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-84053fa4e9919017211c4e4862ca46ba183a16a58d34f18e10ab33a83f5827e0 | Calculate the Fourier series of the function $\sin{(x-1)}\cos{2x}$. | [
"-\\frac{\\sin{1}}{2} \\cos{x} - \\frac{\\sin{1}}{2} \\cos{3x} - \\frac{\\cos{1}}{2} \\sin{x} + \\frac{\\cos{1}}{2} \\sin{3x}"
] | [
"Calculate the Fourier series of the function $\\sin{(x-1)}\\cos{2x}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-6e72fb7c690119a999d69eeb778c485833af3e46490f23d8f0349c35a202a1a4 | Calculate the Fourier series of the function $|\sin{(x)}|$. | [
"\\( \\frac{2}{\\pi} + \\sum_{n=1}^{\\infty} \\frac{2}{\\pi} \\left[ \\frac{1 - (-1)^{n+1}}{2n+1} + \\frac{(-1)^{n} - 1}{2n-1} \\right] \\cos(2nx) \\)"
] | [
"Calculate the Fourier series of the function $|\\sin{(x)}|$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 8
} | 16 | 0 | 0 |
omr-ws-3bf5bd269f29e5cb3255d51af4693fb6b29a6973633c2ec16d46e3f162ed841c | Calculate the Fourier transform of the discrete-time signal \( x(n) = \left(\frac{1}{2}\right)^{|n - 1|} + \delta(n + 2) - \delta(n - 2) \). | [
"\\frac{3 e^{-j\\omega}}{5 - 4 \\cos \\omega} + 2j \\sin(2\\omega)"
] | [
"Calculate the Fourier transform of the discrete-time signal \\( x(n) = \\left(\\frac{1}{2}\\right)^{|n - 1|} + \\delta(n + 2) - \\delta(n - 2) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c7f97d2773018ae5d90c75b174e46514f5efa361f4dc177beae9e4f497a063e5 | Calculate the Fourier transform of the function \( f(x) = \frac{x}{(x^2 + a^2)(x^2 + b^2)} \). | [
" \\frac{i\\pi \\, \\text{sign}(\\omega)}{a^2 - b^2} \\left( e^{-a |\\omega|} - e^{-b |\\omega|} \\right) "
] | [
"Calculate the Fourier transform of the function \\( f(x) = \\frac{x}{(x^2 + a^2)(x^2 + b^2)} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b83c2d472c91f38a6dbf18f7fd4f8df3606dfbef61eef52d2494a2d38c33d05 | Calculate the Fourier transform of the function \( f_a(x) = \begin{cases} 1 - \frac{|x|}{a}, & |x| < a \\ 0, & |x| \geq a \end{cases} \). | [
"\\(\\frac{4\\sin^2\\left(\\frac{a\\omega}{2}\\right)}{2\\pi a\\omega^2}\\)"
] | [
"Calculate the Fourier transform of the function \\( f_a(x) = \\begin{cases} 1 - \\frac{|x|}{a}, & |x| < a \\\\ 0, & |x| \\geq a \\end{cases} \\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 7
} | 10 | 0 | 0 |
omr-ws-a8ac9f3b4062e23c15c7d8ed9ef8aeb56b77c2ed31cb91ca7cf1de9bec767231 | Calculate the Fourier transform of the function \(\frac{J_1(a\sqrt{x^2 + b^2})}{\sqrt{x^2 + b^2}}\), where \(J_1\) is the first-order Bessel function of the first kind. Specifically, evaluate the integral \[ \int_{-\infty}^{\infty} \frac{J_1(a\sqrt{x^2 + b^2})}{\sqrt{x^2 + b^2}} \exp(-i2\pi wx) \, dx. \] | [
"\\frac{2 \\sin\\left(b \\sqrt{a^2 - (2\\pi w)^2}\\right)}{\\sqrt{a^2 - (2\\pi w)^2}}"
] | [
"Calculate the Fourier transform of the function \\(\\frac{J_1(a\\sqrt{x^2 + b^2})}{\\sqrt{x^2 + b^2}}\\), where \\(J_1\\) is the first-order Bessel function of the first kind. Specifically, evaluate the integral\n\n\\[\n\\int_{-\\infty}^{\\infty} \\frac{J_1(a\\sqrt{x^2 + b^2})}{\\sqrt{x^2 + b^2}} \\exp(-i2\\pi wx)... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-c492791769542039b5419e6cb8f28f77e1fb9c2f001a087cfd17bedab8d1e945 | Calculate the Frenet-Serret formulas for the curve given by $$\alpha(t)=(2\sin t,\cos t,0).$$ | [
"\\(\\begin{cases}{\\bf t}' = 2(t')^3 \\bf n \\\\ {\\bf n}' = -2(t')^3 \\bf t \\\\ {\\bf b}' = 0\\end{cases}\\) with \\(t' = \\frac{1}{\\sqrt{3\\cos^2 t + 1}}\\)"
] | [
"Calculate the Frenet-Serret formulas for the curve given by $$\\alpha(t)=(2\\sin t,\\cos t,0).$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 2
} | 14 | 0 | 0 |
omr-ws-d6301620456790f727b9f7780ddf712f41a839f383a45f61bfdd599c9ba1e2d2 | Calculate the Galois group \(\text{Gal}(E/\mathbb{Q})\) where \(E\) is the splitting field of \( f(x) = x^4 + 1 \) over \(\mathbb{Q}\). | [
"\\(V_4\\) or \\(\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/2\\mathbb{Z}\\)"
] | [
"Calculate the Galois group \\(\\text{Gal}(E/\\mathbb{Q})\\) where \\(E\\) is the splitting field of \\( f(x) = x^4 + 1 \\) over \\(\\mathbb{Q}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 29
} | 43 | 0 | 0 |
omr-ws-61e633f7e0e4d78bc5145e4744ba50f7e53ce84e3ef5106e1610cd54bd6ae6c8 | Calculate the Galois group of the polynomial $X^6 - 4X^3 + 1$ over $\mathbb{Q}$. | [
"\\( D_6 \\)"
] | [
"Calculate the Galois group of the polynomial $X^6 - 4X^3 + 1$ over $\\mathbb{Q}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 11
} | 28 | 0 | 0 |
omr-ws-84cdeeb86a0911e54fc0f7e3a4db3d511516f207e9cb47b77214e481d5857bc6 | Calculate the Galois group of the polynomial $X^{8}-3$ over $\mathbb{Q}$. | [
"\\mathbb{Z}/8\\mathbb{Z} \\rtimes (\\mathbb{Z}/8\\mathbb{Z})^\\times"
] | [
"Calculate the Galois group of the polynomial $X^{8}-3$ over $\\mathbb{Q}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5cef7ee9c3c0406b9009d6da1b085d337254c4c315e4b4a7e0fa5694e0dcc4d | Calculate the Galois group over $\mathbb{F}_3$ of the polynomial $f(x) = x^3 + 2x + 1$. | [
"\\( C_3 \\)"
] | [
"Calculate the Galois group over $\\mathbb{F}_3$ of the polynomial $f(x) = x^3 + 2x + 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-ae2adc740842673b75a23d0cdb5e2a849e2f81b637c0c97cd6099898aeebeacf | Calculate the Laplace transform of $\ln x$ and express your answer in terms of $s$, where $\gamma$ is the Euler-Mascheroni constant. | [
" -\\frac{\\gamma + \\ln s}{s} "
] | [
"Calculate the Laplace transform of $\\ln x$ and express your answer in terms of $s$, where $\\gamma$ is the Euler-Mascheroni constant."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25ef11ff288a2e65c4ba3769446cc0f75fa0492545e0b76a3ddf50de47bded2d | Calculate the Laplace transform of \( J_0(t) \) and express your answer in terms of \( s \). What is the simplified form of \( L(J_0(t)) \) for \( s > 1 \)? | [
"\\frac{1}{\\sqrt{s^2 + 1}}"
] | [
"Calculate the Laplace transform of \\( J_0(t) \\) and express your answer in terms of \\( s \\). What is the simplified form of \\( L(J_0(t)) \\) for \\( s > 1 \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-77a79eacdc0173d1f830e8f87cbd12282d551acd8df3bf78b78d322a0ec9462b | Calculate the Laplace transform of the function $\int_{t}^{\infty}\frac{e^{-x}}{x}dx$. | [
"\\(\\frac{\\log(s+1)}{s}\\)"
] | [
"Calculate the Laplace transform of the function $\\int_{t}^{\\infty}\\frac{e^{-x}}{x}dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-e9f689de3b58b7ce01313d939f9de7f0ced1a209b278373f906042989f4143f4 | Calculate the Laplacian of the vector function $\vec{F}(r)$, where $r = \sqrt{x^2 + y^2 + z^2}$, i.e., find $\nabla^2 \vec{F}(r)$. | [
"\\nabla^2 \\vec{F}(r) = \\vec{F}''(r) + \\frac{2}{r} \\vec{F}'(r)"
] | [
"Calculate the Laplacian of the vector function $\\vec{F}(r)$, where $r = \\sqrt{x^2 + y^2 + z^2}$, i.e., find $\\nabla^2 \\vec{F}(r)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-d063ec45bc2a1ebf86c405b50b8f905666de49069cb8271f50496ab327f547b4 | Calculate the Laurent series for the function \( f(z) = \frac{1}{\sqrt{(z-1)(z+1)}} \) for \( |z| > 1 \). | [
"\\(\\sum_{n = 0}^{\\infty}\\frac {(2n)!}{(2^n n!)^2 z^{2n + 1}}\\)"
] | [
"Calculate the Laurent series for the function \\( f(z) = \\frac{1}{\\sqrt{(z-1)(z+1)}} \\) for \\( |z| > 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 19
} | 33 | 0 | 0 |
omr-ws-5f67405c3c0fb7f950ae3f04fe5c5fa98a3b05a701c7f14505c694650fcd425a | Calculate the Lebesgue Constant for equally spaced points in $C[-1,1]$ where $x_i = i/10$ for $i = -10, \ldots, 10$. | [
"12883"
] | [
"Calculate the Lebesgue Constant for equally spaced points in $C[-1,1]$ where $x_i = i/10$ for $i = -10, \\ldots, 10$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e56c4de8bd4674938b4dcc5a566080d47a189a1f8576db165d05e96fcb290d7 | Calculate the Newton-Cotes quadrature and estimate the error for the integral $\int_{-2}^2 x^2e^x \, dx$ using Simpson's rule ($n=2$). | [
"\\(\\frac{8}{3} \\left[ e^{-2} + e^2 \\right]\\)"
] | [
"Calculate the Newton-Cotes quadrature and estimate the error for the integral $\\int_{-2}^2 x^2e^x \\, dx$ using Simpson's rule ($n=2$)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-90e3b9a08765cc55ee0be59fb4fd162c0b15b92e66f1ee3e8ddca1933fe6de52 | Calculate the Newton-Cotes quadrature and estimate the error for the integral $\int_{-2}^2 x^2e^x \, dx$ using the 3/8 rule ($n=3$). | [
"\\(2e^{-2} + \\frac{2}{3} e^{-\\frac{2}{3}} + \\frac{2}{3} e^{\\frac{2}{3}} + 2e^2\\)"
] | [
"Calculate the Newton-Cotes quadrature and estimate the error for the integral $\\int_{-2}^2 x^2e^x \\, dx$ using the 3/8 rule ($n=3$)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea2f94e49e1bafff3b472f27c1d9cfff61c747448caf2471e16229c617c5a9ec | Calculate the Newton-Cotes quadrature and estimate the error for the integral $\int_{-2}^2 x^2e^x \, dx$ using the Trapezoid rule ($n=1$). | [
"\\(8 \\left[ e^{-2} + e^{2} \\right]\\) and \\(-96e^2\\)"
] | [
"Calculate the Newton-Cotes quadrature and estimate the error for the integral $\\int_{-2}^2 x^2e^x \\, dx$ using the Trapezoid rule ($n=1$)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca6c1d7a83802e7dd11917e36e56dbef371daa3a2c0794c92b818a45d2132005 | Calculate the Pearson Product Moment Correlation Coefficient \( r \) for the following set of coordinates: - 10 coordinates of (2, 3) - 20 coordinates of (-1, -4) - 30 coordinates of (5, -2) - 40 coordinates of (-3, 1) | [
"\\(-0.333\\)"
] | [
"Calculate the Pearson Product Moment Correlation Coefficient \\( r \\) for the following set of coordinates:\n- 10 coordinates of (2, 3)\n- 20 coordinates of (-1, -4)\n- 30 coordinates of (5, -2)\n- 40 coordinates of (-3, 1)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-24ddee21cf85dc807023761c7fa81be5b5163da1cd3e5195b13b29b4334503b9 | Calculate the \( m \)-th power of the \( n \times n \) matrix \( A \) given by: \[ A = \begin{bmatrix} 1 & 1 & 1 & \cdots & 1 \\ 0 & 1 & 1 & \cdots & 1 \\ 0 & 0 & 1 & \cdots & 1 \\ \vdots & \vdots & \vdots & \cdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ \end{bmatrix} \] | [
"\n\\begin{bmatrix}\n\\dbinom{m-1}{0} & \\dbinom{m}{1} & \\dbinom{m+1}{2} & \\cdots & \\dbinom{m+n-2}{n-1} \\\\\n0 & \\dbinom{m-1}{0} & \\dbinom{m}{1} & \\cdots & \\dbinom{m+n-3}{n-2} \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & \\dbinom{m-1}{0}\n\\end{bmatrix}\n"
] | [
"Calculate the \\( m \\)-th power of the \\( n \\times n \\) matrix \\( A \\) given by:\n\\[ A = \\begin{bmatrix}\n1 & 1 & 1 & \\cdots & 1 \\\\\n0 & 1 & 1 & \\cdots & 1 \\\\\n0 & 0 & 1 & \\cdots & 1 \\\\\n\\vdots & \\vdots & \\vdots & \\cdots & \\vdots \\\\\n0 & 0 & 0 & \\cdots & 1 \\\\\n\\end{bmatrix} \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-73a8e590683260899a6531aec409f89595114be716ffe3ffe663cbac365ed98c | Calculate the \( n \)-th derivative of the function \( f(x) = (x^n - 1)^n \). | [
"\\( n! \\cdot n^n \\cdot x^{n(n-1)} \\)."
] | [
"Calculate the \\( n \\)-th derivative of the function \\( f(x) = (x^n - 1)^n \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-665b2173df3947ff36a7e36820b4ed84621c64873c4275646ff96c066fe78bb7 | Calculate the \( n \)-th derivative of the function \( f(x) = \frac{1}{4 + x^2} \). | [
"\\( f^{(n)}(x) = \\frac{(-1)^n\\,n!}{2} \\frac{\\sin\\Big((n+1)\\operatorname{arctan\\frac{2}{x}\\Big)}}{(\\sqrt{x^2+4})^{n+1}} \\)"
] | [
"Calculate the \\( n \\)-th derivative of the function \\( f(x) = \\frac{1}{4 + x^2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 4
} | 6 | 0 | 0 |
omr-ws-6a3091fb86d73d13448c2331167aa5b84a949e33009cfaf8cbd1337df12fbb49 | Calculate the \( n \)-th derivative of the function \( k(x) = \frac{\sin(bx + c)}{e^{ax}} \). | [
"\\((-1)^n \\left( \\sqrt{a^2 + b^2} \\right)^n \\sin(bx + c - nd) e^{-ax}\\)"
] | [
"Calculate the \\( n \\)-th derivative of the function \\( k(x) = \\frac{\\sin(bx + c)}{e^{ax}} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-0fafd3b9dda40e501846e75322f9e3bfb0de07b17e456cce0e9c3a01e154f293 | Calculate the alternating sum $2^{100} - 2^{99} + 2^{98} - \cdots + 2^0$. | [
"\\(\\frac{2^{101} + 1}{3}\\)"
] | [
"Calculate the alternating sum $2^{100} - 2^{99} + 2^{98} - \\cdots + 2^0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-7600650b5640c4e234390dcfb2677e8c60ce617424909d50da4965670145665a | Calculate the altitude $(h_S)$ of a pyramid with vertices $S (0, 6, 4)$, $A (3, 5, 3)$, $B ( - 2 , 11,-5 ) $, $C (1, - 1 , 4)$. | [
"3"
] | [
"Calculate the altitude $(h_S)$ of a pyramid with vertices $S (0, 6, 4)$, $A (3, 5, 3)$, $B ( - 2 , 11,-5 ) $, $C (1, - 1 , 4)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e02187b123e0ef151e390c709d6bd4c57f6ab6fd129a01256995c444723940c | Calculate the amount of fluid passing through the triangle with vertices (1,0,0), (0,2,0), and (0,0,3) when the flow vector is given as \( \mathbf{F} = (x, x, x) \). | [
"\\(\\frac{11}{6}\\)"
] | [
"Calculate the amount of fluid passing through the triangle with vertices (1,0,0), (0,2,0), and (0,0,3) when the flow vector is given as \\( \\mathbf{F} = (x, x, x) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-76d73d493d6284e749fbcb9d6a546941f8ad047216d850ed6b20148ea6fba8a5 | Calculate the amount of interest on a loan of $3,200 at 6% interest for 60 days using the ordinary interest method. | [
"32"
] | [
"Calculate the amount of interest on a loan of $3,200 at 6% interest for 60 days using the ordinary interest method."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39cc25fba4b93e190a787f3fdaec93e66112e12821d231c0559fc54b66ba3c06 | Calculate the angle $\angle A_{44}A_{45}A_{46}$ in a regular 72-gon. | [
"175^\\circ"
] | [
"Calculate the angle $\\angle A_{44}A_{45}A_{46}$ in a regular 72-gon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-8723d28eb4901376f2dbfc13419935961e904e0922b4efae4b7798f22dbd7b07 | Calculate the angle $\theta$ such that: \[ \cot(\theta) = \frac{\sqrt{3} \cos(20^\circ)}{2 + \sqrt{3} \cos(10^\circ) + \cos(40^\circ)} \] | [
"\\(70^\\circ\\)"
] | [
"Calculate the angle $\\theta$ such that:\n\\[ \\cot(\\theta) = \\frac{\\sqrt{3} \\cos(20^\\circ)}{2 + \\sqrt{3} \\cos(10^\\circ) + \\cos(40^\\circ)} \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-81ede8c6a939a534fbc01a0eb6887ba7e5816c016068676b335ce4abf880ab13 | Calculate the angle at which the root-locus of the polynomial \( p(s) = s^3 + 8s^2 + (K - 65)s + 2K \) crosses the imaginary axis, given that it crosses at points \( s = \pm j\sqrt{\frac{65}{3}} \) for \( K = \frac{260}{3} \). | [
"\\( 126.56^\\circ \\)"
] | [
"Calculate the angle at which the root-locus of the polynomial \\( p(s) = s^3 + 8s^2 + (K - 65)s + 2K \\) crosses the imaginary axis, given that it crosses at points \\( s = \\pm j\\sqrt{\\frac{65}{3}} \\) for \\( K = \\frac{260}{3} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 5
} | 18 | 0 | 0 |
omr-ws-12ae8edeb871798fe7f7368d547bec2f1309c90d1861fb4ae61cdb1bce46c6a4 | Calculate the angle between the following vectors: (a) ${\bf a} = (6, - 2 , - 3 ) ,\ {\bf b} = (5, 0, 0)$; (b) ${\bf a}= (2, - 4 , 5),\ {\bf b} = (0, 2, 0)$; (c) ${\bf a} = ( - 2 , 6, - 3 ) ,\ {\bf b} = (0, 0, -3 ) $; (d) ${\bf a} = ( - 4 , - 6 , 2),\ {\bf b} = (4, 0, 0)$; (e) ${\bf a} = (3, - 2 , 6),\ {\bf b} = (0, - ... | [
"(a) $\\arccos \\frac{6}{7}$; (b) $\\arccos \\left(-\\frac{4}{3\\sqrt 5}\\right)$; (c) $\\arccos \\frac{3}{7}$; (d) $\\arccos \\left(-\\frac{2}{\\sqrt {14}}\\right)$; (e) $\\arccos \\frac{2}{7}$; (f) $\\arccos \\left(-\\frac{2}{3\\sqrt 5}\\right)$"
] | [
"Calculate the angle between the following vectors: (a) ${\\bf a} = (6, - 2 , - 3 ) ,\\ {\\bf b} = (5, 0, 0)$; (b) ${\\bf a}= (2, - 4 , 5),\\ {\\bf b} = (0, 2, 0)$; (c) ${\\bf a} = ( - 2 , 6, - 3 ) ,\\ {\\bf b} = (0, 0, -3 ) $; (d) ${\\bf a} = ( - 4 , - 6 , 2),\\ {\\bf b} = (4, 0, 0)$; (e) ${\\bf a} = (3, - 2 , 6),... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6069a0e132d1686c1cf9b80ac5f498a9800c240dba834c7523f332b9cf43df0 | Calculate the angle between the vectors ${\bf a}=2{\bf i}+{\bf j}$ and ${\bf b}={\bf i}-2{\bf j}$ and find the lengths of the diagonals of the parallelogram constructed on these vectors as on sides. | [
"$90^\\circ$, $\\sqrt {10}$"
] | [
"Calculate the angle between the vectors ${\\bf a}=2{\\bf i}+{\\bf j}$ and ${\\bf b}={\\bf i}-2{\\bf j}$ and find the lengths of the diagonals of the parallelogram constructed on these vectors as on sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da65f966817a9130e8b7227b7880bbdebedd034af6c3ef35abfba5e0f2079962 | Calculate the angle created by a median line and the height from a vertex of a triangle, given that the measures of the other two angles of the triangle are 60 degrees and 80 degrees. | [
"\\( 11^\\circ 20' 17.22'' \\)"
] | [
"Calculate the angle created by a median line and the height from a vertex of a triangle, given that the measures of the other two angles of the triangle are 60 degrees and 80 degrees."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 21
} | 23 | 0 | 0 |
omr-ws-33eb8b7b57e0cbaebfd05514411a69c901329fdd37e75a1c5d47958506d2a5b0 | Calculate the angle that the plane $x + 3y + z = 3$ forms with the $xy$, $xz$, and $yz$ planes. | [
"\\(\\theta_{xy} = \\cos^{-1} \\left( \\frac{1}{\\sqrt{11}} \\right)\\), \\(\\theta_{xz} = \\cos^{-1} \\left( \\frac{3}{\\sqrt{11}} \\right)\\), \\(\\theta_{yz} = \\cos^{-1} \\left( \\frac{1}{\\sqrt{11}} \\right)\\)"
] | [
"Calculate the angle that the plane $x + 3y + z = 3$ forms with the $xy$, $xz$, and $yz$ planes."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 30
} | 33 | 0 | 0 |
omr-ws-34598360d293fc0fb1b81e5ff3c384ff18b03d1bb64f1a73d6eea377173c5e83 | Calculate the angles $AEB$ and $AFB$ when points $C$ and $D$ lie on a semicircle of diameter $AB$ and arc $CD$ is always constant and equal to $70^\circ$. | [
"\\( \\angle AEB = 55^\\circ \\), \\( \\angle AFB = 125^\\circ \\)"
] | [
"Calculate the angles $AEB$ and $AFB$ when points $C$ and $D$ lie on a semicircle of diameter $AB$ and arc $CD$ is always constant and equal to $70^\\circ$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-0c48d8b04ba83472e8a8b09ad1d620b777536a810ad2dd004c5b09a24124e33e | Calculate the angles of an isosceles triangle if its height drawn to the base is half the bisector of the angle at the base. | [
"108°, 36°, 36°"
] | [
"Calculate the angles of an isosceles triangle if its height drawn to the base is half the bisector of the angle at the base."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-9d46998c6587bd8e03888d247cf7208320bec3b16e4476b17f092dd4a86ffc49 | Calculate the angles of an obtuse triangle with sides AB = 7k, BC = 4k, and AC = 4k + 1, where k is a positive real constant. The radius of the circumscribed circle is 1 unit, and the circumcenter lies on the line supporting the bisector of side BC. Additionally, the angles CBA and BAC differ by 7º. | [
"113^\\circ"
] | [
"Calculate the angles of an obtuse triangle with sides AB = 7k, BC = 4k, and AC = 4k + 1, where k is a positive real constant. The radius of the circumscribed circle is 1 unit, and the circumcenter lies on the line supporting the bisector of side BC. Additionally, the angles CBA and BAC differ by 7º."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-27072ae61f452fd62d635cd8b69fe4acc24a6efa7a0a461be47f11a9102b7ea7 | Calculate the angles that \( (3, 4, 1) \) makes with the coordinate axes. | [
"\\(\\alpha = \\cos^{-1}\\left(\\frac{3}{\\sqrt{26}}\\right)\\), \\(\\beta = \\cos^{-1}\\left(\\frac{4}{\\sqrt{26}}\\right)\\), \\(\\gamma = \\cos^{-1}\\left(\\frac{1}{\\sqrt{26}}\\right)\\)"
] | [
"Calculate the angles that \\( (3, 4, 1) \\) makes with the coordinate axes."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 30
} | 43 | 0 | 0 |
omr-ws-29be43886df164a9c7012c4ece46ead8258471932b9f2b2a1df311bc808486de | Calculate the antiderivative of \( \int x^4 e^{x^4} \, dx \). | [
"\\(\\frac{x}{4\\sqrt[4]{-x^{4}}}\\Gamma\\left(1+\\frac{1}{4},-x^{4}\\right)\\)"
] | [
"Calculate the antiderivative of \\( \\int x^4 e^{x^4} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa4346d7f68e2f1d42ee6fae61bc49f955a92921393780767c53910b6bf64d4b | Calculate the antiderivative of the function $\int \frac{\sqrt{x^4+1}}{x^4-1} \, dx$. | [
"\\text{The antiderivative cannot be expressed in terms of elementary functions.}"
] | [
"Calculate the antiderivative of the function $\\int \\frac{\\sqrt{x^4+1}}{x^4-1} \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-943ebc167ebda43cef8aa5fe7657d03177b15e51c62776518c9d68073a0ba507 | Calculate the antiderivative of the function \( f(x) = \frac{2(n+1)x^n(x^{n+1}-a) + nx^{n+1} + 2a^2x + a}{x^{2n+2} - 2ax^{n+1} + a^2x^2 + a^2} \). | [
"\\ln\\left|x^{2n+2} - 2ax^{n+1} + a^2x^2 + a^2\\right| + \\frac{1}{a} \\arctan\\left( \\frac{x^{n+1} - a}{ax} \\right) + C"
] | [
"Calculate the antiderivative of the function \\( f(x) = \\frac{2(n+1)x^n(x^{n+1}-a) + nx^{n+1} + 2a^2x + a}{x^{2n+2} - 2ax^{n+1} + a^2x^2 + a^2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-50c8655a6002018f2c359fd621850ad2f1e1cd3edfbaa3b85ed387167396f601 | Calculate the antiderivative of the function \( g(x) = \frac{a + bx^n}{x + cx^{2n+1}} \). | [
"a \\ln|x| - \\frac{a}{2n} \\ln\\left(1 + c x^{2n}\\right) + \\frac{b}{n \\sqrt{c}} \\arctan\\left(x^n \\sqrt{c}\\right) + C"
] | [
"Calculate the antiderivative of the function \\( g(x) = \\frac{a + bx^n}{x + cx^{2n+1}} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-d9913bec9649ac3fc807717c7cca175c9de05e719255d83e9be4b364d78d7005 | Calculate the antiderivatives of the function \( f:\left( 0,\frac{\pi}{2}\right)\longrightarrow\mathbb{R} \) defined by \( f(x)=\frac{x^2+8}{\left( x^2-16\right)\sin x+8x\cos x} \). | [
"\\(\\ln|x\\sin(x/2) + 4\\cos(x/2)| - \\ln|x\\cos(x/2) - 4\\sin(x/2)| + C\\)"
] | [
"Calculate the antiderivatives of the function \\( f:\\left( 0,\\frac{\\pi}{2}\\right)\\longrightarrow\\mathbb{R} \\) defined by \\( f(x)=\\frac{x^2+8}{\\left( x^2-16\\right)\\sin x+8x\\cos x} \\)."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-b131500839bba51a546063f02d359a5b7168012db78943c029f3a5494af48f80 | Calculate the appropriate test statistic for the hypothesis test that Toms shoes last more than 60 days, given a sample mean of 67.3 days, a standard deviation of 12 days, and a sample size of 15 shoes. | [
"2.35"
] | [
"Calculate the appropriate test statistic for the hypothesis test that Toms shoes last more than 60 days, given a sample mean of 67.3 days, a standard deviation of 12 days, and a sample size of 15 shoes."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-812cb0c02d7b72379d75c8f284c0103834c737947abfc67326d1b40391caa343 | Calculate the approximate value of $\ln(1.1)$ with an error less than $10^{-4}$. | [
"\\(0.0953083\\)"
] | [
"Calculate the approximate value of $\\ln(1.1)$ with an error less than $10^{-4}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-8ba7beeca7617efdd144cdd4c9f7dcb190afd8d145bc95998c6a58e26d8f9cbe | Calculate the arc distance from the origin to the point \((x(t_1), y(t_1))\) along the exponential spiral \(r = ae^{ct}\) (where \(a > 0\) and \(c > 0\)). | [
"\\(\\frac{a \\sqrt{ c^2 + 1 }}{c} \\left( e^{ct_1} - 1 \\right)\\)"
] | [
"Calculate the arc distance from the origin to the point \\((x(t_1), y(t_1))\\) along the exponential spiral \\(r = ae^{ct}\\) (where \\(a > 0\\) and \\(c > 0\\))."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 7
} | 10 | 0 | 0 |
omr-ws-3953fb374bb8cbfb0e83e99e928441e5352cbd4aa80be348726b395db00f731d | Calculate the arc length of $y = 2a \sin(x) \tan(x)$ from $x = 0$ to $x = \frac{\pi}{3}$. | [
"2a",
"\\int_{0}^{\\frac{\\pi}{3}} \\sqrt{1 + \\frac{4a^2 \\sin^2(x) (2 - \\sin^2(x))^2}{\\cos^4(x)}} \\, dx"
] | [
"Calculate the arc length of $y = 2a \\sin(x) \\tan(x)$ from $x = 0$ to $x = \\frac{\\pi}{3}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-0a9f082b51c260517c5fc0cabcfc2592eecc6b7befa1f9045e23bf107aae4a86 | Calculate the arc length of \( \cos^2\left(\frac{x}{2}\right) \) from \( x = a \) to \( x = b \). | [
"E\\left(b \\left| -\\frac{1}{4}\\right.\\right) - E\\left(a \\left| -\\frac{1}{4}\\right.\\right)"
] | [
"Calculate the arc length of \\( \\cos^2\\left(\\frac{x}{2}\\right) \\) from \\( x = a \\) to \\( x = b \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-7dd46591067faf717c9f06f4f5982f5acaf7f79d24d16b31755ab90d028ea1b9 | Calculate the arc length of \( \cos^4\left(\frac{x}{4}\right) \) from \( x = a \) to \( x = b \). | [
"\\int_{a}^{b} \\sqrt{1 + \\cos^6\\left(\\frac{x}{4}\\right) \\sin^2\\left(\\frac{x}{4}\\right)} \\, dx"
] | [
"Calculate the arc length of \\( \\cos^4\\left(\\frac{x}{4}\\right) \\) from \\( x = a \\) to \\( x = b \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7bd94040cdc9277023a464c458f5383ab094864fc91aaed66c130934627d836a | Calculate the arc length of \( a \cdot \tan(x) \sin(x) \) from \( x = 0 \) to \( x = \frac{\pi}{3} \). | [
"a \\left( \\sqrt{3} - \\ln(2 - \\sqrt{3}) \\right)"
] | [
"Calculate the arc length of \\( a \\cdot \\tan(x) \\sin(x) \\) from \\( x = 0 \\) to \\( x = \\frac{\\pi}{3} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-f57875532ee07c232437531d3b2e59b1da964b652eb66452d4ece9f340577c21 | Calculate the arc length of the curve $y^3 = x^2$ from $x = 0$ to $x = 1$. | [
"\\(\\frac{13\\sqrt{13} - 8}{27}\\)"
] | [
"Calculate the arc length of the curve $y^3 = x^2$ from $x = 0$ to $x = 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d907f7f2230e88e08df968a4f7b053ddc9d4255aefd1913858bca3a6f3f4e447 | Calculate the arc length of the equation \( y = \frac{4}{5}x^{\frac{5}{4}} \) on the interval \([0, 4]\). Given: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx \] With: \[ \frac{dy}{dx} = x^{\frac{1}{4}} \] Thus: \[ L = \int_{0}^{4} \sqrt{1 + (x^{\frac{1}{4}})^2} \, dx = \int_{0}^{4} \sqrt{1 + x^{\fr... | [
"\\(\\frac{16\\sqrt{3}}{5} + \\frac{8}{15}\\)"
] | [
"Calculate the arc length of the equation \\( y = \\frac{4}{5}x^{\\frac{5}{4}} \\) on the interval \\([0, 4]\\).\n\nGiven:\n\\[ L = \\int_{a}^{b} \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2} \\, dx \\]\n\nWith:\n\\[ \\frac{dy}{dx} = x^{\\frac{1}{4}} \\]\n\nThus:\n\\[ L = \\int_{0}^{4} \\sqrt{1 + (x^{\\frac{1}{4}})^2... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-6eb861018b9edf6133084d962e30a0c3ec5f1e93c9e9e320be12cdf45874c5a1 | Calculate the area $A$ of the region $D$ in the plane $Oxy$ between the parabola $y=x^2$ and the lines $y=2$ and $y=2x.$ | [
"\\dfrac{4\\sqrt{2} - 3}{3}"
] | [
"Calculate the area $A$ of the region $D$ in the plane $Oxy$ between the parabola $y=x^2$ and the lines $y=2$ and $y=2x.$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-8fc16fee1c2213086fa268514242a674a7ca17d25abbaf380a55fc5d862c6643 | Calculate the area \( S \) of triangle \( ABC \) as a function of its side lengths \( a \), \( b \), and \( c \). | [
"\\( S = \\sqrt{\\frac{a+b+c}{2} \\left(\\frac{a+b+c}{2} - a\\right) \\left(\\frac{a+b+c}{2} - b\\right) \\left(\\frac{a+b+c}{2} - c\\right)} \\)"
] | [
"Calculate the area \\( S \\) of triangle \\( ABC \\) as a function of its side lengths \\( a \\), \\( b \\), and \\( c \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-e0ac61eb7c676fb7671725df1d96863abdbe51ca0a16ffa4726c9e07d64da45a | Calculate the area and angles of a rhombus given that its perimeter is $36$ and the sum of its diagonals is $20$. | [
"19"
] | [
"Calculate the area and angles of a rhombus given that its perimeter is $36$ and the sum of its diagonals is $20$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8368c7adf2b7c01d8b537a8b19af933c60a38456caa55eb388ca6256b1f9c7a8 | Calculate the area between the curves \( y^2 = 4px \) and \( 16py^2 = 5(x - p)^3 \) for \( p \in (0, \infty) \). | [
"\\(\\frac{104p^2}{3\\sqrt{5}}\\)"
] | [
"Calculate the area between the curves \\( y^2 = 4px \\) and \\( 16py^2 = 5(x - p)^3 \\) for \\( p \\in (0, \\infty) \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-17b67b1ccd07aba13587759f319e099b1c4302b8c5fee93729252853ba0ec44f | Calculate the area between the graph of the function \( f(x) = \frac{x^6}{(1 + x^4)^{1/2}} \) and the x-axis from \( x = 0 \) to \( x = 1 \). | [
"\\dfrac{\\sqrt{2} - 1}{3}"
] | [
"Calculate the area between the graph of the function \\( f(x) = \\frac{x^6}{(1 + x^4)^{1/2}} \\) and the x-axis from \\( x = 0 \\) to \\( x = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-be358569b30e0ebc0d530acded42e17924b190808fee03f9a816d5843c7a363d | Calculate the area bounded by the curve given by the parametric equations \( x = a \cos^3(t) \) and \( y = b \sin^3(t) \). | [
"\\( \\frac{3}{8} \\pi a b \\)"
] | [
"Calculate the area bounded by the curve given by the parametric equations \\( x = a \\cos^3(t) \\) and \\( y = b \\sin^3(t) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-8908ed1c79b52057041ecdc5555ce7f7b0879d09d8b60becb8d8b8d16a9b1c45 | Calculate the area bounded by the curves $y^{2}=\frac{x^{3}}{2\lambda-x}$, $x=\frac{\lambda}{2}$, and $x=2\lambda$, where $\lambda > 0$ is a real parameter. | [
"\\( 2\\lambda^{2} \\left( \\pi + \\frac{7\\sqrt{3}}{8} \\right) \\)"
] | [
"Calculate the area bounded by the curves $y^{2}=\\frac{x^{3}}{2\\lambda-x}$, $x=\\frac{\\lambda}{2}$, and $x=2\\lambda$, where $\\lambda > 0$ is a real parameter."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 21
} | 41 | 0 | 0 |
omr-ws-295bada3ce0fc4f6ac6fc71eb3ade551fa905f642f8ceebb705de16c5b2c87c0 | Calculate the area bounded by the curves \( x = \frac{1}{2} \), \( x = 2 \), \( y = \log_e x \), and \( y = 2^x \) and express your answer in the form \(\frac{A - \sqrt{B}}{\log_e 2} - \frac{C}{D} \log_e 2 + \frac{E}{F}\), where \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) are integers. Find the value of \(A + B + C + ... | [
"18"
] | [
"Calculate the area bounded by the curves \\( x = \\frac{1}{2} \\), \\( x = 2 \\), \\( y = \\log_e x \\), and \\( y = 2^x \\) and express your answer in the form \\(\\frac{A - \\sqrt{B}}{\\log_e 2} - \\frac{C}{D} \\log_e 2 + \\frac{E}{F}\\), where \\(A\\), \\(B\\), \\(C\\), \\(D\\), \\(E\\), and \\(F\\) are integer... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5bfe86c93d489ab7f3300afae7dc52b4b5ffb91c108cffa61bc77882ef854b50 | Calculate the area covered by the conic \( y^2 = 3x^2 + 2x + 4 \), the x-axis, and the line \( x = 3 \). | [
"\\dfrac{5\\sqrt{37}}{3} + \\dfrac{11\\sqrt{3}}{18} \\ln\\left( \\dfrac{10 + \\sqrt{111}}{\\sqrt{11}} \\right)"
] | [
"Calculate the area covered by the conic \\( y^2 = 3x^2 + 2x + 4 \\), the x-axis, and the line \\( x = 3 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-f8bbcd1a8f2b4fe8e8864856a9b4b002247347f2310310cf417a11f7435f42d9 | Calculate the area covered by the straight line \(x - y = 3\) and the curve \(y^2 = 4x\). | [
"\\(\\frac{64}{3}\\)"
] | [
"Calculate the area covered by the straight line \\(x - y = 3\\) and the curve \\(y^2 = 4x\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-6c7e24e5377b1037be50e1c8846edbce26f6868237a7bf32da67d602adae791a | Calculate the area enclosed by the curve given by \( x^4 = a^2(x^2 - y^2) \). | [
"\\dfrac{4a^2}{3}"
] | [
"Calculate the area enclosed by the curve given by \\( x^4 = a^2(x^2 - y^2) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4533265abd875ac04b7a114a242cd70371b8ddb57d44910f13977576f5cadd25 | Calculate the area formed by the centers of all possible inner contact circles of a rectangle with side lengths $|AB| = 12$ and $|BC| = 9$, where the radii of the circles satisfy $2 \le r \le 3$. | [
"22"
] | [
"Calculate the area formed by the centers of all possible inner contact circles of a rectangle with side lengths $|AB| = 12$ and $|BC| = 9$, where the radii of the circles satisfy $2 \\le r \\le 3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-f1ec1fd7b3bbee1df8c05c10e2dff2d16f92488e4795bbe0d0fe5ea8a7b019c2 | Calculate the area formed by the centers of all possible outer contact circles of a rectangle with side lengths $|AB| = 12$ and $|BC| = 9$, where the radii of the circles satisfy $2 \le r \le 3$. | [
"62"
] | [
"Calculate the area formed by the centers of all possible outer contact circles of a rectangle with side lengths $|AB| = 12$ and $|BC| = 9$, where the radii of the circles satisfy $2 \\le r \\le 3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 17,
"genselect": 5
} | 22 | 0 | 0 |
omr-ws-751c4e70d684cfe24ae143857de480eef6f5f932c2c8f1a9bbe76f0d0b16e453 | Calculate the area of a deltoid with sides 8 cm and 15 cm and an angle of 150 degrees between them. | [
"30 cm²"
] | [
"Calculate the area of a deltoid with sides 8 cm and 15 cm and an angle of 150 degrees between them."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-240e1b2729d40dd71af133bcef8d7a86fe1ee96e495142a4a34df12dae52d638 | Calculate the area of an isosceles trapezoid if its altitude is $h$ and one of the legs can be seen from the center of the circumscribed circle at an angle of $60^\circ$. | [
"$h^2\\sqrt 3$"
] | [
"Calculate the area of an isosceles trapezoid if its altitude is $h$ and one of the legs can be seen from the center of the circumscribed circle at an angle of $60^\\circ$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-102cfa0194164bac83adef39cff34d5ce954ac44b1e4febe20ea11adb97f5397 | Calculate the area of intersection between the parabola \( y^2 = -(x-2) \) and the absolute value function \( y = |x| \). | [
"\\(\\frac{13}{6}\\)"
] | [
"Calculate the area of intersection between the parabola \\( y^2 = -(x-2) \\) and the absolute value function \\( y = |x| \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 2,
"tir": 19
} | 26 | 0 | 0 |
omr-ws-8f5a51a8579f56bd19ed79885e2a66401b6f11c1f69d65056771a8c41d83f40d | Calculate the area of quadrilateral $\overline{CIEJ}$ in a cube $\overline{ABCDEFGH}$ with edge length $2$, where $I$ is the midpoint of $\overline{DH}$ and $J$ is the midpoint of $\overline{BF}$. | [
"$\\sqrt{24}$"
] | [
"Calculate the area of quadrilateral $\\overline{CIEJ}$ in a cube $\\overline{ABCDEFGH}$ with edge length $2$, where $I$ is the midpoint of $\\overline{DH}$ and $J$ is the midpoint of $\\overline{BF}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-3201985e15cd56f8289bd36c557d613290c357da7b540030f53dca536879bfa2 | Calculate the area of the common part of two rhombi, the diagonals of the first rhombus being equal to 2 and 3, and the second rhombus being obtained by a rotation of the first through $90^\circ$ about its center. | [
"$\\frac{12}{5}$"
] | [
"Calculate the area of the common part of two rhombi, the diagonals of the first rhombus being equal to 2 and 3, and the second rhombus being obtained by a rotation of the first through $90^\\circ$ about its center."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d20b59b5f302018d0fcbf57717cb2604414dc1ec3a11593e4b9eac23df038667 | Calculate the area of the conical face $T$ defined by the region above the cone $z=\sqrt{x^{2}+y^{2}}$ and inside the cylinder $x^{2}+z^{2} = a^{2}$, where $a$ is a positive constant. The area is given by the integral $T= \iint_T | \varphi_u \times \varphi_v| \, du \, dv$. | [
"\\pi a^2"
] | [
"Calculate the area of the conical face $T$ defined by the region above the cone $z=\\sqrt{x^{2}+y^{2}}$ and inside the cylinder $x^{2}+z^{2} = a^{2}$, where $a$ is a positive constant. The area is given by the integral $T= \\iint_T | \\varphi_u \\times \\varphi_v| \\, du \\, dv$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f770dcd013b419ef02279c71f607558411eabf23641d03d8271049c70388565a | Calculate the area of the figure defined by the inequalities $9x^4 + x^2y^2 + y^4 \leq 3x^2 + y^2$, $0 \leq x$, and $0 \leq y$. | [
"\\(\\frac{\\pi}{6}\\)."
] | [
"Calculate the area of the figure defined by the inequalities $9x^4 + x^2y^2 + y^4 \\leq 3x^2 + y^2$, $0 \\leq x$, and $0 \\leq y$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68e1e089107cb8a56ae1b84247196bde8111f37cbe7e928668d48f21f67f7a3e | Calculate the area of the largest possible circle that can be inscribed in a quadrilateral with side lengths AB = 14 cm, BC = 9 cm, CD = 7 cm, and DA = 12 cm. | [
"\\( 24\\pi \\)"
] | [
"Calculate the area of the largest possible circle that can be inscribed in a quadrilateral with side lengths AB = 14 cm, BC = 9 cm, CD = 7 cm, and DA = 12 cm."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 21
} | 44 | 0 | 0 |
omr-ws-f40004dcbed72d3e6f4f7f02ca0b978eb7d3305eb02be83daa4e74e4bdcc1f4c | Calculate the area of the part of the paraboloid \(4z = x^2 + y^2\) which is cut out by the parabolic cylinder \(z = y^2\) and the plane \(z = 3\). | [
"\\( \\frac{56\\pi}{9} \\)"
] | [
"Calculate the area of the part of the paraboloid \\(4z = x^2 + y^2\\) which is cut out by the parabolic cylinder \\(z = y^2\\) and the plane \\(z = 3\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 22
} | 36 | 0 | 0 |
omr-ws-b2c7a25df3d1d35e61850eabe8435173d1013955b98e4a1275435343f5b61d83 | Calculate the area of the part of the plane $Ax + By + Cz + D = 0$ which lies inside the cylinder $x^2 + y^2 = 1$. Assume $C \ne 0$. | [
"\\(\\pi \\frac{\\sqrt{A^2 + B^2 + C^2}}{|C|}\\)"
] | [
"Calculate the area of the part of the plane $Ax + By + Cz + D = 0$ which lies inside the cylinder $x^2 + y^2 = 1$. Assume $C \\ne 0$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-5448eb29dc26497da95a61ad71da002eb28ce81da008ff0ac8c8758ef1eaed21 | Calculate the area of the pentagon $ABCDE$ given that $AB = BC$, $CD = DE$, $\angle ABC = 120^\circ$, $\angle CDE = 60^\circ$, and $BD = 2$. | [
"\\(\\sqrt{3}\\)"
] | [
"Calculate the area of the pentagon $ABCDE$ given that $AB = BC$, $CD = DE$, $\\angle ABC = 120^\\circ$, $\\angle CDE = 60^\\circ$, and $BD = 2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 4,
"tir": 5
} | 23 | 0 | 0 |
omr-ws-8c56a8b11e9feb1100e6d77b6d9d3620969a4eba52646d4d6cdea41590c8b3c3 | Calculate the area of the pentagon $ABCDE$ given the following conditions: - $\angle ABC = \angle BCD = \angle CDE = 90^\circ$. - $CD > AB$. - $AB = 28$, $BC = 15$, $DE = 10$, and $EA = 13$. | [
"570"
] | [
"Calculate the area of the pentagon $ABCDE$ given the following conditions:\n- $\\angle ABC = \\angle BCD = \\angle CDE = 90^\\circ$.\n- $CD > AB$.\n- $AB = 28$, $BC = 15$, $DE = 10$, and $EA = 13$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-42d322d92b32ff7602ec6dbf13fd024fa130b5197c1142c9af059308e8774020 | Calculate the area of the plane figure bounded by the curve \( x^6 + y^6 = x^4 y \) where \( y \geq 0 \). | [
"\\(\\frac{\\pi}{12}\\)\n\nExplanation:\nThe area calculated in the forum post is for the region where \\(x \\geq 0\\) and \\(y \\geq 0\\). Since the figure is symmetric with respect to the \\(y\\)-axis, the total area is twice the calculated area. Therefore, the total area is \\(2 \\times \\frac{\\pi}{24} = \\frac... | [
"Calculate the area of the plane figure bounded by the curve \\( x^6 + y^6 = x^4 y \\) where \\( y \\geq 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 19
} | 37 | 0 | 0 |
omr-ws-2b93b01d3b9897753ac60a23664be5c2b14eb7aea7f4680f32123f8f490de485 | Calculate the area of the region \( A \subset \mathbb{R}^2 \) of points \( (x, y) \) such that \( x^2 + y^2 \geqslant 1 \) and \( (x - 1)^2 + y^2 \leqslant 1 \). Also, sketch this region. | [
"\\dfrac{\\pi}{3} + \\dfrac{\\sqrt{3}}{2}"
] | [
"Calculate the area of the region \\( A \\subset \\mathbb{R}^2 \\) of points \\( (x, y) \\) such that \\( x^2 + y^2 \\geqslant 1 \\) and \\( (x - 1)^2 + y^2 \\leqslant 1 \\). Also, sketch this region."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c442ea303718b90fccefb285907b03e5d124b2d27e89863003d7339fe7290cf | Calculate the area of the region bounded by the curve \( ax^2 + bxy + cy^2 = 1 \) if the curve is an ellipse. | [
"\\( \\frac{2\\pi}{\\sqrt{4ac - b^2}} \\)"
] | [
"Calculate the area of the region bounded by the curve \\( ax^2 + bxy + cy^2 = 1 \\) if the curve is an ellipse."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-0633ddaa82fbfed38182d4af692ceb45910f596acd0d3fa24b6a3207addad9aa | Calculate the area of the region bounded by the curves \( y(x) = 3x + 0.5 \) and \( g(x) = \sqrt{1 - x^2} \), and the positive x and y axes. | [
"\\dfrac{100\\pi - 42 - \\sqrt{39}}{400}"
] | [
"Calculate the area of the region bounded by the curves \\( y(x) = 3x + 0.5 \\) and \\( g(x) = \\sqrt{1 - x^2} \\), and the positive x and y axes."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-fad48347f64455bc33029677cb80cdb30e51cb84054fbaf52a86604af73d3c44 | Calculate the area of the region bounded by the equation \(x^4 + y^4 = x^2 + y^2\). | [
"\\(\\pi\\sqrt{2}\\)"
] | [
"Calculate the area of the region bounded by the equation \\(x^4 + y^4 = x^2 + y^2\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-fb43e762c97ecf1e2922d06d49437f2899c7f843111343fb0e91ce857e96a4f0 | Calculate the area of the region bounded by the lines \(x = y\), \(x = 2y\), \(x + y = a\), and \(x + 3y = a\). | [
"\\(\\frac{7a^2}{120}\\)"
] | [
"Calculate the area of the region bounded by the lines \\(x = y\\), \\(x = 2y\\), \\(x + y = a\\), and \\(x + 3y = a\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-db66e11fcbd161e98c810266d101b5872780fea4a8f9479d2bc51ad17db09deb | Calculate the area of the region covered by a radar with a range of 50 miles, and divide the result by 100. | [
"78"
] | [
"Calculate the area of the region covered by a radar with a range of 50 miles, and divide the result by 100."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-0b1a6c1128a5740a6856e2fee6b36c1ad11ab7019d7d531539f1d23d6af329b5 | Calculate the area of the region in the plane determined by the inequality $|x| + |y| + |x + y| \le 2$. | [
"3"
] | [
"Calculate the area of the region in the plane determined by the inequality $|x| + |y| + |x + y| \\le 2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-c594084286623e5dc2f61fe2b1d73b41034c1a09e30fbd49ed0a7ed1d689d996 | Calculate the area of the region that lies inside the circle given by the equation \(x^2 + y^2 = 8y\) and outside the circle given by the equation \(x^2 + y^2 = 9\). | [
"\\(\\frac{7\\pi}{4}\\)"
] | [
"Calculate the area of the region that lies inside the circle given by the equation \\(x^2 + y^2 = 8y\\) and outside the circle given by the equation \\(x^2 + y^2 = 9\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de30dc4b1871e75169b90af1ca60cc3766511c1858e2d373f5270481246b9195 | Calculate the area of the regular hexagon $ABCDEF$ with vertices $A$ at $(2, 1)$ and $B$ at $(4, 2)$. | [
"\\(\\frac{15\\sqrt{3}}{2}\\)"
] | [
"Calculate the area of the regular hexagon $ABCDEF$ with vertices $A$ at $(2, 1)$ and $B$ at $(4, 2)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-3389c3691c7528305e13349d93ada94c83726390448e27bf2455b54c6b1e3e11 | Calculate the area of the shark's fin falcata, which is bounded by the portion of the circle with radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)... | [
"\\(\\frac{9\\pi}{8}\\)"
] | [
"Calculate the area of the shark's fin falcata, which is bounded by the portion of the circle with radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\\tfrac{3}{2}$ and center $(0,\\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 6,
"tir": 25
} | 43 | 0 | 0 |
omr-ws-c95ca202de4fe73c02705647d60c685a1b79a6eb4ca87feb89df814d5b8a708a | Calculate the area of the sphere given by $z=\sqrt{9-x^2-y^2}$ that is cut by the cylinder $x^2+y^2=5$. | [
"\\(6\\pi\\)"
] | [
"Calculate the area of the sphere given by $z=\\sqrt{9-x^2-y^2}$ that is cut by the cylinder $x^2+y^2=5$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-9ef8b76c9615a528c9266dec43ef742cfc065ba5b18b5ee9f672879d2e9d1dab | Calculate the area of the surface delimited by the graphs of the functions $9x^2 + 16y^2 - 24xy - 20x - 15y = 0$ and $2x - 11y + 15 = 0$. | [
"\\(\\frac{32}{3}\\)"
] | [
"Calculate the area of the surface delimited by the graphs of the functions $9x^2 + 16y^2 - 24xy - 20x - 15y = 0$ and $2x - 11y + 15 = 0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 6,
"tir": 25
} | 74 | 0 | 0 |
omr-ws-b1ce7ed17390a48c65cc3f0ec969b12f4643fbeb94ea1c2c2feb861dd9fbb156 | Calculate the area of the surface generated when the arch of the parabola from the point (1,1) to (2,4) is rotated around the axis. | [
"\\dfrac{\\pi}{6} \\left(17\\sqrt{17} - 5\\sqrt{5}\\right)"
] | [
"Calculate the area of the surface generated when the arch of the parabola from the point (1,1) to (2,4) is rotated around the axis."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-b5b26c8a3560efd082d63e7da9e5b789983206d5dd6d76bf94f8beca453becf8 | Calculate the area of the surface of revolution obtained by rotating the curve given by \( y = 4 \sqrt{x} \) around the \( x \)-axis for \( \frac{1}{4} \leq x \leq 4 \). | [
"\\( 8\\pi \\left( \\frac{32 \\sqrt{2}}{3} - \\frac{17 \\sqrt{17}}{12} \\right) \\)"
] | [
"Calculate the area of the surface of revolution obtained by rotating the curve given by \\( y = 4 \\sqrt{x} \\) around the \\( x \\)-axis for \\( \\frac{1}{4} \\leq x \\leq 4 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-31065522534bb69a8748f839e8a857bb3306b7ebf5da331d9e5e1e060c388170 | Calculate the area of the trail left by a wood stick of length 1 sliding from a wall. The function describing the trail is given by \(x^{2/3} + y^{2/3} = 1\). Evaluate the integral \(\int_{0}^{1} (1 - x^{2/3})^{3/2} \, dx\). | [
"\\(\\frac{3\\pi}{32}\\)"
] | [
"Calculate the area of the trail left by a wood stick of length 1 sliding from a wall. The function describing the trail is given by \\(x^{2/3} + y^{2/3} = 1\\). Evaluate the integral \\(\\int_{0}^{1} (1 - x^{2/3})^{3/2} \\, dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-fc18777b30d11225d42d999d26f022b98a2e7dbfadc8a967a42522e6a97becee | Calculate the area of the triangle $CDE$ in terms of $a$ in the given configuration where an equilateral triangle $ABC$ of side $a$ has a right isosceles triangle $ACD$ constructed on its exterior with $\angle CAD = 90^\circ$. | [
"\\(\\frac{a^2(1+\\sqrt{3})}{2}\\)"
] | [
"Calculate the area of the triangle $CDE$ in terms of $a$ in the given configuration where an equilateral triangle $ABC$ of side $a$ has a right isosceles triangle $ACD$ constructed on its exterior with $\\angle CAD = 90^\\circ$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-5e40faebfc49207474fe6a5b235a7760ef1d21ec03db9a3310e19f99ed73a8bb | Calculate the area of the triangle formed by the orthocenter (H), the circumcenter (O), and the incenter (I) of a triangle ABC. | [
"\\dfrac{1}{4}"
] | [
"Calculate the area of the triangle formed by the orthocenter (H), the circumcenter (O), and the incenter (I) of a triangle ABC."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-e22ccc675404be44695438010cb6f46625b2920a8f2dbadfb068ecf1d2c67473 | Calculate the area of triangle $ABC$, given that $BD = 4$, $DE = 2$, $EC = 6$, and $BF = FC = 3$. | [
"\\(18\\sqrt{3}\\)"
] | [
"Calculate the area of triangle $ABC$, given that $BD = 4$, $DE = 2$, $EC = 6$, and $BF = FC = 3$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-6ab338e013ff6bfc9ea7846312489094b8d416c864dffa444ad1fc92f6156afa | Calculate the areas of the figures bounded by the indicated lines. $2yx=16+x^2,\ y=5$. | [
"$15-16\\ln 2$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $2yx=16+x^2,\\ y=5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e1042ef0f689741a1862c50b1177084d4a502ae8d550143a53c78ae8473cd9e7 | Calculate the areas of the figures bounded by the indicated lines. $3y=-x^2+8x-7,\ y+1=4/(x-3)$. | [
"$9-8\\ln 2$"
] | [
"Calculate the areas of the figures bounded by the indicated lines. $3y=-x^2+8x-7,\\ y+1=4/(x-3)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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