prompt list | ground_truth string |
|---|---|
[
{
"content": "For which values of \\(\\alpha\\) does the series \\(\\sum_{n=1}^\\infty \\frac{1}{(n^{\\alpha}(n^{1-\\alpha}+1))^{\\frac{1}{3}}}\\) converge?",
"role": "user"
}
] | \alpha > 3 |
[
{
"content": "Evaluate the contour integral \\( \\int_{|z|=2} \\frac{1}{z^3+z^2+z+1} \\, dz \\), where the contour \\(|z|=2\\) is traversed counterclockwise.",
"role": "user"
}
] | 0 |
[
{
"content": "Let $\\alpha$ and $\\beta$ be conjugate complex numbers such that $\\frac{\\alpha}{\\beta^2}$ is a real number and $|\\alpha - \\beta| = 2 \\sqrt{3}.$ Find $|\\alpha|.$",
"role": "user"
}
] | 2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.306 kg.m/s |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.63 mm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.64 J |
[
{
"content": "Find the minimum value of the expression \\( S = \\vec{a} \\cdot \\vec{b} + 2 \\vec{a} \\cdot \\vec{c} + \\vec{b} \\cdot \\vec{c} \\) given that \\(|\\vec{a}| = 1\\), \\(|\\vec{b}| = 2\\), and \\(|\\vec{c}| = 3\\).",
"role": "user"
}
] | -10 |
[
{
"content": "Which condition guarantees that for a non-empty subset $A$ of $\\mathbb{R}^n$ and a point $x \\in \\mathbb{R}^n$, there exists a point $y_0 \\in A$ such that $d(x, y_0) = d(x, A)$? Choose the correct option:\n\n1. $A$ is a non-empty subset of $\\mathbb{R}^n$.\n2. $A$ is a non-empty closed subset o... | 2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | TC = 3L + 128
TFC = 128
TVC = 3L
ATC = 3 + 128/L
AFC = 128/L
AVC = 3
MC = 3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $3.29 |
[
{
"content": "Find the maximum value of \\(|f(z)|\\) over the contour \\(C: |z| = 3\\) for the complex function \\(f(z) = \\frac{1}{z^2 - i}\\) using the ML inequality.",
"role": "user"
}
] | \dfrac{1}{8} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | k=k1k2/(k1+k2) |
[
{
"content": "Determine whether there exist infinitely many positive integers $k$ such that $1093$ divides $3^k - 1$. If so, specify the conditions that $k$ must satisfy for $3^k - 1$ to be divisible by $1093$.",
"role": "user"
}
] | k \equiv 0 \pmod{7} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 14.405 minutes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | +5.4*10^9N |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.064g |
[
{
"content": "Find the norm of the vector \\( v \\) given the following conditions: \\( \\angle uv = \\frac{\\pi}{4} \\), \\( ||u|| = 3 \\), and \\( (u-v) \\perp u \\).",
"role": "user"
}
] | 3\sqrt{2} |
[
{
"content": "Evaluate the limit: \\( \\lim_{x\\to \\pi/2}\\cos(x)^{2x-\\pi} \\)",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Copper |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.58 m/s^2 |
[
{
"content": "Calculate the total number of distinct points of intersection possible among 4 distinct parabolas, 5 distinct circles, and 3 distinct lines in the $x-y$ plane.",
"role": "user"
}
] | 181 |
[
{
"content": "Determine whether the sequence \\((x_n)_{n=1}^\\infty\\), where \\(x_n\\) is the fractional part of the harmonic series \\(1 + \\frac{1}{2} + \\dots + \\frac{1}{n}\\), is dense in the interval \\((0,1)\\). The fractional part of a number \\(y\\) is denoted by \\(y - \\lfloor y \\rfloor\\).",
"... | Yes |
[
{
"content": "Consider a subset $F$ of $[0,1]$ constructed similarly to the Cantor set, but with each interval removed at the $n$th iteration having a length of $\\frac{\\alpha}{3^{n}}$, where $0<\\alpha<1$. Show that the total length of $F$ is $1-\\alpha$ by taking the limit as $n \\to \\infty$ of the length o... | 1 - \alpha |
[
{
"content": "Find the coefficient of $x^2$ in the expansion of $(1+2x-3x^2+4x^3)^{-1/2}$.",
"role": "user"
}
] | 3 |
[
{
"content": "The ratio of $x+2$ to $2x+2$ is equal to the ratio of $4x+3$ to $7x+3$. What is the product of all real x which satisfy this statement?",
"role": "user"
}
] | 0 |
[
{
"content": "Find a non-recursive formula for the sequence \\( s_n \\) defined by the recursive relation \\( s_{n+1} = (n+2)s_n + n! \\) with the initial condition \\( s_1 = 1 \\).",
"role": "user"
}
] | n \cdot n! |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2CuFeS_2 + 3O_2 -> 2CuS + 2FeO + 2SO_2 |
[
{
"content": "Does there exist a topological space $X$ that is not contractible, but whose singular chain complex $C_*(X)$ is contractible?",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | {
"Date": [
"March 1",
"March 16",
"March 31"
],
"Particulars": [
"Opening Stock",
"Purchase",
"Sales"
],
"Unit": [
10,
20,
25
],
"Rate": [
60,
70,
"10*60+15*70"
],
"FIFO": [
600,
1400,
1650
],
"Closing inventory": [
null,
null,
350
]
}
|
[
{
"content": "Evaluate the contour integral \\( \\int_{\\varGamma}\\frac{3e^{z}}{1-e^{z}}dz \\), where \\( \\varGamma \\) is the contour \\( |z|=4\\pi/3 \\).",
"role": "user"
}
] | -6\pi i |
[
{
"content": "Evaluate the polynomial \\(x^{24} + 16x^{12} + 1\\) at all 48th roots of unity. How many distinct values are obtained?",
"role": "user"
}
] | 4 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.297 |
[
{
"content": "Find all integer pairs \\((x, y)\\) that satisfy the equation \\(4y^2 = x^7 + 1\\).",
"role": "user"
}
] | (-1, 0) |
[
{
"content": "Find the value of $5^{2003} \\pmod{1001}$ given the following congruences: \\[ 5^{2003} \\equiv 3 \\pmod{7} \\] \\[ 5^{2003} \\equiv 4 \\pmod{11} \\] \\[ 5^{2003} \\equiv 8 \\pmod{13} \\] Use the Chinese Remainder Theorem to solve the problem.",
"role": "user"
}
] | 983 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1161.46 |
[
{
"content": "In a zero-dimensional topology, where the space has a base of sets that are both open and closed, is every open set also a closed set?",
"role": "user"
}
] | No |
[
{
"content": "Suppose that $a$ and $b$ are integers such that $$3b = 8 - 2a.$$How many of the first six positive integers must be divisors of $2b + 12$?",
"role": "user"
}
] | 3 |
[
{
"content": "In the context of Fourier series, can the coefficients of the full Fourier series always be obtained from the complex form coefficients \\( C_n \\)? Answer \"yes\" or \"no.\"",
"role": "user"
}
] | yes |
[
{
"content": "A circle is drawn on a board with its center marked, along with an inscribed quadrilateral and a circle inscribed within the quadrilateral, also with its center marked. After erasing the quadrilateral (while retaining one vertex) and the inscribed circle (while retaining its center), reconstruct o... | B |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3.38e-7 m |
[
{
"content": "Determine whether a non-Hermitian matrix of complex numbers can have real eigenvalues.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the number of functions \\( f: A \\rightarrow A \\) where \\( A = \\{1, 2, \\ldots, 2014\\} \\), satisfying the following conditions:\n1. There exists a \\( k \\in A \\) such that \\( f \\) is non-decreasing on \\( \\{1, 2, \\ldots, k\\} \\) and non-increasing on \\( \\{k, k+1, \\ldots, ... | 2^{2012} |
[
{
"content": "Let $A$ be an $n \\times n$ matrix with entries in $\\mathbb{C}$. Suppose that the characteristic polynomial of $A$ is the same as the minimal polynomial of $A$, given by \\(p_{A}(t) = m_{A}(t) = (t - \\lambda)^{n}\\). Determine the number of linearly independent eigenvectors for the eigenvalue $\... | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1031.33 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | The six resonance structures involve the movement of pi electrons within the benzene ring and the interaction with the lone pairs on the oxygen atom of the methoxy group and the oxygen atom of the aldehyde group. The image provided in the response illustrates these structures. |
[
{
"content": "Calculate the sum of the series \\( \\sum_{r=0}^{n-1}\\cos^2(\\theta+ \\alpha r ) \\), where \\( \\alpha \\) is the exterior angle of a regular polygon with \\( n \\) sides, for any real number \\( \\theta \\).",
"role": "user"
}
] | \dfrac{n}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 10400 |
[
{
"content": "Determine if the linear operator $T$ is bounded on the space of polynomials, where for a polynomial $p(x) = a_0 + a_1 x + a_2 x^2 + \\dots + a_n x^n$, the operator is defined as $T(p)(x) = a_0 + a_1 x^2 + a_2 x^4 + \\dots + a_n x^{2n}$. The norm on this space is given by $||p|| = \\sup \\{|p(t)| :... | Yes |
[
{
"content": "Calculate the limit: \\[ \\lim_{n \\to \\infty}\\sum_{k=1}^n \\frac {(n+1)^k} {n^{k+1}} \\]",
"role": "user"
}
] | e - 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3.03 x 10^-19 J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1349.61 |
[
{
"content": "Determine whether the subset \\( H = \\{ f \\in E \\mid \\exists n_f \\in \\mathbb{N}_* : f = 0 \\text{ on } [-n_f, +n_f] \\} \\) is a vector subspace of the vector space \\( E \\) of all functions from \\( \\mathbb{R} \\) to \\( \\mathbb{R} \\). Justify your answer rigorously.",
"role": "user... | H |
[
{
"content": "Let $O$ and $H$ be the circumcenter and orthocenter of triangle $ABC$, respectively. Let $a$, $b$, and $c$ denote the side lengths, and let $R$ denote the circumradius. Find $OH^2$ if $R = 7$ and $a^2 + b^2 + c^2 = 29$.",
"role": "user"
}
] | 412 |
[
{
"content": "Calculate the value of the integral \\( \\int_{\\gamma} |z|^2(\\bar{z} + i) \\, dz \\), where \\( \\gamma \\) is the circle defined by \\( |z - i| = 1 \\), traced counterclockwise. Use the parameterization \\( z(t) = \\cos(t) + i(1 + \\sin(t)) \\) for \\( t \\in [0, 2\\pi] \\).",
"role": "user... | 4\pi i |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.13m |
[
{
"content": "Calculate \\( \\lim \\limits_{x \\to 0} \\ x^2\\Big(1+2+3+\\ldots+\\Big[\\frac{1}{\\vert x \\vert}\\Big]\\Big) \\), where \\([.]\\) represents the greatest integer function.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Rent Receipt/ Payment Interest Revenue/ Expense Reduction of Principal Receivable/ Liability
1/1/2014 $87,000.00
12/31/2014 $34,373.77 $7,830.00 $26,543.77 $60,456.23
12/31/2015 $34,373.77 $5,441.06 $28,932.71 $31,523.52
12/31/2016 $34,360.64 $2,837.12 $31,523.52 $(0.00) |
[
{
"content": "Evaluate the line integral \\(\\oint_C {\\bf B}\\cdot d{\\bf x}\\) for the vector field \\({\\bf B}({\\bf x})=\\rho^{-1}e_{\\phi}\\) in cylindrical coordinates, where \\(\\rho \\neq 0\\). The curve \\(C\\) is the circle defined by \\(z=0, \\rho=1\\), and \\(0\\le \\phi \\le 2\\pi\\).",
"role":... | 2\pi |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1149.53 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.36e-2 |
[
{
"content": "Given the equations:\n\\[ a^2x^3 + b^2y^3 + c^2z^3 = p^5 \\]\n\\[ ax^2 = by^2 = cz^2 \\]\n\\[ \\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = \\frac{1}{p} \\]\nfind \\(\\sqrt{a} + \\sqrt{b} + \\sqrt{c}\\) in terms of \\(p\\).",
"role": "user"
}
] | \sqrt{p} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 46.89 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $17,860.42 |
[
{
"content": "Find one triple of integers \\((\\ell, m, n)\\) such that \\(2 < \\ell < m < n\\) and the sums \\(\\sum_{k=1}^{\\ell} k\\), \\(\\sum_{k=\\ell+1}^{m} k\\), and \\(\\sum_{k=m+1}^{n} k\\) form a geometric sequence.",
"role": "user"
}
] | (5, 9, 14) |
[
{
"content": "Let \\(X\\) and \\(Y\\) be random variables with the joint probability density function given by:\n\\[\nf_{X,Y}(x,y) = \\begin{cases} \n0.25ye^{-y} & \\text{if } 0 \\leq |x| \\leq y \\\\\n0 & \\text{otherwise}\n\\end{cases}\n\\]\nFind the conditional density \\(f_{X|Y = y}(x)\\) and identify the c... | 4 |
[
{
"content": "Compute the limit \\( \\lim_{x\\to0}\\frac{\\sin(x^2+\\frac{1}{x})-\\sin\\frac{1}{x}}{x} \\) using L'Hôpital's rule.",
"role": "user"
}
] | 0 |
[
{
"content": "Given that $a, b, c, x, y, z \\in \\mathbb{R}^+$ and $ab + bc + ca = xy + yz + zx = 1$, find the minimum value of the expression $ay + az + bz + bx + cx + cy$. It is known that when $a = b = c = x = y = z = \\frac{1}{\\sqrt{3}}$, the value is 2, and as $a = b = x = y \\to 1$ and $c = z \\to 0$, th... | 2 |
[
{
"content": "Determine whether an isogeny between two principally polarized abelian varieties that respects the polarization is necessarily an isomorphism.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 10.0 cm |
[
{
"content": "Evaluate the expression \\((x^2 - \\frac{1}{27})\\frac{d^2y}{dx^2} + x\\frac{dy}{dx}\\) given the differential equation \\(y^3 - y = 2x\\) and the derivatives \\(\\frac{d^2y}{dx^2} = \\frac{-24y}{(3y^2-1)^3}\\) and \\(\\frac{dy}{dx} = \\frac{2}{3y^2-1}\\).",
"role": "user"
}
] | \dfrac{y}{9} |
[
{
"content": "Calculate the work done by the vector field \\( \\vec{F}(x,y,z)=(xz,yz+x^2yz+y^3z+yz^5,2z^4) \\) along the edge of the surface \\( \\Sigma \\) defined by \\( (x^2+y^2+z^4)e^{y^2}=1, x\\geq 0 \\), oriented such that its normal at the point \\( (1,0,0) \\) is \\( \\vec{N} = (1,0,0) \\).",
"role"... | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 15.07% |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.6e-3 moles |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Nitrobenzene > Phenol >Tolulene |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 13.04 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.8e3 cm/s |
[
{
"content": "Compute the ratio of variances between the estimators \\(\\hat{\\beta_1} = (n+1)X_{\\min}\\) and \\(\\hat{\\beta_2} = \\frac{n+1}{n}X_{\\max}\\), where \\(X_1,X_2,...,X_n\\) are i.i.d. random variables following a uniform distribution on the interval \\([0,\\beta]\\). Determine how this ratio depe... | n^2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 58 m |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 7.47% |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 300690 |
[
{
"content": "Find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that for all real numbers \\( x, y, z \\), the following holds: If \\( x^3 + f(y) \\cdot x + f(z) = 0 \\), then \\( f(x)^3 + y \\cdot f(x) + z = 0 \\).",
"role": "user"
}
] | f(x) = x |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 7.06e-7 M |
[
{
"content": "Evaluate the limit \\( \\displaystyle \\lim_{x \\to 0} \\frac{\\tan(x^2)-\\sin(x^2)}{x^6} \\) using L'Hopital's rule or Taylor series expansion.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.9124 |
[
{
"content": "Given positive numbers \\( p, q, a, b, c \\) where \\( p \\neq q \\), if \\( p, a, q \\) is a geometric sequence, and \\( p, b, c, q \\) is an arithmetic sequence, determine the nature of the roots of the quadratic equation \\( b x^{2}-2 a x+c=0 \\). Choose one of the following options:\n\n(A) The... | A |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 14.24 degrees |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2112 years |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 77056420 |
[
{
"content": "Karlson has 47 ones written on a board. Each minute, he erases two arbitrary numbers and writes their sum on the board. He then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies Karlson could eat in 47 minutes?",
"role": "user"
}
] | 1081 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 193 nm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.6e5 N/m |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 15 kJ |
[
{
"content": "Calculate the limit: \\[ \\lim_{n \\to \\infty }n\\int_{0}^{\\pi}\\left \\{ x \\right \\}^{n}dx \\] where \\( \\{ x \\} \\) denotes the fractional part of \\( x \\).",
"role": "user"
}
] | 3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3.6e-3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 56 pN/C |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | The torque for the rod rotating about the centre is 1.4Nm
The torque for the rod rotating about one end is 5.6Nm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $1000 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 32.4 g |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to +\\infty}|u_n(x) - u(x)| \\) for the sequence of functions \\( u_n(x) = \\sqrt{\\left(\\frac{2}{3}x + \\frac{1}{\\sqrt[3]{n^2}}\\right)^3} - \\frac{1}{n} \\) and \\( u(x) = \\sqrt{\\left(\\frac{2}{3}x\\right)^3} \\).",
"role": "user"
}
] | +\infty |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 30 pF |
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