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-cbe7d8524e81af015254cfd0d8ae62722b467c5197a1e386d994d1103c4a4fc7 | Use calculus to find the behavior of the function \[y = e^x \sin{x} \ \ \ \ \ \ \ -\infty < x < +\infty\] and sketch the graph of the function for $-2\pi \le x \le 2\pi$. Show clearly the locations of the maxima, minima, and points of inflection in your graph. | [
"\\text{Graph sketched with maxima, minima, and inflection points at specified } x \\text{-values.}"
] | [
"Use calculus to find the behavior of the function\n\\[y = e^x \\sin{x} \\ \\ \\ \\ \\ \\ \\ -\\infty < x < +\\infty\\]\nand sketch the graph of the function for $-2\\pi \\le x \\le 2\\pi$. Show clearly the locations of the maxima, minima, and points of inflection in your graph."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-e3be1fdf8f1074a1a84d1054bf9f63a2fa2360484bffb57f85c70b42270ed450 | Use calculus to find the net force of the water pressure on a sphere and then use Archimedes' Principle to determine the buoyancy force the sphere receives. (Hint: convert to polar coordinates) | [
"\\rho g V"
] | [
"Use calculus to find the net force of the water pressure on a sphere and then use Archimedes' Principle to determine the buoyancy force the sphere receives. (Hint: convert to polar coordinates)"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-77909ab87c75fb6adcf5a02838b501710c5e4704591bdb73eea134150d1d51ef | Use calculus to find the number of real solutions of the equation \( e^x + x^2 = 2\cos(x) \). | [
"2"
] | [
"Use calculus to find the number of real solutions of the equation \\( e^x + x^2 = 2\\cos(x) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-bfa643b80b08d5a30254e6b5b432b64ec7863834edb18b5b9556eb52e6b93538 | Use complex exponentials to compute the Laplace transform of $\sin(at)$. | [
"\\dfrac{a}{s^2 + a^2}"
] | [
"Use complex exponentials to compute the Laplace transform of $\\sin(at)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4bdd3dc18c660fbc71407c1d86f372b72651d6ec2402763d61ec490c58154c74 | Use complex exponentials to find the Laplace transform of $t \sin (at)$. | [
"\\(\\frac{2sa}{(s^2 + a^2)^2}\\)"
] | [
"Use complex exponentials to find the Laplace transform of $t \\sin (at)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ae4bfc0f229316b2d1b1b803811488a315987b103391c4603c9fa37e5056e962 | Use convolution and Laplace transforms to find the Laplace transform of the solution \( Y(s) = \mathcal{L}\{y(t)\} \) for the initial value problem defined for \( t \geq 0 \): \[ \frac{dy}{dt} + 4 \int_0^t y(t-w)e^{-4w} \, dw = 7, \quad y(0) = 0 \] | [
"\\( \\frac{7(s+4)}{s(s+2)^2} \\)"
] | [
"Use convolution and Laplace transforms to find the Laplace transform of the solution \\( Y(s) = \\mathcal{L}\\{y(t)\\} \\) for the initial value problem defined for \\( t \\geq 0 \\):\n\n\\[\n\\frac{dy}{dt} + 4 \\int_0^t y(t-w)e^{-4w} \\, dw = 7, \\quad y(0) = 0\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-1eb5d630078d970d11b858daae022630cde80860371ebb901b4352cdc769f036 | Use cylindrical coordinates to find the volume of the solid enclosed by the sphere \(x^2 + y^2 + z^2 = 9\), the plane \(z = 0\), and the cylinder \(x^2 + y^2 = 3y\). | [
"\\(9\\pi\\)"
] | [
"Use cylindrical coordinates to find the volume of the solid enclosed by the sphere \\(x^2 + y^2 + z^2 = 9\\), the plane \\(z = 0\\), and the cylinder \\(x^2 + y^2 = 3y\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 4,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-1a97418e6cef4583a2fd58b2c1de3e94ea6673df78d611f79c1d0c5a6829dc3e | Use cylindrical shells to compute the volume of the solid formed by revolving the region bounded by \( y = x \) and \( y = x^2 - 2 \) about the line \( x = 3 \). | [
"\\dfrac{45}{2}\\pi"
] | [
"Use cylindrical shells to compute the volume of the solid formed by revolving the region bounded by \\( y = x \\) and \\( y = x^2 - 2 \\) about the line \\( x = 3 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e5ca8eace3c600a59083d324acc3507436b8b29c0a2d74c4bd43fa6d6d9fbfed | Use cylindrical shells to find the volume of the solid obtained by revolving around the y-axis the region bounded by the curves \( y^2 = 8x \) and \( x = 2 \). | [
"\\(\\frac{64\\pi}{5}\\)"
] | [
"Use cylindrical shells to find the volume of the solid obtained by revolving around the y-axis the region bounded by the curves \\( y^2 = 8x \\) and \\( x = 2 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d37c49ff2fbd1e13fb261a375e4703db1f536fe1929f2f82db17546e4f6efaa8 | Use determinants to determine the number of solutions for the system of linear equations: \[ 3x - y = -6 \] \[ 2x = 3y + 2 \] | [
"1"
] | [
"Use determinants to determine the number of solutions for the system of linear equations:\n\\[ 3x - y = -6 \\]\n\\[ 2x = 3y + 2 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-908701e007006e436f2eb4b2ede171ad19b9aae5b95217887de0d906ca8912f4 | Use determinants to determine the number of solutions for the system: \[ 3x - y = 7 \] \[ 6x - 2y = 10 \] | [
"0"
] | [
"Use determinants to determine the number of solutions for the system:\n\\[ 3x - y = 7 \\]\n\\[ 6x - 2y = 10 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-53d46538a6db85f2010f0768c90eb0afcfb771147e6f8f682e1279b8d294e73b | Use difference quotients with \( \Delta x = 0.1 \) and \( \Delta y = 0.1 \) to estimate \( f_x(1, 6) \) and \( f_y(1, 6) \) for the function \( f(x, y) = e^{-x} \sin(y) \). | [
"0.3578"
] | [
"Use difference quotients with \\( \\Delta x = 0.1 \\) and \\( \\Delta y = 0.1 \\) to estimate \\( f_x(1, 6) \\) and \\( f_y(1, 6) \\) for the function \\( f(x, y) = e^{-x} \\sin(y) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-dd2fb8390c08b2fadb197fdb00e4a40dd3e0d687da333b0e6c2adee25f0eb1f4 | Use differential calculus to calculate the error in the volume of a conical tank with height and volume both equal to 4 cm, given that each measurement has a possible error of 0.03 cm. | [
"0.28"
] | [
"Use differential calculus to calculate the error in the volume of a conical tank with height and volume both equal to 4 cm, given that each measurement has a possible error of 0.03 cm."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-2d5c7f6c1065d2aa34a50d14e1e0d3d056115e9c7357914d27268c2cc3419bc2 | Use differentials (linear approximation) to estimate \(\cos(-2)\). Base the approximation at \(x = \frac{2\pi}{3}\). | [
"\\(-0.41825\\)"
] | [
"Use differentials (linear approximation) to estimate \\(\\cos(-2)\\). Base the approximation at \\(x = \\frac{2\\pi}{3}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 18
} | 31 | 0 | 0 |
omr-ws-fd9e24064a9c0095066fe36be48c843bc881d0a0b03eae779c58e260da0d26ec | Use differentials to estimate the possible resulting error in the calculated work done by a constant force $\vec{F}$ on an object that moves a distance $S$ along a straight line at an angle $\theta$ with the direction of the force. The force $\vec{F}$ is measured to be 10N with a possible error of 0.1N; the distance $S... | [
"\\( 5.05\\sqrt{3} - \\frac{25\\pi}{9} \\)"
] | [
"Use differentials to estimate the possible resulting error in the calculated work done by a constant force $\\vec{F}$ on an object that moves a distance $S$ along a straight line at an angle $\\theta$ with the direction of the force. The force $\\vec{F}$ is measured to be 10N with a possible error of 0.1N; the dis... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7da8fb2850a9d3b32ed076dc2e782d859b092be0410ffb03410742115081ac51 | Use each of the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly twice to form distinct prime numbers whose sum is as small as possible. What must this minimal sum be? | [
"477"
] | [
"Use each of the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly twice to form distinct prime numbers whose sum is as small as possible. What must this minimal sum be?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-aa3cf42b8a47188e2bc21401c2ee6700ba408d85b7aba4847f8a31f4b3228d4a | Use each of the digits 3, 4, 6, 8 and 9 exactly once to create the greatest possible five-digit multiple of 6. What is that multiple of 6? | [
"98,634"
] | [
"Use each of the digits 3, 4, 6, 8 and 9 exactly once to create the greatest possible five-digit multiple of 6. What is that multiple of 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-daa1522df4ce214b00b200867c82b614a070a34c1168c4cf0d98c469683e8e63 | Use each of the five digits $2, 4, 6, 7$ and $9$ only once to form a three-digit integer and a two-digit integer which will be multiplied together. What is the three-digit integer that results in the greatest product? | [
"762"
] | [
"Use each of the five digits $2, 4, 6, 7$ and $9$ only once to form a three-digit integer and a two-digit integer which will be multiplied together. What is the three-digit integer that results in the greatest product?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-119dd2c90dd2b3a9c548ab18398c2028d9afc5ed797af0671b1b1c057ec3778d | Use each of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once to form two different integers such that one integer is twice as big as the other. What are the two numbers? | [
"6729 and 13458"
] | [
"Use each of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once to form two different integers such that one integer is twice as big as the other. What are the two numbers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 31
} | 31 | 0 | 0 |
omr-ws-030d4a7fb6ecd35a85b55891573263917fcc56f0c83c2ee50379e43686ff1686 | Use each of the numbers 1, 3, 4, and 6 exactly once with any of the four basic math operations (+, -, *, /) to total 24. Each number must be used once and only once. | [
"\\(\\frac{6}{1-\\frac{3}{4}} = 24\\)"
] | [
"Use each of the numbers 1, 3, 4, and 6 exactly once with any of the four basic math operations (+, -, *, /) to total 24. Each number must be used once and only once."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-84cc2b06619b7c0a04881c04d21d17492cbb3ac60b06cf103378584de21f997a | Use elementary transformations to the matrix \[ \begin{bmatrix} \lambda - 13 & -16 & -16 \\ 5 & \lambda + 7 & 6 \\ 5 & 8 & \lambda + 7 \end{bmatrix} \] to get its diagonal form. | [
"\\(\\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & \\lambda^3 + \\lambda^2 - 21\\lambda - 13 \\end{bmatrix}\\)"
] | [
"Use elementary transformations to the matrix \n\n\\[ \\begin{bmatrix} \\lambda - 13 & -16 & -16 \\\\ 5 & \\lambda + 7 & 6 \\\\ 5 & 8 & \\lambda + 7 \\end{bmatrix} \\]\n\nto get its diagonal form."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 11
} | 34 | 0 | 0 |
omr-ws-850093a88f2338b6f47cd43794c20dc52afdaba5e422c3065531e516e01d2391 | Use extrapolation to derive an $O(h^3)$ formula for $f'(x_0)$ given the forward difference formula: \[ f'(x_0) = \frac{1}{h} [f(x_0 + h) - f(x_0)] - \frac{h}{2} f''(x_0) - \frac{h^2}{6} f'''(x_0) + O(h^3). \] | [
"\\frac{f(x_0 + h) - 12f(x_0 + \\frac{h}{2}) + 32f(x_0 + \\frac{h}{4}) - 21f(x_0)}{3h}"
] | [
"Use extrapolation to derive an $O(h^3)$ formula for $f'(x_0)$ given the forward difference formula:\n\\[ f'(x_0) = \\frac{1}{h} [f(x_0 + h) - f(x_0)] - \\frac{h}{2} f''(x_0) - \\frac{h^2}{6} f'''(x_0) + O(h^3). \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-64ed8453b0b541f91a6176350f3a44ef1261ad93d59161659be323def8d86c98 | Use four $\frac{1}{4}$s in an expression that evaluates to 4. The expression must be of the form $$\frac14\circ\frac14\circ\frac14\circ\frac14$$ where $\circ$ can represent one of the following operations: multiplication ($\times$), division ($\div$), addition (+), subtraction (-), and exponentiation ($\wedge$). You ma... | [
"\\(\\frac{\\frac{1}{4}^{\\frac{1}{4}-\\frac{1}{4}}}{\\frac{1}{4}}\\)"
] | [
"Use four $\\frac{1}{4}$s in an expression that evaluates to 4. The expression must be of the form $$\\frac14\\circ\\frac14\\circ\\frac14\\circ\\frac14$$ where $\\circ$ can represent one of the following operations: multiplication ($\\times$), division ($\\div$), addition (+), subtraction (-), and exponentiation ($... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 16
} | 30 | 0 | 0 |
omr-ws-ebdfd88e812d79550544c155f3cb09d5bd1e23f9a6c5f61cda8d051b577631f8 | Use generating functions to evaluate the sum \( \sum_k \binom{n+k}{2k}2^{n-k} \). | [
"\\dfrac{2^{2n+1} + 1}{3}"
] | [
"Use generating functions to evaluate the sum \\( \\sum_k \\binom{n+k}{2k}2^{n-k} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-b587cee2a907365ee39392416c51b6f4efd3ce72060eee0a5559a5ecadb6e47e | Use generating functions to find the number of compositions of $n$ into $k$ summands and then sum over $k$ to get $p_n = 2^{n-1}$. | [
"2^{n-1}"
] | [
"Use generating functions to find the number of compositions of $n$ into $k$ summands and then sum over $k$ to get $p_n = 2^{n-1}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-bb91fc64d17c1f0f7381c917555c52a9653a9ff91e924c55cea3e6f151d9245a | Use generating functions to find the number of positive integer solutions to the equation $3x + 4y + 5z = 100$. | [
"74"
] | [
"Use generating functions to find the number of positive integer solutions to the equation $3x + 4y + 5z = 100$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-704cbfe8d3f8c4d9de8df4f4a72bb3a3d81c85f20b1cb6c6259cdcdb974eb45f | Use generating functions to solve the recurrence relation given by \[ \left\{\begin{matrix} a_0=1 \\ a_n + 3a_{n-1} = 4n^2 - 2n + 2^n \end{matrix}\right., \quad n \geq 1 \] | [
"a_n = n(n + 1) + \\frac{2}{5} \\cdot 2^n + \\frac{3}{5} \\cdot (-3)^n"
] | [
"Use generating functions to solve the recurrence relation given by\n\\[\n\\left\\{\\begin{matrix}\na_0=1 \\\\ \na_n + 3a_{n-1} = 4n^2 - 2n + 2^n\n\\end{matrix}\\right., \\quad n \\geq 1\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a0cccd86fe509a99b49c868d42c13158fbd9039f0c66e252ed2b4725e3ed5f8d | Use geometrical methods to locate (or describe) the centroid of the edges of tetrahedron $ABCD$. | [
"\\(\\frac{1}{4} (A + B + C + D)\\)"
] | [
"Use geometrical methods to locate (or describe) the centroid of the edges of tetrahedron $ABCD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 20
} | 39 | 0 | 0 |
omr-ws-d3f21f980833e5186f6b61242f6fc15c72784c7d7d15f046ed0db1a3f4027564 | Use half-angle formulas to find the exact value of $\sec \frac{15\pi}{8}$. | [
"\\(\\frac{2}{\\sqrt{\\sqrt{2} + 2}}\\)"
] | [
"Use half-angle formulas to find the exact value of $\\sec \\frac{15\\pi}{8}$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-4dc17d1d0f010e6ccdfa6d5b8b7500501fda2057f88cb360856693739cde6780 | Use half-angle identities to evaluate the integral $\int \cos^2(x) \, dx$. | [
"\\frac{1}{2}x + \\frac{1}{4}\\sin(2x) + C"
] | [
"Use half-angle identities to evaluate the integral $\\int \\cos^2(x) \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e71118861c3f3987f749dcb70f3d5678e88ed5e28299ca863964995e9bfd9f3 | Use implicit differentiation to find \( \frac{dy}{dx} \) if \( x^y = y^x \). | [
"\\( \\frac{y(y - x\\ln(y))}{x(x - y\\ln(x))} \\)"
] | [
"Use implicit differentiation to find \\( \\frac{dy}{dx} \\) if \\( x^y = y^x \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 21
} | 35 | 0 | 0 |
omr-ws-30a4e3b3864ea83832a1039dc0de6e0f42f3d6e9f07e68f043ba3e82327fbf7a | Use implicit differentiation to find an equation of the tangent line to the curve $x^2 - xy^3 + y^5 = 81$ at the point where $x = 9$. | [
"\\( x = 9 \\), \\( y = \\frac{1}{18}x + \\frac{15}{6} \\), \\( y = -\\frac{5}{18}x + \\frac{15}{6} \\)"
] | [
"Use implicit differentiation to find an equation of the tangent line to the curve $x^2 - xy^3 + y^5 = 81$ at the point where $x = 9$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 8
} | 10 | 0 | 0 |
omr-ws-f21e8f70c598961e17f962fe5c86cb7f41717d38e5c72405440abdbf69d9fa15 | Use implicit differentiation to find an equation of the tangent line to the curve \( y \sin(8x) = x \cos(2y) \) at the point \( \left( \frac{\pi}{2}, \frac{\pi}{4} \right) \). | [
"\\( y = -2x + \\frac{5 \\pi}{4} \\)"
] | [
"Use implicit differentiation to find an equation of the tangent line to the curve \\( y \\sin(8x) = x \\cos(2y) \\) at the point \\( \\left( \\frac{\\pi}{2}, \\frac{\\pi}{4} \\right) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-00ba619875c8863b569f356bd148ccf4e77a5fd484329c497dac8494073941ae | Use implicit differentiation to find an equation of the tangent line to the curve \((x^2 + y^2)^3 = 8r^2 y^2\) at the point \((1, -1)\). | [
"y = 3x - 4"
] | [
"Use implicit differentiation to find an equation of the tangent line to the curve \\((x^2 + y^2)^3 = 8r^2 y^2\\) at the point \\((1, -1)\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ccbfa0e57dd8316d74efcc7e7925d811cd132299909ad01647ac448122ac4315 | Use implicit differentiation to find the slope of the tangent line to the curve \(2(x^2 + y^2)^2 = 25(x^2 - y^2)\) at the point \((-3, -1)\). | [
"-\\dfrac{9}{13}"
] | [
"Use implicit differentiation to find the slope of the tangent line to the curve \\(2(x^2 + y^2)^2 = 25(x^2 - y^2)\\) at the point \\((-3, -1)\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-edced6d013b1b59e79215988b02c9ecc8e722085058af7560e9510fb18290c36 | Use implicit differentiation to find the slope of the tangent line to the curve \(\sqrt{x + 3y} + \sqrt{3xy} = 12.8\) at the point \((3, 7)\). | [
"\\dfrac{12\\sqrt{42} - 161}{51}"
] | [
"Use implicit differentiation to find the slope of the tangent line to the curve \\(\\sqrt{x + 3y} + \\sqrt{3xy} = 12.8\\) at the point \\((3, 7)\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-c11fe5c2394726d8a92e83903dae7baa6108ee246aecf4ef1aa029c6aabf6a96 | Use index arithmetic to find the value of \( x \) in the congruence \( 7^x \equiv 6 \pmod{17} \). | [
"\\( x = 6 \\)"
] | [
"Use index arithmetic to find the value of \\( x \\) in the congruence \\( 7^x \\equiv 6 \\pmod{17} \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-02197940f50dc4c42032cad96b85ec50ef8a6949da960f814e4c6ffb33e889c7 | Use index arithmetic to find the value of \( x \) in the congruence \( x^8 \equiv 5 \pmod{14} \). | [
"No solution."
] | [
"Use index arithmetic to find the value of \\( x \\) in the congruence \\( x^8 \\equiv 5 \\pmod{14} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-733d3795fa3d5727516bddf148f2a63c4a56d37cfb3e77325ba78248a27f222a | Use integration to determine the volume of the solid which lies above the region between the curves \( y = x^2 \) and \( y = -x^2 \) for \( 0 \leq x \leq 1 \), and whose cross-sections perpendicular to the x-axis are semicircles with bases lying between these curves. | [
"\\dfrac{\\pi}{10}"
] | [
"Use integration to determine the volume of the solid which lies above the region between the curves \\( y = x^2 \\) and \\( y = -x^2 \\) for \\( 0 \\leq x \\leq 1 \\), and whose cross-sections perpendicular to the x-axis are semicircles with bases lying between these curves."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-481e9c6853fb6490964c69bf8e761913b8cabe33b0ae7dce1064b95d5d6659af | Use l'Hopital's rule to find the limit \( \lim_{x \to 0^+} (e^{5/x} - 3x)^{x/2} \). | [
"\\( e^{\\frac{5}{2}} \\)"
] | [
"Use l'Hopital's rule to find the limit \\( \\lim_{x \\to 0^+} (e^{5/x} - 3x)^{x/2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5d251066c34546627445fd4b142bd2e42cef3523af60de359b88f290bdbd1d33 | Use limits to find the derivative of $\frac{\sqrt{x-1}}{\sqrt{x+1}}$. | [
"\\dfrac{1}{(x+1)\\sqrt{x^2 -1}}"
] | [
"Use limits to find the derivative of $\\frac{\\sqrt{x-1}}{\\sqrt{x+1}}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-503e842395e85906cb2d0cd02ec62669c26d5644f8681ab5e41119c661818adc | Use line integrals to find the area of the region bounded by the graphs of $y=x^2$ and $y=5x$. | [
"\\dfrac{125}{6}"
] | [
"Use line integrals to find the area of the region bounded by the graphs of $y=x^2$ and $y=5x$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-140c1bbfdaf373e26fb76ae35e1e32e177a9bbe96f7c11385a1e66b2264a875a | Use linear regression to find \( C \) as a function of \( N \) for the following data: \[ \text{Number of Yearbooks Sold (N)} \quad 15 \quad 76 \quad 124 \quad 255 \quad 312 \] \[ \text{Production Cost (C in $)} \quad 682.50 \quad 1262.00 \quad 1718.00 \quad 2962.50 \quad 3504.00 \] | [
"C = 9.5N + 540"
] | [
"Use linear regression to find \\( C \\) as a function of \\( N \\) for the following data:\n\\[ \\text{Number of Yearbooks Sold (N)} \\quad 15 \\quad 76 \\quad 124 \\quad 255 \\quad 312 \\]\n\\[ \\text{Production Cost (C in $)} \\quad 682.50 \\quad 1262.00 \\quad 1718.00 \\quad 2962.50 \\quad 3504.00 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-95aace90c485cb678608b4597da3eb90a907c0a4960cbbedb02a4ed6e78d4010 | Use linear regression to find \( S \) as a function of \( N \) for the following data: \[ \text{Number of Yearbooks Sold (N)} \quad 15 \quad 76 \quad 124 \quad 255 \quad 312 \] \[ \text{Sales Revenue (S in $)} \quad 187.50 \quad 950.00 \quad 1550.00 \quad 3187.50 \quad 3900.00 \] | [
"S = 12.5N"
] | [
"Use linear regression to find \\( S \\) as a function of \\( N \\) for the following data:\n\\[ \\text{Number of Yearbooks Sold (N)} \\quad 15 \\quad 76 \\quad 124 \\quad 255 \\quad 312 \\]\n\\[ \\text{Sales Revenue (S in $)} \\quad 187.50 \\quad 950.00 \\quad 1550.00 \\quad 3187.50 \\quad 3900.00 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e04620744f32c9a47033193d01c78843f58403c893dcc8b215379c4ccd4083ba | Use linear regression to find the best fit line for the following data table: \[ \text{A) } x \quad 1.5 \quad 0.8 \quad 3.5 \quad 8.5 \quad 6.8 \quad 7.5 \] \[ \text{Y} \quad 7.0 \quad 5.9 \quad 11.9 \quad 24.5 \quad 21.5 \quad 22.5 \] | [
"y = 3.504 + 2.527x"
] | [
"Use linear regression to find the best fit line for the following data table:\n\\[ \\text{A) } x \\quad 1.5 \\quad 0.8 \\quad 3.5 \\quad 8.5 \\quad 6.8 \\quad 7.5 \\]\n\\[ \\text{Y} \\quad 7.0 \\quad 5.9 \\quad 11.9 \\quad 24.5 \\quad 21.5 \\quad 22.5 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-0da1609854a7eb5519479b95a918f43f06e749602f6455486b43debfc52cc8cc | Use linear regression to find the best fit line for the following data table: \[ \text{B) } x \quad 12 \quad 9 \quad 3 \quad 11 \quad 7 \quad 2 \] \[ \text{Y} \quad 8 \quad 45 \quad 120 \quad 20 \quad 70 \quad 133 \] | [
"Y = -12.504x + 157.695"
] | [
"Use linear regression to find the best fit line for the following data table:\n\\[ \\text{B) } x \\quad 12 \\quad 9 \\quad 3 \\quad 11 \\quad 7 \\quad 2 \\]\n\\[ \\text{Y} \\quad 8 \\quad 45 \\quad 120 \\quad 20 \\quad 70 \\quad 133 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51321bcdafe0d7ed22ef55efae885a5213f1d4c86624c73701fe6200cd897146 | Use logarithmic differentiation to find \(\dfrac{d}{dx} \left( \sqrt[4]{\dfrac{x^3-1}{x^2+1}} \right)\). | [
"\\(\\sqrt[4]{\\dfrac{x^3-1}{x^2+1}} \\cdot \\frac{1}{4} \\left( \\frac{3x^2}{x^3-1} - \\frac{2x}{x^2+1} \\right)\\)"
] | [
"Use logarithmic differentiation to find \\(\\dfrac{d}{dx} \\left( \\sqrt[4]{\\dfrac{x^3-1}{x^2+1}} \\right)\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 5,
"tir": 10
} | 21 | 0 | 0 |
omr-ws-c81ae56208827230afb08d16eef0e4723a6bb8ca2cf00834ccd6a7dc8b6fdb7f | Use long division to divide $3x^4-5x^2+3$ by $x+2$. | [
"\\(3x^3 - 6x^2 + 7x - 14 + \\frac{31}{x+2}\\)"
] | [
"Use long division to divide $3x^4-5x^2+3$ by $x+2$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-77e4c65c871e7e8be2689153e23d4897e53e68e54d5165818d4428f9b89532bc | Use matrices to find the area of a figure with vertices (7,0), (-4,-1), (-3,2), and (-6,3). | [
"18"
] | [
"Use matrices to find the area of a figure with vertices (7,0), (-4,-1), (-3,2), and (-6,3)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-868e75a72ef086e191085816e88b1eca578ff04c0c578970523b7c141a6663a2 | Use matrix algebra to solve the following system of equations: \[ 3x + 2y = -2 \] \[ x + 4y = 6 \] | [
"(-2, 2)"
] | [
"Use matrix algebra to solve the following system of equations:\n\\[ 3x + 2y = -2 \\]\n\\[ x + 4y = 6 \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-de06536aaefdcd15ffcf9d52fce30d12d318ef5fae0c407c82737462a123840a | Use matrix algebra to solve the following system of equations: \[ 5x + y = 13 \] \[ 3x + 2y = 5 \] | [
"(3, -2)"
] | [
"Use matrix algebra to solve the following system of equations:\n\\[ 5x + y = 13 \\]\n\\[ 3x + 2y = 5 \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-1e05c1a782eb1a0e5aad58946b76b740a203ed5c05156e13a85fbbc5d3f88a9f | Use numbers $1$ to $49$ to fill in the boxes in the magic square such that all five numbers in every row, column, and diagonal sum to the same value. What is the number that should be filled in box $X$? | [
"25"
] | [
"Use numbers $1$ to $49$ to fill in the boxes in the magic square such that all five numbers in every row, column, and diagonal sum to the same value. What is the number that should be filled in box $X$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 1,
"tir": 5
} | 16 | 0 | 0 |
omr-ws-efe16f1039b2b438545fe8062d7e36f4e7ca71f1f35001445f4314785be15824 | Use one determinant to express the following product: \[ \begin{vmatrix} a & -a & a & a \\ -b & b & b & b \\ c & c & -c & c \\ d & d & d & -d \end{vmatrix} \cdot \begin{vmatrix} a & b \\ c & d \end{vmatrix} \] | [
"-16abc(ad-bc)"
] | [
"Use one determinant to express the following product:\n\\[ \\begin{vmatrix} a & -a & a & a \\\\ -b & b & b & b \\\\ c & c & -c & c \\\\ d & d & d & -d \\end{vmatrix} \\cdot \\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-b403099b203a9dbac57a10f5ac411b1ad21d4d1ac36b78c752d043c7b12ee1cc | Use parameter differentiation by considering $\text{I}(a) = \int_0^{1}\frac{x^a-1}{\log x}\,dx$. Then find $\text{I}'(a) = \frac{1}{a+1}$. | [
"\\(\\frac{1}{a+1}\\)"
] | [
"Use parameter differentiation by considering $\\text{I}(a) = \\int_0^{1}\\frac{x^a-1}{\\log x}\\,dx$. Then find $\\text{I}'(a) = \\frac{1}{a+1}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-95713b86b561c8dd42338c48f51bdf563ab9b30017893e4f9038595ae7d49f66 | Use part (b) to determine how many terms \( M \) are needed so that: \[ \sum_{n=1}^{M} \frac{1}{n} > 50. \] | [
"\\( 7.9129 \\times 10^{21} \\)"
] | [
"Use part (b) to determine how many terms \\( M \\) are needed so that:\n\\[ \\sum_{n=1}^{M} \\frac{1}{n} > 50. \\]"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f7aff0e7b92a44cac9c8b6f7818a7286e621c48ddeb9bf9ea89f3b17ad0a4665 | Use part (b) to write \( z^5 - 32 \) in the form \( (z - 2)(z^2 - 4 \cos(m)z + 4)(z^2 - 4 \cos(n)z + 4) \), where you should state the values of \( m \) and \( n \). | [
"\\( m = \\frac{2\\pi}{5}, n = \\frac{4\\pi}{5} \\)"
] | [
"Use part (b) to write \\( z^5 - 32 \\) in the form \\( (z - 2)(z^2 - 4 \\cos(m)z + 4)(z^2 - 4 \\cos(n)z + 4) \\), where you should state the values of \\( m \\) and \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 23
} | 30 | 0 | 0 |
omr-ws-2ce8bf0a48ee4b133b7023f01045dc7d8161345a9cab293376010fafade55dc0 | Use part (i) to find the greatest value of \( xy^{2} \) in the region of the \( x-y \) plane given by \( x \geq 0 \), \( y \geq 0 \), and \( x + 2y \leq 3 \). For what values of \( x \) and \( y \) is this greatest value achieved? | [
"1 (achieved at \\( x = 1 \\) and \\( y = 1 \\))"
] | [
"Use part (i) to find the greatest value of \\( xy^{2} \\) in the region of the \\( x-y \\) plane given by \\( x \\geq 0 \\), \\( y \\geq 0 \\), and \\( x + 2y \\leq 3 \\). For what values of \\( x \\) and \\( y \\) is this greatest value achieved?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a73a8b5ce05f89d8d66ee965949d11899dd4578f9292ca90b4a200520ce84d6c | Use parts (a) and (b) to express \(\pi\) in terms of sums of powers of \(\frac{1}{2}\) and \(\frac{1}{3}\). Add terms in this sum until you get an approximation that equals \(\pi\) to 5 decimal places. How many terms do you need? | [
"8"
] | [
"Use parts (a) and (b) to express \\(\\pi\\) in terms of sums of powers of \\(\\frac{1}{2}\\) and \\(\\frac{1}{3}\\). Add terms in this sum until you get an approximation that equals \\(\\pi\\) to 5 decimal places. How many terms do you need?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-e7ee9a303d00a7dd2d451a9ba67c1c033bea1567b76c6e169cac055ddd08d896 | Use parts (a) and (b) to express \(\pi\) in terms of sums of powers of \(\frac{1}{2}\) and powers of \(\frac{1}{3}\). Add terms in this sum until you get an approximation that equals \(\pi\) to 5 decimal places. How many terms do you need? | [
"\\(\\pi \\approx 3.14159\\)"
] | [
"Use parts (a) and (b) to express \\(\\pi\\) in terms of sums of powers of \\(\\frac{1}{2}\\) and powers of \\(\\frac{1}{3}\\). Add terms in this sum until you get an approximation that equals \\(\\pi\\) to 5 decimal places. How many terms do you need?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e0ba07b997bda689c87a2db6be94defc26e56ee1c5717521870105f269d048a | Use parts (i) and (ii) to solve the equation $y^{3} - 3y - \sqrt{2}=0$, giving your answers in surd form. | [
"\\(\\frac{\\sqrt{3} + 1}{2\\sqrt{2}}, -\\frac{\\sqrt{2}}{2}, -\\frac{\\sqrt{3} - 1}{2\\sqrt{2}}\\)"
] | [
"Use parts (i) and (ii) to solve the equation $y^{3} - 3y - \\sqrt{2}=0$, giving your answers in surd form."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8de0d6716f12a56c11a4d343a1ab06d3b54c624fe18766e295e8df7963741253 | Use parts (ii) and (iii) to deduce the sum to infinity of the series \[ \frac{1}{1 \times 2} + \frac{1}{3 \times 2^{3}} + \frac{1}{5 \times 2^{5}} + \cdots. \] | [
"\\(\\frac{1}{2} \\ln 3\\)."
] | [
"Use parts (ii) and (iii) to deduce the sum to infinity of the series\n\\[ \\frac{1}{1 \\times 2} + \\frac{1}{3 \\times 2^{3}} + \\frac{1}{5 \\times 2^{5}} + \\cdots. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-edf8657d94fecf56e86c42b46007f3ccaad0009e516054d634967cba8e54ea07 | Use parts (iii) and (iv) to deduce the sum to infinity of the series \[ \frac{1}{1 \times 2} + \frac{1}{3 \times 2^{3}} + \frac{1}{5 \times 2^{5}} + \cdots. \] | [
"\\(\\frac{1}{2} \\ln 3\\)"
] | [
"Use parts (iii) and (iv) to deduce the sum to infinity of the series\n\n\\[ \\frac{1}{1 \\times 2} + \\frac{1}{3 \\times 2^{3}} + \\frac{1}{5 \\times 2^{5}} + \\cdots. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e628f98c49c85cc58711d981ed389a895411f0ade53511c2c9805e0870cabd10 | Use place value to figure out $90000 + 9000 + 900 + 90 + 9 + 1$. | [
"100000"
] | [
"Use place value to figure out $90000 + 9000 + 900 + 90 + 9 + 1$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-a113332ebaf803d15b82238b40ee8841acea666e8e7ac4b89a38519dbc0a7cdf | Use polar coordinates to find the volume of the solid above the cone \( z = \sqrt{x^2 + y^2} \) and below the sphere \( x^2 + y^2 + z^2 = 1 \). | [
"\\dfrac{\\pi}{3} \\left(2 - \\sqrt{2}\\right)"
] | [
"Use polar coordinates to find the volume of the solid above the cone \\( z = \\sqrt{x^2 + y^2} \\) and below the sphere \\( x^2 + y^2 + z^2 = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c00e7f541204c4524e8b432ea8b6ab8516700534d831bd8cdc4e0a7a781c4f3b | Use polar coordinates to find the volume of the solid enclosed by the hyperboloid $-x^2 - y^2 + z^2 = 13$ and the plane $z = 4$. | [
"\\(\\frac{\\pi(-92 + 26\\sqrt{13})}{3}\\)"
] | [
"Use polar coordinates to find the volume of the solid enclosed by the hyperboloid $-x^2 - y^2 + z^2 = 13$ and the plane $z = 4$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-66bfa1e6f4e0122ae08838e365565ad030aa9b65d5ea2e7269aa332b3aa2ca81 | Use power series to compute $\int_0^{\frac{\pi}{2}} \ln \sin x \, dx$. | [
"\\(-\\frac{\\pi}{2} \\ln 2\\)"
] | [
"Use power series to compute $\\int_0^{\\frac{\\pi}{2}} \\ln \\sin x \\, dx$."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-2ce95964c7960d693587447938f79115821e65df44238694a30ed8a4c3ef4b08 | Use power series to find a formula for the sequence defined by: \[ a_0 = 1, \quad a_1 = 1, \quad a_{n+2} - 4a_{n+1} + 4a_n = 0 \] | [
"\\( a_n = \\left(1 - \\frac{n}{2}\\right)2^n \\)"
] | [
"Use power series to find a formula for the sequence defined by:\n\\[ a_0 = 1, \\quad a_1 = 1, \\quad a_{n+2} - 4a_{n+1} + 4a_n = 0 \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-4a2e31b131cf81e7a15f601f4cf82c2acb704e3eb156a9361a581fa154b7f432 | Use power series to solve the differential equation $xy'' + y' + (y^2 - x^2)y = 0$. | [
"0"
] | [
"Use power series to solve the differential equation $xy'' + y' + (y^2 - x^2)y = 0$."
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 23
} | 23 | 0 | 0 |
omr-ws-5f918da53582150e99f3c49dbb3716bdec106001170fae84e1be2854b92ca026 | Use power series to solve the differential equation $y'' + y' = x + y^2$. | [
"y = a_0 + a_1 x + \\frac{a_0^2 - a_1}{2} x^2 + \\frac{1 - a_0^2 + 2 a_0 a_1 + a_1}{6} x^3 + \\cdots",
"y(x) = a_0 + a_1 x + \\frac{a_0^2 - a_1}{2}x^2 + \\frac{1 - a_0^2 + a_1(2a_0 + 1)}{6}x^3 + \\cdots"
] | [
"Use power series to solve the differential equation $y'' + y' = x + y^2$."
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 29,
"genselect": 1
} | 30 | 0 | 0 |
omr-ws-76ee72c5516b43496402cb07c51671ac88050250aea3dd041a0a5925fae80e00 | Use power-reducing formulas to rewrite the expression \((\sin x)^8\) in terms of the first power of the cosine. | [
"\\dfrac{35 - 56 \\cos 2x + 28 \\cos 4x - 8 \\cos 6x + \\cos 8x}{128}"
] | [
"Use power-reducing formulas to rewrite the expression \\((\\sin x)^8\\) in terms of the first power of the cosine."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e85703fb8c74a517cbcebae346907fbb35a449b9e9fe30859eb997ef077d0478 | Use quaternions to find the 30º counterclockwise rotation of the vector \((2, 6, 1)\) around the axis directed by \((2, -1, 2)\). | [
"\\left( \\dfrac{6\\sqrt{3} - 13}{6}, \\dfrac{9\\sqrt{3} + 1}{3}, \\dfrac{3\\sqrt{3} + 14}{6} \\right)"
] | [
"Use quaternions to find the 30º counterclockwise rotation of the vector \\((2, 6, 1)\\) around the axis directed by \\((2, -1, 2)\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-4ec5c2a19f7f635ac054e213665ef0a3feac6909894e0f1a24587869b2df624e | Use separation of variables to find an expression for \( v(t) \) in terms of \( t \), where \( v(t) \) is the velocity of a skydiver at time \( t \) seconds, given the differential equation \(\frac{dv}{dt} = -2v - 32\) and the initial condition \( v(0) = -50 \). | [
"\\( v(t) = -16 - 34e^{-2t} \\)"
] | [
"Use separation of variables to find an expression for \\( v(t) \\) in terms of \\( t \\), where \\( v(t) \\) is the velocity of a skydiver at time \\( t \\) seconds, given the differential equation \\(\\frac{dv}{dt} = -2v - 32\\) and the initial condition \\( v(0) = -50 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-3a5b1fe4c1e4cd817d4446eebed9deeabbfd592abc3c926443d8d5447875e0bc | Use some of the digits 1-9 without repeating in the expression $\sqrt{\boxed{\;}\,\boxed{\;}\,\boxed{\;}}+\sqrt{\boxed{\;}\,\boxed{\;}}+\sqrt{\boxed{\;}}$. What is the closest to $0$ it can get? | [
"18.66"
] | [
"Use some of the digits 1-9 without repeating in the expression $\\sqrt{\\boxed{\\;}\\,\\boxed{\\;}\\,\\boxed{\\;}}+\\sqrt{\\boxed{\\;}\\,\\boxed{\\;}}+\\sqrt{\\boxed{\\;}}$. What is the closest to $0$ it can get?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-361668f9228b7996af9fa98ee1eade6a342bfbe80d77380aa2aee8ab1cc5d936 | Use spherical coordinates to find the mass of the solid bounded below by the cone \( z = \sqrt{x^2 + y^2} \) and above by the sphere \( x^2 + y^2 + z^2 = 9 \), given that the density is \( \rho(x, y, z) = x^2 + y^2 + z^2 \). | [
"\\(\\frac{486 - 243\\sqrt{2}}{5}\\pi\\)"
] | [
"Use spherical coordinates to find the mass of the solid bounded below by the cone \\( z = \\sqrt{x^2 + y^2} \\) and above by the sphere \\( x^2 + y^2 + z^2 = 9 \\), given that the density is \\( \\rho(x, y, z) = x^2 + y^2 + z^2 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b6d3ed82485c03afac342f944f343ad943f030cc4816a43de9cdfea896cf8b06 | Use sum or difference identities to solve the equation exactly: \(\cos 3x \cos x = \sin 3x \sin x\). | [
"\\( x = \\frac{\\pi}{8} + \\frac{k\\pi}{4} \\)"
] | [
"Use sum or difference identities to solve the equation exactly: \\(\\cos 3x \\cos x = \\sin 3x \\sin x\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-edac4f8c9b55c6f1c01b96531df75268619a99e2077ec68cbb4e6e99cceea40e | Use sum or difference identities to solve the equation exactly: \(\cos 3x \cos x = \sin 3x \sin x\). Hint: Use the identity \(\cos(a + b) = \cos a \cos b - \sin a \sin b\) with \(a = 3x\) and \(b = x\). | [
"\\(x = \\frac{\\pi}{8} + \\frac{k\\pi}{4}\\) for integer \\(k\\)."
] | [
"Use sum or difference identities to solve the equation exactly: \\(\\cos 3x \\cos x = \\sin 3x \\sin x\\). Hint: Use the identity \\(\\cos(a + b) = \\cos a \\cos b - \\sin a \\sin b\\) with \\(a = 3x\\) and \\(b = x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-7c48667de7e959688ceb30e20a17371f168cd7df8c266938df229cb91495bfcc | Use synthetic division to solve the division problem \(\frac{x^3 - 4x^2 - 19x - 9}{x + 3}\). | [
"x^2 - 7x + 2 - \\frac{15}{x + 3}"
] | [
"Use synthetic division to solve the division problem \\(\\frac{x^3 - 4x^2 - 19x - 9}{x + 3}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd8bc56c16059e394887266ade9b1cc784ee5c4014358c421bdc7d3815f469cf | Use tangent line approximation to find the approximate value of $\cos\bigg(90^\circ+0.02\bigg)$. | [
"-0.2"
] | [
"Use tangent line approximation to find the approximate value of $\\cos\\bigg(90^\\circ+0.02\\bigg)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae3437df3e0c75098692b8c4c7023d6eecd668b47935ea1d93c23f446722bf8c | Use the "method of Frobenius" to find series solutions about the regular singular point at $x=0$ for the differential equation $4x^2\frac{d^2y}{dx^2}+2x\frac{dy}{dx}+xy=0.$ | [
"\\( y(x) = A \\cos \\sqrt{x} + B \\sin \\sqrt{x} \\)"
] | [
"Use the \"method of Frobenius\" to find series solutions about the regular singular point at $x=0$ for the differential equation $4x^2\\frac{d^2y}{dx^2}+2x\\frac{dy}{dx}+xy=0.$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 1,
"tir": 8
} | 22 | 0 | 0 |
omr-ws-d1143d4ea995e321fe25cb8be0d865bc03730fb03f4f8ca42c9697f628bee556 | Use the Beta function to express the definite integral \(\int_0^1 \frac{1}{\sqrt{1 + x^4}} \, dx\). | [
"\\(\\frac{1}{8}B\\left(\\frac{1}{4}, \\frac{1}{4}\\right)\\)"
] | [
"Use the Beta function to express the definite integral \\(\\int_0^1 \\frac{1}{\\sqrt{1 + x^4}} \\, dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 18
} | 32 | 0 | 0 |
omr-ws-1f14bc63a68e446d14f2daad17bc26ea58c2701b09faae74a576e6a32ab50680 | Use the Binomial series to expand the function $\frac{3}{(4+x)^3}$ as a power series. Express the result in the form $f(x) = \sum_{n=0} ^{\infty}( ... )$. | [
"$\\sum_{n=0}^\\infty\\frac{3(-1)^n(n+2)(n+1)}{2^{2n+7}}x^n$"
] | [
"Use the Binomial series to expand the function $\\frac{3}{(4+x)^3}$ as a power series. Express the result in the form $f(x) = \\sum_{n=0} ^{\\infty}( ... )$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-1b4fe1d788fa115b2ec25ac3c32baeeaa3cffb0af2c3c74cdebed2442f448828 | Use the Cauchy Integral Formula to compute the integral $\int_C \frac{\bar{z}}{z^2-1} dz$ where $C: |z| = 2$. | [
"0"
] | [
"Use the Cauchy Integral Formula to compute the integral $\\int_C \\frac{\\bar{z}}{z^2-1} dz$ where $C: |z| = 2$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1b22367c8e0152a33401d3b420dc3ef6d7cf53d6c6759c4bb195848ff44ad259 | Use the Cauchy-Riemann relations to determine where \( f \) is analytic and find \( f'(z) \). Given \( u_x = 1 \), \( u_y = -\sin(y) \), \( v_x = 1 \), and \( v_y = -\cos(x) \). Determine the points where these relations hold and where \( f \) is analytic. | [
"\\( f'(z) = 1 - i \\) at isolated points \\( y = -\\frac{\\pi}{2} + 2k\\pi \\) and \\( x = -\\pi + 2k\\pi \\), but \\( f \\) is nowhere analytic."
] | [
"Use the Cauchy-Riemann relations to determine where \\( f \\) is analytic and find \\( f'(z) \\). Given \\( u_x = 1 \\), \\( u_y = -\\sin(y) \\), \\( v_x = 1 \\), and \\( v_y = -\\cos(x) \\). Determine the points where these relations hold and where \\( f \\) is analytic."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-885ed9193cf95d97acac307f4d791c6072fb29c5ea9a8337b3e1c9a664a64063 | Use the Central Limit Theorem to obtain a theoretical upper bound for the probability that the average of the squares of 900 x-coordinates, which are uniformly distributed, is greater than or equal to 0.36. | [
"0.0037"
] | [
"Use the Central Limit Theorem to obtain a theoretical upper bound for the probability that the average of the squares of 900 x-coordinates, which are uniformly distributed, is greater than or equal to 0.36."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-6c0558f4edd5e2df905fa34a44f74f73e570ab3c21ca15b1e584d13549b08513 | Use the Discrete Fourier Transform (DFT) to approximate the value of the Fourier Transform of the signal \( f(t) \) at the frequencies \( \omega = 0, 1, 2, 3 \), where \[ f(t) = \begin{cases} 0 & \text{if } t < 0 \\ t^4 e^{-t} & \text{if } t \geq 0 \end{cases} \] | [
"0.0758 + 0.00288i"
] | [
"Use the Discrete Fourier Transform (DFT) to approximate the value of the Fourier Transform of the signal \\( f(t) \\) at the frequencies \\( \\omega = 0, 1, 2, 3 \\), where\n\n\\[ f(t) = \\begin{cases} \n0 & \\text{if } t < 0 \\\\\nt^4 e^{-t} & \\text{if } t \\geq 0 \n\\end{cases} \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-7d0c9025f6a8c1a131ae90d2882ad99c8bca8f54a8983ff4421b927393d93455 | Use the Euclidean Algorithm to find all integers \( p \) and \( q \) such that \( 599p + 53q = 1 \). Express your answers in terms of the integer parameter \( t \). | [
"\\( p = 10 + 53t \\), \\( q = -113 - 599t \\)"
] | [
"Use the Euclidean Algorithm to find all integers \\( p \\) and \\( q \\) such that \\( 599p + 53q = 1 \\). Express your answers in terms of the integer parameter \\( t \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 30
} | 42 | 0 | 0 |
omr-ws-24266844ca7bb4ffff907e5e947a7358ee4feace420ab4fcdf829fcb84b037aa | Use the Euclidean Algorithm to find integers \( x \) and \( y \) such that \( 74x + 58y = 14 \). | [
"\\( (77, -98) \\)"
] | [
"Use the Euclidean Algorithm to find integers \\( x \\) and \\( y \\) such that \\( 74x + 58y = 14 \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 9
} | 20 | 0 | 0 |
omr-ws-059e07738c9d904ef79433dc10132b20965067624062b03a3d5d66de2bb7647e | Use the Euclidean algorithm to find the greatest common divisor (gcd) of 1800 and 3345. | [
"15"
] | [
"Use the Euclidean algorithm to find the greatest common divisor (gcd) of 1800 and 3345."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-72b3e88ceb8eea58d9f0776e488d89640fe0706526e82b2b3eb174b5ef977ede | Use the Euclidean algorithm to find the greatest common divisor (gcd) of 182 and 445. | [
"1"
] | [
"Use the Euclidean algorithm to find the greatest common divisor (gcd) of 182 and 445."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85abf17f3015e9418ceb62f955e35b68adfa10de8fc25e04b072a9237c381283 | Use the Euler method to find the approximate values of \( y_1, y_2, y_3, \) and \( y_4 \) for the differential equation \( y' = -2 - 3x - 2y \) with initial condition \( y(0) = 2 \) and step size \( \Delta x = 0.5 \). | [
"\\( y_1 = 0, y_2 = -\\frac{7}{4}, y_3 = -\\frac{5}{2}, y_4 = -\\frac{13}{4} \\)"
] | [
"Use the Euler method to find the approximate values of \\( y_1, y_2, y_3, \\) and \\( y_4 \\) for the differential equation \\( y' = -2 - 3x - 2y \\) with initial condition \\( y(0) = 2 \\) and step size \\( \\Delta x = 0.5 \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-b8462a2f5d2d99acb6534a4d8daac60ee15e78bf1fa9f48b7a0475fe6f15a524 | Use the Fast Exponential algorithm to determine $38^{67} \mod 77$. | [
"3"
] | [
"Use the Fast Exponential algorithm to determine $38^{67} \\mod 77$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6aaf57af56eaa0f4d487a5ac0664b39b58f58ffeb63f12bbfb81b6d15a63d88d | Use the Fourier transform of the function \( f(x) \) to compute the integral: \[ \int_{0}^{\infty} \frac{\sin (a t) \cos (x t)}{t} \, dt \] | [
"\n\\begin{cases}\n\\displaystyle \\frac{\\pi}{2} & \\text{if } |x| < a, \\\\\n\\displaystyle \\frac{\\pi}{4} & \\text{if } |x| = a, \\\\\n0 & \\text{if } |x| > a.\n\\end{cases}\n"
] | [
"Use the Fourier transform of the function \\( f(x) \\) to compute the integral:\n\\[ \\int_{0}^{\\infty} \\frac{\\sin (a t) \\cos (x t)}{t} \\, dt \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-31ec84cacf9abab0d400319906db1525f420fc41117a958d6dc011c182edad4f | Use the Frobenius method to find the basis of solutions for the differential equation \( xy'' - (x+1)y' + y = 0 \). | [
"\\( y = \\alpha e^{x} + \\beta (-1 - x), \\quad \\alpha, \\beta \\in \\mathbb{R} \\)"
] | [
"Use the Frobenius method to find the basis of solutions for the differential equation \\( xy'' - (x+1)y' + y = 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-3fc9b249e4607883cf605bb9ba45433250c481d95aa34b5c351eb7eabebdffcf | Use the Fundamental Theorem of Calculus to find $\int_S K ~ds$ where $S$ represents the half-ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ with $z \geq 0$. | [
"2\\pi"
] | [
"Use the Fundamental Theorem of Calculus to find $\\int_S K ~ds$ where $S$ represents the half-ellipsoid $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1$ with $z \\geq 0$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-aaf85ae0bce4ac24a937c84a6c632d7425243e64209b0b1dcb85d3f77c9728b0 | Use the Fundamental Theorem of Calculus to find $\int_S K ~ds$ where $S$ represents the half-hyperboloid $x^2+y^2-z^2=1$ with $z \leq 0$. | [
"2\\pi"
] | [
"Use the Fundamental Theorem of Calculus to find $\\int_S K ~ds$ where $S$ represents the half-hyperboloid $x^2+y^2-z^2=1$ with $z \\leq 0$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9011ea448f5bd90dc8aca20e105f1a44b941d99c326bcaf477d1f65e49517999 | Use the Fundamental Theorem of Calculus to find $\int_S K ~ds$ where $S$ represents the surface of revolution of the curve $z=\cos{x}$, $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$, around the z-axis. | [
"\\( 4\\pi \\)"
] | [
"Use the Fundamental Theorem of Calculus to find $\\int_S K ~ds$ where $S$ represents the surface of revolution of the curve $z=\\cos{x}$, $-\\frac{\\pi}{2} \\leq x \\leq \\frac{\\pi}{2}$, around the z-axis."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-e9b99f0632f70aa92153ee7d24a7eb9b1b638a5bfb013eb24244df5ffa395730 | Use the Fundamental Theorem of Calculus to find the derivative of the function: \[ f(x) = \int_{0}^{x^2} \sin(t^3) \, dt \] | [
"\\( 2x \\sin(x^6) \\)"
] | [
"Use the Fundamental Theorem of Calculus to find the derivative of the function:\n\\[ f(x) = \\int_{0}^{x^2} \\sin(t^3) \\, dt \\]"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-7ff5d7bf3efeb03c0b4b95ce41b86ec7eb4667453338d366fd1640d8592f5bc1 | Use the Gauss method to find the inverse of matrix \( A \). | [
"\\begin{pmatrix} -2 & 1 \\\\ \\frac{3}{2} & -\\frac{1}{2} \\end{pmatrix}"
] | [
"Use the Gauss method to find the inverse of matrix \\( A \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-9aadd4bbb8962d225bf48ee1e7f3e3f1b970055d2db1c983e68db750b62b9acb | Use the Gaussian algorithm to solve the following systems of linear equations. $\begin{array}[t]{lll} 10x-9z=19,\medskip \\ 8x-y=10,\medskip \\ y-12z=10. \end{array}$ | [
"$(1,-2,-1)$"
] | [
"Use the Gaussian algorithm to solve the following systems of linear equations. $\\begin{array}[t]{lll} 10x-9z=19,\\medskip \\\\ 8x-y=10,\\medskip \\\\ y-12z=10. \\end{array}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37c60ccac2172948633fca676da85fa9d49b3187a4759d1bdba7b6304d47d54b | Use the Gaussian algorithm to solve the following systems of linear equations. $\begin{array}[t]{lll} 2x+y+z=7,\medskip \\ x+2y+z=8,\medskip \\ x+y+2z=9. \end{array}$ | [
"$(1,2,3)$"
] | [
"Use the Gaussian algorithm to solve the following systems of linear equations. $\\begin{array}[t]{lll} 2x+y+z=7,\\medskip \\\\ x+2y+z=8,\\medskip \\\\ x+y+2z=9. \\end{array}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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