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-9c93c20fb845d21db29c8c24ac0e1f00911261ca182c82379a12089b4a0d37f3 | Calculate $$ \lim_{x\to\ -1} \frac{n-\arccos x} {\sqrt{1+x}}, \forall n\in \mathbb{N}^{*}.$$ | [
"+\\infty"
] | [
"Calculate $$ \\lim_{x\\to\\ -1} \\frac{n-\\arccos x} {\\sqrt{1+x}}, \\forall n\\in \\mathbb{N}^{*}.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-b697b0c04efa6a4ac2590a8b085c6be3e44749300972bbfe66843ea747fc452c | Calculate $$(2+ \sqrt{3})^3 (3- \sqrt{2})^4 + (2 - \sqrt{2})^3(3+ \sqrt{2})^4$$ | [
"\\(9836 - 30\\sqrt{6}\\)"
] | [
"Calculate $$(2+ \\sqrt{3})^3 (3- \\sqrt{2})^4 + (2 - \\sqrt{2})^3(3+ \\sqrt{2})^4$$"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad7f6298a358058d7e021b1fcdd7b661cde696a093815e574f3250c5da0ea9e9 | Calculate $$(f^{2^{2023}})^{(100)}(3) \mod (2^{89}-1)$$ where $f(x)=\dfrac{x^2-x}{2}$, $g^{2}=g\circ g$, and $g^{(2)}=g''$. | [
"0"
] | [
"Calculate $$(f^{2^{2023}})^{(100)}(3) \\mod (2^{89}-1)$$ where $f(x)=\\dfrac{x^2-x}{2}$, $g^{2}=g\\circ g$, and $g^{(2)}=g''$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-2c93e8f79c64a56468390529e3e2e266ba1a635bae7a5089bd3dc371ba708fc1 | Calculate $$(f^{2^{2023}})^{(2^{2023})}(3) \mod (2^{89}-1)$$ where $f(x)=\dfrac{x^2-x}{2}$, $g^{2}=g\circ g$, and $g^{(2)}=g''$. | [
"0"
] | [
"Calculate $$(f^{2^{2023}})^{(2^{2023})}(3) \\mod (2^{89}-1)$$ where $f(x)=\\dfrac{x^2-x}{2}$, $g^{2}=g\\circ g$, and $g^{(2)}=g''$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-034345e960d4e44162682f59793e0a3ae1ae030e88d8a7b20314f72913ea2733 | Calculate $$L = \lim _{n \rightarrow \infty} \int _0 ^1 \left( \frac{k}{\sqrt[n]{x} +k -1} \right)^n dx$$ | [
"\\(\\frac{k}{k-1}\\)"
] | [
"Calculate\n$$L = \\lim _{n \\rightarrow \\infty} \\int _0 ^1 \\left( \\frac{k}{\\sqrt[n]{x} +k -1} \\right)^n dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-e05dc53e2474deb183b08a76cd220709c476e680ec2a84476c750e0c9d58a156 | Calculate $$U=\sum \limits _{k \geq 1,\gcd(k,30)=1} \dfrac {1}{k^2}.$$ | [
"\\(\\frac{8\\pi^2}{75}\\)"
] | [
"Calculate $$U=\\sum \\limits _{k \\geq 1,\\gcd(k,30)=1} \\dfrac {1}{k^2}.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-c7388eef5efc330e616c14fb0f3bb2668911e57d8a3b6ae1756fb0c0ae8b8e54 | Calculate $$\cot^4\frac{\pi}{7}+\cot^4\frac{2\pi}{7}+\cot^4 \frac{3 \pi}{7}$$ | [
"19"
] | [
"Calculate $$\\cot^4\\frac{\\pi}{7}+\\cot^4\\frac{2\\pi}{7}+\\cot^4 \\frac{3 \\pi}{7}$$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-4b500ecf03016c8358808f4df780f9416a6aed5a60e8e38890f6c9f6774fbc1d | Calculate $$\iiint_V \left(x^2+y^2+z^2 \right) dxdydz,$$ where $V$ is defined by the conditions $1 \le x^2+y^2+z^2 \le 4$ and $x^2+y^2 \le z^2$. | [
"\\dfrac{62\\pi}{5} \\left(2 - \\sqrt{2}\\right)"
] | [
"Calculate\n$$\\iiint_V \\left(x^2+y^2+z^2 \\right) dxdydz,$$\nwhere $V$ is defined by the conditions $1 \\le x^2+y^2+z^2 \\le 4$ and $x^2+y^2 \\le z^2$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67771432ae7e2571fc97ca7ef58913d8b45d1f36d9d5ba33a6ab5a523d5ae040 | Calculate $$\iiint_V\frac{xyz}{x^2+y^2}\,dxdydz,$$ where $V$ is the domain bounded by the surface $(x^2+y^2+z^2)^2=a^2xy$ and the plane $z=0$. | [
"\\dfrac{a^4}{144}"
] | [
"Calculate \n$$\\iiint_V\\frac{xyz}{x^2+y^2}\\,dxdydz,$$\nwhere $V$ is the domain bounded by the surface $(x^2+y^2+z^2)^2=a^2xy$ and the plane $z=0$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ff93416a72902eaa789bca558759a5e1dc55d7064991b28acaa56ed843c861d2 | Calculate $$\int_{0}^{1}\frac{\ln(x)\ln(1-x)}{1+x^{2}}dx$$ where $Li_{k}(x)=\sum_{n=1}^{+\infty}\frac{x^{n}}{n^{k}}$ is the Polylogarithm function for $|x|\leq 1$ and $k \geq 2$. | [
"\\frac{\\pi^3}{64} - \\frac{1}{2}G\\ln 2"
] | [
"Calculate $$\\int_{0}^{1}\\frac{\\ln(x)\\ln(1-x)}{1+x^{2}}dx$$ where $Li_{k}(x)=\\sum_{n=1}^{+\\infty}\\frac{x^{n}}{n^{k}}$ is the Polylogarithm function for $|x|\\leq 1$ and $k \\geq 2$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-8b2d246037e5e64e71e0e380e345d1d788b5595ba3a4e5216a4d18206c478565 | Calculate $$\left \lfloor{\sqrt{1}} \right \rfloor + \left \lfloor{\sqrt{2}} \right \rfloor + \left \lfloor{\sqrt{3}} \right \rfloor + \dots + \left \lfloor{\sqrt{2021}} \right \rfloor .$$ | [
"59598"
] | [
"Calculate $$\\left \\lfloor{\\sqrt{1}} \\right \\rfloor + \\left \\lfloor{\\sqrt{2}} \\right \\rfloor + \\left \\lfloor{\\sqrt{3}} \\right \\rfloor + \\dots + \\left \\lfloor{\\sqrt{2021}} \\right \\rfloor .$$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-084f9783d3998540f081a40561bc6932bf66f97e033504328f640724a4832e77 | Calculate $$\left( \frac{1}{2015} + \frac{2}{2015} \right) + \left( \frac{2}{2015} + \frac{3}{2015} \right) + \left( \frac{3}{2015} + \frac{4}{2015} \right) + \cdots + \left( \frac{2013}{2015} + \frac{2014}{2015} \right).$$ | [
"2013"
] | [
"Calculate $$\\left( \\frac{1}{2015} + \\frac{2}{2015} \\right) + \\left( \\frac{2}{2015} + \\frac{3}{2015} \\right) + \\left( \\frac{3}{2015} + \\frac{4}{2015} \\right) + \\cdots + \\left( \\frac{2013}{2015} + \\frac{2014}{2015} \\right).$$"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-fbbf8b2e7afe3c2ee757f03d69f24e8f4fcf48dba68bbc2be6085fb96aae705f | Calculate $$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+\frac{a_1x+a_3x^3+\cdots+a_{2n-1}x^{2n-1}}{n} \right)^ndx}$$ where $\left(a_n\right)_{n\geq1}$ is a sequence of nonnegative real numbers which converges to $a \in \mathbb{R}$. | [
"1 + a"
] | [
"Calculate $$\\lim \\limits_{n\\to \\infty}\\sqrt[n]{\\int \\limits_0^1 \\left(1+\\frac{a_1x+a_3x^3+\\cdots+a_{2n-1}x^{2n-1}}{n} \\right)^ndx}$$ where $\\left(a_n\\right)_{n\\geq1}$ is a sequence of nonnegative real numbers which converges to $a \\in \\mathbb{R}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-bc619340d460fa42df4e7394c68e0088e94c3ffb701ee184cd22c32e49b72f1b | Calculate $$\lim \limits_{n\to \infty}\sqrt[n]{\int \limits_0^1 \left(1+a_nx^n \right)^ndx}$$ where $\left(a_n\right)_{n\geq1}$ is a sequence of nonnegative real numbers which converges to $a \in \mathbb{R}$. | [
"1 + a"
] | [
"Calculate $$\\lim \\limits_{n\\to \\infty}\\sqrt[n]{\\int \\limits_0^1 \\left(1+a_nx^n \\right)^ndx}$$ where $\\left(a_n\\right)_{n\\geq1}$ is a sequence of nonnegative real numbers which converges to $a \\in \\mathbb{R}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-82b220f389313538491bf0a4e959a80efe817a14235a13912186cbe874c7956b | Calculate $$\lim _{n \rightarrow \infty} n \left[ L - \int _0 ^1 \left( \frac{k}{\sqrt[n]{x} + k -1} \right) ^n dx \right]$$ where $$L = \frac{k}{k-1}.$$ | [
"\\(\\frac{k}{(k-1)^2}\\)"
] | [
"Calculate\n$$\\lim _{n \\rightarrow \\infty} n \\left[ L - \\int _0 ^1 \\left( \\frac{k}{\\sqrt[n]{x} + k -1} \\right) ^n dx \\right]$$\nwhere\n$$L = \\frac{k}{k-1}.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 10
} | 22 | 0 | 0 |
omr-ws-6b7a6dad4f4950e00ce201bea45f4677bc3273136b90d281a9933690add77c33 | Calculate $$\lim\limits_{n\to+\infty} \frac{n! e^n}{n^n}$$ | [
"\\(\\infty\\)"
] | [
"Calculate $$\\lim\\limits_{n\\to+\\infty} \\frac{n! e^n}{n^n}$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-4d9d38124f9e8b16a3033880bb826b8421db39807f7e606ae60444753ac54ba6 | Calculate $$\lim\limits_{n\to+\infty} \frac{n^n}{n! e^n}$$ | [
"0"
] | [
"Calculate $$\\lim\\limits_{n\\to+\\infty} \\frac{n^n}{n! e^n}$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b82f3743c056fd13c45d1f45f748a18461eccc4fc65fa8fe003346974ad69316 | Calculate $$\lim_{n \to \infty} (n!)^{1/n}.$$ | [
"\\(\\infty\\)"
] | [
"Calculate $$\\lim_{n \\to \\infty} (n!)^{1/n}.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-b59265fbcabeb8c4a5e735b6d62eb777fe0b550c1b5b9ffc031aa87ae7a8b953 | Calculate $$\lim_{n \to \infty} \frac{1}{n}(n!)^{1/n}.$$ | [
"\\(\\frac{1}{e}\\)"
] | [
"Calculate $$\\lim_{n \\to \\infty} \\frac{1}{n}(n!)^{1/n}.$$"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-2d6c462d2581807041f36b01939004968d9375d3d64894195d82ef427fc00b9c | Calculate $$\lim_{n \to \infty} \int_0 ^1 \frac{(1-x^n)^3}{(1+x^3)^n}dx.$$ | [
"0"
] | [
"Calculate $$\\lim_{n \\to \\infty} \\int_0 ^1 \\frac{(1-x^n)^3}{(1+x^3)^n}dx.$$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e18ddb0dfb98685c18615512347e0128f6157586bc1bdf07ac29a7bf1ba0332b | Calculate $$\lim_{n \to \infty}\sup B_{n} \text{ and } \lim_{n\to \infty}\inf B_{n}, \quad \text{where} \quad B_{n}=(-\infty,b_{n}], \quad b_{2n}=1+\frac{1}{2n}, \quad b_{2n-1}=-1-\frac{1}{2n+1}.$$ | [
"$\\liminf B_n = (-\\infty, -1)$, $\\limsup B_n = (-\\infty, 1]$"
] | [
"Calculate $$\\lim_{n \\to \\infty}\\sup B_{n} \\text{ and } \\lim_{n\\to \\infty}\\inf B_{n}, \\quad \\text{where} \\quad B_{n}=(-\\infty,b_{n}], \\quad b_{2n}=1+\\frac{1}{2n}, \\quad b_{2n-1}=-1-\\frac{1}{2n+1}.$$"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-5b005e328653e64f5a3f8d78e58dbda90f9da4cfe791859b3ad71aebf259ee6d | Calculate $$\lim_{n\rightarrow \infty }\sum_{k=1}^{n }\frac{1}{n+k+b+\sqrt{n^2+kn+a}}$$ where $a$ and $b$ are two nonnegative real numbers. | [
"\\(2 \\ln \\frac{1+\\sqrt2}{2}\\)"
] | [
"Calculate\n$$\\lim_{n\\rightarrow \\infty }\\sum_{k=1}^{n }\\frac{1}{n+k+b+\\sqrt{n^2+kn+a}}$$\nwhere $a$ and $b$ are two nonnegative real numbers."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-df3bb0f8371da1a3c90006eec824f5ff66818060a62da51b179453df98f492e5 | Calculate $$\lim_{n\to \infty} \sin(a\sqrt{4n^{2}+5n+1}),$$ where $a\in\mathbb{R}$ and $n\in \mathbb{Z}^{+}.$ | [
"\\(\\sin \\left( \\frac{5a}{4} \\right)\\) if \\( a \\equiv 0 (\\pi) \\), otherwise the limit does not exist."
] | [
"Calculate $$\\lim_{n\\to \\infty} \\sin(a\\sqrt{4n^{2}+5n+1}),$$ where $a\\in\\mathbb{R}$ and $n\\in \\mathbb{Z}^{+}.$"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-cde4d35e06f44e2e885f96821608df1a9a260a939a44927121cdbdaf083d6e5e | Calculate $$\lim_{n\to \infty}\sup A_{n} \text{ and } \lim_{n\to \infty} \inf A_{n}, \quad \text{where} \quad A_{n}=p_{n}\mathbb{N}, \quad (p_{n})_{n=1}^{\infty} \quad \text{is the sequence of primes}.$$ | [
"\\(\\liminf A_{n} = \\emptyset, \\limsup A_{n} = \\emptyset\\)."
] | [
"Calculate $$\\lim_{n\\to \\infty}\\sup A_{n} \\text{ and } \\lim_{n\\to \\infty} \\inf A_{n}, \\quad \\text{where} \\quad A_{n}=p_{n}\\mathbb{N}, \\quad (p_{n})_{n=1}^{\\infty} \\quad \\text{is the sequence of primes}.$$"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-08659199dc5feb57a4f8af893a51edb4c8982f21a210b59fde7d4096915a96a0 | Calculate $$\lim_{n\to \infty}\sup C_{n} \text{ and } \lim_{n\to \infty}\inf C_{n}, \quad \text{where} \quad C_{2n}=\left[0,3+\frac{1}{3n}\right), \quad C_{2n+1}=\left(-1-\frac{1}{3n},2\right].$$ | [
"$\\lim_{n\\to \\infty}\\sup C_{n} = [-1,3], \\lim_{n\\to \\infty}\\inf C_{n} = [0,2]$."
] | [
"Calculate $$\\lim_{n\\to \\infty}\\sup C_{n} \\text{ and } \\lim_{n\\to \\infty}\\inf C_{n}, \\quad \\text{where} \\quad C_{2n}=\\left[0,3+\\frac{1}{3n}\\right), \\quad C_{2n+1}=\\left(-1-\\frac{1}{3n},2\\right].$$"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-a1a680feff1147afd324501cd0eabf15184dcbc9385921b1ed84042f5b8b0316 | Calculate $$\lim_{n\to\infty}\int_1^2\sqrt[n]{\lfloor x^n\rfloor}\,dx$$ where $\lfloor x\rfloor$ denotes the floor of $x\in\mathbb{R}$. | [
"\\(\\frac{3}{2}\\)"
] | [
"Calculate $$\\lim_{n\\to\\infty}\\int_1^2\\sqrt[n]{\\lfloor x^n\\rfloor}\\,dx$$ where $\\lfloor x\\rfloor$ denotes the floor of $x\\in\\mathbb{R}$."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b2e378aa39f08b48902efd06afd0edc7eada08c8e4e35ccf988d34364cfddbcd | Calculate $$\lim_{n\to\infty}\left(\sqrt[n+1]{\Gamma\left(\frac{(n+1)a+1}a\right)}-\sqrt[n]{\Gamma\left(\frac{na+1}a\right)}\right),$$ where $\Gamma$ is the Euler gamma function and $a \geq 1$ is an integer. | [
"\\dfrac{1}{e}"
] | [
"Calculate\n$$\\lim_{n\\to\\infty}\\left(\\sqrt[n+1]{\\Gamma\\left(\\frac{(n+1)a+1}a\\right)}-\\sqrt[n]{\\Gamma\\left(\\frac{na+1}a\\right)}\\right),$$\nwhere $\\Gamma$ is the Euler gamma function and $a \\geq 1$ is an integer."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2b19907f4778bd904c0922ea0445392c40c26fb8e15078635cc2d45e06d102ca | Calculate $$\lim_{n\to\infty}{\sqrt n\cdot\int_{0}^{\frac{\pi}2}{\sqrt{a\sin^{2n}{x}+b\cos^{2n}{x}}\cdot dx}}$$ where \(a, b \geq 0\). | [
"\\((\\sqrt{a}+\\sqrt{b})\\sqrt{\\frac{\\pi}{2}}\\)"
] | [
"Calculate\n$$\\lim_{n\\to\\infty}{\\sqrt n\\cdot\\int_{0}^{\\frac{\\pi}2}{\\sqrt{a\\sin^{2n}{x}+b\\cos^{2n}{x}}\\cdot dx}}$$\nwhere \\(a, b \\geq 0\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-801d31619daadce0bb6601da282f05684c371f7eba680e14718f19662ae25cf4 | Calculate $$\lim_{x \to a}\frac{x^{x^a}-a^{a^a}}{x-a}$$ | [
"\\( a^{a^a + a - 1}(a \\ln(a) + 1) \\)"
] | [
"Calculate $$\\lim_{x \\to a}\\frac{x^{x^a}-a^{a^a}}{x-a}$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-7c6c82b99dd381067eb7d4521724ac37b83e4e2901729bea923b35a437e137fb | Calculate $$\lim_{x\to 1^{-}} \frac {\cos^{-1} x^{n+1}}{\cos^{-1} x^{n}}$$ | [
"\\(\\sqrt{\\frac{n+1}{n}}\\)"
] | [
"Calculate\n$$\\lim_{x\\to 1^{-}} \\frac {\\cos^{-1} x^{n+1}}{\\cos^{-1} x^{n}}$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3632d082fc5e1dd7d48269819ad8e4a41e33fed55d7551bf9bdf6061b71bbc84 | Calculate $$\lim_{x\to\infty} (a^x+b^x)^{1/x}$$ | [
"a"
] | [
"Calculate\n$$\\lim_{x\\to\\infty} (a^x+b^x)^{1/x}$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2a656a684bafe1031920ceb138418fad44d11176ef8658603aa72163a7e9f79a | Calculate $$\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right).$$ | [
"\\(\\frac{1}{16}\\)"
] | [
"Calculate $$\\sum \\limits_{n=1}^{\\infty} \\left(n+\\frac{1}{4}-\\zeta(3)-\\zeta(5)-\\cdots -\\zeta(2n+1)\\right).$$"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 28
} | 47 | 0 | 0 |
omr-ws-616c976de99a8cfa65dec97655461827ca2a073547d1248135ce975deb8ebd95 | Calculate $$\sum\limits_{k=0}^{n} \Bigg((-1)^k\cdot \begin{pmatrix}n \\ k \end{pmatrix} \cdot k^{n+2}\Bigg).$$ | [
"(-1)^n \\cdot \\dfrac{n! \\cdot n(n + 1)(n + 2)(3n + 1)}{24}"
] | [
"Calculate \n$$\\sum\\limits_{k=0}^{n} \\Bigg((-1)^k\\cdot \\begin{pmatrix}n \\\\ k \\end{pmatrix} \\cdot k^{n+2}\\Bigg).$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7e3229fcca8a003c4739ffa56073c5e1131a155c099db8b7664fdaed35eb4960 | Calculate $$\sum_{d|16!,d>0}\frac{\sqrt{16!}}{d+\sqrt{16!}}$$ | [
"3360"
] | [
"Calculate $$\\sum_{d|16!,d>0}\\frac{\\sqrt{16!}}{d+\\sqrt{16!}}$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf72ab7f15eb27e3b14c52d26f2038f325dcf7fac1330ace4b3aaa09631917d8 | Calculate $$\sum_{j=0}^{n}\left( {n \choose j}\left( \sum_{i=0}^{n} {j \choose i}8^i\right)\right)$$ | [
"\\(10^n\\)"
] | [
"Calculate $$\\sum_{j=0}^{n}\\left( {n \\choose j}\\left( \\sum_{i=0}^{n} {j \\choose i}8^i\\right)\\right)$$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-588d584c331589beb7dda041902a1317151331cb30f91eff407f4434f7073dc5 | Calculate $$\sum_{k=1}^{2016}\frac{k^2+k+1}{(k+1)!}-\sum_{k=1}^{2017}(-1)^k\cdot \frac{k^2+k+1}{k!}.$$ | [
"1"
] | [
"Calculate $$\\sum_{k=1}^{2016}\\frac{k^2+k+1}{(k+1)!}-\\sum_{k=1}^{2017}(-1)^k\\cdot \\frac{k^2+k+1}{k!}.$$"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-73eb750590b58a37e26752f4979c968669551728cfe9d780d3de9ac45e3f4c50 | Calculate $(-1)^{(e+\pi)}$. | [
"\\(\\cos((e+\\pi)\\pi) + i\\sin((e+\\pi)\\pi)\\)"
] | [
"Calculate $(-1)^{(e+\\pi)}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-a0d4027dff1e0064f4636ac69b336289e3f4f85c438a04bd6d31101c405e2a93 | Calculate $(-1)^{(i+\pi)}$. | [
"\\( e^{-\\pi} \\cdot (\\cos(\\pi^2) + i\\sin(\\pi^2)) \\)"
] | [
"Calculate $(-1)^{(i+\\pi)}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 13
} | 61 | 0 | 0 |
omr-ws-859238d7fc5da1e013b79e1663ff08c2db0102f6ecaf1c023e2cedb06e727015 | Calculate $(-1)^{(i+e)}$. | [
"\\( e^{-\\pi} \\left( \\cos(\\pi e) + i \\sin(\\pi e) \\right) \\)"
] | [
"Calculate $(-1)^{(i+e)}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-a6a465dea36fd9c973798baf81a352c5e317f79cae360141e2c85b5f4a86a2a8 | Calculate $(-1)^{47} + 2^{(3^3+4^2-6^2)}$. | [
"127"
] | [
"Calculate $(-1)^{47} + 2^{(3^3+4^2-6^2)}$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-355f21f4cd0325f8fb1a5602607c819a9273918a8a3c255369ce32eeb1996316 | Calculate $(\cos(2\pi/7))^{1/3} + (\cos(4\pi/7))^{1/3} + (\cos(6\pi/7))^{1/3}$. | [
"\\sqrt[3]{\\frac{1}{2}}"
] | [
"Calculate $(\\cos(2\\pi/7))^{1/3} + (\\cos(4\\pi/7))^{1/3} + (\\cos(6\\pi/7))^{1/3}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-9158012d4b13c5b38c489e00eee6dc3c15bc66001f09a18193e7135cb2a0ddbb | Calculate $1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 + 9 + \ldots - 96 + 97 + 98 + 99 + 100$. | [
"202"
] | [
"Calculate $1 + 2 + 3 + 4 - 5 - 6 - 7 - 8 + 9 + \\ldots - 96 + 97 + 98 + 99 + 100$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-1a545785f6582dd351b24834b347c4eb9a89e23ba85c96f600e2ac193ff03b34 | Calculate $1+5\sin 2\alpha -3\cos ^{-1}2\alpha $ if $\tan \alpha =-2$. | [
"2"
] | [
"Calculate $1+5\\sin 2\\alpha -3\\cos ^{-1}2\\alpha $ if $\\tan \\alpha =-2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ace66562bb77639ed956645ee634a5b771403342e7cdf6be44e6ed3d8f54589c | Calculate $1+\sum_{k=0}^{m-1}\binom{m}{k}S_{n}(k)$ where $S_{n}(k)=\sum_{j=1}^{n}j^k$. | [
"\\((n+1)^m\\)"
] | [
"Calculate $1+\\sum_{k=0}^{m-1}\\binom{m}{k}S_{n}(k)$ where $S_{n}(k)=\\sum_{j=1}^{n}j^k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-482fa036b157b0059c2aacea6452cf656e22e9c997022c61ea69ecacb53fc710 | Calculate $11^{644} \bmod{645}$. | [
"1"
] | [
"Calculate $11^{644} \\bmod{645}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14aab518e73d0bc8b9a709ad4c0e41b55c4427dcd870626693bdad06cee0d5b1 | Calculate $1357^{2468}$ and express the answer in three significant digits. For example, the format should be $1.23 \times 10^3$. | [
"\\(1.61 \\times 10^{7731}\\)"
] | [
"Calculate $1357^{2468}$ and express the answer in three significant digits. For example, the format should be $1.23 \\times 10^3$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 2,
"tir": 21
} | 26 | 0 | 0 |
omr-ws-366f45bf157c788baf812a6cf167e4b74a1dbda9030485e64cb54e591071c1c7 | Calculate $17^{-1} \bmod{113}$. | [
"20"
] | [
"Calculate $17^{-1} \\bmod{113}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-dd19afa9fb59dc7a233c8e463f9a1fc05ea263aee91753d4912b61e8d1e7201c | Calculate $37^{230} \bmod{113}$. | [
"50"
] | [
"Calculate $37^{230} \\bmod{113}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-e210daadb8bc47c62eede295c90ff0b3f92c6dbce3416050da8c0cb1d397cd75 | Calculate $37^{380} \bmod{63}$. | [
"46"
] | [
"Calculate $37^{380} \\bmod{63}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-1b647475d5c24030b81e237cc515dade22000e826feb3c27f44cd883f81dfb46 | Calculate $42^{1003} \mod{2489}$. | [
"1776"
] | [
"Calculate $42^{1003} \\mod{2489}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 27
} | 40 | 0 | 0 |
omr-ws-6e347b67d6938f0b47b9079e4355f3e30ed9ad7ad588b69cfbe8ec4b77ec524d | Calculate $5 \cdot 8! \cdot (10^9 - 1) \mod 77$. | [
"49"
] | [
"Calculate $5 \\cdot 8! \\cdot (10^9 - 1) \\mod 77$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-4cbef669522e2c77ce1527c28083c28ff3c42186c2b20b4da9570d9b6c1c9bf5 | Calculate $5\cdot5! + 4\cdot4!+4!$. | [
"720"
] | [
"Calculate $5\\cdot5! + 4\\cdot4!+4!$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4bfd9674946280989ab718d58e4aad0db6ed1ad88c0a75353f556abcd4dcbd5d | Calculate $6 \cdot 8\frac{1}{3}$. | [
"50"
] | [
"Calculate $6 \\cdot 8\\frac{1}{3}$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4e7d5f3551d461de0a4729c33de84ebac75345286358de4339c600ccdcce872 | Calculate $6!-5\cdot5!-5!$. | [
"0"
] | [
"Calculate $6!-5\\cdot5!-5!$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-18cc9cd964a875ba264bbb0eaed01a133de0104ffb9c2b629c81c2b20de7924d | Calculate $6\div2(1+2)$. | [
"9"
] | [
"Calculate $6\\div2(1+2)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b4717632f3736623f698def89c8000b605b7d041361bfd71319357c7bd8eed6d | Calculate $7^4 - 7^3 - 7^2$ without using a calculator. | [
"2009"
] | [
"Calculate $7^4 - 7^3 - 7^2$ without using a calculator."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-9b7acb3e96ee342e1040c339245870e99f1410f3150d15146012ce1f2f931be7 | Calculate $7^{35461} \bmod{297}$. | [
"271"
] | [
"Calculate $7^{35461} \\bmod{297}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8034aa0ece21b7747c6a16b8f679a5078114bcba78291f30880f91c76fa10dc6 | Calculate $7^{644} \bmod{645}$. | [
"436"
] | [
"Calculate $7^{644} \\bmod{645}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-9d460fa224d7ce1adc2097b683c8ff6e5d3470f6701352ed5eeb8fc04876bbba | Calculate $9^^8^^7^^7^^6^^5^^4^^3^^2 \mod 9!$. | [
"45441"
] | [
"Calculate $9^^8^^7^^7^^6^^5^^4^^3^^2 \\mod 9!$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-4fa8d179966b2080db36769add243e5866c2ede239721a399eddc2e2ceaaad2c | Calculate $D('appeases')$, the number of permutations of the letters of the word 'appeases' such that no letter is in its original place. | [
"297"
] | [
"Calculate $D('appeases')$, the number of permutations of the letters of the word 'appeases' such that no letter is in its original place."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-f79b1887d4392609808eb27e51916adf19fe66e7e89a0021acdd5a07f79068ec | Calculate $Der_k(B,B)$ where $k$ is a field, $A=k[x,y]$, $I=(x^2,xy,y^2)$, and $B=A/I$. | [
"$k^4$"
] | [
"Calculate $Der_k(B,B)$ where $k$ is a field, $A=k[x,y]$, $I=(x^2,xy,y^2)$, and $B=A/I$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-d81de41ac501656cf3999a70f66cc9c52c4bf1198b0b5b207c7924b7ba77a6c0 | Calculate $E((1+\sqrt{2})^{2^{2^{2018}}}) \mod (2^{89}-1)$, where $E(x)$ is the integer part of $x$. | [
"1"
] | [
"Calculate $E((1+\\sqrt{2})^{2^{2^{2018}}}) \\mod (2^{89}-1)$, where $E(x)$ is the integer part of $x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-f5f49b1ac26277938491b87dc6fc66f2b6f2165c8aa19753caad86d2cbe27c11 | Calculate $F_G \mod (2^{89}-1)$, where $(F_n)_n$ is the Fibonacci sequence and $G$ is the Graham number. | [
"0"
] | [
"Calculate $F_G \\mod (2^{89}-1)$, where $(F_n)_n$ is the Fibonacci sequence and $G$ is the Graham number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 3
} | 17 | 0 | 0 |
omr-ws-1a1b6fb17945e3062c01facefb6e321f7a81e9441e98fb2cb2b2c6384afc469b | Calculate $H^*(\mathbb{RP}^n, \mathbb{Z}_2)$. | [
"\\mathbb{Z}_2[x]/(x^{n+1})"
] | [
"Calculate $H^*(\\mathbb{RP}^n, \\mathbb{Z}_2)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1d9924d12b09de8944af4376747f9455c8cc6b50c06d2d8d29bbf49df90e864 | Calculate $L(x_0, x_1, ..., x_n; \frac{x^{n+1}}{x+1})$. | [
"\\frac{x^{n+1} - \\frac{\\prod_{i=0}^n (x - x_i)}{\\prod_{i=0}^n (x_i + 1)}}{x + 1}"
] | [
"Calculate $L(x_0, x_1, ..., x_n; \\frac{x^{n+1}}{x+1})$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-ba2e1d3d8f888d9634f8cc9fdae5d2e1a0c4741fb7b99f82c7efc2cfb67e6628 | Calculate $L(x_0, x_1, ..., x_n; \frac{x^{n+1}}{x-1})$. | [
"\\dfrac{x^{n+1} - \\prod\\limits_{i=0}^n \\dfrac{x - x_i}{1 - x_i}}{x - 1}"
] | [
"Calculate $L(x_0, x_1, ..., x_n; \\frac{x^{n+1}}{x-1})$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-fae45a2d74cfe2149f7033f5ac593171101b3266368a962efb1886006be6a5d7 | Calculate $P(Z < 1.42)$ for a normal distribution with $\mu = 76$ and $\sigma = 7$. | [
"0.9222"
] | [
"Calculate $P(Z < 1.42)$ for a normal distribution with $\\mu = 76$ and $\\sigma = 7$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-3b895a397bccbb90a553915d6bc47be102408a5003fbc402705b6d15da58c777 | Calculate $P(|X-\mu_{X}|\geqslant k\sigma_{X})$ for $X\sim Exp(\lambda)$ and compare the answer to the bound given by Chebychev's inequality. | [
"\\( e^{-(k+1)} \\) if \\( k > 1 \\), otherwise \\( 1 - \\frac{2\\sinh(k)}{e} \\)"
] | [
"Calculate $P(|X-\\mu_{X}|\\geqslant k\\sigma_{X})$ for $X\\sim Exp(\\lambda)$ and compare the answer to the bound given by Chebychev's inequality."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-519578f4b668df63b7c76a28b51550b934d088f90f05d173b39084d10307faed | Calculate $P(|X-\mu_{X}|\geqslant k\sigma_{X})$ for $X\sim U(0,1)$ and compare the answer to the bound given by Chebychev's inequality. | [
"\\( P(|X-\\mu_X|\\geqslant k\\sigma_X) = 1 - \\frac{k}{\\sqrt{3}} \\) for \\( k < \\sqrt{3} \\) and \\( 0 \\) for \\( k \\geq \\sqrt{3} \\)."
] | [
"Calculate $P(|X-\\mu_{X}|\\geqslant k\\sigma_{X})$ for $X\\sim U(0,1)$ and compare the answer to the bound given by Chebychev's inequality."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 22
} | 36 | 0 | 0 |
omr-ws-cfad006c132f12ec8dbb0a0afe3ef645ff2d45caa3e8f095aa8227149ce8e275 | Calculate $S(3, 6)$ where $S(m, n)$ is the number of $(x_1, x_2, \ldots, x_n)$ satisfying: (i) $0 \leq x_1 \leq x_2 \leq \ldots \leq x_n \leq n$ (ii) $3 \mid (x_1 + x_2 + \ldots + x_n)$ | [
"312"
] | [
"Calculate $S(3, 6)$ where $S(m, n)$ is the number of $(x_1, x_2, \\ldots, x_n)$ satisfying:\n(i) $0 \\leq x_1 \\leq x_2 \\leq \\ldots \\leq x_n \\leq n$\n(ii) $3 \\mid (x_1 + x_2 + \\ldots + x_n)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-45507d3cb940d26fe4bdd0921b8df847b76537da4c25f7cfe0a0ff0fdb12bdeb | Calculate $S(47, 2021)$ where $S(m, n)$ is the number of $(x_1, x_2, \ldots, x_n)$ satisfying: (i) $0 \leq x_1 \leq x_2 \leq \ldots \leq x_n \leq n$ (ii) $47 \mid (x_1 + x_2 + \ldots + x_n)$ | [
"\\dfrac{1}{47} \\dbinom{4042}{2021}"
] | [
"Calculate $S(47, 2021)$ where $S(m, n)$ is the number of $(x_1, x_2, \\ldots, x_n)$ satisfying:\n(i) $0 \\leq x_1 \\leq x_2 \\leq \\ldots \\leq x_n \\leq n$\n(ii) $47 \\mid (x_1 + x_2 + \\ldots + x_n)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-0d797653298bac649049a04f53bc6895ed7d7767f03413adb3db668e1e120078 | Calculate $[(12^{12} \div 12^{11})^2 \cdot 4^2] \div 2^4$. | [
"144"
] | [
"Calculate $[(12^{12} \\div 12^{11})^2 \\cdot 4^2] \\div 2^4$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87d4e5f4d13be3f6615dadfe7da81547e35b5e70a3f1288d8c3e4ac34560e7cb | Calculate $\Psi(s,t)=E(e^{i s V_t})$ where $(V_t)$ is the process defined by $dV_t=\lambda(\theta-V_t)dt+\nu \sqrt{V_t}dW_t$. | [
"\\Psi(s, t) = \\left(1 - \\frac{\\nu^2 i s}{2\\lambda}(1 - e^{-\\lambda t})\\right)^{\\frac{2\\lambda\\theta}{\\nu^2}} \\exp\\left( \\frac{i s V_0 e^{-\\lambda t}}{1 - \\frac{\\nu^2 i s}{2\\lambda}(1 - e^{-\\lambda t})} \\right)"
] | [
"Calculate $\\Psi(s,t)=E(e^{i s V_t})$ where $(V_t)$ is the process defined by $dV_t=\\lambda(\\theta-V_t)dt+\\nu \\sqrt{V_t}dW_t$."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-374759e5ff8ccbf3e8d8b2a5f396db952629a2e80b511df87039158796747dc1 | Calculate $\arccos(\cos 10)$. | [
"\\(4\\pi - 10\\)"
] | [
"Calculate $\\arccos(\\cos 10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-88a96f9ecea03086d62a1ed3cf7b8ac99e96ea24ed7ec2e9690ae13ce235628f | Calculate $\arcsin(\sin 6)$. | [
"\\(6 - 2\\pi\\)"
] | [
"Calculate $\\arcsin(\\sin 6)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-9e4eded37826caca86fcfbb70b68d9ef0aeadd9d4777c6bc5b7b0b6674313ebd | Calculate $\arcsin(\sin(\pi^2))$. | [
"3\\pi - \\pi^2",
"\\(3\\pi - \\pi^2\\)"
] | [
"Calculate $\\arcsin(\\sin(\\pi^2))$."
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 29
} | 61 | 0 | 0 |
omr-ws-81e966cafc7d8e30e88341426b9d0240509960128c4750c3641ac0b5f5123b8c | Calculate $\binom{2010}{1}+\binom{2009}{2}+\cdots+\binom{1006}{1005}$. | [
"$F_{2012} - 1$"
] | [
"Calculate $\\binom{2010}{1}+\\binom{2009}{2}+\\cdots+\\binom{1006}{1005}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 12
} | 58 | 0 | 0 |
omr-ws-9a75438cf7e3a41c5abd09012bbfe91ea3cce9525e26c6099d0c6128ec655289 | Calculate $\binom{99}{19}$ modulo 4. | [
"2"
] | [
"Calculate $\\binom{99}{19}$ modulo 4."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-82139df20ccc952a9078b8ab4b0794b670f9494c69ad9a9d49d03abf26114540 | Calculate $\cos (\theta -\varphi )$ if $\cos \theta +\cos \varphi =a$, $\sin \theta -\sin \varphi =b$, $a^2+b^2\ne 0$. | [
"$\\frac{a^2-b^2}{a^2+b^2}$"
] | [
"Calculate $\\cos (\\theta -\\varphi )$ if $\\cos \\theta +\\cos \\varphi =a$, $\\sin \\theta -\\sin \\varphi =b$, $a^2+b^2\\ne 0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d7f9813efd1acf96ce692e2d36ff2996ff4f0877a75af352b4d0540f004cd426 | Calculate $\cos(8x)$ given that $\tan(x) \cdot \tan(2x) = \cot(2x) \cdot \cot(3x)$. | [
"-1"
] | [
"Calculate $\\cos(8x)$ given that $\\tan(x) \\cdot \\tan(2x) = \\cot(2x) \\cdot \\cot(3x)$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-526e83725d4963a6e869ab3c9ecd8a8f4fd17fee0c6d6d19698efb6cfc1f1e60 | Calculate $\cos\frac{\pi}{17}\cos\frac{2\pi}{17}\cos\frac{4\pi}{17}\cos\frac{8\pi}{17}$. | [
"\\(\\frac{1}{16}\\)"
] | [
"Calculate $\\cos\\frac{\\pi}{17}\\cos\\frac{2\\pi}{17}\\cos\\frac{4\\pi}{17}\\cos\\frac{8\\pi}{17}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-994653898d8df798cf475eac87c6e0a6d52ab2c40c0cb7921283dd3540910aef | Calculate $\cos\frac{\pi}{7} \cdot \cos\frac{2\pi}{7} \cdot \cos\frac{4\pi}{7}$. | [
"\\(-\\frac{1}{8}\\)"
] | [
"Calculate $\\cos\\frac{\\pi}{7} \\cdot \\cos\\frac{2\\pi}{7} \\cdot \\cos\\frac{4\\pi}{7}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-ffcf9b0d4d6a1f81e5da07cb382a7b7c9a95988f2a02935f162b1f6c6182d58d | Calculate $\cos\left(\frac{4\pi}{11}\right) + \cos\left(\frac{7\pi}{11}\right)$ using the winding function. | [
"0"
] | [
"Calculate $\\cos\\left(\\frac{4\\pi}{11}\\right) + \\cos\\left(\\frac{7\\pi}{11}\\right)$ using the winding function."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9248fb0496546e58aaae49021008d3356546e5bd0d799959f4c06d96dbf2063e | Calculate $\cos{15^\circ}$ and $\cos{75^\circ}.$ | [
"\\(\\cos 15^\\circ = \\frac{\\sqrt{6} + \\sqrt{2}}{4}\\), \\(\\cos 75^\\circ = \\frac{\\sqrt{6} - \\sqrt{2}}{4}\\)"
] | [
"Calculate $\\cos{15^\\circ}$ and $\\cos{75^\\circ}.$"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-b3401749a13ff0259ce6941d4e394283a8f7897e2f1fea75c4ac978d863a3a01 | Calculate $\displaystyle \int_0^1 x^{n-1}\ln x\,dx$ for a positive integer $n$. | [
"\\(-\\frac{1}{n^2}\\) for \\(n \\ge 2\\) and \\(-1\\) for \\(n = 1\\)."
] | [
"Calculate $\\displaystyle \\int_0^1 x^{n-1}\\ln x\\,dx$ for a positive integer $n$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf170ec86a37c979285e196217d3ac1631acb08d2590afec2b5f39378c364f44 | Calculate $\displaystyle \lim_{n\to \infty}\int_{0}^{\frac{\pi}{2}}\sqrt[n]{\sin^{n}(x)+\cos^{n}(x)}dx$. | [
"\\(\\sqrt{2}\\)"
] | [
"Calculate $\\displaystyle \\lim_{n\\to \\infty}\\int_{0}^{\\frac{\\pi}{2}}\\sqrt[n]{\\sin^{n}(x)+\\cos^{n}(x)}dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1d495786fc5881c85342c403ec833ed89413952aabbfba82ff716d809fc7b914 | Calculate $\displaystyle \lim_{n\to \infty}n^{2}\left(\int_{0}^{\frac{\pi}{2}}\sqrt[n]{\sin^{n}(x)+\cos^{n}(x)}dx-\lim_{n\to \infty} \int_{0}^{\frac{\pi}{2}}\sqrt[n]{\sin^{n}(x)+\cos^{n}(x)}dx\right)$. | [
"\\dfrac{\\sqrt{2}\\, \\pi^2}{24}"
] | [
"Calculate $\\displaystyle \\lim_{n\\to \\infty}n^{2}\\left(\\int_{0}^{\\frac{\\pi}{2}}\\sqrt[n]{\\sin^{n}(x)+\\cos^{n}(x)}dx-\\lim_{n\\to \\infty} \\int_{0}^{\\frac{\\pi}{2}}\\sqrt[n]{\\sin^{n}(x)+\\cos^{n}(x)}dx\\right)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-885bde6202daacdcdf83108fe3e57870e132b6e5b1c0e5def2208dd1cb4f8ba4 | Calculate $\displaystyle \sum^{3n-1}_{r=0}(-1)^r\binom{6n}{2r+1}3^r$. | [
"0"
] | [
"Calculate $\\displaystyle \\sum^{3n-1}_{r=0}(-1)^r\\binom{6n}{2r+1}3^r$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0d5f9921f1bade86a2caddd86dd860b3a6a42ab7016a409b7e34da45ad790a1b | Calculate $\displaystyle \sum_{0\le k\le n}(-1)^k\binom{2n+1}{k}(2n+1-2k)^{2l+1}$, where $l \in \mathbb{Z}_{\ge 0}$. | [
"0"
] | [
"Calculate $\\displaystyle \\sum_{0\\le k\\le n}(-1)^k\\binom{2n+1}{k}(2n+1-2k)^{2l+1}$, where $l \\in \\mathbb{Z}_{\\ge 0}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-5265d1d06ea3aa53a6e61997308fbfdb1d0abc5ee676be6e3103c38621e47eff | Calculate $\displaystyle \sum_{i=0}^{2022} \left[ (-1)^i i \binom{4046}{2023-i} \right]$. | [
"\\(-\\binom{4044}{2022}\\)"
] | [
"Calculate $\\displaystyle \\sum_{i=0}^{2022} \\left[ (-1)^i i \\binom{4046}{2023-i} \\right]$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-9d2cf722417c9b1651719fca7ba9b3790acae65234b6ebf74a4bcd71ac770e1d | Calculate $\displaystyle \sum_{n=0}^{\infty}(-1)^n\left(\frac{1}{(n+1)^2}-\frac{1}{(n+2)^2}+\frac{1}{(n+3)^2}-\dots \right).$ | [
"\\(\\log 2\\)"
] | [
"Calculate $\\displaystyle \\sum_{n=0}^{\\infty}(-1)^n\\left(\\frac{1}{(n+1)^2}-\\frac{1}{(n+2)^2}+\\frac{1}{(n+3)^2}-\\dots \\right).$"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 21
} | 44 | 0 | 0 |
omr-ws-1c71cf22a4a5ed0ecc7f0a4ce44c9b5f9727ca380a847751beea91bdbcc4041c | Calculate $\displaystyle\int\limits_1^2 x^{\mathrm{e}^x} \mathrm{\,d}x$. | [
"24.6"
] | [
"Calculate $\\displaystyle\\int\\limits_1^2 x^{\\mathrm{e}^x} \\mathrm{\\,d}x$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-26f7e5f1016178ac1f075e8d952d358f687fae70fa40ab7110168ebf5129a032 | Calculate $\displaystyle\lim_{x \to 0} \dfrac{1}{[x]}$, where $[.]$ denotes the greatest integer function. Determine whether the limit exists and, if so, what its value is. | [
"-1"
] | [
"Calculate $\\displaystyle\\lim_{x \\to 0} \\dfrac{1}{[x]}$, where $[.]$ denotes the greatest integer function. Determine whether the limit exists and, if so, what its value is."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-d829ccc9b4e360097c4cec7b87c9dc112a0d6c65ac30436e827c0d80f16d74b0 | Calculate $\displaystyle\lim_{x\to 0}\ \left[\int_0^{x} \frac{(1+\sin t)^2}{x}\ \text{d}t\right]$. | [
"1"
] | [
"Calculate $\\displaystyle\\lim_{x\\to 0}\\ \\left[\\int_0^{x} \\frac{(1+\\sin t)^2}{x}\\ \\text{d}t\\right]$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-7de815045a640e61cd65dffba4584aebf4fb3b63f7768ec2355376070b0a01dd | Calculate $\displaystyle{\sum_{n=1}^\infty\left(\lfloor\sqrt[n]{2010}\rfloor-1\right)}$ where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$. | [
"2077"
] | [
"Calculate $\\displaystyle{\\sum_{n=1}^\\infty\\left(\\lfloor\\sqrt[n]{2010}\\rfloor-1\\right)}$ where $\\lfloor x\\rfloor$ is the largest integer less than or equal to $x$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-7e303ae9a624c38f465d40649294f6f34392c4a2c95f243987ae7265c73a7aff | Calculate $\frac{(1-\sqrt{3}i)^{100}}{2^{100}}$. | [
"\\(\\frac{-1 + \\sqrt{3}i}{2}\\)"
] | [
"Calculate $\\frac{(1-\\sqrt{3}i)^{100}}{2^{100}}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-d0d846ec4d9bafc7c6f9ec9f2fc19ae664a5a82eae687c10a045c5d7f946d6cb | Calculate $\frac{1}{2} \cdot \frac{3}{5} \cdot \frac{7}{11}$. | [
"\\frac{21}{110}"
] | [
"Calculate $\\frac{1}{2} \\cdot \\frac{3}{5} \\cdot \\frac{7}{11}$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1edaaff87998d6aebd0aa57e53526b78eaba36008ccae3833dca48fe8d9a8b6b | Calculate $\frac{1}{4} \cdot \frac{2}{5} \cdot \frac{3}{6} \cdot \frac{4}{7} \cdots \frac{49}{52} \cdot \frac{50}{53}$. Express your answer as a common fraction. | [
"\\frac{1}{23426}"
] | [
"Calculate $\\frac{1}{4} \\cdot \\frac{2}{5} \\cdot \\frac{3}{6} \\cdot \\frac{4}{7} \\cdots \\frac{49}{52} \\cdot \\frac{50}{53}$. Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7364beb09501dfa7dfbb679db27a4841bd02fe207570ad05f55bb27d52c02c45 | Calculate $\frac{3 \cdot 5! + 15\cdot 4!}{6!}$ | [
"1"
] | [
"Calculate $\\frac{3 \\cdot 5! + 15\\cdot 4!}{6!}$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e8d9dee85094264c8cce398f4b75e679a2b0fa94fe6f4a394e217e0d44f2328e | Calculate $\frac{6^k}{(3^k-2^k)(3^{k+1}-2^{k+1})}$. | [
"\\dfrac{2^k}{3^k - 2^k} - \\dfrac{2^{k+1}}{3^{k+1} - 2^{k+1}}"
] | [
"Calculate $\\frac{6^k}{(3^k-2^k)(3^{k+1}-2^{k+1})}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
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