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-58048e1353f157a306247bc3f96be5d735b9a18a76b530637b1b9738bcb116f0 | Urn A contains 6 red and 4 green marbles. Urn B contains 7 red and 3 green marbles. If 2 marbles are drawn at random from each urn, what is the probability that from these drawn, there will be 2 red and 2 green marbles? Write the answer in the form \( \frac{a}{b} \) where \( a \) and \( b \) are relatively prime whole ... | [
"\\(\\frac{1}{3}\\)"
] | [
"Urn A contains 6 red and 4 green marbles. Urn B contains 7 red and 3 green marbles. If 2 marbles are drawn at random from each urn, what is the probability that from these drawn, there will be 2 red and 2 green marbles? Write the answer in the form \\( \\frac{a}{b} \\) where \\( a \\) and \\( b \\) are relatively ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-2bec2d469f1db00dff67b8f01ff669adfddcce3a2f53945ebab253c0ecc33b81 | Urn I and urn II each have $1$ red and $4$ black balls. Urn III has $R$ red and $B$ black balls. Three balls are drawn at random from urn I and placed in urn II. Then three balls are drawn at random from urn II and placed in urn III. Finally, one ball is drawn at random from urn III. (a) Find the relation between $R$ a... | [
"R > B + \\dfrac{9}{5}"
] | [
"Urn I and urn II each have $1$ red and $4$ black balls. Urn III has $R$ red and $B$ black balls. Three balls are drawn at random from urn I and placed in urn II. Then three balls are drawn at random from urn II and placed in urn III. Finally, one ball is drawn at random from urn III.\n(a) Find the relation between... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-23a3d0ccd103fa0476bd95d421af54de721345cf75781bb347435bd11313f46d | Usain Bolt is running at a speed of 36 km/hr on a 200-meter-long linear runway. When he reaches the halfway point (100 meters), a cheetah is released from a cage 300 meters away from him and chases him at a speed of 120 km/hr. Will Usain Bolt reach the finish line before the cheetah catches him? | [
"No, Usain Bolt will not reach the finish line before the cheetah catches him."
] | [
"Usain Bolt is running at a speed of 36 km/hr on a 200-meter-long linear runway. When he reaches the halfway point (100 meters), a cheetah is released from a cage 300 meters away from him and chases him at a speed of 120 km/hr. Will Usain Bolt reach the finish line before the cheetah catches him?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-14122331559c49607814942df64aa860f00bc5e60c6283af2d33089962c8cf61 | Usain runs with distances increasing by 30 meters each minute, starting at 70 meters in the first minute. Harry starts running 3 minutes later, covering distances increasing by 50 meters each minute, starting at 100 meters in the first minute. How many minutes did Harry run before catching up with Usain? What were thei... | [
"Harry ran for 9 minutes before catching up with Usain. Their respective speeds at the moment Harry caught up with Usain were 430 meters per minute for Usain and 500 meters per minute for Harry."
] | [
"Usain runs with distances increasing by 30 meters each minute, starting at 70 meters in the first minute. Harry starts running 3 minutes later, covering distances increasing by 50 meters each minute, starting at 100 meters in the first minute. How many minutes did Harry run before catching up with Usain? What were... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-83f5cc359b7822f9884be759ff8e99783a37295b467eaae6d3dbbb52d517ed5b | Use $\log_{10} 2 = 0.301$, $\log_{10} 3 = 0.477$, and $\log_{10} 7 = 0.845$ to find the value of $\log_{10} (10!)$. Round the fractional part of the result to the nearest tenth, rounding up if the fractional part is 0.5 or higher, and rounding down if it is less than 0.5. | [
"6.56"
] | [
"Use $\\log_{10} 2 = 0.301$, $\\log_{10} 3 = 0.477$, and $\\log_{10} 7 = 0.845$ to find the value of $\\log_{10} (10!)$. Round the fractional part of the result to the nearest tenth, rounding up if the fractional part is 0.5 or higher, and rounding down if it is less than 0.5."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 6
} | 12 | 0 | 0 |
omr-ws-ebd80485d56fd599c8c8e4296cd3d0bcc38fcd0301a1aa0a8c9c5ab8f15149c1 | Use $s_n$ and integral bounds to find the smallest interval that contains the sum $s$ of the series $\sum_{k=1}^{\infty} \frac{1}{2^k k!}$. If the midpoint $s_n*$ of this interval is used to approximate $s$, how large should $n$ be chosen to ensure that the error is less than $0.001$? | [
"4"
] | [
"Use $s_n$ and integral bounds to find the smallest interval that contains the sum $s$ of the series $\\sum_{k=1}^{\\infty} \\frac{1}{2^k k!}$. If the midpoint $s_n*$ of this interval is used to approximate $s$, how large should $n$ be chosen to ensure that the error is less than $0.001$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-62075498531172ca49d7c127b80e4cd15000213c33177b0b40b6e9c5c96fc412 | Use $s_n$ and integral bounds to find the smallest interval that contains the sum $s$ of the series $\sum_{k=1}^{\infty} \frac{1}{k^3}$. If the midpoint $s_n*$ of this interval is used to approximate $s$, how large should $n$ be chosen to ensure that the error is less than $0.001$? | [
"23"
] | [
"Use $s_n$ and integral bounds to find the smallest interval that contains the sum $s$ of the series $\\sum_{k=1}^{\\infty} \\frac{1}{k^3}$. If the midpoint $s_n*$ of this interval is used to approximate $s$, how large should $n$ be chosen to ensure that the error is less than $0.001$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c0753118f5ff567b4cef693b2f84980f475e5b863572dae6d4fcd443f89220df | Use 20 unit cubes to form a solid. Assign each unit cube an integer between 1 and 8 (inclusive). For any surface of the figure, the eight assigned values of the cubes in this surface must all be different. How many ways are there to assign the numbers to satisfy this condition? | [
"40320"
] | [
"Use 20 unit cubes to form a solid. Assign each unit cube an integer between 1 and 8 (inclusive). For any surface of the figure, the eight assigned values of the cubes in this surface must all be different. How many ways are there to assign the numbers to satisfy this condition?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-9abfb083e16dde2bdd8ad1e48c63e69335e449501f5943b795c2d1083654646b | Use 36 small equilateral triangle tiles with side length 1 to form a layout where 30 points are marked as good points. Each small triangle is colored either red or blue. How many coloring methods are there such that for any good point, the number of red triangles with it as an apex equals the number of blue triangles? ... | [
"64"
] | [
"Use 36 small equilateral triangle tiles with side length 1 to form a layout where 30 points are marked as good points. Each small triangle is colored either red or blue. How many coloring methods are there such that for any good point, the number of red triangles with it as an apex equals the number of blue triang... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-310e18b1765bce987679bd34482a58cd60829205c694739f7d209f7b5b6215b3 | Use Cramer's rule to solve the following system of equations: \[ 2x - y = 4 \] \[ 3x + y - z = 10 \] \[ y + z = 3 \] | [
"z = 1"
] | [
"Use Cramer's rule to solve the following system of equations:\n\\[ 2x - y = 4 \\]\n\\[ 3x + y - z = 10 \\]\n\\[ y + z = 3 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-c73c6346aa7a71a69b50e76df2f966d5b6ab3e8ca67fb446ea54fa00b5bd2107 | Use Cramer's rule to solve the system of linear equations: \[ 2x - y = 4 \] \[ 3x + y - z = 10 \] \[ y + z = 3 \] | [
"z = 1"
] | [
"Use Cramer's rule to solve the system of linear equations:\n\\[ 2x - y = 4 \\]\n\\[ 3x + y - z = 10 \\]\n\\[ y + z = 3 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-3bb6d6d2da05362333a7f6d78fa5509783286f483c776d513c2f31f0f7b1d409 | Use Cramer's rule to solve the system of linear equations: \[ x + 2y = 7 \] \[ 3x - 9 = -9 \] | [
"\\( x = 0, y = \\frac{7}{2} \\)"
] | [
"Use Cramer's rule to solve the system of linear equations:\n\\[ x + 2y = 7 \\]\n\\[ 3x - 9 = -9 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 23
} | 30 | 0 | 0 |
omr-ws-11f82fa73ca30aa8812a4f50c7cf8275d2110acae74cdb10a8643c294f2f2043 | Use De Moivre's Theorem to express $\cos^4{x}$ in terms of $\cos{kx}$ and $\sin{kx}$, where $k = 1,2,3$. | [
"\\(\\frac{1}{16} \\left( \\cos{4x} + 4\\cos{2x} + 6 \\right)\\)"
] | [
"Use De Moivre's Theorem to express $\\cos^4{x}$ in terms of $\\cos{kx}$ and $\\sin{kx}$, where $k = 1,2,3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-c7449b1f60a393ecd06d06ac845d8b05d79afea2edc4b5dfa459a26254520b7b | Use De Moivre's Theorem to express $\sin^4{x}$ in terms of $\cos{kx}$ and $\sin{kx}$, where $k = 1,2,3$. | [
"\\(\\sin^4{x} = \\frac{1}{8}\\cos{4x} - \\frac{1}{2}\\cos{2x} + \\frac{3}{8}\\)"
] | [
"Use De Moivre's Theorem to express $\\sin^4{x}$ in terms of $\\cos{kx}$ and $\\sin{kx}$, where $k = 1,2,3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-963c5806fa32ccaf499119c0f99772466ffc3bf817c5f23e8d6e7588a617d3ba | Use De Moivre's theorem to find $\tan 4\theta$ in terms of $\tan \theta$. | [
"\\(\\dfrac {4\\tan \\theta - 4\\tan^3 \\theta} {1 - 6\\tan^2 \\theta + \\tan^4 \\theta}\\)"
] | [
"Use De Moivre's theorem to find $\\tan 4\\theta$ in terms of $\\tan \\theta$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-31fb2b9dbfe22a7c79a0894619d41371f1ca1bd9a9da5cfed379d8cb2894de4e | Use De Moivre's theorem to find the cube roots of $-2-2i$ in modulus-argument form. | [
"\\(\\left( \\sqrt{2}, \\frac{5\\pi}{12} \\right), \\left( \\sqrt{2}, \\frac{13\\pi}{12} \\right), \\left( \\sqrt{2}, \\frac{21\\pi}{12} \\right)\\)"
] | [
"Use De Moivre's theorem to find the cube roots of $-2-2i$ in modulus-argument form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 30
} | 35 | 0 | 0 |
omr-ws-86ae35c476655f3be30337170a21a3ab7ba917e994791183d0a20a6c47e6d159 | Use De Moivre's theorem to solve the equation: \[ (x-2)^2(x^2+2x+4)^2 + 12 = 0 \] | [
"The solutions are the cube roots of \\( 8 + 2i\\sqrt{3} \\) and \\( 8 - 2i\\sqrt{3} \\)."
] | [
"Use De Moivre's theorem to solve the equation:\n\\[ (x-2)^2(x^2+2x+4)^2 + 12 = 0 \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 35,
"tir": 10
} | 45 | 0 | 0 |
omr-ws-17fbd98b2808a824c4f80db3a4cc62f08007e700afb218ab05f98f32cc3e114f | Use Descartes' Rule of Signs to find the number of positive and negative zeros of the function \( f(x) = x^3 + 5x^2 + 8x + 6 \). | [
"Positive roots: 0, Negative roots: 1"
] | [
"Use Descartes' Rule of Signs to find the number of positive and negative zeros of the function \\( f(x) = x^3 + 5x^2 + 8x + 6 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 1,
"tir": 22
} | 34 | 0 | 0 |
omr-ws-67d9f3ad27edbc6ef19c54d9ed9d31998868e1851030d87de371d89a63d0b37d | Use Dijkstra's algorithm to find the shortest distance from vertex 5 to all other vertices. Show the final spanning tree. | [
"0"
] | [
"Use Dijkstra's algorithm to find the shortest distance from vertex 5 to all other vertices. Show the final spanning tree."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-a106bd15787016ab8788e57102a74f5e25d58f22fc28323b8589ceafda0ead7f | Use Euclid's algorithm to calculate the GCD of \(5 + 12i\) and \(7 - 10i\) in \(\mathbb{Z}[i]\). Then express this GCD as a linear combination of these two numbers. | [
"i = (1 + 5i)(5 + 12i) + (5 + 2i)(7 - 10i)"
] | [
"Use Euclid's algorithm to calculate the GCD of \\(5 + 12i\\) and \\(7 - 10i\\) in \\(\\mathbb{Z}[i]\\). Then express this GCD as a linear combination of these two numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-5b35093231acfa7e491f224ecbd7bc5009386d607ed0c17964dd705fd898533c | Use Euclid's algorithm to calculate the greatest common divisor (gcd) of \(5 + 12i\) and \(7 - 10i\) in \(\mathbb{Z}[i]\). Then, express this gcd as a linear combination of these two numbers. | [
"\\(i = (7 - 10i)(5 + 2i) + (5 + 12i)(1 + 5i)\\)"
] | [
"Use Euclid's algorithm to calculate the greatest common divisor (gcd) of \\(5 + 12i\\) and \\(7 - 10i\\) in \\(\\mathbb{Z}[i]\\). Then, express this gcd as a linear combination of these two numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 2,
"tir": 21
} | 32 | 0 | 0 |
omr-ws-0729dd45f5d994728455a28112ed79ed7f4326a376afec2d524baad97786d09a | Use Euler's formula, $e^{it} = \cos t + i \sin t$, to find the two complex square roots of $-i$ by writing $-i$ as a complex exponential. | [
"\\(\\frac{1 - i}{\\sqrt{2}}\\) and \\(\\frac{-1 + i}{\\sqrt{2}}\\)"
] | [
"Use Euler's formula, $e^{it} = \\cos t + i \\sin t$, to find the two complex square roots of $-i$ by writing $-i$ as a complex exponential."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 32
} | 43 | 0 | 0 |
omr-ws-6c460bee46f3bab9e1bc6722bd9770730b9b1658896898a2477e1e2a49b3127a | Use Euler's identity to evaluate the product \[ \prod_{n=1}^{\infty}\left(1 - \frac{1}{a^n}\right) \] for complex \( a \) such that \( |a| < 1 \). The identity is given by \[ \prod_{n=1}^{\infty}\left(1 - \frac{1}{a^n}\right) = \sum_{-\infty}^{+\infty} (-1)^n a^{-\frac{1}{2}(3n^2 + n)}. \] | [
"\\sum_{n=-\\infty}^{\\infty} (-1)^n a^{-\\frac{1}{2}(3n^2 + n)}"
] | [
"Use Euler's identity to evaluate the product\n\\[ \\prod_{n=1}^{\\infty}\\left(1 - \\frac{1}{a^n}\\right) \\]\nfor complex \\( a \\) such that \\( |a| < 1 \\). The identity is given by\n\\[ \\prod_{n=1}^{\\infty}\\left(1 - \\frac{1}{a^n}\\right) = \\sum_{-\\infty}^{+\\infty} (-1)^n a^{-\\frac{1}{2}(3n^2 + n)}. \\]... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a0cb90e27663f5f468a5b9b48edd663d0ea1ad8512efa010d303f8dc7b096131 | Use Euler's method with step size $h=10^{-2}$ to find an approximation of the solution to the Cauchy problem $$\begin{cases}x'=e^{-x^{2}}+t^{2}\\x(0)=1 \end{cases}$$ at the nodes $$t_{j}=jh, \quad j=1,\ldots,50.$$ | [
"1.196"
] | [
"Use Euler's method with step size $h=10^{-2}$ to find an approximation of the solution to the Cauchy problem \n$$\\begin{cases}x'=e^{-x^{2}}+t^{2}\\\\x(0)=1 \\end{cases}$$\nat the nodes $$t_{j}=jh, \\quad j=1,\\ldots,50.$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-5e82594c57948d470987c5b307e922b9c40131eeb4a5ded53d518e28a2f42cc9 | Use FOIL to multiply \((8x - 1)(2x + 5)\). | [
"16x^2 + 38x - 5"
] | [
"Use FOIL to multiply \\((8x - 1)(2x + 5)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-16688b662295e7c291106c63b6492f3f5bd03b1a0a79b5805de378535f1fc3f9 | Use Fermat's Little Theorem to find the smallest odd prime number which divides \( 1155^{55} - 1 \). | [
"31"
] | [
"Use Fermat's Little Theorem to find the smallest odd prime number which divides \\( 1155^{55} - 1 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 5
} | 53 | 0 | 0 |
omr-ws-8c3b0c072ba67ea888ee459184af00b627521eee14ff18ff0da21995f3210d06 | Use Fermat's Little Theorem to find the value of $2^{10} + 4^{20} + 6^{30} + 12^{40} \mod 13$. | [
"4"
] | [
"Use Fermat's Little Theorem to find the value of $2^{10} + 4^{20} + 6^{30} + 12^{40} \\mod 13$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f78d2fdf8d04f03152e33f2ed6d032e78ffcfc7aa99348336eeaa791b18b040c | Use Fermat's little theorem to evaluate $\epsilon_3(X^{231} + 3X^{117} - 2X^{53} + 1)$, where $\epsilon_a:\mathbb{Z}_5[X] \to \mathbb{Z}_5, X \mapsto a$ is the evaluation homomorphism. | [
"1"
] | [
"Use Fermat's little theorem to evaluate $\\epsilon_3(X^{231} + 3X^{117} - 2X^{53} + 1)$, where $\\epsilon_a:\\mathbb{Z}_5[X] \\to \\mathbb{Z}_5, X \\mapsto a$ is the evaluation homomorphism."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de482275668d0855cde02de5d4400ed0277d8ea25e124a6f1a991ea8dcd73d5f | Use Feynman's trick to evaluate the integral $\int_{0}^{1}\bigg(x\log(x)\bigg)^4dx$. | [
"\\(\\frac{24}{3125}\\)"
] | [
"Use Feynman's trick to evaluate the integral $\\int_{0}^{1}\\bigg(x\\log(x)\\bigg)^4dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-8e79c5e6654180f2efd68064bc4a7ceff391f7154e809f9402f3931e752719a7 | Use Gauss Lemma to calculate the Legendre Symbol $ \left(\frac{6}{13}\right)$. Given the definition from Ireland and Rosen: Let $ \pm m_t$ be the least residue of $ ta$, where $ m_t$ is positive. As $ t$ ranges between 1 and $ \frac{(p-1)}{2}$, $ n$ is the number of minus signs that occur in this way. Calculate $ \left... | [
"1"
] | [
"Use Gauss Lemma to calculate the Legendre Symbol $ \\left(\\frac{6}{13}\\right)$. \n\nGiven the definition from Ireland and Rosen: Let $ \\pm m_t$ be the least residue of $ ta$, where $ m_t$ is positive. As $ t$ ranges between 1 and $ \\frac{(p-1)}{2}$, $ n$ is the number of minus signs that occur in this way. Cal... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8e9109c2a88ebdfe85c6e68e7936ca1e2bfd8f7ec6b86143db345fbbb81083ea | Use Gauss's Lemma to solve \(\left(\frac{a+5}{37}\right)\). | [
"-1"
] | [
"Use Gauss's Lemma to solve \\(\\left(\\frac{a+5}{37}\\right)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-13b242b2dfa78a535bca1e5d5cdf71865853e40646fa5904e569f0785239fa56 | Use Gaussian elimination to solve the following system of linear equations: \[ \begin{cases} -x + y + 2z = 0 \\ 3x + 2y + z = 0 \\ 2x + 5y + 3z = 0 \end{cases} \] | [
"(x, y, z) = (0, 0, 0)"
] | [
"Use Gaussian elimination to solve the following system of linear equations:\n\\[\n\\begin{cases}\n-x + y + 2z = 0 \\\\\n3x + 2y + z = 0 \\\\\n2x + 5y + 3z = 0\n\\end{cases}\n\\]"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-9cc46ef76504fa08840f85c8d9e0e4ab0bddfa5491fc772270ff5fe3a7fd1e85 | Use Green's Theorem to evaluate the line integral $$\oint_{C}(y^2 dx + x dy),$$ where the curve \( C \) has the vector equation \(\alpha(t) = 2(\cos(t))^3 \mathbf{i} + 2(\sin(t))^3 \mathbf{j}\) for \(0 \leq t \leq 2\pi\). | [
"\\(\\frac{3\\pi}{2}\\)"
] | [
"Use Green's Theorem to evaluate the line integral $$\\oint_{C}(y^2 dx + x dy),$$ where the curve \\( C \\) has the vector equation \\(\\alpha(t) = 2(\\cos(t))^3 \\mathbf{i} + 2(\\sin(t))^3 \\mathbf{j}\\) for \\(0 \\leq t \\leq 2\\pi\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-784f42e185c78b97684909d41500de5249152a37d3bd2f84c9db49697d8ac92e | Use Green's Theorem to find the area of the region between the graphs of \( y = x^2 \) and \( y = x^3 \). | [
"\\(\\frac{1}{12}\\)"
] | [
"Use Green's Theorem to find the area of the region between the graphs of \\( y = x^2 \\) and \\( y = x^3 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-2c2444145630c498f21d39277eb67f680d10aed475280f94a71977abc06d1c2c | Use Green's theorem to calculate the outwards flux for the vector field \( \mathbf{F}(x,y) = \arctan\left(\frac{y}{x}\right) \mathbf{i} + \ln(x^2 + y^2) \mathbf{j} \) over the curve \( C \) defined by the boundary of the region in polar coordinates given by \( 1 \leq r \leq 2 \) and \( 0 \leq \theta \leq \pi \). | [
"2"
] | [
"Use Green's theorem to calculate the outwards flux for the vector field \\( \\mathbf{F}(x,y) = \\arctan\\left(\\frac{y}{x}\\right) \\mathbf{i} + \\ln(x^2 + y^2) \\mathbf{j} \\) over the curve \\( C \\) defined by the boundary of the region in polar coordinates given by \\( 1 \\leq r \\leq 2 \\) and \\( 0 \\leq \\t... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-17491d790a6fb6dad780f0b48f63d5255d3e50a98e282fbceb453726e33f9962 | Use Jensen's inequality to maximize the expression \(2\sin(60^\circ - c) + \sin(2c - 60^\circ)\) and find the value of \(c\). | [
"\\( c = 40^\\circ \\)"
] | [
"Use Jensen's inequality to maximize the expression \\(2\\sin(60^\\circ - c) + \\sin(2c - 60^\\circ)\\) and find the value of \\(c\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-b0491af187c87fd35472c2e149fa062d08419507833dd2271740b79f79f3c9b8 | Use Kruskal's algorithm to find a Minimum Cost Spanning Tree (MCST) in the graph G. | [
"\\text{Graph details required}"
] | [
"Use Kruskal's algorithm to find a Minimum Cost Spanning Tree (MCST) in the graph G."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-77b5190ba38f1d9be58c003d8306d091ef942c6351532d66ee187296c4eaa186 | Use Lagrange multipliers to find the dimensions of a rectangular box of maximum volume that can be inscribed (with edges parallel to the coordinate axes) in the ellipsoid \(\left( \frac{x^2}{a^2} \right) + \left( \frac{y^2}{b^2} \right) + \left( \frac{z^2}{c^2} \right) = 1\). | [
"\\( \\frac{a}{\\sqrt{3}}, \\frac{b}{\\sqrt{3}}, \\frac{c}{\\sqrt{3}} \\)"
] | [
"Use Lagrange multipliers to find the dimensions of a rectangular box of maximum volume that can be inscribed (with edges parallel to the coordinate axes) in the ellipsoid \\(\\left( \\frac{x^2}{a^2} \\right) + \\left( \\frac{y^2}{b^2} \\right) + \\left( \\frac{z^2}{c^2} \\right) = 1\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 5
} | 8 | 0 | 0 |
omr-ws-185f430189f609d99f1bffbf27e42e12a45284993b1b6b672a39b97801a8802a | Use Lagrange multipliers to find the maximum and minimum values of \( f(x, y) = \frac{1}{x} + \frac{1}{y} \) subject to the constraint \( \frac{1}{x^2} + \frac{1}{y^2} = 1 \). | [
"Maximum value \\( 2\\sqrt{2} \\), Minimum value \\( -2\\sqrt{2} \\)"
] | [
"Use Lagrange multipliers to find the maximum and minimum values of \\( f(x, y) = \\frac{1}{x} + \\frac{1}{y} \\) subject to the constraint \\( \\frac{1}{x^2} + \\frac{1}{y^2} = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1272afaa246269b00b695f8509b4d541af9f9210a8ea55804b75fb6078b0e136 | Use Lagrange multipliers to find the maximum and minimum values of \( f(x, y, z) = x^2 + 2y^2 + 3z^2 \) subject to the constraints \( x + y + z = 1 \) and \( x - y + 2z = 2 \). | [
"\\dfrac{33}{23}"
] | [
"Use Lagrange multipliers to find the maximum and minimum values of \\( f(x, y, z) = x^2 + 2y^2 + 3z^2 \\) subject to the constraints \\( x + y + z = 1 \\) and \\( x - y + 2z = 2 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-685a50c1b760fd1ce13682f54f1acac2caaa9efe91a202943cc3a496d64dfb3c | Use Lagrange multipliers to find the minimum value of \(a + b + c\) given the constraint \(a^2 + b^2 + c^2 + abc = 4\) where \(a, b, c\) are real numbers. | [
"-6"
] | [
"Use Lagrange multipliers to find the minimum value of \\(a + b + c\\) given the constraint \\(a^2 + b^2 + c^2 + abc = 4\\) where \\(a, b, c\\) are real numbers."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 6,
"tir": 17
} | 60 | 0 | 0 |
omr-ws-b7f4a6396e3f9a58ebb461a4f38e4c40662cd0920d4d105799f7739776334031 | Use Lagrange multipliers to find the point closest to the origin on the intersection of the planes \( x + 2y + z = 5 \) and \( 2x + y - z = 1 \). | [
"\\left( \\dfrac{1}{3}, \\dfrac{5}{3}, \\dfrac{4}{3} \\right)"
] | [
"Use Lagrange multipliers to find the point closest to the origin on the intersection of the planes \\( x + 2y + z = 5 \\) and \\( 2x + y - z = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3307d0602b901862aaae5c667bc0d5475e737d4225c4a21570ffd5e37ec3f4d8 | Use Lagrange multipliers to identify the critical points of \( f(x, y, z) = x + y + z \) subject to the constraints \( x^2 - z^2 = 4 \) and \( z + 2y = 1 \). | [
"\\left( -\\frac{4\\sqrt{3}}{3}, \\frac{3 - 2\\sqrt{3}}{6}, \\frac{2\\sqrt{3}}{3} \\right)"
] | [
"Use Lagrange multipliers to identify the critical points of \\( f(x, y, z) = x + y + z \\) subject to the constraints \\( x^2 - z^2 = 4 \\) and \\( z + 2y = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-736439f8b68e7e2d62bd580d5db2a24d9ab1bbc680c0f1063827782a5b7940b4 | Use Lagrange's error bound to find the maximum error of the approximation $\ln(1+x) \approx x - \frac{x^2}{2}$ when $|x| \leq 0.1$. | [
"0.000457"
] | [
"Use Lagrange's error bound to find the maximum error of the approximation $\\ln(1+x) \\approx x - \\frac{x^2}{2}$ when $|x| \\leq 0.1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 14
} | 24 | 0 | 0 |
omr-ws-aaa7b681d5fccb091709bc8d8e02a1f545404f520f91c18512f9bb6d1150a17c | Use Laplace transforms to solve the differential equation $y'''(t) + 8y(t) = 32t^3 - 16t + 1$ with initial conditions $y(0) = y'(0) = y''(0) = 0$. | [
"\\( y(t) = \\frac{9}{4}e^{t}\\cos(\\sqrt{3}t) + e^{t}\\sin(\\sqrt{3}t) + \\frac{5}{8}e^{-2t} + 4t^{3} - 2t - \\frac{23}{8} \\)"
] | [
"Use Laplace transforms to solve the differential equation $y'''(t) + 8y(t) = 32t^3 - 16t + 1$ with initial conditions $y(0) = y'(0) = y''(0) = 0$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 18
} | 35 | 0 | 0 |
omr-ws-bdb2f1e944244702feaab53cd9e9e7b4f38727dba262c1aa64c579a91a27e76b | Use Mackey's irreducibility criterion for semidirect products \( G = A \rtimes H \) (\( A \) abelian and normal) to find all irreducible representations of \( G = \left\{ \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \mid a \in F_{p}^{*}, b \in F_{p} \right\} \), where \( F_p \) is the finite field with \( p \) elements... | [
" \\text{There are } p \\text{ irreducible representations: } p-1 \\text{ of degree 1 and 1 of degree } p-1 "
] | [
"Use Mackey's irreducibility criterion for semidirect products \\( G = A \\rtimes H \\) (\\( A \\) abelian and normal) to find all irreducible representations of \\( G = \\left\\{ \\begin{pmatrix} a & b \\\\ 0 & 1 \\end{pmatrix} \\mid a \\in F_{p}^{*}, b \\in F_{p} \\right\\} \\), where \\( F_p \\) is the finite fi... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-9a45f66b9b4e5322c1bbc2171ca8f46cea289f58f19916b22ff7c2be59942ecf | Use Mackey's irreducibility criterion for semidirect products \( G = A \rtimes H \) (\( A \) abelian and normal) to find all irreducible representations of \( G = \left\{ \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \mid a \in \mathbb{F}_{p}^{*}, b \in \mathbb{F}_{p} \right\} \), where \( \mathbb{F}_p \) is the finite ... | [
" \\text{There are } p-1 \\text{ one-dimensional representations and one } (p-1)\\text{-dimensional irreducible representation.} "
] | [
"Use Mackey's irreducibility criterion for semidirect products \\( G = A \\rtimes H \\) (\\( A \\) abelian and normal) to find all irreducible representations of \\( G = \\left\\{ \\begin{pmatrix} a & b \\\\ 0 & 1 \\end{pmatrix} \\mid a \\in \\mathbb{F}_{p}^{*}, b \\in \\mathbb{F}_{p} \\right\\} \\), where \\( \\ma... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f205062deeb1384fd1ec8fe7f04553c007969ccb57ad980cc3ca86a112b7d2df | Use Maclaurin's theorem with the remainder to expand $\ln(1+x)$. | [
"\\(\\sum_{k=0}^{\\infty} (-1)^{k} \\frac{x^{k+1}}{k+1}\\)"
] | [
"Use Maclaurin's theorem with the remainder to expand $\\ln(1+x)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 3
} | 8 | 0 | 0 |
omr-ws-72bff8a3fa4bb486d37ef481a610caf30150010d716c3275dd4db0a95aff92c1 | Use Newton's Method to find the root of the function $f(x) = x^3 - 7x^2 + 8x - 3$ with the initial guess $x_0 = 5$. | [
"\\( 5.685779596779870267465811683852026406949078859865979427099129309657045987820655044891898594792426507 \\)"
] | [
"Use Newton's Method to find the root of the function $f(x) = x^3 - 7x^2 + 8x - 3$ with the initial guess $x_0 = 5$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-0601342981347ea8cd9e14792fd7c28e36dcbebd89a429060fc592610c014c36 | Use Newton's method to estimate all real solutions of the equation \(x^3 + x - 1 = 0\) accurate to 6 decimal places. Determine an appropriate initial value \(x_1\) to start the iteration. | [
"\\( x_1 = \\frac{1}{2} \\)"
] | [
"Use Newton's method to estimate all real solutions of the equation \\(x^3 + x - 1 = 0\\) accurate to 6 decimal places. Determine an appropriate initial value \\(x_1\\) to start the iteration."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-07bea5aed6d3bf856298704496f98e815e661bf95ba32c8c922443e30e9ac763 | Use Newton's method to find all solutions of the equation $\frac{x}{x^2+1} = \sqrt{2-x}$ correct to eight decimal places. | [
"1.82210736"
] | [
"Use Newton's method to find all solutions of the equation $\\frac{x}{x^2+1} = \\sqrt{2-x}$ correct to eight decimal places."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-6b879941aed16ee0c31b908aad432b54aed9c334e886d86511843653689def6e | Use Newton's method to find all solutions of the equation $x^5 - 5x^4 + x^3 - x^2 - x + 8 = 0$ correct to eight decimal places. | [
"4.82999050"
] | [
"Use Newton's method to find all solutions of the equation $x^5 - 5x^4 + x^3 - x^2 - x + 8 = 0$ correct to eight decimal places."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-264c6fb5fe9a161ec081495cdd195d3afc0c92dbd34de6c0db1589d1fafa260c | Use Newton's method to find as many roots as possible of the nonlinear system: \[ x + y + z = 6 \] \[ x^2 + y^2 + z^2 = 14 \] \[ x^3 + y^3 + z^3 = 37 \] At each iteration step, solve the equation \( Jd = -f \) where \( d = x^{(k+1)} - x^{(k)} \) in MATLAB. | [
"(1.258, 1.605, 3.137)"
] | [
"Use Newton's method to find as many roots as possible of the nonlinear system:\n\\[ x + y + z = 6 \\]\n\\[ x^2 + y^2 + z^2 = 14 \\]\n\\[ x^3 + y^3 + z^3 = 37 \\]\nAt each iteration step, solve the equation \\( Jd = -f \\) where \\( d = x^{(k+1)} - x^{(k)} \\) in MATLAB."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-93cca14658d8a2e42934dbe43951a8b09fc185f6006e05c81bb4c8b61dce06e7 | Use Newton's method to find the two real solutions of the equation \( x^4 - 2x^3 - x^2 - 2x + 2 = 0 \). | [
"2.573"
] | [
"Use Newton's method to find the two real solutions of the equation \\( x^4 - 2x^3 - x^2 - 2x + 2 = 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-02cc0daba58d7ae752b04bbb0d5545f7fcb740ff42b73ead96206744fd152ab6 | Use Newton's method with the initial approximation $x_1 = 2$ to find $x_3$, the third approximation to the solution of the equation $\frac{2}{x} - x^2 + 1 = 0$. Round your answer to four decimal places. | [
"1.5214"
] | [
"Use Newton's method with the initial approximation $x_1 = 2$ to find $x_3$, the third approximation to the solution of the equation $\\frac{2}{x} - x^2 + 1 = 0$. Round your answer to four decimal places."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-969bcba932a077205f1f6e436e8df802ce92df68dff211fe8d770509b1a370db | Use Parseval's identity to find the value of $\int_{-\infty}^\infty \left(\frac{\sin(x)}{x}\right)^4 \, dx$ for the function $f(t) = (1 - |t|)_{\chi[-1,1]}$. | [
"\\(\\frac{2}{\\pi^2}\\)"
] | [
"Use Parseval's identity to find the value of $\\int_{-\\infty}^\\infty \\left(\\frac{\\sin(x)}{x}\\right)^4 \\, dx$ for the function $f(t) = (1 - |t|)_{\\chi[-1,1]}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b1507fee28372230646fd8959b401950a38281e12078c9b1e3810a486677986 | Use Rodrigues' formula to find the Legendre polynomials \( P_6(x) \), \( P_7(x) \), and \( P_8(x) \). Rodrigues' formula is given by: \[ P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n \] | [
"P_8(x) = \\frac{1}{128} \\left(6435x^8 - 12012x^6 + 6930x^4 - 1260x^2 + 35\\right)"
] | [
"Use Rodrigues' formula to find the Legendre polynomials \\( P_6(x) \\), \\( P_7(x) \\), and \\( P_8(x) \\).\n\nRodrigues' formula is given by:\n\\[ P_n(x) = \\frac{1}{2^n n!} \\frac{d^n}{dx^n} (x^2 - 1)^n \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a02236801aba2438dabbc66405e4904ce23e603e3cf3cc8f0d614ee2f8aa546d | Use Sarrus' expansion to simplify the determinant \[ \left|\begin{array}{ccc}ax-by-cz&ay+bx&cx+az\\ay+bx&by-cz-ax&bz+cy\\cx+az&bz+cy&cz-ax-by\end{array}\right|. \] | [
"\\((a^{2}+b^{2}+c^{2})(x^{2}+y^{2}+z^{2})(ax+by+cz)\\)"
] | [
"Use Sarrus' expansion to simplify the determinant \\[ \\left|\\begin{array}{ccc}ax-by-cz&ay+bx&cx+az\\\\ay+bx&by-cz-ax&bz+cy\\\\cx+az&bz+cy&cz-ax-by\\end{array}\\right|. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 13
} | 25 | 0 | 0 |
omr-ws-00a64c3ac3af9711978f1274099af3f58717bbc6f914403a1e487b92bc89bade | Use Simpson's rule with \( n = 4 \) subintervals to estimate the work done, \( w \), by a gas expanding from 2 cm\(^3\) to 10 cm\(^3\), given by the integral \( w = \int_{2}^{10} 500V^{-1.4} \, dV \). | [
"727.59 cm\\(^3\\)"
] | [
"Use Simpson's rule with \\( n = 4 \\) subintervals to estimate the work done, \\( w \\), by a gas expanding from 2 cm\\(^3\\) to 10 cm\\(^3\\), given by the integral \\( w = \\int_{2}^{10} 500V^{-1.4} \\, dV \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dfd9456a6551a090aed64e1bc311efdad4cf983194a9eca206bfcf5a53cc34c4 | Use Stirling's approximation to simplify \((2n-1)!! = \frac{(2n)!}{2^n n!}\) and calculate \(\sqrt[n+1]{(2n+1)!!}\). | [
"\\(\\frac{2n}{e}\\)"
] | [
"Use Stirling's approximation to simplify \\((2n-1)!! = \\frac{(2n)!}{2^n n!}\\) and calculate \\(\\sqrt[n+1]{(2n+1)!!}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 5,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-f1804ab3b4b527e84e03ff3850033878826e0ef2563d9da4d5632332dd129a14 | Use Stokes' Theorem to evaluate the double integral over S of the curl of vector F dot dvectorS, where vector F(x,y,z) = <e^(xy)cos(z), x^2z, xy> and S is the hemisphere x = (1 - y^2 - z^2)^(1/2), oriented in the direction of the positive x-axis. The curl of vector F is given by <x - x^2, -y - e^(xy)sin(z), xe^y cos(z)... | [
"0"
] | [
"Use Stokes' Theorem to evaluate the double integral over S of the curl of vector F dot dvectorS, where vector F(x,y,z) = <e^(xy)cos(z), x^2z, xy> and S is the hemisphere x = (1 - y^2 - z^2)^(1/2), oriented in the direction of the positive x-axis. The curl of vector F is given by <x - x^2, -y - e^(xy)sin(z), xe^y c... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-751d7b9bbad5a0bec60e1fc435b1e877e37ab289c8ab7537fecadb92e4bf5fc3 | Use Stokes' Theorem to evaluate the flux of the curl of the vector field \( F = 2z\textbf{i} + 3x\textbf{j} + 5y\textbf{k} \) across the surface \( S \) given by \( \textbf{r}(r, \theta) = (r \cos \theta)\textbf{i} + (r \sin \theta)\textbf{j} + (4 - r^2)\textbf{k} \) for \( 0 \leq r \leq 2 \) and \( 0 \leq \theta \leq ... | [
"12\\pi"
] | [
"Use Stokes' Theorem to evaluate the flux of the curl of the vector field \\( F = 2z\\textbf{i} + 3x\\textbf{j} + 5y\\textbf{k} \\) across the surface \\( S \\) given by \\( \\textbf{r}(r, \\theta) = (r \\cos \\theta)\\textbf{i} + (r \\sin \\theta)\\textbf{j} + (4 - r^2)\\textbf{k} \\) for \\( 0 \\leq r \\leq 2 \\)... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-d78dc93b42ce5eecc575c8859e626e36f411e65f6dd152847cd90c29d912b6ad | Use Stokes' theorem to evaluate the line integral $\left|\oint _L(y^2-z)dx+(z-x^2)dy+(x+2y)dz\right|$, where $L$ is the curve defined by $\frac{x^2}{4}+y^2=1$ and $x+y+z=2$. | [
"\\(\\frac{188}{15}\\)"
] | [
"Use Stokes' theorem to evaluate the line integral $\\left|\\oint _L(y^2-z)dx+(z-x^2)dy+(x+2y)dz\\right|$, where $L$ is the curve defined by $\\frac{x^2}{4}+y^2=1$ and $x+y+z=2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e9791e3de6564bf8195eef391f37b7d11b9144a77fdfee9b52d473e851ac3ea | Use Stokes' theorem to evaluate the line integral of the vector field \( F(x,y,z) = (x + y^2) \mathbf{i} + (y + z^2) \mathbf{j} + (z + x^2) \mathbf{k} \) over the curve \( C \), which is the boundary of the triangle with vertices (1,0,0), (0,1,0), and (0,0,1). | [
"-1"
] | [
"Use Stokes' theorem to evaluate the line integral of the vector field \\( F(x,y,z) = (x + y^2) \\mathbf{i} + (y + z^2) \\mathbf{j} + (z + x^2) \\mathbf{k} \\) over the curve \\( C \\), which is the boundary of the triangle with vertices (1,0,0), (0,1,0), and (0,0,1)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-546947e4bf2744ddc4780eebb6137779c8202c616dfdf76fd55cda2af433f7dc | Use Taylor approximation to estimate \(\sin(0.3)\) with an error less than 0.01. Determine if 0.3 is in degrees or radians. | [
"0.3"
] | [
"Use Taylor approximation to estimate \\(\\sin(0.3)\\) with an error less than 0.01. Determine if 0.3 is in degrees or radians."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-c8cb850ff42a6abbed26475853cd6c4751e23b4b9ab90aa4763d28ec79677c2f | Use Taylor expansions to find a cubic polynomial that approximates $\sec\left(\frac{\pi}{3} + x\right)$ when $x$ is small. | [
"\\(2 + 2\\sqrt{3}x + 7x^2 + \\frac{23\\sqrt{3}}{3}x^3\\)"
] | [
"Use Taylor expansions to find a cubic polynomial that approximates $\\sec\\left(\\frac{\\pi}{3} + x\\right)$ when $x$ is small."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-3b85e513b1e75e2ff3c8e208c21ba078cf780d7c963fce30ae356f117fddcf43 | Use Taylor expansions to find a cubic polynomial that approximates $\sin^{-1}\left(\frac{1}{2} + x\right)$ when $x$ is small. | [
"\\(\\frac{\\pi}{6} + \\frac{2\\sqrt{3}}{3}x + \\frac{2\\sqrt{3}}{9}x^2 + \\frac{16\\sqrt{3}}{81}x^3\\)"
] | [
"Use Taylor expansions to find a cubic polynomial that approximates $\\sin^{-1}\\left(\\frac{1}{2} + x\\right)$ when $x$ is small."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2d138cb656abdb53c60ad523dc9f188bccc6dc7cc4527b54057d2481cf04a0b7 | Use Taylor expansions to find a cubic polynomial that approximates $\sqrt{9 + x}$ when $x$ is small. | [
"\\(3 + \\frac{27}{2}x + \\frac{729}{8}x^2 + \\frac{10935}{16}x^3\\)"
] | [
"Use Taylor expansions to find a cubic polynomial that approximates $\\sqrt{9 + x}$ when $x$ is small."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f3234c3637c712895b16d9f1e0bf3336eefc3367b52ad6345a16d25956abfe0 | Use Taylor expansions to find a cubic polynomial that approximates $\tan\left(\frac{\pi}{4} + x\right)$ when $x$ is small. | [
"\\(1 + 2x + 2x^2 + \\frac{8}{3}x^3\\)"
] | [
"Use Taylor expansions to find a cubic polynomial that approximates $\\tan\\left(\\frac{\\pi}{4} + x\\right)$ when $x$ is small."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-ea6974e84ec4cf55d8e2f7f6d84dfd4715ac53c48b68af38896d1b918dbfb580 | Use Taylor's Theorem to determine how many terms of the series for a given function are required to approximate the function to within five decimal places of precision (i.e., an error of less than \(10^{-5}\)). | [
"9"
] | [
"Use Taylor's Theorem to determine how many terms of the series for a given function are required to approximate the function to within five decimal places of precision (i.e., an error of less than \\(10^{-5}\\))."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cdc9b43066afaa539100a31d43287e643edcad63e25f85dfea65b7f342fb6b56 | Use Taylor's theorem to determine the minimum degree of a Maclaurin polynomial that will approximate $\cos(2)$ to within $0.0001$ of its true value. | [
"10"
] | [
"Use Taylor's theorem to determine the minimum degree of a Maclaurin polynomial that will approximate $\\cos(2)$ to within $0.0001$ of its true value."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c851c49fb53f7e26ffb4bf140b5c5dde0480a33c315115f70eedf98cf5cdaab8 | Use \( \frac{d}{dx} \ln (2x + \sqrt{4x^2 + 1}) \) and \( \frac{d}{dx}(x \sqrt{4x^2 + 1}) \) to evaluate \( \int_0^1 \sqrt{4x^2 + 1} \, dx \). | [
"\\(\\frac{\\sqrt{5}}{2} + \\frac{1}{4} \\ln (2 + \\sqrt{5})\\)"
] | [
"Use \\( \\frac{d}{dx} \\ln (2x + \\sqrt{4x^2 + 1}) \\) and \\( \\frac{d}{dx}(x \\sqrt{4x^2 + 1}) \\) to evaluate \\( \\int_0^1 \\sqrt{4x^2 + 1} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-b1b8773b2c38adeed42c6915472bdb3c7bb2a3bdf0d80b07670b4863f9eceb18 | Use a Maclaurin Series to approximate the integral $\int_{0}^{0.2} \sqrt[3]{1+x^4} \ \mathrm{d}x$ to three decimal-place accuracy. | [
"0.200"
] | [
"Use a Maclaurin Series to approximate the integral $\\int_{0}^{0.2} \\sqrt[3]{1+x^4} \\ \\mathrm{d}x$ to three decimal-place accuracy."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-b9866594c5d5689236303251f51eaeb2189c5d6ba78bab86002a5d04465241c3 | Use a Maclaurin polynomial for the function \( f(x) = \arccos(0.5 + x) \) to estimate \( \arccos(0.52) \) correct to 6 decimal places. | [
"1.023945"
] | [
"Use a Maclaurin polynomial for the function \\( f(x) = \\arccos(0.5 + x) \\) to estimate \\( \\arccos(0.52) \\) correct to 6 decimal places."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-a37401513459434b4f2dfebd0f38c3e2ba7f1b1c03413a4b7b5d57a51e991482 | Use a Taylor Polynomial for \(\sin(x)\) and apply Lagrange's form of the remainder to approximate \(\sin\left(\frac{17\pi}{36}\right)\) to 4 decimal place accuracy. | [
"0.9962"
] | [
"Use a Taylor Polynomial for \\(\\sin(x)\\) and apply Lagrange's form of the remainder to approximate \\(\\sin\\left(\\frac{17\\pi}{36}\\right)\\) to 4 decimal place accuracy."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bf56d3ac9a00f7203c31c8011bb4dea8314da0dcd7751f695fec70cd7cab788f | Use a calculator to approximate two values of \(\theta\) (where \(0 \leq \theta < 2\pi\)) that satisfy the equation \(\cos \theta = 0.9848\). Round to three decimal places. | [
"\\(0.175\\) and \\(6.108\\)"
] | [
"Use a calculator to approximate two values of \\(\\theta\\) (where \\(0 \\leq \\theta < 2\\pi\\)) that satisfy the equation \\(\\cos \\theta = 0.9848\\). Round to three decimal places."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 30
} | 30 | 0 | 0 |
omr-ws-5180baaf6080554d4e8eaaa772009467876466612604bf056c143a93afd2487a | Use a compass and a ruler to construct a triangle given the intersection point of its medians, the orthocenter, and one of the vertices. | [
"\\triangle ABC"
] | [
"Use a compass and a ruler to construct a triangle given the intersection point of its medians, the orthocenter, and one of the vertices."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-39d6621aafdbad4630841f6bbf6e1cf7dab7e1ccb0939add109c1d0fcdc9bb78 | Use a compass and a ruler to reconstruct triangle $ABC$ given its vertex $A$, the midpoint of the side $BC$, and the foot of the perpendicular drawn from point $B$ to the bisector of the angle $\angle BAC$. | [
"\\triangle ABC"
] | [
"Use a compass and a ruler to reconstruct triangle $ABC$ given its vertex $A$, the midpoint of the side $BC$, and the foot of the perpendicular drawn from point $B$ to the bisector of the angle $\\angle BAC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-288a284ba413301c12a91af3a0b1afaaed4c4a2da194ede7fb6def3cc13dfeaf | Use a compass and straightedge to construct five circles of equal radius, the centers of which lie on the diagonals of a given square, such that four circles touch two sides of the square and the fifth circle touches the other four circles. | [
"\\frac{\\sqrt{2} - 1}{2} \\times \\text{side length of the square}"
] | [
"Use a compass and straightedge to construct five circles of equal radius, the centers of which lie on the diagonals of a given square, such that four circles touch two sides of the square and the fifth circle touches the other four circles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-a9ef9f070f92af2175d3d66a4c78e5d864c2b0175e0c457d278cdb721dd6006e | Use a compass and straightedge to draw a straight line that divides the area of an arbitrary mushroom-gon (a convex figure consisting of two segments and an arc of a circle) in half. | [
"\\text{Draw a line from the midpoint of the circular arc to the midpoint of the line segment connecting the endpoints of the arc.}"
] | [
"Use a compass and straightedge to draw a straight line that divides the area of an arbitrary mushroom-gon (a convex figure consisting of two segments and an arc of a circle) in half."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-720629811a303b33dfcea2348abed3d6fc957b4b54b7766cdb45cd5263255645 | Use a double integral in polar coordinates to find the volume of the solid enclosed by the paraboloid \( z = 4 - x^2 - y^2 \) and \( z = 0 \). | [
"\\( 8\\pi \\)"
] | [
"Use a double integral in polar coordinates to find the volume of the solid enclosed by the paraboloid \\( z = 4 - x^2 - y^2 \\) and \\( z = 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-9689b20a9692fd84d816bfca419c51eb7472bde2e6a33b9fcb76352c228d0561 | Use a graphing utility to graph the function \( f(x) = x^4 - 4x^2 + k \) for different values of \( k \). Find values of \( k \) such that the zeros of \( f \) satisfy the specified characteristics. a) Four real zeros b) Two real zeros each of multiplicity 2 c) Two real zeros and two complex roots d) Four complex zeros | [
"a) \\( 0 \\leq k \\leq 4 \\), b) \\( k = 4 \\), c) \\( k < 0 \\), d) \\( k > 4 \\)"
] | [
"Use a graphing utility to graph the function \\( f(x) = x^4 - 4x^2 + k \\) for different values of \\( k \\). Find values of \\( k \\) such that the zeros of \\( f \\) satisfy the specified characteristics.\na) Four real zeros\nb) Two real zeros each of multiplicity 2\nc) Two real zeros and two complex roots\nd) F... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-ef012ccd789e23f7e3f3b7549530a181b2d2372b1de7d4070311cab6f6d295bb | Use a line integral to find the area bounded by the curves $y=\frac{x^3}{8}$ and $y=4\sqrt{x}$. | [
"\\dfrac{40}{3}"
] | [
"Use a line integral to find the area bounded by the curves $y=\\frac{x^3}{8}$ and $y=4\\sqrt{x}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0086dd377525dd7f75575972441d1c2e1d5afe85ca79b5329c38189560d75b70 | Use a linear approximation to estimate the value of \(\log(998)\). What is the (approximate) error in your estimate? | [
"\\(\\frac{1}{500 \\ln(10)} \\approx 0.0009\\)"
] | [
"Use a linear approximation to estimate the value of \\(\\log(998)\\). What is the (approximate) error in your estimate?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a405017ad810c59e4a4c45decff64018d5d764ad35ce1f5c4abd66d8fe078933 | Use a recursive definition to solve the problem of finding the number of ways to tile a \(2 \times n\) rectangle with \(1 \times 2\) and \(2 \times 1\) tiles. | [
"F_{n+1}"
] | [
"Use a recursive definition to solve the problem of finding the number of ways to tile a \\(2 \\times n\\) rectangle with \\(1 \\times 2\\) and \\(2 \\times 1\\) tiles."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-8561835765154e8b4f2727dc0cfa6f632161c83eadd994822cea8bd2a03de466 | Use a similar substitution to find an expression for \[ \int \frac{\cosh x}{\cosh 2x} \ dx. \] | [
"\\(\\frac{1}{\\sqrt{2}} \\arctan(\\sqrt{2} \\sinh x) + C\\)"
] | [
"Use a similar substitution to find an expression for\n\n\\[\n\\int \\frac{\\cosh x}{\\cosh 2x} \\ dx.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-fbd037f9ab35876744621593209263da13370b73356249cc3c7a0906f151c2dd | Use a single equation to denote that at least two of the numbers $a, b, c$ are equal to zero. | [
"$|ab|+|bc|+|ca|=0$"
] | [
"Use a single equation to denote that at least two of the numbers $a, b, c$ are equal to zero. "
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f811b0f20ac31c4b2fb67b3b832b26515997d3c19f5d8010557a3c0f57b278a0 | Use a suitable substitution to solve the differential equation $y(xy+1)+x(1+xy+x^2y^2)y'=0$. | [
"\\(x^2y^2(A+2\\ln y)=2xy+1\\)"
] | [
"Use a suitable substitution to solve the differential equation $y(xy+1)+x(1+xy+x^2y^2)y'=0$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9bb8e20e9ddaa8a037f06a87a6797618fe753608c49e3dc2928bddcd5d3ae500 | Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first four hours the tickets were on sale. | [
"155.25"
] | [
"Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first four hours the tickets were on sale."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-51d11b0b2c14497fb322044f5ce6a44ac529882341f7fe60127a700e7e436e69 | Use a triple integral to define the volume under the surface \( z = 8 - 2x^2 - 2y^2 \) and over the \( xy \)-plane. | [
"\\( 16\\pi \\)"
] | [
"Use a triple integral to define the volume under the surface \\( z = 8 - 2x^2 - 2y^2 \\) and over the \\( xy \\)-plane."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-7af350c4bbbd9830ccfb1b3a161610c465f055c2ef520bd0bcf8ad4029ebae70 | Use a triple integral to find the volume of the solid bounded by the cylinder \(x = y^2\) and the planes \(z = 0\) and \(x + z = 1\). | [
"\\(\\frac{8}{15}\\)"
] | [
"Use a triple integral to find the volume of the solid bounded by the cylinder \\(x = y^2\\) and the planes \\(z = 0\\) and \\(x + z = 1\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3e4bef44687a676f9e643771fe008cc7ff3c38684c268f663197a524937df42c | Use a triple integral to find the volume of the solid bounded by the elliptic cylinder $4x^2 + z^2 = 4$ and the planes $y = 0$ and $y = z + 2$. | [
"\\(4\\pi\\)"
] | [
"Use a triple integral to find the volume of the solid bounded by the elliptic cylinder $4x^2 + z^2 = 4$ and the planes $y = 0$ and $y = z + 2$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-fde32d259f7475715a1009a4506e7460d104475d272922a7820d02ce1ed2b15a | Use a triple integral to find the volume of the solid enclosed between the surfaces $z=3x^2+y^2$ and $z=16-x^2-y^2$. | [
"\\(32\\pi\\sqrt{2}\\)"
] | [
"Use a triple integral to find the volume of the solid enclosed between the surfaces $z=3x^2+y^2$ and $z=16-x^2-y^2$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-a04dae131352e755bf9f2789267572bef2660dcf22e537da37299c39d3114767 | Use algebra to find each point at which the line \( x - 2y = -5 \) intersects the circle \( x^2 + y^2 = 20 \). | [
"\\((2-\\sqrt{3}, -1-2\\sqrt{3})\\) and \\((2+\\sqrt{3}, -1+2\\sqrt{3})\\)"
] | [
"Use algebra to find each point at which the line \\( x - 2y = -5 \\) intersects the circle \\( x^2 + y^2 = 20 \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-033d73b1979fcfd1bdee444eabe1acb74a8be3100db102185b8d8fc872289a0d | Use all 10 digits 0-9 once each to make a 10-digit number that is divisible by 11. | [
"1234756809"
] | [
"Use all 10 digits 0-9 once each to make a 10-digit number that is divisible by 11."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-25a04d45461a2b0d714c660f4ab9932baad5534b743c5b37119fa1cb65302a90 | Use an appropriate series to evaluate $\int_0^{0.1} x^2e^{-x^2}dx$ correct to four decimal places. | [
"-0.0002"
] | [
"Use an appropriate series to evaluate $\\int_0^{0.1} x^2e^{-x^2}dx$ correct to four decimal places."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7082e68b89a22ad017af79b6ccfc652ccba29a50b246d56c007de8a45ef639fe | Use an identity of the form \( \mathsf{PAP}^{T} = \mathsf{D} \) where \( \mathsf{D} \) is a diagonal matrix, to find \( \mathsf{A}^{-1} \). | [
"\\mathsf{P}^{\\top} \\mathsf{D}^{-1} \\mathsf{P}"
] | [
"Use an identity of the form \\( \\mathsf{PAP}^{T} = \\mathsf{D} \\) where \\( \\mathsf{D} \\) is a diagonal matrix, to find \\( \\mathsf{A}^{-1} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f0b201c4be6e8f914dcb729199ec3f094c02b51505100712a89d8b6daaa1aebd | Use both cylindrical and spherical coordinates to find the volume of the solid \( E \) that lies above the cone \( z = \sqrt{x^2 + y^2} \) and below the sphere \( x^2 + y^2 + z^2 = 16 \). | [
"\\(\\frac{128\\pi - 32\\pi\\sqrt{2}}{3}\\)"
] | [
"Use both cylindrical and spherical coordinates to find the volume of the solid \\( E \\) that lies above the cone \\( z = \\sqrt{x^2 + y^2} \\) and below the sphere \\( x^2 + y^2 + z^2 = 16 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 4
} | 16 | 0 | 0 |
omr-ws-92bad92aeccd13154fe21331aca19c1f6fac850f374b09e989315a5bccaa3bb9 | Use calculus to analyze the behavior of the function \( y = e^x \sin x \) for \( -\infty < x < +\infty \) and sketch its graph for \( -2\pi \leq x \leq 2\pi \). | [
"y = e^x \\sin x"
] | [
"Use calculus to analyze the behavior of the function \\( y = e^x \\sin x \\) for \\( -\\infty < x < +\\infty \\) and sketch its graph for \\( -2\\pi \\leq x \\leq 2\\pi \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-08bc386223f29b7753bf7725102e7fa164d811efdc87c9a2c082d7d40494f790 | Use calculus to find the behavior of the function \( y = e^x \sin{x} \) for \( -\infty < x < +\infty \). Sketch the graph of the function for \( -2\pi \le x \le 2\pi \) and clearly indicate the locations of the maxima, minima, and points of inflection. | [
"Maxima at \\( x = \\frac{3\\pi}{4} \\), Minima at \\( x = \\frac{7\\pi}{4} \\), Points of inflection at \\( x = -\\frac{3\\pi}{2}, -\\frac{\\pi}{2}, \\frac{\\pi}{2}, \\frac{3\\pi}{2} \\)"
] | [
"Use calculus to find the behavior of the function \\( y = e^x \\sin{x} \\) for \\( -\\infty < x < +\\infty \\). Sketch the graph of the function for \\( -2\\pi \\le x \\le 2\\pi \\) and clearly indicate the locations of the maxima, minima, and points of inflection."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 8
} | 9 | 0 | 0 |
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