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-39e8a2feb0f8ad6b1cbee7f0ebdb81737d72949f6e3049bd9e81d29b0e73fe31 | Use the Gaussian algorithm to solve the following systems of linear equations. $\begin{array}[t]{lll} 3x-4y+5z=18,\medskip \\ 2x+4y-3z=26,\medskip \\ x-6y+8z=0. \end{array}$ | [
"$(8,4,2)$"
] | [
"Use the Gaussian algorithm to solve the following systems of linear equations. $\\begin{array}[t]{lll} 3x-4y+5z=18,\\medskip \\\\ 2x+4y-3z=26,\\medskip \\\\ x-6y+8z=0. \\end{array}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2a9a6cec582883b20ee52ad1f66c2041112718f8de4494872fae7233d74b8a37 | Use the Gaussian algorithm to solve the following systems of linear equations. $\begin{array}[t]{lll} \frac{x}{a^3}-\frac{y}{a^2}+\frac{z}{a}=1,\medskip \\ \frac{x}{b^3}-\frac{y}{b^2}+\frac{z}{b}=1,\medskip \\ \frac{x}{c^3}-\frac{y}{c^2}+\frac{z}{c}=1. \end{array}$ | [
"$(abc,ab+ac+bc,a+b+c)$"
] | [
"Use the Gaussian algorithm to solve the following systems of linear equations. $\\begin{array}[t]{lll} \\frac{x}{a^3}-\\frac{y}{a^2}+\\frac{z}{a}=1,\\medskip \\\\ \\frac{x}{b^3}-\\frac{y}{b^2}+\\frac{z}{b}=1,\\medskip \\\\ \\frac{x}{c^3}-\\frac{y}{c^2}+\\frac{z}{c}=1. \\end{array}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-353b05e39f098dc184a44d3ed6dcb0c63667af69e0b842377ca421f8509914bb | Use the Gaussian algorithm to solve the following systems of linear equations. $\begin{array}[t]{lll} x+2y+z+7=0,\medskip \\ 2x+y-z-1=0,\medskip \\ 3x-y+2z-2=0. \end{array}$ | [
"$(1,-3,-2)$"
] | [
"Use the Gaussian algorithm to solve the following systems of linear equations. $\\begin{array}[t]{lll} x+2y+z+7=0,\\medskip \\\\ 2x+y-z-1=0,\\medskip \\\\ 3x-y+2z-2=0. \\end{array}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92c404bc32b88b1e3a9e8163ec2d2fc2d457f18d03c9d204706c7a5150eee035 | Use the Gaussian algorithm to solve the following systems of linear equations. $\begin{array}[t]{lll} x+a^2y+b^2z=0,\medskip \\ x+ay+bz=0,\medskip \\ x+y+z=1. \end{array}$ | [
"$\\left(-\\frac{ab}{(b-1)(1-a)},\\frac{b}{(a-1)(b-a)},\\frac{a}{(b-1)(b-a)}\\right)$"
] | [
"Use the Gaussian algorithm to solve the following systems of linear equations. $\\begin{array}[t]{lll} x+a^2y+b^2z=0,\\medskip \\\\ x+ay+bz=0,\\medskip \\\\ x+y+z=1. \\end{array}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2eab14d0555aa0a2f139c64dd06a57905035c0a1b7d31aa19151cc9b6cd0331e | Use the Gram-Schmidt process to determine an orthonormal basis for $\mathbb{R}^3$ starting from the standard basis, using the inner product defined as: \[ \langle \mathbf{v}, \mathbf{w} \rangle = 2v_1w_1 + 2v_2w_2 + 2v_3w_3 - v_1w_2 - v_2w_1 - v_2w_3 - v_3w_2. \] | [
"\\left\\{ \\left( \\frac{\\sqrt{2}}{2}, 0, 0 \\right), \\left( \\frac{\\sqrt{6}}{6}, \\frac{\\sqrt{6}}{3}, 0 \\right), \\left( \\frac{\\sqrt{3}}{6}, \\frac{\\sqrt{3}}{3}, \\frac{\\sqrt{3}}{2} \\right) \\right\\}"
] | [
"Use the Gram-Schmidt process to determine an orthonormal basis for $\\mathbb{R}^3$ starting from the standard basis, using the inner product defined as:\n\\[\n\\langle \\mathbf{v}, \\mathbf{w} \\rangle = 2v_1w_1 + 2v_2w_2 + 2v_3w_3 - v_1w_2 - v_2w_1 - v_2w_3 - v_3w_2.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-6e857f2e4f2610feca4bc54f8a29d3c1ee23929bf78adfc7a4f00fb94955a4e2 | Use the Kronecker theorem for the function \( f(x) = x^4 - 5x^2 + 6 \) over the field \( \mathbb{Q} \). | [
"\\mathbb{Q}(\\sqrt{2}, \\sqrt{3})"
] | [
"Use the Kronecker theorem for the function \\( f(x) = x^4 - 5x^2 + 6 \\) over the field \\( \\mathbb{Q} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30cd9fb975b0aeb435fbbb270c2a29dcb36fde6d5dcbf0ece5e0ec2263d1968d | Use the LLL algorithm to reduce the lattice with basis \( v_1 = (20, 16, 3) \), \( v_2 = (15, 0, 10) \), and \( v_3 = (0, 18, 9) \). | [
"(-5, 2, 16)"
] | [
"Use the LLL algorithm to reduce the lattice with basis \\( v_1 = (20, 16, 3) \\), \\( v_2 = (15, 0, 10) \\), and \\( v_3 = (0, 18, 9) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-d26eb7012e7fc09aeb4c40e961fc845f3803e009297a813634cc4fdf1d5220a2 | Use the Lagrange multiplier method to find the conditions for criticality of the distance between points on the parabola given by the equation \( y - x^2 = 1 \) and the straight line given by the equation \( y - ax + a^2 = 0 \), where \( a \) is a real number. | [
"x = \\frac{a}{2}"
] | [
"Use the Lagrange multiplier method to find the conditions for criticality of the distance between points on the parabola given by the equation \\( y - x^2 = 1 \\) and the straight line given by the equation \\( y - ax + a^2 = 0 \\), where \\( a \\) is a real number."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 2
} | 15 | 0 | 0 |
omr-ws-21cb29f42ac8e2da67b8a6410fd7cea428915fba1bf37fa36910605315a73e6a | Use the MacLaurin Series for $\sin x$ to generate the first four non-zero terms of the MacLaurin Series for $f(x) = x^2 \cos(\pi x)$. What is the coefficient of the $x^4$ term in this series? | [
"-\\dfrac{\\pi^2}{2}"
] | [
"Use the MacLaurin Series for $\\sin x$ to generate the first four non-zero terms of the MacLaurin Series for $f(x) = x^2 \\cos(\\pi x)$. What is the coefficient of the $x^4$ term in this series?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-750dd612a6627bda87b9635c372e57890e0c3d5622b56a90d46abbd253f281f5 | Use the Maclaurin polynomial of degree $3$ to estimate the greatest value of $\theta$ for which the construction is accurate to within $5\%$. | [
"1"
] | [
"Use the Maclaurin polynomial of degree $3$ to estimate the greatest value of $\\theta$ for which the construction is accurate to within $5\\%$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-3203689d287a66b9342e8c2ee7f7b1c04392309796d0dfe40b7b4ca9808add45 | Use the Mayer Vietoris sequence to compute the homology groups of a sphere with \( n \) handles. | [
"\\( H^0 = \\mathbb{Z}, H^1 = \\mathbb{Z}^{2n}, H^2 = \\mathbb{Z}, H^i = 0 \\text{ for } i > 2 \\)"
] | [
"Use the Mayer Vietoris sequence to compute the homology groups of a sphere with \\( n \\) handles."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 7
} | 13 | 0 | 0 |
omr-ws-3d885a624dcfe8564935acfcfb08dc017d364f5e134cc4a6b799a3234507b951 | Use the Mean Value Theorem to find the value of \( c \) in the interval \((-1, 1)\) that satisfies the equation \( f'(c) = \frac{f(1) - f(-1)}{1 - (-1)} \) for the function \( f(x) = \sin^{-1}(x) \). | [
"\\( c = \\pm \\sqrt{1 - \\frac{4}{\\pi^2}} \\)"
] | [
"Use the Mean Value Theorem to find the value of \\( c \\) in the interval \\((-1, 1)\\) that satisfies the equation \\( f'(c) = \\frac{f(1) - f(-1)}{1 - (-1)} \\) for the function \\( f(x) = \\sin^{-1}(x) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-cc30df0bdd12fa1c49116a8cfc92fd16dee0088126ac0a00e60692603c4f6b9d | Use the Newton-Raphson method to find an iterative formula for finding \( \frac{1}{N} \). | [
"x_{n+1} = x_n (2 - N x_n)"
] | [
"Use the Newton-Raphson method to find an iterative formula for finding \\( \\frac{1}{N} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2d47808a198181c910958b0d9a38f26d01b297a5336b685d45e28b5cfe585009 | Use the Newton-Raphson method to find, to 4 decimal places, a closer approximation to the root of $z^{3} - 4z + 5 = 0$ near to $1.2 + 0.7i$. | [
"1.1845 + 0.7258i"
] | [
"Use the Newton-Raphson method to find, to 4 decimal places, a closer approximation to the root of $z^{3} - 4z + 5 = 0$ near to $1.2 + 0.7i$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da2f3ba5341146a40d8ac6ac8d5400a0e495d60b4ab303562760aa7f955015c6 | Use the Newton-Raphson method, with a first approximation \( x_1 = 2 \), to find the next three approximations \( x_2 \), \( x_3 \), and \( x_4 \) to a root of \( f(x) = x \tanh x - 2 = 0 \). Give the answers correct to 4 decimal places. | [
"\\( x_2 = 2.0651 \\), \\( x_3 = 2.0653 \\), \\( x_4 = 2.0653 \\)"
] | [
"Use the Newton-Raphson method, with a first approximation \\( x_1 = 2 \\), to find the next three approximations \\( x_2 \\), \\( x_3 \\), and \\( x_4 \\) to a root of \\( f(x) = x \\tanh x - 2 = 0 \\). Give the answers correct to 4 decimal places."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-f0a0d40917c84475a554fb5fdb6ebf9aa364feb6f5dd91a99b729f5d163cae10 | Use the Residue theorem to evaluate the integral $\int_{- \infty}^{ + \infty} \frac{x^{2m}}{1+x^{2n}} \, dx$, where $m$ and $n$ are integers with $m < n$. | [
"\\dfrac{\\pi}{n \\sin\\left( \\dfrac{(2m + 1)\\pi}{2n} \\right)}"
] | [
"Use the Residue theorem to evaluate the integral $\\int_{- \\infty}^{ + \\infty} \\frac{x^{2m}}{1+x^{2n}} \\, dx$, where $m$ and $n$ are integers with $m < n$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f3e6fd6b952e736686be97098e7ee40cc9a598fa8140cbcc076ab2bc6dee45a5 | Use the Residue theorem to evaluate the integral $\int_{0}^{2 \pi } {\cos}^{2n} x \, dx$. | [
"\\(\\frac{\\binom{2n}{n} 2\\pi}{2^{2n}}\\)"
] | [
"Use the Residue theorem to evaluate the integral $\\int_{0}^{2 \\pi } {\\cos}^{2n} x \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-9433494270cbbb1008e7c1fa984f11acc7a206565d8f38f64b482eac48ec9da8 | Use the Runge-Kutta method with step sizes \( h = 0.1 \) and \( h = 0.05 \) to approximate the values \( x(1) \) and \( y(1) \) for the initial value problem given by: \[ x' = -x - y - (1 + t^3)e^{-t}, \quad x(0) = 0, \] \[ y' = -x - y - (t - 3t^2)e^{-t}, \quad y(0) = 1. \] Compare the approximations with the actual va... | [
"0.5666"
] | [
"Use the Runge-Kutta method with step sizes \\( h = 0.1 \\) and \\( h = 0.05 \\) to approximate the values \\( x(1) \\) and \\( y(1) \\) for the initial value problem given by:\n\\[ x' = -x - y - (1 + t^3)e^{-t}, \\quad x(0) = 0, \\]\n\\[ y' = -x - y - (t - 3t^2)e^{-t}, \\quad y(0) = 1. \\]\nCompare the approximati... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-2e93c02853a5ae13f9f16ee1f2b59a3dbf4eef1a7507d42361eede67e6c13cc2 | Use the Simplex method to maximize the function $x_1 + x_2 + x_3$ subject to the constraints: \[ \begin{aligned} &x_1 + 2x_2 + 3x_3 \leq 5, \\ &3x_1 + x_2 + 2x_3 \leq 4, \\ &2x_1 + 3x_2 + x_3 \leq 3, \\ &x_1 + x_2 + x_3 \geq 2. \end{aligned} \] | [
"2"
] | [
"Use the Simplex method to maximize the function $x_1 + x_2 + x_3$ subject to the constraints:\n\\[\n\\begin{aligned}\n&x_1 + 2x_2 + 3x_3 \\leq 5, \\\\\n&3x_1 + x_2 + 2x_3 \\leq 4, \\\\\n&2x_1 + 3x_2 + x_3 \\leq 3, \\\\\n&x_1 + x_2 + x_3 \\geq 2.\n\\end{aligned}\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-6827fcf2c44be44a3b18634f0dffc7292e2680d2f069cafd11a3682402c9ec10 | Use the Smith normal form to create the Jordan normal form for the matrix \(\begin{pmatrix} -1 & 2 & -1 & -2 \\ -1 & 1 & 0 & -1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 1 & -1 \end{pmatrix}\). | [
"\\begin{pmatrix} i & 1 & 0 & 0 \\\\ 0 & i & 0 & 0 \\\\ 0 & 0 & -i & 1 \\\\ 0 & 0 & 0 & -i \\end{pmatrix}"
] | [
"Use the Smith normal form to create the Jordan normal form for the matrix \\(\\begin{pmatrix} -1 & 2 & -1 & -2 \\\\ -1 & 1 & 0 & -1 \\\\ 0 & 0 & 1 & -2 \\\\ 0 & 0 & 1 & -1 \\end{pmatrix}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-b689596983989769708e8e92636db554572548dc6c38b5d8ce80c3420bcc027f | Use the Taylor Series found in part (a) to find the limit as \( x \) approaches 0 of \( \frac{1 - x^2 - f(x)}{x^4} \). | [
"-\\frac{1}{2}"
] | [
"Use the Taylor Series found in part (a) to find the limit as \\( x \\) approaches 0 of \\( \\frac{1 - x^2 - f(x)}{x^4} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 19
} | 33 | 0 | 0 |
omr-ws-3ee49438111b52a67e3daf995c2daae888b7b7d495c704a61cc01715e285dc90 | Use the Taylor expansions to calculate $e^{i \epsilon A}$ in closed form, where \[ A = \begin{bmatrix} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{bmatrix} \] and $\epsilon$ is a small positive real number. | [
"\\( I \\cos \\epsilon + i A \\sin \\epsilon \\)"
] | [
"Use the Taylor expansions to calculate $e^{i \\epsilon A}$ in closed form, where\n\\[ A = \\begin{bmatrix} \n\\cos \\theta & \\sin \\theta \\\\\n\\sin \\theta & -\\cos \\theta\n\\end{bmatrix} \\]\nand $\\epsilon$ is a small positive real number."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-6f9941a7f5fd349978425913157028f798f9dcef3648c80e9b1add88358d3571 | Use the Taylor series of $arctan(x)$ for $-1 \leq x \leq 1$ to obtain the first five elements of the Taylor series for $arctan\left(\frac{1-x}{1+x}\right)$. | [
"\\(-\\frac{\\pi}{4} + x - \\frac{x^3}{3} + \\frac{x^5}{5}\\)"
] | [
"Use the Taylor series of $arctan(x)$ for $-1 \\leq x \\leq 1$ to obtain the first five elements of the Taylor series for $arctan\\left(\\frac{1-x}{1+x}\\right)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-781c9580d02acb31460f7f7f1de68a44a55a939fe76d08bf2a9af1f7e44010f5 | Use the Trapezium Rule with $n$ subintervals to approximate $ \int_{0}^{a} x^2 \, dx $. Find the limit of this approximation as $n$ approaches infinity and express your answer in terms of $a$. | [
"\\dfrac{a^3}{3}"
] | [
"Use the Trapezium Rule with $n$ subintervals to approximate $ \\int_{0}^{a} x^2 \\, dx $. Find the limit of this approximation as $n$ approaches infinity and express your answer in terms of $a$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e2da1349e730f8557c95c1bff7b4cc4465859fa6a1117253e0f3473126510b9 | Use the base conversion formula to solve the expression $\log_2 20 = \frac{\log_4 20}{\log_4 2}$. | [
"\\log_2 20"
] | [
"Use the base conversion formula to solve the expression $\\log_2 20 = \\frac{\\log_4 20}{\\log_4 2}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-941853ae267c4cab1226a97a96c23774eebb9eb219b75f760b60ce714196db99 | Use the binomial expansion for \( n = \frac{1}{3} \) to approximate \( \sqrt[3]{26} \). | [
"\\(\\frac{80}{27}\\)"
] | [
"Use the binomial expansion for \\( n = \\frac{1}{3} \\) to approximate \\( \\sqrt[3]{26} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 13
} | 27 | 0 | 0 |
omr-ws-9425baa7be239bb12d20e13a980dde1d9d2eb86501cfc4b107937e7eed9902d4 | Use the binomial theorem to find the sum of the infinite series \(1 - \frac{1}{6} + \frac{(1 \cdot 3)}{2!} \left(\frac{1}{6}\right)^2 - \frac{(1 \cdot 3 \cdot 5)}{3!} \left(\frac{1}{6}\right)^3 + \cdots\). | [
"\\( \\sqrt{\\frac{6}{5}} \\)"
] | [
"Use the binomial theorem to find the sum of the infinite series \\(1 - \\frac{1}{6} + \\frac{(1 \\cdot 3)}{2!} \\left(\\frac{1}{6}\\right)^2 - \\frac{(1 \\cdot 3 \\cdot 5)}{3!} \\left(\\frac{1}{6}\\right)^3 + \\cdots\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e76da53c0cfe97a4153220a27e99b24d952dd19d5c191b843dc1935793fd424 | Use the change of variables between $\{S_n\}$ and $\{Y_n\}$ to compute the probability of $S_1, S_2, S_3 \geq 0$. | [
"\\dfrac{3}{8}"
] | [
"Use the change of variables between $\\{S_n\\}$ and $\\{Y_n\\}$ to compute the probability of $S_1, S_2, S_3 \\geq 0$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e72d6c7789b8113f066cca91847773825e21eb01ccdd914f0e013edd96c7d916 | Use the convolution theorem and Laplace transforms to solve the following inverse Laplace transform: \[ \mathcal{L}^{-1} \left\{ \frac{s}{(s-1)(s+1)} \right\} \] | [
"\\cosh(t)"
] | [
"Use the convolution theorem and Laplace transforms to solve the following inverse Laplace transform:\n\n\\[ \\mathcal{L}^{-1} \\left\\{ \\frac{s}{(s-1)(s+1)} \\right\\} \\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e6a096779ea2480630f7ea6724b1c56bf1469587417347f3c8c1a25504628317 | Use the convolution theorem to find the convolution of $\sin(3t)$ and $t$. | [
"\\dfrac{t}{3} - \\dfrac{\\sin(3t)}{9}"
] | [
"Use the convolution theorem to find the convolution of $\\sin(3t)$ and $t$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4188949f856cd86bb0cef6513cd885fcd2daeb608b605438f9b504686d42d00 | Use the convolution theorem to find the convolution of $e^{2t}$ and $e^{3t}$. | [
"e^{3t} - e^{2t}"
] | [
"Use the convolution theorem to find the convolution of $e^{2t}$ and $e^{3t}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7663f58035ce243a9b9373d5ef3ec97ef73a25e556e35b0e4677661c82a2e961 | Use the definition of an ellipse to find an equation of an ellipse with foci at (-1,1) and (1,-1) and the difference of focal radii equal to 2. | [
"xy = -\\frac{1}{2}"
] | [
"Use the definition of an ellipse to find an equation of an ellipse with foci at (-1,1) and (1,-1) and the difference of focal radii equal to 2."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-31f998ad5027f29644ee47f77d020b3460ad253d5ad921787fb8193abde2e63e | Use the definition of an ellipse to find an equation of an ellipse with foci at (1,1) and (-1,-1) and the sum of focal radii equal to 3. | [
"\\(\\sqrt{(x-1)^2 + (y-1)^2} + \\sqrt{(x+1)^2 + (y+1)^2} = 3\\)"
] | [
"Use the definition of an ellipse to find an equation of an ellipse with foci at (1,1) and (-1,-1) and the sum of focal radii equal to 3."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d743592c5b704b8d6290aa19e7e682407eabfcbfa0286fa05f72512e6f32a01b | Use the definition of the definite integral to calculate the area bounded by the curves $y=\ln x$, $y=-\frac{x}{2}+2$, and the $x$-axis. Assume the $x$-coordinate of the intersection point is $2.3183$. | [
"1.619"
] | [
"Use the definition of the definite integral to calculate the area bounded by the curves $y=\\ln x$, $y=-\\frac{x}{2}+2$, and the $x$-axis. Assume the $x$-coordinate of the intersection point is $2.3183$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-a24fb58b2926624fa5e11b284e7f5ef707b3b2289e387376d5beb65a23cb3279 | Use the definition of the signed curvature $\kappa = \frac{f''(x)}{(1+(f'(x))^{2})^{\frac{3}{2}}}$ to determine all curves for which the signed curvature is a non-zero constant. What is the geometrical significance of $\kappa$ for these curves? | [
"$\\kappa = \\frac{1}{r}$"
] | [
"Use the definition of the signed curvature $\\kappa = \\frac{f''(x)}{(1+(f'(x))^{2})^{\\frac{3}{2}}}$ to determine all curves for which the signed curvature is a non-zero constant. What is the geometrical significance of $\\kappa$ for these curves?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 4
} | 17 | 0 | 0 |
omr-ws-4f62ad3cfc9d8d25d6510e8108a13978bad8687482da19b9924fe49430cf5b20 | Use the definition of uniform continuity to determine whether the function $f(x) = x \sin \frac{1}{x}$ is uniformly continuous on the interval $(0, \pi)$. | [
"The function \\( f(x) = x \\sin \\frac{1}{x} \\) is uniformly continuous on the interval \\( (0, \\pi) \\)."
] | [
"Use the definition of uniform continuity to determine whether the function $f(x) = x \\sin \\frac{1}{x}$ is uniformly continuous on the interval $(0, \\pi)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 26
} | 40 | 0 | 0 |
omr-ws-c6f6eec3ea5e9ecb803e70984b12fe2cd158ab5d9144f063a11e6a4ba05cf0b7 | Use the definition to differentiate the function \( f(x) = \sqrt{5x} \). | [
"\\( \\frac{\\sqrt{5}}{2\\sqrt{x}} \\)"
] | [
"Use the definition to differentiate the function \\( f(x) = \\sqrt{5x} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-08818c1c6f71d89dfb25e036cec0f7531ee8cce095e4123e91a3b98691072ebc | Use the derivative \[ \frac{d^{2n}}{dz^{2n}}\left\{\frac{1}{\sqrt{(1-z)(1+z)}}\right\} \] to find the Laurent Series coefficients for the function \( f(z) = \frac{1}{\sqrt{(z-1)(z+1)}} \) for \( |z| > 1 \). | [
"\\(\\frac{(2n)!}{(2^n n!)^2} \\frac{1}{z^{2n+1}}\\)"
] | [
"Use the derivative \n\\[ \\frac{d^{2n}}{dz^{2n}}\\left\\{\\frac{1}{\\sqrt{(1-z)(1+z)}}\\right\\} \\]\nto find the Laurent Series coefficients for the function \\( f(z) = \\frac{1}{\\sqrt{(z-1)(z+1)}} \\) for \\( |z| > 1 \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-c5ab089d784c4b4d6a2d493c281c756029792e017f44f62b6bd19004347ec16b | Use the digits 2, 0, 1, and 6 exactly once, along with the operations +, -, x, ÷, sqrt (square root), ^ (raise to a power), “!” (factorial), combinations, and permutations, to write expressions for the counting numbers 1 through 100. The decimal point may be used, and multi-digit numbers such as 20 or 201 may be used. ... | [
"10"
] | [
"Use the digits 2, 0, 1, and 6 exactly once, along with the operations +, -, x, ÷, sqrt (square root), ^ (raise to a power), “!” (factorial), combinations, and permutations, to write expressions for the counting numbers 1 through 100. The decimal point may be used, and multi-digit numbers such as 20 or 201 may be u... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-8181d908df62c0b687325b38734822ce405221d33cd556cb53322590c8665262 | Use the distance formula to determine if \(PQ = QR\) given the points \(P(4, -4)\), \(Q(2, -6)\), and \(R(-1, 3)\). | [
"\\(\\frac{-31}{3}\\)"
] | [
"Use the distance formula to determine if \\(PQ = QR\\) given the points \\(P(4, -4)\\), \\(Q(2, -6)\\), and \\(R(-1, 3)\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16210d70cb954be21f21443b0c0c6b3f846331e7a310ffa66f76a6f4547c0286 | Use the distance formula to find a polynomial equation with integer coefficients for the set of coplanar points such that for each point \((x, y)\), its distance from the point \((-4, -1)\) is \(\frac{2}{3}\) times its distance from the line \( x = -5 \). Identify the conic. | [
"5x^2 + 9y^2 + 32x + 18y + 53 = 0"
] | [
"Use the distance formula to find a polynomial equation with integer coefficients for the set of coplanar points such that for each point \\((x, y)\\), its distance from the point \\((-4, -1)\\) is \\(\\frac{2}{3}\\) times its distance from the line \\( x = -5 \\). Identify the conic."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-0e54aae363335c0bac2826d2cd63b526b967e94a9c0a226209bba842e0071c6b | Use the divergence theorem to find the volume of the region inside the surface \( S \) in \( \mathbb{R}^3 \) defined by the parametric equations: \[ r = 2 + \cos t + \cos u \] \[ \theta = t \quad \text{with} \quad 0 \leq t \leq 2\pi \] \[ z = \sin u \quad \text{with} \quad 0 \leq u \leq 2\pi \] | [
"4\\pi^2"
] | [
"Use the divergence theorem to find the volume of the region inside the surface \\( S \\) in \\( \\mathbb{R}^3 \\) defined by the parametric equations:\n\\[ r = 2 + \\cos t + \\cos u \\]\n\\[ \\theta = t \\quad \\text{with} \\quad 0 \\leq t \\leq 2\\pi \\]\n\\[ z = \\sin u \\quad \\text{with} \\quad 0 \\leq u \\leq... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c324f986f8edaedf72fcdba54c18f470f691796d0db8cd35d097811486f83f1 | Use the divergence theorem to find the volume of the region inside the surface \( S \) in \( \mathbb{R}^3 \) defined by the parametric equations: \[ r = 2 + \cos t + \cos u \] \[ \theta = t, \quad 0 \leq t \leq 2\pi \] \[ z = \sin u, \quad 0 \leq u \leq 2\pi \] | [
"4\\pi^2"
] | [
"Use the divergence theorem to find the volume of the region inside the surface \\( S \\) in \\( \\mathbb{R}^3 \\) defined by the parametric equations:\n\\[ r = 2 + \\cos t + \\cos u \\]\n\\[ \\theta = t, \\quad 0 \\leq t \\leq 2\\pi \\]\n\\[ z = \\sin u, \\quad 0 \\leq u \\leq 2\\pi \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-6968ddc496e994717f01aa4c42744ab8fd2158f858c600bd1ebdba58aec7d97a | Use the equality \[ \frac{{\arctan x}}{x} = \int\limits_0^1 {\frac{{dy}}{{1 + {x^2}{y^2}}}} \] to calculate the integral \[ \int\limits_0^1 {\frac{{\arctan x}}{x}\frac{{dx}}{{\sqrt {1 - {x^2}} }}}. \] | [
"\\dfrac{\\pi}{2} \\ln(1 + \\sqrt{2})"
] | [
"Use the equality\n\\[\n\\frac{{\\arctan x}}{x} = \\int\\limits_0^1 {\\frac{{dy}}{{1 + {x^2}{y^2}}}}\n\\]\nto calculate the integral\n\\[\n\\int\\limits_0^1 {\\frac{{\\arctan x}}{x}\\frac{{dx}}{{\\sqrt {1 - {x^2}} }}}.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36f11efc8bd261ab52d44d11ac13f667ce5f0158a6960ff142e99c3c80012cf6 | Use the equation $\det(I+uv^T)=1\pm v^Tu$ to find the eigenvalues of $uv^T$. | [
"\\( 0 \\) (with multiplicity \\( n-1 \\)) and \\( v^Tu \\) (if \\( v^Tu \\neq 0 \\))"
] | [
"Use the equation $\\det(I+uv^T)=1\\pm v^Tu$ to find the eigenvalues of $uv^T$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 23
} | 31 | 0 | 0 |
omr-ws-ca468ee9d614da52a4aa1ef204ec1b2869ae13d315b65dcdca5d9b74c1eb36ec | Use the equation from part a to find at what time(s) a rider would be 10 feet to the left or at the center during the first rotation. | [
"\\( t = \\frac{60}{\\pi} \\cos^{-1} \\left(-\\frac{1}{6}\\right), \\frac{60}{\\pi} \\left(2\\pi - \\cos^{-1} \\left(-\\frac{1}{6}\\right)\\right), 30, 90 \\)"
] | [
"Use the equation from part a to find at what time(s) a rider would be 10 feet to the left or at the center during the first rotation."
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2fe507494022d0d48eda54834e13f4b6e155dcfe19348b5db6aae1c1c0f1fffb | Use the existence and uniqueness theorem to find all solutions of the differential equation \( x'' + 4x = 0 \). | [
"\\( x(t) = C_1 \\cos(2t) + C_2 \\sin(2t) \\)"
] | [
"Use the existence and uniqueness theorem to find all solutions of the differential equation \\( x'' + 4x = 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-35bf80904ae53fbfac7b7f7fdd7cb6961f3fe66a4cb856c8632b5ed03c4eeff2 | Use the exponential generating function to find the number of n-digit words from the alphabet {0,1,2} in which none of the digits appears exactly three times. | [
"3^n - 3 \\cdot 2^{n-3} \\binom{n}{3} + 60 \\binom{n}{6} - 1680 \\cdot \\delta_{n,9}"
] | [
"Use the exponential generating function to find the number of n-digit words from the alphabet {0,1,2} in which none of the digits appears exactly three times."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-bc1e903ff542d8c7bbd6955686f97cfba96e81964c804122df3c31fee2fae0e1 | Use the expression $\left(fg\right)^{n}=\sum_{k=0}^{n}\binom{n}{k}f^{(k)}g^{(n-k)}$ with the formula $f'(x)(x^{2}+1)=1$ to calculate $arctan^{(n)}(0)$. | [
"$\\arctan^{(n)}(0) = \\begin{cases} (-1)^k (2k)! & \\text{if } n = 2k + 1 \\\\ 0 & \\text{if } n = 2k \\end{cases}$"
] | [
"Use the expression $\\left(fg\\right)^{n}=\\sum_{k=0}^{n}\\binom{n}{k}f^{(k)}g^{(n-k)}$ with the formula $f'(x)(x^{2}+1)=1$ to calculate $arctan^{(n)}(0)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 26
} | 38 | 0 | 0 |
omr-ws-6f2c6396e4caf2616b029a1d3c0ca059bb72abd999932925cb4e1461d116b30d | Use the expression $f(t) = t + A \sin t + B \cos t$ to find $A$ and $B$ by substituting for $f(t)$ and $f(x)$ in the equation \[ f(t) = t + \int_{0}^{\pi} f(x) \sin (x+t) \, dx, \] and equating coefficients of $\sin t$ and $\cos t$. | [
"\\( A = -2, B = 0 \\)"
] | [
"Use the expression $f(t) = t + A \\sin t + B \\cos t$ to find $A$ and $B$ by substituting for $f(t)$ and $f(x)$ in the equation\n\n\\[ f(t) = t + \\int_{0}^{\\pi} f(x) \\sin (x+t) \\, dx, \\]\n\nand equating coefficients of $\\sin t$ and $\\cos t$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 19
} | 19 | 0 | 0 |
omr-ws-d6b1b893fd08ef8bb61dd708e0c8ef12a57c6a90ebdef3dbbb08037640b95721 | Use the expressions for $\sin x$ and $\cos x$ in terms of $\tan \frac{1}{2}x$ to find five different sets of Pythagorean triples. (A Pythagorean triple is a set of numbers $a, b, c$ such that $a^{2} + b^{2} = c^{2}$.) | [
"\\((4, 3, 5), (6, 8, 10), (8, 15, 17), (10, 24, 26), (12, 35, 37)\\)"
] | [
"Use the expressions for $\\sin x$ and $\\cos x$ in terms of $\\tan \\frac{1}{2}x$ to find five different sets of Pythagorean triples. (A Pythagorean triple is a set of numbers $a, b, c$ such that $a^{2} + b^{2} = c^{2}$.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8c8c1236b903b88e1caed2a878560ac45026cf9e9e008f26bbcd1c1e4f6de613 | Use the extended Euclidean algorithm to find \( x \) such that \( x \equiv 3 \pmod{5} \) and \( x \equiv 2 \pmod{7} \). | [
"23"
] | [
"Use the extended Euclidean algorithm to find \\( x \\) such that \\( x \\equiv 3 \\pmod{5} \\) and \\( x \\equiv 2 \\pmod{7} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-664cb85b27ff4fcae8d9eddcf0c74e83e6e084a004d5d98eddc73211cbabe933 | Use the fact \( 1 + \frac{p}{2} \leq \frac{e^p - 1}{p} \leq 1 + \frac{p}{2} + p^2 \) for \( 0 < p \leq 1 \) to find the limit \( \lim_{p \to +0} (t_p - s_p) \). | [
"0"
] | [
"Use the fact \\( 1 + \\frac{p}{2} \\leq \\frac{e^p - 1}{p} \\leq 1 + \\frac{p}{2} + p^2 \\) for \\( 0 < p \\leq 1 \\) to find the limit \\( \\lim_{p \\to +0} (t_p - s_p) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-565a0c419d568735a7c79db76379e99dba397f50847d45388d94fb0a8b67dfc3 | Use the fact that $ax^2+bx+c$ can be written as $(x-\frac{-b+\sqrt{b^2-4ac}}{2a})(x-\frac{-b-\sqrt{b^2-4ac}}{2a})=0$ to discuss the graphs of the possible solutions of the inequality $ax^2+bx+c\le0$. | [
"(-\\infty, \\infty)"
] | [
"Use the fact that $ax^2+bx+c$ can be written as $(x-\\frac{-b+\\sqrt{b^2-4ac}}{2a})(x-\\frac{-b-\\sqrt{b^2-4ac}}{2a})=0$ to discuss the graphs of the possible solutions of the inequality $ax^2+bx+c\\le0$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-abed3b1119cb653f4cffbb79c112a5ac27a43759473e67ace374a2130a1af441 | Use the fact that \( \cos(x) = \frac{e^{ix} + e^{-ix}}{2} \) to deduce a formula for \( \cos^n(x) \). | [
"\\( \\cos^n(x) = \\frac{1}{2^n} \\sum_{k=0}^{n} \\binom{n}{k} e^{ix(2k - n)} \\)"
] | [
"Use the fact that \\( \\cos(x) = \\frac{e^{ix} + e^{-ix}}{2} \\) to deduce a formula for \\( \\cos^n(x) \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-ec8b37877dcd1d768bb267c7348f0e2ca1aafacca1e3487523657033d8979b39 | Use the fact that \( x + a \) is always a factor of \( x^3 + a^3 \) to find a formula for the sum of perfect cubes. | [
"\\( x^3 + a^3 = (x + a)(x^2 - ax + a^2) \\)"
] | [
"Use the fact that \\( x + a \\) is always a factor of \\( x^3 + a^3 \\) to find a formula for the sum of perfect cubes."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-c97d8c0713a40b3afe710201262b968d39d04697d6750772462cdcd34581006b | Use the fact that \(\int_0^1 \frac{1}{1+x^2} \, dx = \frac{\pi}{4}\) to find the value of \(\zeta(4)\). | [
"\\dfrac{\\pi^4}{90}"
] | [
"Use the fact that \\(\\int_0^1 \\frac{1}{1+x^2} \\, dx = \\frac{\\pi}{4}\\) to find the value of \\(\\zeta(4)\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6d446a76764c5ee96766951164fff2320f1eef932ed79645f0fe853c64db3506 | Use the factor theorem to factor the polynomial \(x^3 - 6x^2 + 6x - 5\). | [
"\\((x - 5)(x^2 + x + 1)\\)"
] | [
"Use the factor theorem to factor the polynomial \\(x^3 - 6x^2 + 6x - 5\\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-314d0e8388fda6291d829f129966dcff3ceb137f7811ed15420be65f16f5b558 | Use the factorization \(x^{4} + x^{2} +1 = (x^{2} - \sqrt{3} i x -1)(x^{2} + \sqrt{3} i x -1)\) to express \(\frac{1}{n^{4} + n^{2} + 1}\) in partial fractions. | [
"\\(\\frac{\\frac{i}{\\sqrt{3}(\\sqrt{3}i+1)}}{n+\\frac{\\sqrt{3}i+1}{2}}+ \\frac{\\frac{i}{\\sqrt{3}(\\sqrt{3}i-1)}}{n+\\frac{-\\sqrt{3}i+1}{2}}+ \\frac{\\frac{-i}{\\sqrt{3}(\\sqrt{3}i-1)}}{n+\\frac{\\sqrt{3}i-1}{2}}+ \\frac{\\frac{-i}{\\sqrt{3}(\\sqrt{3}i+1)}}{n-\\frac{\\sqrt{3}i+1}{2}}\\)"
] | [
"Use the factorization \\(x^{4} + x^{2} +1 = (x^{2} - \\sqrt{3} i x -1)(x^{2} + \\sqrt{3} i x -1)\\) to express \\(\\frac{1}{n^{4} + n^{2} + 1}\\) in partial fractions."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee2e55f2308dd25d00abdb395a5caac4348fa44b0978d94d84709db2849e22ac | Use the factorization from part (a) to calculate \(\sum_{k=1}^{n} \frac{k}{k^4 + k^2 + 1}\). | [
"\\(\\frac{1}{2} \\left(1 - \\frac{1}{n^2 + n + 1}\\right)\\)"
] | [
"Use the factorization from part (a) to calculate \\(\\sum_{k=1}^{n} \\frac{k}{k^4 + k^2 + 1}\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-db10728e9bd82418d7a0e0629279c90d87ea1cc2d6f0ffe9b6ab78cdfcbeedaf | Use the factorization method from the previous problem to factorize \( 3805 = 168^2 + 99^2 = 156^2 + 117^2 \). | [
"\\( 3805 = 5 \\cdot 761 \\)"
] | [
"Use the factorization method from the previous problem to factorize \\( 3805 = 168^2 + 99^2 = 156^2 + 117^2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 26
} | 40 | 0 | 0 |
omr-ws-d5b69658b6a63fbb16e62c7a3febcf9c44dcf840ced618fe527657dc7d89be83 | Use the formal definition of the limit to find the largest value of $\delta$ that satisfies $\epsilon = 1$ for $\lim_{x \to 5}(x^{2} -15x+50)$. | [
"\\(\\frac{5 - \\sqrt{21}}{2}\\)"
] | [
"Use the formal definition of the limit to find the largest value of $\\delta$ that satisfies $\\epsilon = 1$ for $\\lim_{x \\to 5}(x^{2} -15x+50)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-66c451e8961d04c4bcfca8d5b3ed948a79d2a584b7856c7c5166f09de9b7b3eb | Use the formula $$\sum_{a+b+2c=k}(-1)^{b+c}\begin{pmatrix} 65+a\\ a\end{pmatrix}\begin{pmatrix} 15+b \\ b \end{pmatrix} \begin{pmatrix} 15+c \\ c\end{pmatrix}$$ to determine the number of ways to make one dollar in change using fifty commemorative quarters and sixteen different presidential dollar coins. | [
"17"
] | [
"Use the formula\n$$\\sum_{a+b+2c=k}(-1)^{b+c}\\begin{pmatrix} 65+a\\\\ a\\end{pmatrix}\\begin{pmatrix} 15+b \\\\ b \\end{pmatrix} \\begin{pmatrix} 15+c \\\\ c\\end{pmatrix}$$\nto determine the number of ways to make one dollar in change using fifty commemorative quarters and sixteen different presidential dollar c... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-6f480db423de31932adf93f5265ee53c9a63383a057d4322ef3bb7702416e45a | Use the formula $\frac{\pi^{2}}{6}=1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots$ to compute the value of $\pi$ using an Excel spreadsheet. | [
"3.1416"
] | [
"Use the formula $\\frac{\\pi^{2}}{6}=1+\\frac{1}{2^{2}}+\\frac{1}{3^{2}}+\\frac{1}{4^{2}}+\\cdots$ to compute the value of $\\pi$ using an Excel spreadsheet."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-e077c1dff9d2a10094d5d1384a46ed41da256a5cd640467c106651d65246ddb1 | Use the formula $\frac{\pi}{2}=1+\frac{1}{3}(1+\frac{2}{5}(1+\frac{3}{7}(1+\cdots$ to compute the value of $\pi$ using an Excel spreadsheet. | [
"3.141592654"
] | [
"Use the formula $\\frac{\\pi}{2}=1+\\frac{1}{3}(1+\\frac{2}{5}(1+\\frac{3}{7}(1+\\cdots$ to compute the value of $\\pi$ using an Excel spreadsheet."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cbaaa18245dfee562184716f48f57a36fd62e6a6e3f39f658bf81aeb1035bf99 | Use the formula $\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots$ to compute the value of $\pi$ using an Excel spreadsheet. | [
"\\pi"
] | [
"Use the formula $\\frac{\\pi}{4}=1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+\\cdots$ to compute the value of $\\pi$ using an Excel spreadsheet."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-79cd7f2e46946c8d7a672b6de58d9a4342efb6a18427578a6a15e371eddd0d46 | Use the formula \( N(P) = 6066 - 31 \cos 2P \) to find the latitude \( P \) where the nautical mile is 6051 feet. | [
"\\( P \\approx 30.53^\\circ \\)"
] | [
"Use the formula \\( N(P) = 6066 - 31 \\cos 2P \\) to find the latitude \\( P \\) where the nautical mile is 6051 feet."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-0a340feafc2aac57f40adce4c532a37b04d7de81ff9ee20e0e5ad46cda448a3d | Use the formula \( SA = \int \int \sqrt{1 + \left(\frac{dz}{dr}\right)^2 + \frac{1}{r^2} \left(\frac{dz}{d\theta}\right)^2} \, dA \) to find the area of the spiral ramp defined by \( z = 0 \), \( 0 \leq r \leq 1 \), and \( 0 \leq \theta \leq \pi \). Hint: Use the integration formula \( \int \sqrt{u^2 + a^2} \, du = \fr... | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"Use the formula \\( SA = \\int \\int \\sqrt{1 + \\left(\\frac{dz}{dr}\\right)^2 + \\frac{1}{r^2} \\left(\\frac{dz}{d\\theta}\\right)^2} \\, dA \\) to find the area of the spiral ramp defined by \\( z = 0 \\), \\( 0 \\leq r \\leq 1 \\), and \\( 0 \\leq \\theta \\leq \\pi \\). Hint: Use the integration formula \\( \... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 4,
"tir": 27
} | 37 | 0 | 0 |
omr-ws-3157ef7383c3bd61e264eb0ae6c8bae6f6d6a1a4e9f79b5687ce5bf83eb8f658 | Use the formula \( \sum_{n \leq x} \tau(n) = x \log x + (2\gamma - 1)x + q(x) \), where \( x \geq 1 \) and \( |q(x)| \leq 4x^{1/2} \), to show that \( \sum_{n \leq 100} \tau(n) = 482 \). How can we use that formula given that \( 4\sqrt{100} > 1 \)? | [
"482"
] | [
"Use the formula \\( \\sum_{n \\leq x} \\tau(n) = x \\log x + (2\\gamma - 1)x + q(x) \\), where \\( x \\geq 1 \\) and \\( |q(x)| \\leq 4x^{1/2} \\), to show that \\( \\sum_{n \\leq 100} \\tau(n) = 482 \\). How can we use that formula given that \\( 4\\sqrt{100} > 1 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5cfa9f3ef3508fb10c60b5621ac5c7bba887a7324ac388bbb63a1da3487e0c24 | Use the formula for \( \binom{-n}{k} \) to find the coefficient of the power series that solves the problem of distributing seven stuffed animals among four children. | [
"120"
] | [
"Use the formula for \\( \\binom{-n}{k} \\) to find the coefficient of the power series that solves the problem of distributing seven stuffed animals among four children."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-9d063314bf87aeef1d7b65dcecac1357ab10b1838f823b2901e420dc5650fc21 | Use the function \( f(t) = \int_{0}^{+\infty} \frac{x - \sin x}{x^3} e^{-xt} \, dx \). | [
"\\frac{\\pi}{4}(1 - t^2) + \\frac{(t^2 - 1)}{2} \\arctan t + t \\ln\\left( \\frac{t}{\\sqrt{1 + t^2}} \\right) - \\frac{t}{2}"
] | [
"Use the function \\( f(t) = \\int_{0}^{+\\infty} \\frac{x - \\sin x}{x^3} e^{-xt} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-f00f8c1a84c96b5993e586e32e7b73f330260ec1a17d86ee29ce5bc0ac6c970c | Use the gamma function to find the value of the integral $\int_{0}^{\frac{\pi}{2}}\sin^4(x)\cos^6(x)dx$. Please explain how you are using the gamma function in your solution. | [
"$\\frac{3}{512}\\pi$"
] | [
"Use the gamma function to find the value of the integral $\\int_{0}^{\\frac{\\pi}{2}}\\sin^4(x)\\cos^6(x)dx$. Please explain how you are using the gamma function in your solution."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-4c8f8a5506c7770fea145a7d361b8471c24aee2d51d8ff29231ec72bdd4807b2 | Use the graph of \( f(x) = x + \frac{x}{\sqrt{x^2+1}} \) to determine the solutions of \( m\sqrt{x^2+1} = x \) for \( m \in \mathbb{R} \). | [
"x = \\dfrac{m}{\\sqrt{1 - m^2}} \\text{ for } |m| < 1"
] | [
"Use the graph of \\( f(x) = x + \\frac{x}{\\sqrt{x^2+1}} \\) to determine the solutions of \\( m\\sqrt{x^2+1} = x \\) for \\( m \\in \\mathbb{R} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-8bcbe70ee7ca0bb73e426f23e76bf35827783fea318c01a3095a1cf26b1f7cfe | Use the hint to compute the following limit: \[ \lim_{n \to 0^+} \int_{n}^{2n} \left( \frac{e^{-x} - 1}{x} + \frac{1}{x} \right) \, dx \] | [
"\\(\\ln(2)\\)"
] | [
"Use the hint to compute the following limit:\n\\[\n\\lim_{n \\to 0^+} \\int_{n}^{2n} \\left( \\frac{e^{-x} - 1}{x} + \\frac{1}{x} \\right) \\, dx\n\\]"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-277a0eda751417b7d2e2a93ca95e71ede30a39ddc33fb8434a49273a5b2fcfdf | Use the identity $\int_{0}^{a} f(x) \ dx = \int_{0}^{a} f(a-x) \ dx$ to evaluate $\int_{0}^{\frac{1}{2} \pi} \frac{\sin x}{\cos x + \sin x}\ dx$. | [
"$\\frac{\\pi}{4}$"
] | [
"Use the identity $\\int_{0}^{a} f(x) \\ dx = \\int_{0}^{a} f(a-x) \\ dx$ to evaluate $\\int_{0}^{\\frac{1}{2} \\pi} \\frac{\\sin x}{\\cos x + \\sin x}\\ dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-799481c52146fe26cda50957a2e460ef623206cfaebd77e5e9db0bde94535dd2 | Use the identity \((f(n) \cdot f(n+3))^2 + (2f(n+1) \cdot f(n+2))^2 = (f(2n+3))^2\) to generate six primitive Pythagorean triples. | [
"(3, 4, 5), (5, 12, 13), (39, 80, 89), (105, 208, 233)"
] | [
"Use the identity \\((f(n) \\cdot f(n+3))^2 + (2f(n+1) \\cdot f(n+2))^2 = (f(2n+3))^2\\) to generate six primitive Pythagorean triples."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5c5fb3c06cc7eb942d76941d947bb253170157507249920ab209579a594f699 | Use the identity \((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\) to formulate and solve a new problem. | [
"(5, 12, 13)"
] | [
"Use the identity \\((x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2\\) to formulate and solve a new problem."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-a2df2076e1539f6d734e0895b4d602a759560fedc3ee4e58c812155d68321764 | Use the integral $\int_{0}^{1} (1-x)^{5} e^{x}\ dx$ to show that $e \approx 2.72$. | [
"\\( e \\approx 2.72 \\)"
] | [
"Use the integral $\\int_{0}^{1} (1-x)^{5} e^{x}\\ dx$ to show that $e \\approx 2.72$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-c6531c3c45d265a09ad756711e0ce5a3dfdcb65ecc4a665c358225934604a3fd | Use the integral $\sinh^{-1} x = \int \frac{1}{\sqrt{1+x^{2}}}\ dx$ to find the expansion of $\sinh^{-1} x$ as the sum of powers of $x$, giving the first three terms and an expression for the general term. | [
"\\( x - \\frac{x^3}{6} + \\frac{3x^5}{40} + \\cdots \\) and the general term is \\( \\frac{(-1)^n (1 \\cdot 3 \\cdot 5 \\cdots (2n-1))}{2^n n! (2n+1)} x^{2n+1} \\)."
] | [
"Use the integral $\\sinh^{-1} x = \\int \\frac{1}{\\sqrt{1+x^{2}}}\\ dx$ to find the expansion of $\\sinh^{-1} x$ as the sum of powers of $x$, giving the first three terms and an expression for the general term."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 25
} | 36 | 0 | 0 |
omr-ws-f5af135eb73e299138738b17fc1b017f7823f4b1f1c957414d9ccb481afc323a | Use the intersect function from your calculator to solve the equation \(3\sqrt{x+4}=x^2-1\). Give the solution and sketch the graph on the axes. | [
"2.99"
] | [
"Use the intersect function from your calculator to solve the equation \\(3\\sqrt{x+4}=x^2-1\\). Give the solution and sketch the graph on the axes."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-cafa2902f371f44bded90b44f4aeb26e68426a59c4ab6f9f516c037ebe30b45f | Use the inverse of the matrix \( A \) to decode the cryptogram below: \[ A = \begin{pmatrix} 1 & -2 & 2 \\ -1 & 1 & 3 \\ 1 & -1 & 4 \end{pmatrix} \] Cryptogram: \( 13, -26, 21, 33, -53, -12, 18, -23, -42, 5, -20, 56, -24, 23, 77 \) | [
"INFORMATION"
] | [
"Use the inverse of the matrix \\( A \\) to decode the cryptogram below:\n\\[ A = \\begin{pmatrix} 1 & -2 & 2 \\\\ -1 & 1 & 3 \\\\ 1 & -1 & 4 \\end{pmatrix} \\]\nCryptogram: \\( 13, -26, 21, 33, -53, -12, 18, -23, -42, 5, -20, 56, -24, 23, 77 \\)"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-24832a6bb13454ae0c9716334ff2093030f84a246598f6fc4d761f20ecc53c44 | Use the iteration formula \( x_0 = 2.1 \) and \( x_{n+1} = \pi - \arcsin\left(\frac{2}{5}x_n\right) \) to find \(\alpha\) correct to two decimal places. | [
"2.12"
] | [
"Use the iteration formula \\( x_0 = 2.1 \\) and \\( x_{n+1} = \\pi - \\arcsin\\left(\\frac{2}{5}x_n\\right) \\) to find \\(\\alpha\\) correct to two decimal places."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-f6b8e431987f825cb4742b29adf3c0db4eba768df5299a45852d694452752d09 | Use the line integral to find the area of the part cut off the vertical square cylinder \( |x| + |y| = a \) by a sphere \( x^2 + y^2 + z^2 = a^2 \). Verify the answer by finding a trivial method to solve this problem. | [
"\\( 2\\pi a^2 \\)"
] | [
"Use the line integral to find the area of the part cut off the vertical square cylinder \\( |x| + |y| = a \\) by a sphere \\( x^2 + y^2 + z^2 = a^2 \\). Verify the answer by finding a trivial method to solve this problem."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 25
} | 46 | 0 | 0 |
omr-ws-969a04eb513d51f5bb46fbf8e3ba932d5f3b6291b32063db4c3ccadcbc2218cd | Use the linear regression method to determine the constants \( A \) and \( B \) for the relationship \( R = A + Bt \) given the data: | Culture | t (hours) | R (units/hour) | |---------|-----------|----------------| | A | 2.9 | 10.0 | | B | 6.1 | 13.1 | | C | 0.7 | 8.1 | | D | 6.9 | 13.4 | | E | 4.8 | 10.8 | | F | 8.9 ... | [
"A = 7.543 \\text{ and } B = 0.813"
] | [
"Use the linear regression method to determine the constants \\( A \\) and \\( B \\) for the relationship \\( R = A + Bt \\) given the data:\n\n| Culture | t (hours) | R (units/hour) |\n|---------|-----------|----------------|\n| A | 2.9 | 10.0 |\n| B | 6.1 | 13.1 |\n| C ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-0d635f5edda4b4d5fecba172b39f5015290bf261843360792dc56538dab03166 | Use the linear system to calculate the profit or loss when 120, 240, 318, and 146 books are sold. Show your calculations. | [
"376"
] | [
"Use the linear system to calculate the profit or loss when 120, 240, 318, and 146 books are sold. Show your calculations."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-d7448aa704a141ffa0d4abbac4d54810d265649937ecbb80b66431f8ee0a97db | Use the linearization of \( f(x) = \sqrt{e^x} \) to approximate \( \frac{1}{e^{0.1}} \). | [
"0.95"
] | [
"Use the linearization of \\( f(x) = \\sqrt{e^x} \\) to approximate \\( \\frac{1}{e^{0.1}} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-85f85061d1ac087b33567c85c642a7e8abdcc5f1f284be6696d1024cfa576d15 | Use the massage tool technique to evaluate the sum \( \sum_{n=1}^{10000} \frac{1}{\sqrt{n}} \). | [
"198.54"
] | [
"Use the massage tool technique to evaluate the sum \\( \\sum_{n=1}^{10000} \\frac{1}{\\sqrt{n}} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-1eb3b8fe81c963d25e5e13927b67e37e2a7641064f13374814f038bce82e1597 | Use the matrix method to solve the following system of equations: \[ x^2 + y^2 + z^2 = 9 \] \[ 2x^2 + 5y^2 + 7z^2 = 52 \] \[ 2x^2 + y^2 - z^2 = 0 \] | [
"\\(x = \\pm 1\\), \\(y = \\pm \\sqrt{3}\\), \\(z = \\pm \\sqrt{5}\\)"
] | [
"Use the matrix method to solve the following system of equations:\n\\[ x^2 + y^2 + z^2 = 9 \\]\n\\[ 2x^2 + 5y^2 + 7z^2 = 52 \\]\n\\[ 2x^2 + y^2 - z^2 = 0 \\]"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6af303bfa81485ef0bf8aa8430df036a97f9fbdebaecf41c568103502e529f7d | Use the matrix of cofactors to calculate the inverse of the matrix \( A \), where \[ A = \begin{pmatrix} \cos(t) & 0 & \sin(t) \\ 0 & 1 & 0 \\ \sin(t) & 0 & \cos(t) \end{pmatrix}. \] | [
"\\[ A^{-1} = \\frac{1}{1-2\\sin^2(t)} \\cdot \\begin{pmatrix} \\cos(t) & 0 & -\\sin(t) \\\\ 0 & 1-2\\sin^2(t) & 0 \\\\ -\\sin(t) & 0 & \\cos(t) \\end{pmatrix}. \\]"
] | [
"Use the matrix of cofactors to calculate the inverse of the matrix \\( A \\), where\n\\[ A = \\begin{pmatrix} \\cos(t) & 0 & \\sin(t) \\\\ 0 & 1 & 0 \\\\ \\sin(t) & 0 & \\cos(t) \\end{pmatrix}. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-aac2d9da88b5b3e6fe454d7a4cbe40e7c5ed9ae3ec36c48060cca8de4f308a1a | Use the method of Frobenius to obtain two linearly independent series solutions about \( x = 0 \) for the differential equation \( 2xy'' + 5y' + xy = 0 \). After making a few substitutions, the equation becomes: \[ 2\sum_{n=0}^{\infty}(n+1)(n+r-1)c_{n}x^{n+r-1} + 5\sum_{n=0}^{\infty}(n+r)c_{n}x^{n+r-1} + \sum_{n=0}^{\i... | [
"\\[ y = \\frac{C_{1}}{x^{\\frac{3}{2}}}\\left(1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{40} + \\cdots \\right) + C_{2}\\left(1 - \\frac{x^{2}}{14} + \\frac{x^{4}}{616} + \\cdots \\right) \\]"
] | [
"Use the method of Frobenius to obtain two linearly independent series solutions about \\( x = 0 \\) for the differential equation \\( 2xy'' + 5y' + xy = 0 \\). \n\nAfter making a few substitutions, the equation becomes:\n\\[ 2\\sum_{n=0}^{\\infty}(n+1)(n+r-1)c_{n}x^{n+r-1} + 5\\sum_{n=0}^{\\infty}(n+r)c_{n}x^{n+r-... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 16
} | 18 | 0 | 0 |
omr-ws-098f773a0bafbdefe65ddf6c10b9f22d28604a687c212737b48d24bf4f2475ac | Use the method of Gauss (Gaussian elimination) to solve the following system of linear equations: \[ 7x + 2y + 3z = 15 \] \[ 5x - 3y + 2z = 15 \] \[ 10x - 11y + 5z = 36 \] | [
"(2, -1, 1)"
] | [
"Use the method of Gauss (Gaussian elimination) to solve the following system of linear equations:\n\\[ 7x + 2y + 3z = 15 \\]\n\\[ 5x - 3y + 2z = 15 \\]\n\\[ 10x - 11y + 5z = 36 \\]"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 26
} | 36 | 0 | 0 |
omr-ws-df1847ce10ae4933c2f4e52993bae945cb64f52b31b0dec98bca241887581772 | Use the method of cylindrical shells to find the volume of the solid generated by rotating the region bounded by $y=x^{3}-x^{2}$, $y=0$, and $x=0$ about the line $x = -1$. | [
"\\(\\frac{4\\pi}{15}\\)"
] | [
"Use the method of cylindrical shells to find the volume of the solid generated by rotating the region bounded by $y=x^{3}-x^{2}$, $y=0$, and $x=0$ about the line $x = -1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-93c2e9530a9cad0b45d5fc9bb1284224a9c0b3792aacf7b0e5fc808d581411d4 | Use the method of differences on the result from part (i) to find the sum of the infinite series $\frac{1}{2}\left(\frac{3}{5}\right)^{2} + \frac{1}{4}\left(\frac{3}{5}\right)^{4} + \frac{1}{6}\left(\frac{3}{5}\right)^{6} + \cdots$. | [
"\\(\\ln\\left(\\frac{5}{4}\\right)\\)"
] | [
"Use the method of differences on the result from part (i) to find the sum of the infinite series $\\frac{1}{2}\\left(\\frac{3}{5}\\right)^{2} + \\frac{1}{4}\\left(\\frac{3}{5}\\right)^{4} + \\frac{1}{6}\\left(\\frac{3}{5}\\right)^{6} + \\cdots$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-d89b082c870a741f685e6c0bffddd34c4fb9c9d9353b6857e2a631b34a3bd329 | Use the method of generating functions to find a closed form expression for the sequence defined by $a_{0}=1$ and $a_{n+1}=\sum_{k=0}^{n}a_{k}a_{n-k}$. | [
"\\( a_n = \\frac{1}{n+1} \\binom{2n}{n} \\)"
] | [
"Use the method of generating functions to find a closed form expression for the sequence defined by $a_{0}=1$ and $a_{n+1}=\\sum_{k=0}^{n}a_{k}a_{n-k}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-5ca5f8d430c7923391dab064aa45c40134dd910b066fead4ddc1071ba831453f | Use the method of variation of parameters to find a particular solution of the differential equation $y'' + 4y = \sin^2 x$. Given: - The complementary solution is $y_c(x) = c_1 \cos 2x + c_2 \sin 2x$. - The particular solution should be found using the method of variation of parameters. | [
"\\( \\frac{1}{8}(1 - x \\sin 2x) \\)"
] | [
"Use the method of variation of parameters to find a particular solution of the differential equation $y'' + 4y = \\sin^2 x$. \n\nGiven:\n- The complementary solution is $y_c(x) = c_1 \\cos 2x + c_2 \\sin 2x$.\n- The particular solution should be found using the method of variation of parameters."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 5
} | 15 | 0 | 0 |
omr-ws-cfa52f71b46664806edb30cc290e0bc6805a62591f4c1d5bcc490bd7ddbf5270 | Use the method of variation of parameters to find a particular solution to the differential equation $y'' - 4y = \sinh 2x$. | [
"\\( \\frac{1}{4}x \\sinh 2x - \\frac{1}{16}\\sinh 2x \\)"
] | [
"Use the method of variation of parameters to find a particular solution to the differential equation $y'' - 4y = \\sinh 2x$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 2,
"tir": 7
} | 18 | 0 | 0 |
omr-ws-361ec0f3e8f9c450d5086ad71f503efbb3b24bf97bae157c6fddda2bde5a825e | Use the midpoint rule to evaluate the integral \(\int_{10}^{17} (-3x - 8x^2) \, dx\) with \(n = 3\). | [
"-10692.759"
] | [
"Use the midpoint rule to evaluate the integral \\(\\int_{10}^{17} (-3x - 8x^2) \\, dx\\) with \\(n = 3\\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-b906a40870d8d2c187fa697873d05ed1a82bc22ab781b32a8519822429ff4edf | Use the more refined formula $N!=N^Ne^{-N}\sqrt{2\pi N}$ to improve the estimates made for $P_{max}$ and compare it with the earlier estimate of $P_{max} \approx 1$. | [
"\\( P_{max} \\approx \\sqrt{\\frac{2}{\\pi N}} \\)"
] | [
"Use the more refined formula $N!=N^Ne^{-N}\\sqrt{2\\pi N}$ to improve the estimates made for $P_{max}$ and compare it with the earlier estimate of $P_{max} \\approx 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 2
} | 19 | 0 | 0 |
omr-ws-0fc37628b40e986aa71e8495ab05b17b0fe7a35f8b285310343c50217302be2b | Use the multinomial theorem to expand $(a+b+c)^5$ and determine each coefficient. | [
"The coefficients are given in the expansion provided in Post 3."
] | [
"Use the multinomial theorem to expand $(a+b+c)^5$ and determine each coefficient."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74a0dd9a83bff3c4b6f182079ddc3b6185f8126e42518fb02868c890c9a6698d | Use the multinomial theorem to expand $(x + y + z)^3$. | [
"\\(x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3y^2x + 3y^2z + 3z^2x + 3z^2y + 6xyz\\)"
] | [
"Use the multinomial theorem to expand $(x + y + z)^3$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
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