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-8e7ca59a3e36ae47e27ba9739efe87fb0390e9f07d652fa1970fc1904e52091b | Calculate the areas of the figures bounded by the indicated lines. $x=1,\ x=2,\ y=0,\ \log _2 x+\log _2 y=0$. | [
"$\\ln 2$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $x=1,\\ x=2,\\ y=0,\\ \\log _2 x+\\log _2 y=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81913d965deeeb54952c46dd27c239451d1ddf929f21f267c07500528c6c0a20 | Calculate the areas of the figures bounded by the indicated lines. $xy=7,\ y=0,\ x=4,\ x=12$. | [
"$7\\ln 3$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $xy=7,\\ y=0,\\ x=4,\\ x=12$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f45b9e7de8d8cddd2b5600b58b6fdd3c21db221a65d5e4cd1d2e3fb8bd8d8295 | Calculate the areas of the figures bounded by the indicated lines. $y=(1/2)^x,\ x-2y+2=0,\ x=2$. | [
"$3\\left(1-\\frac{1}{4\\ln 2}\\right)$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=(1/2)^x,\\ x-2y+2=0,\\ x=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2c3827aca0b24099f121bbf2e24ab2297f9a3a2bb7436986f82321c96de13001 | Calculate the areas of the figures bounded by the indicated lines. $y=-1+8x^2-x^4,\ y=15,\ x=1\ (x\ge 1)$. | [
"$3\\frac{8}{15}$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=-1+8x^2-x^4,\\ y=15,\\ x=1\\ (x\\ge 1)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d51429a969481f99561c71d31849f67738d8645ef90e6da703836615991a041b | Calculate the areas of the figures bounded by the indicated lines. $y=-x^2+\frac{7}{2}x+1,\ y=2^{-x},\ x=2\ (x\le 2)$. | [
"$6\\frac{1}{3}-\\frac{3}{4\\ln 2}$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=-x^2+\\frac{7}{2}x+1,\\ y=2^{-x},\\ x=2\\ (x\\le 2)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5b42f75f6dc4266b9147cbceb5ecf9a2ee53c613c41e6a11b15d8a6a6070fd6 | Calculate the areas of the figures bounded by the indicated lines. $y=1/(1+x^2),\ y=x^2/2$. | [
"$\\frac{\\pi }{2}-\\frac{1}{3}$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=1/(1+x^2),\\ y=x^2/2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27e28b1b3030f0c7fbc9e2c0ab7c1ad2c0a8e22db0a4f10bb8bb40dbd6555c25 | Calculate the areas of the figures bounded by the indicated lines. $y=2^x,\ y=2,\ x=-1$. | [
"$4-\\frac{3}{\\ln 2}$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=2^x,\\ y=2,\\ x=-1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d98b16f92f9123ce0254269c1e44cf48df70ac8f442bf7002f22f9bd48ed988c | Calculate the areas of the figures bounded by the indicated lines. $y=2^x,\ y=4^x,\ x=1$. | [
"$\\frac{1}{2}\\ln 2$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=2^x,\\ y=4^x,\\ x=1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31e2ddedf0c364468e422e57378e9eb360bc9f7cb76342118fb6a0bd3fcd95e6 | Calculate the areas of the figures bounded by the indicated lines. $y=2x^2+1,\ y=x+2,\ y=1.5$. | [
"$\\frac{19}{24}$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=2x^2+1,\\ y=x+2,\\ y=1.5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40bd0742611068f174b302ae9c9b92a530ce2a775bcdaeb3a89d9f39e9802710 | Calculate the areas of the figures bounded by the indicated lines. $y=5/x,\ y=6-x$. | [
"$12-5\\ln 5$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=5/x,\\ y=6-x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40480bc410fc2d1e2c3db47f9276ef1627dd82e08ce1e72454314a152b5d6dec | Calculate the areas of the figures bounded by the indicated lines. $y=\sqrt x,\ y=\sqrt {4-3x},\ y=0$. | [
"8/9"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=\\sqrt x,\\ y=\\sqrt {4-3x},\\ y=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e1582d81d7170f0dddf79344eb1df254d0c4f56d88cd113a98ab086805fbf8dc | Calculate the areas of the figures bounded by the indicated lines. $y=x^2,\ y=1/x,\ y=0,\ x=2$. | [
"$\\frac{1}{3}+\\ln 2$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=x^2,\\ y=1/x,\\ y=0,\\ x=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a874a471b13ee72b30019848d005286ddb19b2622893d4d3af6c89f2c5309007 | Calculate the areas of the figures bounded by the indicated lines. $y=x^3,\ y=1/x,\ x=2$. | [
"$\\frac{15}{4}-\\ln 2$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $y=x^3,\\ y=1/x,\\ x=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5caaf81e1f3f7323342d85bff26a02394a740305941d82044511a3d115e79786 | Calculate the areas of the figures bounded by the indicated lines. Calculate the area of a plane figure bounded by parts of the lines $\max(x,y)=1$ and $x^2+y^2=1$ lying in the first quadrant: $$\max(x,y)= \left\{\begin{array}{lll} x, & \mbox{if} & x\ge y,\medskip \\ y, & \mbox{if} & x<y. \end{array}\right.$$ | [
"$1-\\pi /4$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. Calculate the area of a plane figure bounded by parts of the lines $\\max(x,y)=1$ and $x^2+y^2=1$ lying in the first quadrant: $$\\max(x,y)= \\left\\{\\begin{array}{lll} x, & \\mbox{if} & x\\ge y,\\medskip \\\\ y, & \\mbox{if} & x<y. \\end{array}\\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e8a4b33e1ef8eef04c76af578afe79cf811e74a2ea2229d33486bed076402de | Calculate the areas of the figures bounded by the indicated lines. Find the area of the figure whose set of points satisfies the system of inequalities $$\begin{array}{ll} x^2+y^2\le r^2, & r>0,\medskip \\ x-y\le 0, & y\ge 0. \end{array}$$ | [
"$\\frac{3}{8}\\pi r^2$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. Find the area of the figure whose set of points satisfies the system of inequalities $$\\begin{array}{ll} x^2+y^2\\le r^2, & r>0,\\medskip \\\\ x-y\\le 0, & y\\ge 0. \\end{array}$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bbf8fa9bcdbb6d2d645402e27d1e126e1d8a91a4a565d3460349ce743f9865f1 | Calculate the arithmetic mean of the numbers $n \sin(n)$ for $n = 2, 4, \ldots, 180$ and express your answer in terms of a trigonometric function evaluated at an integer angle in degrees. | [
"\\cot 1^\\circ"
] | [
"Calculate the arithmetic mean of the numbers $n \\sin(n)$ for $n = 2, 4, \\ldots, 180$ and express your answer in terms of a trigonometric function evaluated at an integer angle in degrees."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26eadbd61505f62031b7b273b75edfd0dd12ea18a13a4c88118260d8b4bfe3c1 | Calculate the arithmetic mean, harmonic mean, and geometric mean of \(9^{800}\) and \(7^{900}\). Express your answer in scientific notation with three significant figures. | [
"\\(1.00 \\times 10^{778}\\)"
] | [
"Calculate the arithmetic mean, harmonic mean, and geometric mean of \\(9^{800}\\) and \\(7^{900}\\). Express your answer in scientific notation with three significant figures."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02743bc9fbb657b0d547ed290b0d340167db610ca097516d70df2112ccc153c4 | Calculate the average distance $\langle d \rangle$ between the points in the region $D$ to the line $x+y=0$. The region $D$ is the quarter disk defined by $x^{2}+y^{2}\leq a^{2}$, $x\geq 0$, and $y\geq 0$. Use the mean value theorem to find $\langle d \rangle = \frac{1}{\text{area}(D)} \iint_{D} d(x,y) \, dx \, dy$, wh... | [
"\\(\\frac{4a\\sqrt{2}}{3\\pi}\\)"
] | [
"Calculate the average distance $\\langle d \\rangle$ between the points in the region $D$ to the line $x+y=0$. The region $D$ is the quarter disk defined by $x^{2}+y^{2}\\leq a^{2}$, $x\\geq 0$, and $y\\geq 0$. Use the mean value theorem to find $\\langle d \\rangle = \\frac{1}{\\text{area}(D)} \\iint_{D} d(x,y) \... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-164042f41a1c6808aa83023281e0595d7962d240ce510aaa7013250d14a076cc | Calculate the average of $n\sin n$ for $n=2,4,6,...,180$ and express your answer in terms of a trigonometric function. What is this function evaluated at $1$? | [
"\\cot 1"
] | [
"Calculate the average of $n\\sin n$ for $n=2,4,6,...,180$ and express your answer in terms of a trigonometric function. What is this function evaluated at $1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-75f35785b421019da56438aef7ecc6aa59d7595f55e2bb85d30514556c4d57b8 | Calculate the average speed of the cyclist between the bus stop and the traffic light, given that she travels at a constant speed of $3 \text{ m/s}$ for $20$ seconds and then decelerates at a constant rate of $0.2 \text{ m/s}^2$ until coming to rest. | [
"\\( 2.4 \\text{m/s} \\)"
] | [
"Calculate the average speed of the cyclist between the bus stop and the traffic light, given that she travels at a constant speed of $3 \\text{ m/s}$ for $20$ seconds and then decelerates at a constant rate of $0.2 \\text{ m/s}^2$ until coming to rest."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-32fd1016a311d9dc2d32db8e7ad51ecd59f4e35a5d08110203cf3a819a2a6bd2 | Calculate the average value of the function \( g(x) = \sin^4(2x - \frac{\pi}{6}) \cos(2x - \frac{\pi}{6}) \) over the interval \([0, \frac{\pi}{4}]\). | [
"\\(\\frac{9\\sqrt{3}+1}{80\\pi}\\)"
] | [
"Calculate the average value of the function \\( g(x) = \\sin^4(2x - \\frac{\\pi}{6}) \\cos(2x - \\frac{\\pi}{6}) \\) over the interval \\([0, \\frac{\\pi}{4}]\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-353ef16b12aa33ab96c4ef2888dff4366533bfb03c982e2603d18e22670a04ce | Calculate the cardinality of each fiber of \( f_a \) when it is not a permutation; i.e., the number of elements in the set \( \{ f_a^{-1}(x) \} \) for all \( x \in \mathbb{Z}_m \). | [
"\\gcd(a, m)"
] | [
"Calculate the cardinality of each fiber of \\( f_a \\) when it is not a permutation; i.e., the number of elements in the set \\( \\{ f_a^{-1}(x) \\} \\) for all \\( x \\in \\mathbb{Z}_m \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-eb4816a323b15006111f2854b5b2c9d600942c853c12275fd9faf385cb34fbfd | Calculate the cardinality of the set \( A = \{ n \in \mathbb{N} \mid |\cos(n+1) - \cos(n)| < \frac{1}{2} \} \). | [
"\\(\\aleph_0\\)"
] | [
"Calculate the cardinality of the set \\( A = \\{ n \\in \\mathbb{N} \\mid |\\cos(n+1) - \\cos(n)| < \\frac{1}{2} \\} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 24
} | 38 | 0 | 0 |
omr-ws-bbfad40c7106c6207e4d4f140fa9064c82cc67ce07cf02e570f893c9fd770af9 | Calculate the cardinality of the union of subgroups \( A_n \) of \( (\mathbb{Z}/10^{100}\mathbb{Z}, +) \), where each \( A_n \) is a subgroup of order \( 2^n 5^{100-n} \) for \( n = 0, 1, 2, \ldots, 100 \). | [
"\\dfrac{4 \\cdot 5^{100} - 2^{100}}{3}"
] | [
"Calculate the cardinality of the union of subgroups \\( A_n \\) of \\( (\\mathbb{Z}/10^{100}\\mathbb{Z}, +) \\), where each \\( A_n \\) is a subgroup of order \\( 2^n 5^{100-n} \\) for \\( n = 0, 1, 2, \\ldots, 100 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-3138846bb2a8d230d5928cc7d3e11f79ac1f9caafddde776f1e92d28981653a7 | Calculate the center of rotation for triangle \( \triangle ABC \) with vertices \( A = (-2, 2) \), \( B = (6, -1) \), and \( C = (1, 5) \) after a 90-degree rotation. Show the steps. | [
"\\left( \\dfrac{5}{3}, 2 \\right)"
] | [
"Calculate the center of rotation for triangle \\( \\triangle ABC \\) with vertices \\( A = (-2, 2) \\), \\( B = (6, -1) \\), and \\( C = (1, 5) \\) after a 90-degree rotation. Show the steps."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-0e0d97cde4feccf4ca0ec0c3452a301efb7341710d703efc512a529505d31a47 | Calculate the central angle of a circular sector such that the area of its circular segment is equal to the area of its isosceles triangle. | [
"1.895"
] | [
"Calculate the central angle of a circular sector such that the area of its circular segment is equal to the area of its isosceles triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-0154301f441afcddb70547385790dfa03a6a866cbde9d8aeb2db46c7f6136078 | Calculate the change in the height of the mountain after 100 years, given that it is being uplifted at a rate of 2 meters per 1000 years and eroded at a rate of 2 cm per year. | [
"-1.8 meters"
] | [
"Calculate the change in the height of the mountain after 100 years, given that it is being uplifted at a rate of 2 meters per 1000 years and eroded at a rate of 2 cm per year."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-89bce245e93d3f7b7e79ccfdc2d8d1ed2be6e2b646c0540c66a9680f79c9021e | Calculate the circumradius \( R \) of triangle \( ABC \) as a function of its side lengths \( a \), \( b \), and \( c \). | [
"R = \\dfrac{abc}{4\\sqrt{s(s - a)(s - b)(s - c)}}"
] | [
"Calculate the circumradius \\( R \\) of triangle \\( ABC \\) as a function of its side lengths \\( a \\), \\( b \\), and \\( c \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a8366a6edc775bab28eb1bd84f21eab7a067429bd86ef996946c4d49d3edfd91 | Calculate the class group and class number of $\mathbb{Q}(\sqrt{-17})$. | [
"4"
] | [
"Calculate the class group and class number of $\\mathbb{Q}(\\sqrt{-17})$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 3
} | 24 | 0 | 0 |
omr-ws-8767ec54e7644f6aa4b3f4929bbcdae5585cffc799b3e383c0861801d9056e6f | Calculate the closed form (in terms of $n$ and $\Gamma$) of the expression $I(n,n+2,n,...,n)-I(n,n+1,n+1,n,...,n)$ where $n_0=n_1=...=n_d=n \in \mathbb{N}$, and $I(n_0,...,n_d)$ is defined as: \[ I(n_0,...,n_d) = \int_{0}^{1}\int_{0}^{1-x_1}\int_{0}^{1-x_1-x_2}...\int_{0}^{1-x_1-...-x_d}(1-x_1-x_2-...-x_d)^{n_0}x_1^{n_... | [
"0"
] | [
"Calculate the closed form (in terms of $n$ and $\\Gamma$) of the expression $I(n,n+2,n,...,n)-I(n,n+1,n+1,n,...,n)$ where $n_0=n_1=...=n_d=n \\in \\mathbb{N}$, and $I(n_0,...,n_d)$ is defined as:\n\n\\[ I(n_0,...,n_d) = \\int_{0}^{1}\\int_{0}^{1-x_1}\\int_{0}^{1-x_1-x_2}...\\int_{0}^{1-x_1-...-x_d}(1-x_1-x_2-...-x... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-4d518d5fb881bef2c44d0b60eb88d45992cddee9de8556bc4e327e79c2df3467 | Calculate the closed form of the series $$\sum^{\infty}_{n=1}\frac{1}{n}\cdot e^{\frac{2\pi i n}{3}}.$$ | [
"$-\\log(1 - e^{\\frac{2\\pi i}{3}})$"
] | [
"Calculate the closed form of the series $$\\sum^{\\infty}_{n=1}\\frac{1}{n}\\cdot e^{\\frac{2\\pi i n}{3}}.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 19
} | 36 | 0 | 0 |
omr-ws-cc9a5957b00da6ab08cc57f6388d54a2593fadd58021715318bc35348f429fa8 | Calculate the coefficient for $x^{1997}$ in the expression $\frac{1}{(1-x)(1-x^{10})(1-x^{100})(1-x^{1000})}$. | [
"2650"
] | [
"Calculate the coefficient for $x^{1997}$ in the expression $\\frac{1}{(1-x)(1-x^{10})(1-x^{100})(1-x^{1000})}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-b963f9bafa8592a168187388eb4b01854c9b43c07d6841a867739ec4fa1a95b4 | Calculate the coefficient of $x^6$ in the Taylor series expansion of $\ln \cos x$ about $x = 0$. | [
"-\\dfrac{1}{45}"
] | [
"Calculate the coefficient of $x^6$ in the Taylor series expansion of $\\ln \\cos x$ about $x = 0$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff07d10263f14d6a0b6cfaa61935fa58267999d2626c12467ac70bf6e680f998 | Calculate the coefficient of \(x^{2018}\) in the polynomial \(P(x) = (x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024)\). | [
"512"
] | [
"Calculate the coefficient of \\(x^{2018}\\) in the polynomial \\(P(x) = (x+1)(x^2+2)(x^4+4)\\cdots(x^{1024}+1024)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-7116dfb4b189ef30ea023b6d47c5387d8dc51027e699e65536702668800b1825 | Calculate the coefficients in the asymptotic expansion of $\theta_n$ given by $$\theta_n=\frac{1}{3}+\frac{4}{135n}-\frac{8}{2835n^2}-\frac{16}{8505n^3}+\cdots$$ where $\theta_n$ is defined by $$\frac{1}{2}e^n=\sum^{n-1}_{k=0}\frac{n^k}{k!}+\frac{n^n}{n!}\theta_n \quad (n\geq1), \quad \theta_0=\frac{1}{2}.$$ | [
"\\frac{1}{3} + \\frac{4}{135n} - \\frac{8}{2835n^2} - \\frac{16}{8505n^3} + \\cdots"
] | [
"Calculate the coefficients in the asymptotic expansion of $\\theta_n$ given by\n$$\\theta_n=\\frac{1}{3}+\\frac{4}{135n}-\\frac{8}{2835n^2}-\\frac{16}{8505n^3}+\\cdots$$\nwhere $\\theta_n$ is defined by\n$$\\frac{1}{2}e^n=\\sum^{n-1}_{k=0}\\frac{n^k}{k!}+\\frac{n^n}{n!}\\theta_n \\quad (n\\geq1), \\quad \\theta_0=... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-00d86ddd9c056b108da57cfc625147bbfa9b9f8553ec32f2170f4b37b00a342d | Calculate the coefficients of the terms with exponent 0 mod \( p \) in the expansion of \(\prod_{i=1}^{p-1}(1+x^i)\). | [
"\\dfrac{2^{p-1} + p - 1}{p}"
] | [
"Calculate the coefficients of the terms with exponent 0 mod \\( p \\) in the expansion of \\(\\prod_{i=1}^{p-1}(1+x^i)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c3a7aee0580fa00a58d7f291937079432a40208d2e977bf8f70e822665b34b75 | Calculate the cohomology of the Klein Bottle with coefficients in $\mathbb{F}_2$. | [
"\\( H^0(K; \\mathbb{F}_2) = \\mathbb{F}_2, \\, H^1(K; \\mathbb{F}_2) = \\mathbb{F}_2 \\oplus \\mathbb{F}_2, \\, H^2(K; \\mathbb{F}_2) = 0 \\)"
] | [
"Calculate the cohomology of the Klein Bottle with coefficients in $\\mathbb{F}_2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-489365afbe2ebe8e0ef43e4f5de676514fd3647bb9be62323342a52946a63c7d | Calculate the cokernel of the map $Hom_B(\Omega_{A/k}\otimes_A B,B) \to Hom_B(I/I^2,B)$ where $k$ is a field, $A=k[x,y]$, $I=(x^2,xy,y^2)$, and $B=A/I$. | [
"\\( k^4 \\)"
] | [
"Calculate the cokernel of the map $Hom_B(\\Omega_{A/k}\\otimes_A B,B) \\to Hom_B(I/I^2,B)$ 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"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-93860ccf8ba87457907b4e2f6181f6aa6fa7333b7b12a97cbce090d26c001326 | Calculate the commutator $[\hat{M_x}, \hat{M}^2]$ where \[ \hat{M_x} = ih \left( z \frac{\partial}{\partial y} - y \frac{\partial}{\partial z} \right), \] \[ \hat{M_y} = ih \left( x \frac{\partial}{\partial z} - z \frac{\partial}{\partial x} \right), \] \[ \hat{M_z} = ih \left( y \frac{\partial}{\partial x} - x \frac{\... | [
"0"
] | [
"Calculate the commutator $[\\hat{M_x}, \\hat{M}^2]$ where\n\\[\n\\hat{M_x} = ih \\left( z \\frac{\\partial}{\\partial y} - y \\frac{\\partial}{\\partial z} \\right),\n\\]\n\\[\n\\hat{M_y} = ih \\left( x \\frac{\\partial}{\\partial z} - z \\frac{\\partial}{\\partial x} \\right),\n\\]\n\\[\n\\hat{M_z} = ih \\left( y... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3443d17e56298dd17d92efffe27d351d05cd19669a89be56f88d071fd1a1c4a2 | Calculate the commutator $[\hat{M_y}, \hat{M_z}]$ where \[ \hat{M_y} = ih \left( x \frac{\partial}{\partial z} - z \frac{\partial}{\partial x} \right) \] and \[ \hat{M_z} = ih \left( y \frac{\partial}{\partial x} - x \frac{\partial}{\partial y} \right). \] | [
"\\(-\\hbar^2 \\left( z \\frac{\\partial}{\\partial y} - y \\frac{\\partial}{\\partial z} \\right)\\)"
] | [
"Calculate the commutator $[\\hat{M_y}, \\hat{M_z}]$ where\n\\[\n\\hat{M_y} = ih \\left( x \\frac{\\partial}{\\partial z} - z \\frac{\\partial}{\\partial x} \\right)\n\\]\nand\n\\[\n\\hat{M_z} = ih \\left( y \\frac{\\partial}{\\partial x} - x \\frac{\\partial}{\\partial y} \\right).\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 3
} | 9 | 0 | 0 |
omr-ws-445f345dff76419275f9d6acd36ae46abc17298025cd2ef6db0d9c8f230aee12 | Calculate the complex integral \( I = \int_{|z|=2} \frac{2z+1}{z^2 - 4z + 3} \, dz \). | [
"-3\\pi i"
] | [
"Calculate the complex integral \\( I = \\int_{|z|=2} \\frac{2z+1}{z^2 - 4z + 3} \\, dz \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-374e4a040f967b9ab084f00e18f50278adbf2c18f82ad3cef589aac938f7cfbe | Calculate the complex number \( z \) such that \( |z-1| = |z-1-i| \). | [
"\\( z = a + \\frac{1}{2}i \\) where \\( a \\) is any real number."
] | [
"Calculate the complex number \\( z \\) such that \\( |z-1| = |z-1-i| \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-b2ccd860cc0fafb05dc841a639366edb7c325858a11d0e30ebe71ceef4ab4764 | Calculate the condition number \( K(A) \) of the matrix \( A = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \) using both the infinity norm and the 1 norm, and verify that they are different. Note: The condition number \( K(A) \) for a matrix \( A \) is given by \( K(A) = ||A|| \cdot ||A^{-1}|| \) for a square matrix and ... | [
"\\dfrac{9}{7}"
] | [
"Calculate the condition number \\( K(A) \\) of the matrix \\( A = \\begin{pmatrix} 1 \\\\ 2 \\\\ 3 \\end{pmatrix} \\) using both the infinity norm and the 1 norm, and verify that they are different.\n\nNote: The condition number \\( K(A) \\) for a matrix \\( A \\) is given by \\( K(A) = ||A|| \\cdot ||A^{-1}|| \\)... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-308f359974a22def36e16ec7494d1d430143b068ba12ad51b2fb0b6cc1b90fd7 | Calculate the constant term of \( x^{2013} \) in the expansion of \( (1+x)^{4016} + x(1+x)^{4015} + x^2(1+x)^{4014} + x^3(1+x)^{4013} + \cdots + x^{2003}(1+x)^{2003} \). | [
"\\dbinom{4017}{2013} - \\dbinom{2013}{9}"
] | [
"Calculate the constant term of \\( x^{2013} \\) in the expansion of \\( (1+x)^{4016} + x(1+x)^{4015} + x^2(1+x)^{4014} + x^3(1+x)^{4013} + \\cdots + x^{2003}(1+x)^{2003} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-09482d40ae6a6ce7123bda6d3bb5733531296272c02661506e220d131c6ca9cf | Calculate the constant term of \( x^{2013} \) in the expansion of \( (1+x)^{4016} + x(1+x)^{4015} + x^2(1+x)^{4014} + x^3(1+x)^{4013} + \ldots + x^{2003}(1+x)^{2003} \). | [
"\\(\\binom{4016}{2013}\\)"
] | [
"Calculate the constant term of \\( x^{2013} \\) in the expansion of \\( (1+x)^{4016} + x(1+x)^{4015} + x^2(1+x)^{4014} + x^3(1+x)^{4013} + \\ldots + x^{2003}(1+x)^{2003} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f22a41e7328c185be7c255bc8b5c28775f7fb7047bf27413da280f699bb4a387 | Calculate the constant value of \( x^{2013} \) from the expansion: \( (1+x)^{4016} + x(1+x)^{4015} + x^2(1+x)^{4014} + x^3(1+x)^{4013} + \ldots + x^{2003}(1+x)^{2003} \). | [
"\\dbinom{4017}{2013} - \\dbinom{2013}{9}"
] | [
"Calculate the constant value of \\( x^{2013} \\) from the expansion: \\( (1+x)^{4016} + x(1+x)^{4015} + x^2(1+x)^{4014} + x^3(1+x)^{4013} + \\ldots + x^{2003}(1+x)^{2003} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-1805ee75434c88c8ee7cf6e8dd153376aff639422ac7f1048a1c5194b71131d0 | Calculate the contour integral $ \oint_{|z|=3} \frac{(z-1)^3e^{\frac{1}{1-z}}}{z-2}dz $. | [
"\\(\\frac{2\\pi i}{3}\\)"
] | [
"Calculate the contour integral $ \\oint_{|z|=3} \\frac{(z-1)^3e^{\\frac{1}{1-z}}}{z-2}dz $."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-9516f7d65b11db76ada81e452a8053cb560bb96dae009f349d9b6dde50238930 | Calculate the contour integral $\oint_{|z+1|=2}\frac{\sin (1/z)}{z^3(z+1)(z-2)}dz$. Specifically, determine the residue at $z = 0$. | [
"\\(-\\frac{1}{24} \\sin \\left( \\frac{1}{2} \\right)\\)"
] | [
"Calculate the contour integral $\\oint_{|z+1|=2}\\frac{\\sin (1/z)}{z^3(z+1)(z-2)}dz$. Specifically, determine the residue at $z = 0$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ed38b88d1e34fbe6417d4c0f9fb5dc0aa2ece022da1ec4d588542a61335ea0b0 | Calculate the convex hull \(\textbf{conv} S\) of the set \( S = \left\{ \begin{pmatrix} x \\ y \end{pmatrix} \in \mathbb{R}^{2}: y \geq \frac{1}{1+x^{2}} \right\} \). | [
"\\(\\textbf{conv} S = \\left\\{ \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\in \\mathbb{R}^{2}: y > 0 \\right\\}\\)"
] | [
"Calculate the convex hull \\(\\textbf{conv} S\\) of the set \\( S = \\left\\{ \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\in \\mathbb{R}^{2}: y \\geq \\frac{1}{1+x^{2}} \\right\\} \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-157710dfd3bb757f51cb5681d52ddcf6ec7f3888623515bde7dfff78eca4d021 | Calculate the convolution $f * \rho_\varepsilon$ if $f = \chi_{[a,b]}$ and $\rho_\varepsilon$ is defined as $$\rho_\varepsilon(x)= \left\{ \begin{array}{ll} e^{\frac{1}{x^2 -1}} & \text{if } |x| < 1 \\ 0 & \text{if } |x| \geq 1 \end{array} \right.$$ | [
"(f * \\rho_\\varepsilon)(x) = \\begin{cases}\n\\displaystyle \\int_{\\max(x - b, -1)}^{\\min(x - a, 1)} e^{\\frac{1}{z^2 - 1}} \\, dz & \\text{if } a - 1 < x < b + 1, \\\\\n0 & \\text{otherwise}.\n\\end{cases}"
] | [
"Calculate the convolution $f * \\rho_\\varepsilon$ if $f = \\chi_{[a,b]}$ and $\\rho_\\varepsilon$ is defined as\n$$\\rho_\\varepsilon(x)= \\left\\{ \\begin{array}{ll}\n e^{\\frac{1}{x^2 -1}} & \\text{if } |x| < 1 \\\\\n 0 & \\text{if } |x| \\geq 1\n \\end{array}\n \\right.$$"... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-6a9de07d46ac22a7550ca960780012f0594899d492f64eeafb5d463e590d18ed | Calculate the coordinates of the center of the conic given by the equation $x^2 - 2xy + 2x + 2y + 1 = 0$. | [
"(1, 2)"
] | [
"Calculate the coordinates of the center of the conic given by the equation $x^2 - 2xy + 2x + 2y + 1 = 0$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d0b60c503912617896d79e0252bfb2df2bb6e32c4d8c924b2c2289f1158e9236 | Calculate the coordinates of the vertex $C$ of the equilateral triangle $ABC$ if $A (1, 3)$ and $B (3, 1)$. | [
"$\\left(2+\\sqrt 3,2+\\sqrt 3\\right)$ or $\\left(2-\\sqrt 3,2-\\sqrt 3\\right)$"
] | [
"Calculate the coordinates of the vertex $C$ of the equilateral triangle $ABC$ if $A (1, 3)$ and $B (3, 1)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d654734c6e8f3c89e51ac7c2a01bd02512b2f0b21786ffbfb382e748a2d12f55 | Calculate the coordinates of the vertices $C$ and $D$ of the square $ABCD$ if $A (2, 1)$ and $B (0, 4)$. | [
"$C_1(3,6),\\ D_1(5,3)$ or $C_2(-3,2),\\ D_2(-1,-1)$"
] | [
"Calculate the coordinates of the vertices $C$ and $D$ of the square $ABCD$ if $A (2, 1)$ and $B (0, 4)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f442881aca997577a7fc005dd62c9882387a4e9d82366034f61799f525b31830 | Calculate the cosines of the angles which the following vectors make with the base vectors: (a) ${\bf a}={\bf i}+{\bf j}+{\bf k}$; (b) ${\bf b}=-3{\bf j}-{\bf k}$; (c) ${\bf c}=-5{\bf i}$; (d) ${\bf d}=3{\bf j}+4{\bf k}$. | [
"(a) $\\frac{1}{\\sqrt 3},\\ \\frac{1}{\\sqrt 3},\\ \\frac{1}{\\sqrt 3}$; (b) $0,\\ -\\frac{3}{\\sqrt {10}},\\ -\\frac{1}{\\sqrt {10}}$; (c) $-1,\\ 0,\\ 0$; (d) $0,\\ \\frac{3}{5},\\ \\frac{4}{5}$"
] | [
"Calculate the cosines of the angles which the following vectors make with the base vectors: (a) ${\\bf a}={\\bf i}+{\\bf j}+{\\bf k}$; (b) ${\\bf b}=-3{\\bf j}-{\\bf k}$; (c) ${\\bf c}=-5{\\bf i}$; (d) ${\\bf d}=3{\\bf j}+4{\\bf k}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08775090b60aaec2c087acf66fb98eed2b269bf8eeb510c5a4c57d252976164b | Calculate the cost if the radius of the cylinder is 4 cm, given that the volume is 500 cubic centimeters, the top and bottom cost 6 cents per square centimeter, and the sides cost 4 cents per square centimeter. | [
"\\( 500 + 192\\pi \\) cents"
] | [
"Calculate the cost if the radius of the cylinder is 4 cm, given that the volume is 500 cubic centimeters, the top and bottom cost 6 cents per square centimeter, and the sides cost 4 cents per square centimeter."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7b56c2be3e60f960d987de24d108b126094819c0201050f29899b6fc27ade24 | Calculate the covariance between the values \( X(t) \) and \( X(t+s) \) taken by a Poisson process \( X(t) \) with constant rate \( \lambda \) at the two times \( t \) and \( t+s \), where \( s > 0 \). | [
"\\(\\lambda t\\)"
] | [
"Calculate the covariance between the values \\( X(t) \\) and \\( X(t+s) \\) taken by a Poisson process \\( X(t) \\) with constant rate \\( \\lambda \\) at the two times \\( t \\) and \\( t+s \\), where \\( s > 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-45dd042a24a6c1842a596eb7979712b83fd1ee9e05402a902c2afc5810a08f55 | Calculate the cross-sectional area of a plane that intersects a cube through the midpoints of two edges sharing a common vertex and the midpoint of the edge orthogonal to those edges. The total surface area of the cube is $6 \text{ cm}^2$. | [
"\\(\\frac{7\\sqrt{24}}{11}\\)"
] | [
"Calculate the cross-sectional area of a plane that intersects a cube through the midpoints of two edges sharing a common vertex and the midpoint of the edge orthogonal to those edges. The total surface area of the cube is $6 \\text{ cm}^2$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a0853c77fb6d9678f944ee49f14455553e7d57e7fd7f8c3fe73e9e97562ff730 | Calculate the curl of the vector field $\nabla \times \left(\frac{\vec{p} \times \vec{r}}{r^{3}}\right)$ in both Cartesian and spherical coordinates, where $\vec{p}$ is a constant vector. | [
"\\dfrac{3(\\vec{p} \\cdot \\hat{r})\\hat{r} - \\vec{p}}{r^3}"
] | [
"Calculate the curl of the vector field $\\nabla \\times \\left(\\frac{\\vec{p} \\times \\vec{r}}{r^{3}}\\right)$ in both Cartesian and spherical coordinates, where $\\vec{p}$ is a constant vector."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-76b4b797202f14337ad24e163cbb57b2bfe15ccf6d16ae3b26db1005e9b46d11 | Calculate the curl of the vector function $\vec{F}(r)$, where $r = \sqrt{x^2 + y^2 + z^2}$, i.e., find $\nabla \times \vec{F}(r)$. | [
" \\nabla \\times \\vec{F}(r) = \\frac{\\vec{r}}{r} \\times \\frac{d\\vec{F}}{dr} "
] | [
"Calculate the curl of the vector function $\\vec{F}(r)$, where $r = \\sqrt{x^2 + y^2 + z^2}$, i.e., find $\\nabla \\times \\vec{F}(r)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-d23b9b64e72c7335f6122b93555622b3d9969120244a550fc9e8441e2c72e8d0 | Calculate the curvature of the curve given by \( x^4 = a^2(x^2 - y^2) \). | [
"\\dfrac{a^2 |x| (3a^2 - 2x^2)}{(4x^4 - 5a^2x^2 + 2a^4)^{3/2}}"
] | [
"Calculate the curvature of the curve given by \\( x^4 = a^2(x^2 - y^2) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-92e709782af671fedc3f40a491776d6f9fd12b98b2eac0bd95b40e929e7500d4 | Calculate the curvature of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 69\) at its vertices. | [
"\\dfrac{b}{a^2 \\sqrt{69}}"
] | [
"Calculate the curvature of the ellipse \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 69\\) at its vertices."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-9e303d8eabfe3a5da85f578bbe0f5b9bf2d4f1fdb1fa715293616bae8d770535 | Calculate the curvilinear integral using Stokes' Theorem: \[ \int\limits_\alpha {\left( {y - z} \right)dx + \left( {z - x} \right)dy + \left( {x - y} \right)dz} \] where $\alpha$ is the curve defined by the equations: \[ x^2 + 4y^2 = 1 \quad \text{and} \quad x^2 + y^2 = z. | [
"-\\pi"
] | [
"Calculate the curvilinear integral using Stokes' Theorem:\n\\[\n\\int\\limits_\\alpha {\\left( {y - z} \\right)dx + \\left( {z - x} \\right)dy + \\left( {x - y} \\right)dz}\n\\]\nwhere $\\alpha$ is the curve defined by the equations:\n\\[\nx^2 + 4y^2 = 1 \\quad \\text{and} \\quad x^2 + y^2 = z."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2cfa758e7e1f425ddc92f89f4d3f9afe160abee170f4c4d19ff558ddc0783ec6 | Calculate the definite integral $$\int_0^{\infty} \frac{9\sqrt{3}\,x}{x^3 + 1}\,\frac{\text{d}x}{\sqrt{4\,x^3 + 1}}\,.$$ | [
"\\(2\\pi\\)"
] | [
"Calculate the definite integral\n$$\\int_0^{\\infty} \\frac{9\\sqrt{3}\\,x}{x^3 + 1}\\,\\frac{\\text{d}x}{\\sqrt{4\\,x^3 + 1}}\\,.$$"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 7,
"tir": 25
} | 37 | 0 | 0 |
omr-ws-874dae385115c89bc91ad5acd0ffa0b2f47fa8c82b98616fefb24224ec994d17 | Calculate the definite integral $\int_0^{\frac{\ln 3}{2}} \frac{e^x+1}{e^{2x}+1}dx$. | [
"\\(\\frac{\\pi}{12} + \\frac{1}{2} \\ln \\frac{3}{2}\\)"
] | [
"Calculate the definite integral $\\int_0^{\\frac{\\ln 3}{2}} \\frac{e^x+1}{e^{2x}+1}dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-bf939f9f100656831566cf0171fcaf608271ca228c2e6d555f399e7718ccbec4 | Calculate the definite integral $\int_0^{\frac{\pi}{2}} x\left|\sin ^2x-\frac{1}{2}\right|dx$. | [
"\\(\\frac{\\pi}{8}\\)"
] | [
"Calculate the definite integral $\\int_0^{\\frac{\\pi}{2}} x\\left|\\sin ^2x-\\frac{1}{2}\\right|dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 27
} | 50 | 0 | 0 |
omr-ws-319861411401e502ecf0c5b7aa00d07dac0dd83dcb280e1f56caafc92b4816da | Calculate the definite integral $\int_2^3 \ln(x) \, dx$ using Archimedes' Method of Exhaustion. | [
"3 \\ln(3) - 2 \\ln(2) - 1"
] | [
"Calculate the definite integral $\\int_2^3 \\ln(x) \\, dx$ using Archimedes' Method of Exhaustion."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f170c8e0fe6f32deee23d353bd4a20b3847ad2fbfb70474fc49c1cf90871a376 | Calculate the definite integral $\int_{-i\infty}^{i\infty} e^{b^2} db$. | [
"\\(i\\sqrt{\\pi}\\)"
] | [
"Calculate the definite integral $\\int_{-i\\infty}^{i\\infty} e^{b^2} db$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-786039bbcdf92c8c6d6f8ec34bb3f13bd03a59b8df07589729d1cff57b8d3ac8 | Calculate the definite integral \( \int_{0}^{\frac{\pi}{4}} \arctan \sqrt{\frac{\cos 2x}{2 \cos^2 x}} \, dx \). | [
"\\dfrac{\\pi^2}{32}"
] | [
"Calculate the definite integral \\( \\int_{0}^{\\frac{\\pi}{4}} \\arctan \\sqrt{\\frac{\\cos 2x}{2 \\cos^2 x}} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-117a78aa7089434f29fcef5fc58a2b1ee5a450a5bd051c66e5ea7d5f9063f2ad | Calculate the definite integral \(\int_{0}^{\pi/4} x^3 \tan(x) \, dx\) and express your answer in terms of elementary functions and constants. If the integral cannot be expressed in terms of elementary functions, state "Non-elementary". | [
"Non-elementary"
] | [
"Calculate the definite integral \\(\\int_{0}^{\\pi/4} x^3 \\tan(x) \\, dx\\) and express your answer in terms of elementary functions and constants. If the integral cannot be expressed in terms of elementary functions, state \"Non-elementary\"."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0aad484677868905ecde9812a84b75f0a2350ad22688562767ec81c605545c60 | Calculate the definite integral of the function \( y = |x + 1| \) from the lower bound \(-3\) to the upper bound \(2\). | [
"\\(\\frac{13}{2}\\)"
] | [
"Calculate the definite integral of the function \\( y = |x + 1| \\) from the lower bound \\(-3\\) to the upper bound \\(2\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-5fb04ab0d6939b175fd4b9d6465ed93ec0f24ce52b3cdf187c112f0e4ccae096 | Calculate the density function of the conditional distribution of $(X_1,X_n)$ given that $X_1$ and $X_n$ occupy the positions $3$ and $5$, respectively, in the sorted sequence $T=(X_{(1)},X_{(2)},\cdots, X_{(n)})$ where $X_1,\cdots, X_n$ are independent random samples from the same exponential distribution $\mathcal{E}... | [
"\\frac{n!}{2(n - 5)!} \\lambda^2 (1 - e^{-\\lambda x})^2 (e^{-\\lambda x} - e^{-\\lambda y}) e^{-\\lambda x - \\lambda y (n - 4)}"
] | [
"Calculate the density function of the conditional distribution of $(X_1,X_n)$ given that $X_1$ and $X_n$ occupy the positions $3$ and $5$, respectively, in the sorted sequence $T=(X_{(1)},X_{(2)},\\cdots, X_{(n)})$ where $X_1,\\cdots, X_n$ are independent random samples from the same exponential distribution $\\ma... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c765751e297879582ee6ebc989c878bb93ad6b13b29d69b032fb00784cf38d98 | Calculate the density function of the distribution of the sorted sequence $T=(X_{(1)},X_{(2)},\cdots, X_{(n)})$ where $X_1,\cdots, X_n$ are independent random samples from the same exponential distribution $\mathcal{E}(\lambda)$. | [
"\\( n! \\prod_{i=1}^n f(t_i) \\cdot 1_{\\{t_1 < ... < t_n\\}} \\)"
] | [
"Calculate the density function of the distribution of the sorted sequence $T=(X_{(1)},X_{(2)},\\cdots, X_{(n)})$ where $X_1,\\cdots, X_n$ are independent random samples from the same exponential distribution $\\mathcal{E}(\\lambda)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-da8e130a344f62fc2a2635151e63c32b0cd44c6e73f201a571a7662f4118c209 | Calculate the density of the material of a right circular cylinder with radius $r$ that is floating on water, submerged to a depth of $\frac{r}{2}$. | [
"\\(1000 \\cdot \\frac{4\\pi - 3\\sqrt{3}}{12\\pi}\\) or approximately \\(195.5 \\text{ kg/m}^3\\)"
] | [
"Calculate the density of the material of a right circular cylinder with radius $r$ that is floating on water, submerged to a depth of $\\frac{r}{2}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-84fbc97fe53bfe9be77b8b8e0aa5b23c46595a29669d8a8e3060418686a312b1 | Calculate the depth of the water in a spherical tank of radius 50 feet when it is filled to three-fourths of its total capacity. | [
"67.36 feet"
] | [
"Calculate the depth of the water in a spherical tank of radius 50 feet when it is filled to three-fourths of its total capacity."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-7172552f500ee78b38fd68fff0b45f3a271c546ea992716758ad86f0a1f193c6 | Calculate the derivative $f'(x)$ of the function $f(x) = \log_{x} (\log_{10} x)$ at $x = e$. | [
"\\(\\frac{1+\\ln(\\ln10)}{e}\\)"
] | [
"Calculate the derivative $f'(x)$ of the function $f(x) = \\log_{x} (\\log_{10} x)$ at $x = e$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-8d4890434eaa27a764c2b71d0ceb15189021d7511cc113bc486d142809844ea3 | Calculate the derivative function of \( f(x) = \sin{x} + |\sin{x}| \). | [
"\\( f'(x) = \\begin{cases} \n2 \\cos x & \\text{if } \\sin x > 0 \\\\\n0 & \\text{if } \\sin x < 0 \\\\\n\\text{undefined} & \\text{if } \\sin x = 0 \n\\end{cases} \\)"
] | [
"Calculate the derivative function of \\( f(x) = \\sin{x} + |\\sin{x}| \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 18
} | 21 | 0 | 0 |
omr-ws-8430502b64b7bfd7ac1a68226a3416703ec50858172ec81efb5ee603c96226f8 | Calculate the derivative of $f(x) = [x] \sin^2(\pi x)$. | [
"\\( f'(x) = n \\pi \\sin(2\\pi x) \\) for \\( n \\le x < n+1 \\) where \\( n \\) is an integer."
] | [
"Calculate the derivative of $f(x) = [x] \\sin^2(\\pi x)$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-a76e969855757657d790ec5d47b377ab0ba597163bf453d3a356677e742c9721 | Calculate the derivative of $f(x) = \lvert\sin{x}\rvert \sin{x}$. | [
"\\(\\sin{2x}\\)"
] | [
"Calculate the derivative of $f(x) = \\lvert\\sin{x}\\rvert \\sin{x}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b17c7a5948956f8477b46f358c5e1ac6a5791e82725fb37b6d59c81594f909a8 | Calculate the derivative of $f(x) = \lvert{x}\rvert^3$. | [
"\\( f'(x) = 3x^2 \\text{ for } x \\ge 0 \\text{ and } f'(x) = -3x^2 \\text{ for } x < 0. \\)"
] | [
"Calculate the derivative of $f(x) = \\lvert{x}\\rvert^3$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-f9fbbcd2387b93ba2c58bd6dc7545ee840ae335824cf28a59f0761e48c98d7c3 | Calculate the derivative of $f(x) = x\lvert{x}\rvert$. | [
"\\( f'(x) = \\begin{cases} \n2x & \\text{if } x \\ge 0 \\\\\n-2x & \\text{if } x < 0 \n\\end{cases} \\)"
] | [
"Calculate the derivative of $f(x) = x\\lvert{x}\\rvert$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-d8270f86610649a5050d24253de007c41bf071ebc5ef2030881584edaa667d56 | Calculate the derivative of \( y^{-1} \) with respect to \( x \) at \( x = \frac{\pi}{3} \) for the function \( y = \sin\left(\frac{5x^2 - 6}{x^2 - 2x - 8}\right) \). | [
"353"
] | [
"Calculate the derivative of \\( y^{-1} \\) with respect to \\( x \\) at \\( x = \\frac{\\pi}{3} \\) for the function \\( y = \\sin\\left(\\frac{5x^2 - 6}{x^2 - 2x - 8}\\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-31d3684df9c2f327c4df6e91579ccc7244cd23d4f7e532a3067ef8c9c142433b | Calculate the derivative of the function $f(x) = (\sin x + |\sin x|)^2$. | [
"\\( f'(x) = \\begin{cases} \n0 & \\text{if } x < 0 \\\\\n4 \\sin 2x & \\text{if } x \\ge 0 \n\\end{cases} \\)"
] | [
"Calculate the derivative of the function $f(x) = (\\sin x + |\\sin x|)^2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e56479a59574b0aa97e1c42446416de4e058d2f0475c470b7945092f8e23060b | Calculate the derivative of the function $f(x) = (x - [x]) \sin^2(\pi x)$. | [
"\\( f'(x) = \\pi(x - [x]) \\sin(2\\pi x) + \\sin^2(\\pi x) \\)"
] | [
"Calculate the derivative of the function $f(x) = (x - [x]) \\sin^2(\\pi x)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-89de9949611937eb53ab2e2f550f18100fdfec50f1f7f6cb446a0ca59b919fa9 | Calculate the derivative of the function $y = -9E-05x^6 + 0.0024x^5 - 0.0294x^4 + 0.1285x^3 - 0.0251x^2 + 2.9239x - 7.9929$. | [
" -0.00054x^5 + 0.012x^4 - 0.1176x^3 + 0.3855x^2 - 0.0502x + 2.9239 "
] | [
"Calculate the derivative of the function $y = -9E-05x^6 + 0.0024x^5 - 0.0294x^4 + 0.1285x^3 - 0.0251x^2 + 2.9239x - 7.9929$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e6603d9effdc1c55b7165162218b1a83cbcb1719b188b5841b4ab86ee85040f | Calculate the derivative of the function \( I(a) \) where \[ I(a) = \iiint\limits_{0 \leq x \leq y \leq z \leq a} \frac{dxdydz}{x^3 + y^3 + z^3 + a^3}. \] Find \( I'(a) \). | [
"0"
] | [
"Calculate the derivative of the function \\( I(a) \\) where\n\\[ I(a) = \\iiint\\limits_{0 \\leq x \\leq y \\leq z \\leq a} \\frac{dxdydz}{x^3 + y^3 + z^3 + a^3}. \\]\nFind \\( I'(a) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca03a2796149c9ffc2aaa856d8284e76f9045e20db769695e051677a7df977d4 | Calculate the derivative of the function \( f(x) = (x - [x]) \sin^2(\pi x) \). | [
"\\( f'(x) = \\sin^{2}(\\pi x) + \\pi (x - \\lfloor x \\rfloor) \\sin(2\\pi x) \\)"
] | [
"Calculate the derivative of the function \\( f(x) = (x - [x]) \\sin^2(\\pi x) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 4,
"tir": 23
} | 42 | 0 | 0 |
omr-ws-1bdcdd11f5e1dfd1730fa5eeb7a2d27228c377325f1be9c2b8569ed27a2550bd | Calculate the derivative of the function \( f(x) = \frac{1-x}{(1+x)(1+x^2)} \). | [
"\\(\\frac{2(x^3 - x^2 - x - 1)}{(1+x)^2(1+x^2)^2}\\)"
] | [
"Calculate the derivative of the function \\( f(x) = \\frac{1-x}{(1+x)(1+x^2)} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-28a5b17ce38c9194f1c1a724364871208ed3081b25d0bcfa17a2d8689a018ec6 | Calculate the derivative of the function \( f(x) = x^{2x^{3x^{4x^{5x{\ldots}}}}} \) with respect to \( x \). | [
"f(x) \\left( \\sum_{n=1}^{\\infty} \\frac{(n+1)!}{1} (\\ln x)^{n-1} \\right) \\frac{1}{x}"
] | [
"Calculate the derivative of the function \\( f(x) = x^{2x^{3x^{4x^{5x{\\ldots}}}}} \\) with respect to \\( x \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-132915d0549d9ea3384434219e51c4445803ea234099ae04448d4ee31db2e456 | Calculate the derivative of the function \( y = \cos(\tan(\sqrt{1 + 2x^2})) \). | [
" -\\dfrac{2x \\sin\\left(\\tan\\left(\\sqrt{1 + 2x^2}\\right)\\right) \\sec^2\\left(\\sqrt{1 + 2x^2}\\right)}{\\sqrt{1 + 2x^2}} "
] | [
"Calculate the derivative of the function \\( y = \\cos(\\tan(\\sqrt{1 + 2x^2})) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a4a6ca95fcf6fd685e298f75e8467eded32a8e56b8d7f57545263b0e7a5c86f1 | Calculate the derivative of the function \( y = \sin x \cdot |\cos x| \). | [
"\\( y' = \\cos (2x) \\cdot \\text{sign}(\\cos x) \\) for \\( x \\neq \\left( k + \\frac{1}{2} \\right)\\pi \\)",
"\\cos(2x) \\cdot \\text{sgn}(\\cos x)"
] | [
"Calculate the derivative of the function \\( y = \\sin x \\cdot |\\cos x| \\)."
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 24
} | 56 | 0 | 0 |
omr-ws-f92111ff5fd8d49ede1689350af1960e9bae92722a05e861e435cee891793ea0 | Calculate the derivative of the power sequence \( x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} \) (n times), where \( n \) is a natural number that tends to infinity. | [
"\\( \\frac{f(x)^2}{x(1 - f(x) \\log x)} \\)"
] | [
"Calculate the derivative of the power sequence \\( x^{x^{x^{\\cdot^{\\cdot^{\\cdot}}}}} \\) (n times), where \\( n \\) is a natural number that tends to infinity."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-f4c2e2978c639bb61ae803e9a5c8d1e8698dd6f503a6a5289a4dec16737c37fa | Calculate the derivative, in the sense of distributions, of the function \( f(x) \) defined on \( ]-1,1[ \) as follows: \[ f(x) = \begin{cases} \frac{x}{|x|}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases} \] | [
"\\( 2\\delta_0 \\)"
] | [
"Calculate the derivative, in the sense of distributions, of the function \\( f(x) \\) defined on \\( ]-1,1[ \\) as follows:\n\n\\[ f(x) = \\begin{cases} \n\\frac{x}{|x|}, & \\text{if } x \\neq 0 \\\\\n0, & \\text{if } x = 0 \n\\end{cases} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-241da08026dab342b99c08ee5618e710bf77fa796b8941598f1c06971eb93429 | Calculate the derivatives of the following composite functions. $y=\frac{(\tan^2 x-1)(\tan^4 x+10\tan^2 x+1)}{3\tan^3 x}$. | [
"$\\frac{1}{\\sin ^4 x\\cos ^4 x}$"
] | [
"Calculate the derivatives of the following composite functions. $y=\\frac{(\\tan^2 x-1)(\\tan^4 x+10\\tan^2 x+1)}{3\\tan^3 x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e793a7d35e70adf6fba62d31c8e37f0fa03a9df79593c32acc1dca55c437981 | Calculate the derivatives of the following composite functions. $y=\frac{(a+bx^n)^m}{(a-bx^n)^m}$. | [
"$\\frac{2abmnx^{n-1}(a+bx^n)^{m-1}}{(a-bx^n)^{m+1}}$"
] | [
"Calculate the derivatives of the following composite functions. $y=\\frac{(a+bx^n)^m}{(a-bx^n)^m}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76cd6ce52be8aef7b362c5dcf29ef799cf6bf25e50a0384df856b40edcd9b95d | Calculate the derivatives of the following composite functions. $y=\frac{1}{15}\cos ^3 x(3\cos ^2 x-5)$. | [
"$\\sin ^3 x\\cos ^2 x$"
] | [
"Calculate the derivatives of the following composite functions. $y=\\frac{1}{15}\\cos ^3 x(3\\cos ^2 x-5)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29b4519e3343630b3bc7dae25d064ba05018267366b14ed4a593bf8f1d6f3021 | Calculate the derivatives of the following composite functions. $y=\frac{2+\sqrt x}{2-\sqrt x}$. | [
"$\\frac{2}{\\sqrt 2\\left(2-\\sqrt x\\right)^2}$"
] | [
"Calculate the derivatives of the following composite functions. $y=\\frac{2+\\sqrt x}{2-\\sqrt x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc6e2da36943102fbe27357502a275282a919a96c5d52186db52312f90510751 | Calculate the derivatives of the following composite functions. $y=e^{\sqrt {\ln(ax^2+bx+c)}}$. | [
"$\\frac{e^{\\sqrt {\\ln(ax^2+bx+c)}}(2ax+b)}{2\\sqrt {\\ln(ax^2+bx+c)} (ax^2+bx+c)}$"
] | [
"Calculate the derivatives of the following composite functions. $y=e^{\\sqrt {\\ln(ax^2+bx+c)}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7d99ad9560757e6036a9bbeec926a5621762e3463a31db071a9007ac3ad2202 | Calculate the derivatives of the following functions. $f(x)=\frac{2x\sqrt {\frac{1}{4}\left(\frac{1}{\sqrt x}+\sqrt x\right)^2-1}} {2\sqrt {\frac{1}{4}\left(\frac{1}{\sqrt x}+\sqrt x\right)^2-1} -\frac{1}{2}\left(\sqrt {\frac{1}{x}}-\sqrt x\right)}$. | [
"$f'(x)= \\left\\{\\begin{array}{lll} 2, & x\\in (0,1),\\\\ 2/3, & x\\in (1,\\infty ). \\end{array}\\right.$"
] | [
"Calculate the derivatives of the following functions. $f(x)=\\frac{2x\\sqrt {\\frac{1}{4}\\left(\\frac{1}{\\sqrt x}+\\sqrt x\\right)^2-1}} {2\\sqrt {\\frac{1}{4}\\left(\\frac{1}{\\sqrt x}+\\sqrt x\\right)^2-1} -\\frac{1}{2}\\left(\\sqrt {\\frac{1}{x}}-\\sqrt x\\right)}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13867a63209e8cd8bf25b2bbc0ab0acc56127aa1b016eeaf57d40a92b6d6bc6d | Calculate the derivatives of the following functions. $f(x)=\frac{\sqrt {1+\left(\frac{x^2-1}{2x}\right)^2}}{(x^2+1)\frac{1}{x^2}}$. | [
"$f'(x)= \\left\\{\\begin{array}{rll} -1/2, & x\\in (-\\infty ,0),\\\\ 1/2, & x\\in (0,\\infty ) \\end{array}\\right.$"
] | [
"Calculate the derivatives of the following functions. $f(x)=\\frac{\\sqrt {1+\\left(\\frac{x^2-1}{2x}\\right)^2}}{(x^2+1)\\frac{1}{x^2}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a78d59a8c6ceaa83fa66779deda15320c7c36f9febdf6ca46e39c369cc11f48 | Calculate the derivatives of the following functions. $f(x)=\sqrt {x+2\sqrt {2x-4}}+\sqrt {x-2\sqrt {2x-4}}$. | [
"$f'(x)= \\left\\{\\begin{array}{lll} 0, & x\\in [2,4),\\\\ 1/\\sqrt {x-2}, & x\\in (4,\\infty ). \\end{array}\\right.$"
] | [
"Calculate the derivatives of the following functions. $f(x)=\\sqrt {x+2\\sqrt {2x-4}}+\\sqrt {x-2\\sqrt {2x-4}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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