 | Calculate $\lim_{n\to \infty} \sqrt[n]{{nk}\choose{n}}$, where $k$ is an arbitrarily fixed integer greater than $1$. | [
"\\(\\frac{k^k}{(k-1)^{k-1}}\\)"
] | [
"Calculate $\\lim_{n\\to \\infty} \\sqrt[n]{{nk}\\choose{n}}$, where $k$ is an arbitrarily fixed integer greater than $1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-1f6e64858609e6234c88cf1661f71c2a8a03c80bb71b26e7797eb11181b7ff56 | Calculate $\lim_{n\to \infty}\frac{1}{n}\sqrt[p]{1^p+2^p+...+n^p}$, where $p$ is a natural number and $p>0$. | [
"$\\infty$"
] | [
"Calculate $\\lim_{n\\to \\infty}\\frac{1}{n}\\sqrt[p]{1^p+2^p+...+n^p}$, where $p$ is a natural number and $p>0$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 26
} | 44 | 0 | 0 |
omr-ws-825d8aff403d090340dab473dd953d06d38b01c2b6d1fb2355b4f8942ab7367b | Calculate $\lim_{n\to+\infty}\sqrt[n]{(n!)^2}$. | [
"+\\infty"
] | [
"Calculate $\\lim_{n\\to+\\infty}\\sqrt[n]{(n!)^2}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-717a90a03e3d97ed408295883f2b66a4bb1fb58f2bc5dd1b6af1a2f9be30df7d | Calculate $\lim_{n\to\infty} 2^n f(1)f(2)\cdots f(n)$, where $f(x) = \frac{x+1}{2x+3}$. | [
"0"
] | [
"Calculate $\\lim_{n\\to\\infty} 2^n f(1)f(2)\\cdots f(n)$, where $f(x) = \\frac{x+1}{2x+3}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cfa1c8c2f4b813ee13dbadea509f86cb9458060578ba0a96abe27fdbd4ddc2dd | Calculate $\lim_{n\to\infty} S_n$, where $S_n = \sum_{1\leqslant i<j<k \leqslant n} a^{i+j+k}$ and $|a| < 1$. | [
"\\(\\frac{a^6}{(1-a)(1-a^2)(1-a^3)}\\)"
] | [
"Calculate $\\lim_{n\\to\\infty} S_n$, where $S_n = \\sum_{1\\leqslant i<j<k \\leqslant n} a^{i+j+k}$ and $|a| < 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 30
} | 44 | 0 | 0 |
omr-ws-32548b0192ed2fe95090e49c3be66788b174786411495bfa746bc2f57bc220b3 | Calculate $\lim_{n\to\infty} \dfrac{a_{n}}{b_{n}}$, where $a_{n}=\int_{0}^{\dfrac{\pi}{4}}(\cos 2x)^{n}dx$ and $b_{n}=\int_{0}^{\dfrac{\pi}{4}}(\cos x)^{n}dx$. | [
"\\(\\frac{1}{2}\\)"
] | [
"Calculate $\\lim_{n\\to\\infty} \\dfrac{a_{n}}{b_{n}}$, where $a_{n}=\\int_{0}^{\\dfrac{\\pi}{4}}(\\cos 2x)^{n}dx$ and $b_{n}=\\int_{0}^{\\dfrac{\\pi}{4}}(\\cos x)^{n}dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c21691b7abae7974e70388e8679461703a0043a50fc2fccf74481b2d24970e89 | Calculate $\lim_{n\to\infty} \left\{n^{2}\left(\int_{0}^{1}\sqrt[n]{1+x^{n}}dx -1\right)\right\}$. | [
"\\(\\frac{\\pi^2}{12}\\)"
] | [
"Calculate $\\lim_{n\\to\\infty} \\left\\{n^{2}\\left(\\int_{0}^{1}\\sqrt[n]{1+x^{n}}dx -1\\right)\\right\\}$."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-b45643a16f730f121181cb33a709b73ffc354ce271af26a1222b79df20bfe324 | Calculate $\lim_{n\to\infty}\frac{1^{n}+2^{n}+\cdots +n^{n}}{n^{n}}$ using Cesaro's theorem. | [
"\\dfrac{e}{e - 1}"
] | [
"Calculate $\\lim_{n\\to\\infty}\\frac{1^{n}+2^{n}+\\cdots +n^{n}}{n^{n}}$ using Cesaro's theorem."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-189963818247e177d7fddaa2d552654ade7be708d1c0b790c026f9284cad4211 | Calculate $\lim_{n\to\infty}\frac{1}{n^{3}}\int_{0}^{n^{2}}[\sqrt{x}]dx$, where $n$ is a natural number. | [
"\\(\\frac{2}{3}\\)"
] | [
"Calculate $\\lim_{n\\to\\infty}\\frac{1}{n^{3}}\\int_{0}^{n^{2}}[\\sqrt{x}]dx$, where $n$ is a natural number."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-fa0d6992285c7a5bccda482cad8f5e56145fc3f2a21cbc9db8ba9f87aa15818e | Calculate $\lim_{n\to\infty}\frac{\ln\left(1+2e+4e^{4}+\cdots +2n\cdot e^{n^{2}}\right)}{n^{2}}$ using Cesaro's theorem. | [
"1"
] | [
"Calculate $\\lim_{n\\to\\infty}\\frac{\\ln\\left(1+2e+4e^{4}+\\cdots +2n\\cdot e^{n^{2}}\\right)}{n^{2}}$ using Cesaro's theorem."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ac2ae27942fad97f67fed5a81f5a5faeba31fb1878be46448ce459f5fcab9788 | Calculate $\lim_{n\to\infty}\frac{\sqrt{1}+\sqrt{2}+\cdots +\sqrt{n}}{n\sqrt{n}}$ without using integrals. | [
"\\(\\frac{2}{3}\\)"
] | [
"Calculate $\\lim_{n\\to\\infty}\\frac{\\sqrt{1}+\\sqrt{2}+\\cdots +\\sqrt{n}}{n\\sqrt{n}}$ without using integrals."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 33
} | 48 | 0 | 0 |
omr-ws-810ed1922f635d37468182b839db8360140b8f25b2d889f738eda12fbba31717 | Calculate $\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}$ | [
"\\( e \\)"
] | [
"Calculate $\\lim_{n\\to\\infty}\\frac{n}{\\sqrt[n]{n!}}$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-21f4c1ceeca379263bf6bd4257c315cc15bb9447ffdd3b977d20d70176e45990 | Calculate $\lim_{n\to\infty}\int^1_0 \frac{n y^{n-1}}{1+y}\,dy$. | [
"\\(\\frac{1}{2}\\)"
] | [
"Calculate $\\lim_{n\\to\\infty}\\int^1_0 \\frac{n y^{n-1}}{1+y}\\,dy$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d4e1b652abe92895e790a48ef274d5268af4e9b4d3b59dfc3a664feffd33185f | Calculate $\lim_{n\to\infty}\int_{0}^{1}x^{2}|\sin n\pi x|dx$. | [
"\\(\\frac{2}{3\\pi}\\)"
] | [
"Calculate $\\lim_{n\\to\\infty}\\int_{0}^{1}x^{2}|\\sin n\\pi x|dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-eeb479d58f63a05d32df09380f31885823755ed4e07ec79505283fcd333512a3 | Calculate $\lim_{n\to\infty}\left(\frac{(n!)^2}{n^{2n}}\right)^{\frac{1}{n}}$. | [
"\\( e^{-2} \\)"
] | [
"Calculate $\\lim_{n\\to\\infty}\\left(\\frac{(n!)^2}{n^{2n}}\\right)^{\\frac{1}{n}}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-89d59db1a226b50e77a22b91261cb0010009874e09fbaef177b212b2cd58b05f | Calculate $\lim_{n\to\infty}\left(\frac{1}{n}\sum_{k=1}^{n}\sqrt{1+\frac{k}{n}}\right)$ without using integrals. | [
"$\\dfrac {4\\sqrt{2} - 2} {3}$"
] | [
"Calculate $\\lim_{n\\to\\infty}\\left(\\frac{1}{n}\\sum_{k=1}^{n}\\sqrt{1+\\frac{k}{n}}\\right)$ without using integrals."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-5ca852da381627de235dfea1aa3a4a86e86fd291a28416fa789235c601d7a020 | Calculate $\lim_{n\to\infty}\left(\frac{n!}{n^n \cdot e^{-n}}\right)^{\frac{1}{n}}$. | [
"1"
] | [
"Calculate $\\lim_{n\\to\\infty}\\left(\\frac{n!}{n^n \\cdot e^{-n}}\\right)^{\\frac{1}{n}}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fff2a322b884037dd60d9c8f7808a33bb9276563287ded54c40c880fbf5d2f9b | Calculate $\lim_{n\to\infty}\sqrt[n]{n!}$. | [
"\\(+\\infty\\)"
] | [
"Calculate $\\lim_{n\\to\\infty}\\sqrt[n]{n!}$."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-d35711fc6f62aefb17b8a3a54adeeb48cbb05c452b0643e865140e38580b5d49 | Calculate $\lim_{n\to\infty}\sum_{k=0}^{n}\binom{n}{k}^{-1}$. | [
"2"
] | [
"Calculate $\\lim_{n\\to\\infty}\\sum_{k=0}^{n}\\binom{n}{k}^{-1}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-5ecb0fc70073dc71857c41fca1c745156d52608df102f4eb4f34c1a61df080ad | Calculate $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{\sqrt{2n^{2}-k^{2}}}{n^{2}}$ without using integrals. Is it possible to calculate this limit using the Cesaro theorem? If so, demonstrate the process. | [
"\\dfrac{1}{2} + \\dfrac{\\pi}{4}"
] | [
"Calculate $\\lim_{n\\to\\infty}\\sum_{k=1}^{n}\\frac{\\sqrt{2n^{2}-k^{2}}}{n^{2}}$ without using integrals. Is it possible to calculate this limit using the Cesaro theorem? If so, demonstrate the process."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0e05681002b12610df5ed0371c76e68758433ee895ed2b9abbd1bb3213a9f7c3 | Calculate $\lim_{t \to -\infty}\left[\int_{t}^{0} \frac{2^{x}}{9-4^{x}}\ dx\right]$. | [
"\\(\\frac{1}{6}\\)"
] | [
"Calculate $\\lim_{t \\to -\\infty}\\left[\\int_{t}^{0} \\frac{2^{x}}{9-4^{x}}\\ dx\\right]$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-c4bdd82d3f88491e68262a2604007ea7cae34d43f4a8c753441e0ed08fb1c24e | Calculate $\lim_{x\to 0} \frac{x^3 + \sin(x)\tan(2x)-2\tan(x)}{\arctan^2(x\tan(x))-\tan(x\arctan^2(x))}$. | [
"\\(\\infty\\)"
] | [
"Calculate $\\lim_{x\\to 0} \\frac{x^3 + \\sin(x)\\tan(2x)-2\\tan(x)}{\\arctan^2(x\\tan(x))-\\tan(x\\arctan^2(x))}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-391b5959452d8b54f31907fb3525141dcf1c56f7777082dc156f612f292da16b | Calculate $\lim_{x\to 0} \frac{x^3+\sin(2x)-2\sin(x)}{\arctan(x^3)-(\arctan(x))^3}$. | [
"\\(\\frac{1}{4}\\)"
] | [
"Calculate $\\lim_{x\\to 0} \\frac{x^3+\\sin(2x)-2\\sin(x)}{\\arctan(x^3)-(\\arctan(x))^3}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 27
} | 39 | 0 | 0 |
omr-ws-a420b0ae936fc38bda25eb78ca87157e2ffe0a67844b60064d00d4e6c779f67c | Calculate $\lim_{x\to 0}\left(\frac{e^{x}-1}{1-\cos x}\right)$ without using approximations. | [
"\\(\\infty\\)"
] | [
"Calculate $\\lim_{x\\to 0}\\left(\\frac{e^{x}-1}{1-\\cos x}\\right)$ without using approximations."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-99cc3daa5502b4c8acef333f285c69648fe49638d0147cd30992cac191ed7410 | Calculate $\lim_{x\to a}\frac{x^{a^x}-a^{a^a}}{x-a}$, where $a > 0$. | [
"\\( a^{a^{a}+a}\\left(\\frac{1}{a} + (\\ln a)^2\\right) \\)"
] | [
"Calculate $\\lim_{x\\to a}\\frac{x^{a^x}-a^{a^a}}{x-a}$, where $a > 0$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 15
} | 25 | 0 | 0 |
omr-ws-6a6ebb62f23b15dfd92a0023107e4a511f0db6fd3672b9c9d165cb8fc84b5d3e | Calculate $\lim_{x\to e}\frac{\sqrt{\ln x}-1}{e-x}$ without using L'Hôpital's rule. | [
"\\(-\\frac{1}{2e}\\)"
] | [
"Calculate $\\lim_{x\\to e}\\frac{\\sqrt{\\ln x}-1}{e-x}$ without using L'Hôpital's rule."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-ad9dcd4bf0baca6354af3cc7d2dc74dad5b261505db6ad051bb39922bfc6ef40 | Calculate $\lim_{x\to+\infty}\frac{\sqrt{x}f(x)-x}{\ln x}$, if it exists, where $f(x)$ is defined by the relations $$f^2(x)-\ln f(x)=x$$ and $$f(x)\geq\frac{\sqrt{2}}{2}$$ for $x\geq\frac{1+\ln2}{2}$. | [
"\\(\\frac{1}{4}\\)"
] | [
"Calculate $\\lim_{x\\to+\\infty}\\frac{\\sqrt{x}f(x)-x}{\\ln x}$, if it exists, where $f(x)$ is defined by the relations $$f^2(x)-\\ln f(x)=x$$ and $$f(x)\\geq\\frac{\\sqrt{2}}{2}$$ for $x\\geq\\frac{1+\\ln2}{2}$."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-5794cdb6efd0160448041532624ed179d3305bcf2616448ca366130fc9077148 | Calculate $\lim_{x\to0}\frac{\ln(1+x^{2006})-\ln^{2006}(1+x)}{x^{2007}}$ | [
"1003"
] | [
"Calculate $\\lim_{x\\to0}\\frac{\\ln(1+x^{2006})-\\ln^{2006}(1+x)}{x^{2007}}$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-28dce1e083f216f099d11d4e39cd244a22c7479b6122906ffb23642cee54a32d | Calculate $\lim_{x\to0}\frac{\sin x^n - \sin^n x}{x^{n+2}}$, where $n \in \mathbb{N}$ and $n \ge 2$. | [
"\\(\\frac{n}{6}\\)"
] | [
"Calculate $\\lim_{x\\to0}\\frac{\\sin x^n - \\sin^n x}{x^{n+2}}$, where $n \\in \\mathbb{N}$ and $n \\ge 2$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-e3a8301e07363ebc62e4dc197aa8dcce403a9becd10201c287c3118ae7180b19 | Calculate $\lim_{x\to0}\frac{\tan x^n - \ln^n(1+x)}{x^{n+1}}$, where $n \in \mathbb{N}$. | [
"\\(\\frac{n}{2}\\)"
] | [
"Calculate $\\lim_{x\\to0}\\frac{\\tan x^n - \\ln^n(1+x)}{x^{n+1}}$, where $n \\in \\mathbb{N}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b19a21659fe6b767ddbcb4ddd6dea76777d64d9178584bcc70c51daecfa2d281 | Calculate $\lim_{x\to\infty} \int_0^{x} \arctan(t) \, \mathrm{dt}.$ | [
"\\(\\infty\\)"
] | [
"Calculate $\\lim_{x\\to\\infty} \\int_0^{x} \\arctan(t) \\, \\mathrm{dt}.$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-d245ba3a55a4268b82ccfa9501469e36cb154f978e8ec679641f8170c6c794c0 | Calculate $\liminf a_n$ and $\limsup a_n$ for the sequence defined by $$ a_n=\frac{\phi(2)+\phi(3)+...+\phi(n)}{n^2}. $$ | [
"\\(\\liminf a_n = \\frac{3}{\\pi^2}\\) and \\(\\limsup a_n = \\frac{3}{\\pi^2}\\)."
] | [
"Calculate $\\liminf a_n$ and $\\limsup a_n$ for the sequence defined by\n$$ a_n=\\frac{\\phi(2)+\\phi(3)+...+\\phi(n)}{n^2}. $$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 26
} | 40 | 0 | 0 |
omr-ws-c7d86ca5dc05d94b4322f0950996cabffa9b7223e4b7c55f1682ac1ea00fc215 | Calculate $\mathbb{P}(\{X=1\} \cup \{X=5\})$, where $X$ is the number of heads. | [
"\\dfrac{3}{16}"
] | [
"Calculate $\\mathbb{P}(\\{X=1\\} \\cup \\{X=5\\})$, where $X$ is the number of heads."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b9e0470b30824455747c200a873250117f74e3a3bd32d8f9cff8049fab0422d2 | Calculate $\mathbb{P}[Y_{1}+Y_{3}\leqslant 1]$ where $Y_{1},Y_{2},Y_{3}$ are the order statistics associated with the random variables $X_{1},X_{2},X_{3}$ with distribution ${\rm Unif}(0,1)$. | [
"\\(\\frac{1}{6}\\)."
] | [
"Calculate $\\mathbb{P}[Y_{1}+Y_{3}\\leqslant 1]$ where $Y_{1},Y_{2},Y_{3}$ are the order statistics associated with the random variables $X_{1},X_{2},X_{3}$ with distribution ${\\rm Unif}(0,1)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf66b2da2d5889cb252c4b85e68d18645318f01e0088b36138e27a73c5a8d5f4 | Calculate $\mathop {\lim }\limits_{x \to + \infty } \left( {2x^3 \sin {1 \over x} - x^3 \sin {2 \over x}} \right)$. | [
"1"
] | [
"Calculate $\\mathop {\\lim }\\limits_{x \\to + \\infty } \\left( {2x^3 \\sin {1 \\over x} - x^3 \\sin {2 \\over x}} \\right)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-bb6bb20d2f5ac46534795d443764c592e71e706c88c5295ad6e5866124501c58 | Calculate $\max_{n \in \mathbb{N}}\frac{n+3}{2n^{2}+n+4}$. | [
"\\(\\frac{4}{7}\\)"
] | [
"Calculate $\\max_{n \\in \\mathbb{N}}\\frac{n+3}{2n^{2}+n+4}$."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-897d51f5c9bc500788f6e5ddc3e97bcad6feb4b52d10762648952ae8ec2054e1 | Calculate $\min_{x\in \mathbb{R}}\left(\max\left(2\mid x \mid , \mid 1+x \mid\right)\right)$. | [
"\\(\\frac{2}{3}\\)"
] | [
"Calculate $\\min_{x\\in \\mathbb{R}}\\left(\\max\\left(2\\mid x \\mid , \\mid 1+x \\mid\\right)\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-8042c5eb7d1c4199d15246f7c8faeb113c00030fe101826b96bdf1c99727bd8a | Calculate $\min_{z\in \mathbb{C}}\left(\max\left(\mid 1+z \mid , \mid 1+z^2 \mid\right)\right)$. | [
"1"
] | [
"Calculate $\\min_{z\\in \\mathbb{C}}\\left(\\max\\left(\\mid 1+z \\mid , \\mid 1+z^2 \\mid\\right)\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-c207a90b96afee1e59e8a0820edd997188685f836a50e334c1954a09725b50f3 | Calculate $\oint_\gamma \bar{A} \cdot d\bar{r}$ where $\bar{A} = -y\hat{x} + x\hat{y} + f(z)\hat{z}$ and $\gamma$ is the large circle (base) of the hemisphere lying in the $xy$-plane with positive orientation. | [
"\\(2\\pi R^2\\)"
] | [
"Calculate $\\oint_\\gamma \\bar{A} \\cdot d\\bar{r}$ where $\\bar{A} = -y\\hat{x} + x\\hat{y} + f(z)\\hat{z}$ and $\\gamma$ is the large circle (base) of the hemisphere lying in the $xy$-plane with positive orientation."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 5
} | 22 | 0 | 0 |
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