prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate $|2\\omega^2-4\\omega-30|$ if $\\omega=1-5i$.",
"role": "user"
}
] | 82 |
[
{
"content": "Show that \\(\\lim_{x\\to 0^+}\\frac{x}{a}\\cdot\\left\\lfloor\\frac{b}{x}\\right\\rfloor=\\frac{b}{a}\\).",
"role": "user"
}
] | \dfrac{b}{a} |
[
{
"content": "Let \\( f(t) = 1 - t^2 \\) for \\( |t| < 1 \\) and \\( 0 \\) elsewhere. Compute the Fourier transform of \\( f(t) \\) and use the result to evaluate the integral \\( \\int_{-\\infty}^{\\infty} \\frac{\\sin t - t \\cos t}{t^3} \\, dt \\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Find the number of divisors of the number \\(2^2 \\times 3^3 \\times 5^5 \\times 7^4 \\times 11^3\\) that are of the form \\(6k+1\\), where \\(k\\) is a non-negative integer.",
"role": "user"
}
] | 60 |
[
{
"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... | 65.7 kg |
[
{
"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... | 26.3 cm |
[
{
"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... | 153.14 kJ |
[
{
"content": "The Lucas numbers $L_n$ are given by the recursion $L_0 = 2, L_1 = 1$, and $L_n = L_{n-1} + L_{n-2}$. Find the units digit of $L_{L_{10}}$.",
"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... | 0.587 |
[
{
"content": "Evaluate the limit as \\( n \\to \\infty \\): \\[ \\lim_{n\\to\\infty}\\frac {3^{3n} \\binom{2n}{n}^2} {\\binom{3n}{n} \\binom{6n}{3n}} \\]",
"role": "user"
}
] | 2 |
[
{
"content": "Evaluate the limit: $$\\lim_{b \\to 0^{-}} \\int_{b}^{0} \\frac{1}{\\sqrt{\\sin(b)-\\sin(x)}} \\;dx$$",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the set \\( \\{(x,y) \\in [0,1]^2 : 0 \\leq y \\leq f(x) \\} \\), where \\( f \\) is the Cantor Lebesgue function, is a Borel set in \\( \\mathbb{R}^2 \\).",
"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... | 1.52 |
[
{
"content": "How many normal subgroups does a non-abelian group $G$ of order $21$ have, other than the identity subgroup $\\{e\\}$ and $G$ itself? Choose the correct answer from the options below:\n\nA) 0\nB) 1\nC) 3\nD) 7",
"role": "user"
}
] | B |
[
{
"content": "Find the limit \\( \\lim_{n \\to \\infty} \\frac{(2n^{\\frac{1}{n}}-1)^n}{n^2} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Rationalize the denominator of $\\frac{5}{\\sqrt{125}}$.",
"role": "user"
}
] | \frac{\sqrt{5}}{5} |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0} \\frac{\\int_0^x \\cos(t^2) \\, dt}{x}.$$",
"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... | 393.94K |
[
{
"content": "Consider a length space $M$ and a finite set of geodesics $\\Pi$ in this space, where each geodesic is the unique path between its endpoints. Suppose that any two distinct geodesics $\\pi_1, \\pi_2 \\in \\Pi$ intersect at most at a single point. Is it possible to assign a direction to each geodesi... | Yes |
[
{
"content": "A positive integer is written on each vertex of a pentagon, with each vertex having a different integer. On each side of the pentagon, the least common multiple (LCM) of the numbers at the vertices forming that side is written. If the same number n is written on each side, determine the minimum po... | 30 |
[
{
"content": "Vasya wrote down the non-zero coefficients of an 8th-degree polynomial \\( P(x) \\). He then calculated the derivative of the polynomial and noted its non-zero coefficients, continuing this process until a constant was obtained, which he also noted. What is the minimum number of different numbers ... | 8 |
[
{
"content": "Evaluate the complex integral \\( \\oint g(z) \\, dz \\) where \\( g(z) = \\left(V+\\frac{Vb^2}{z^2}\\right)^2 \\), with \\( V \\) and \\( b \\) being real constants, over a closed contour that does not enclose the origin.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit \\(\\lim_{x\\to \\infty} \\left(\\frac{(2x)!}{x^x \\cdot x!}\\right)^{\\frac{1}{x}}\\).",
"role": "user"
}
] | \dfrac{4}{e} |
[
{
"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... | 30.5 days |
[
{
"content": "Determine whether there exists a continuously differentiable function $f: [-1, 1] \\rightarrow \\mathbb{R}$ such that $f$ converges uniformly to zero but is unbounded in the norm $N_2(f) = |f(0)| + \\sup_{x \\in [-1, 1]} |f'(x)|$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let $X \\sim \\exp(1)$, where the density function is $\\exp(-x)$ for $x \\ge 0$ and zero otherwise. Calculate $E[X^n]$ for every $n \\in \\mathbb{N}$.",
"role": "user"
}
] | n! |
[
{
"content": "Find the equation of the common tangent to the curves \\( y^2 = 8x \\) and \\( xy = -1 \\).",
"role": "user"
}
] | y = x + 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... | 1.076e-7 |
[
{
"content": "Simplify: $x(3x^2-2)-5(x^2-2x+7)$. Express your answer in the form $Ax^3+Bx^2+Cx+D.$",
"role": "user"
}
] | 3x^3-5x^2+8x-35 |
[
{
"content": "Does there exist a Borel set $B \\subseteq E$, where $E \\subseteq \\mathbb{R}$ with $m^*(E) > 0$, such that $m(B) > 0$? Here, $m^*(E)$ denotes the Lebesgue outer measure of $E$, defined as:\n\\[ m^*(E) = \\inf\\left\\{\\sum_{n=1}^\\infty|b_n - a_n|\\, :\\mid\\, E \\subseteq \\bigcup_{n = 1}^\\inf... | Yes |
[
{
"content": "Determine the number of degree \\(d\\) hypersurfaces in \\(\\mathbb{P}^n\\) that contain a generic set of \\(\\binom{d+n}{d}-1\\) points.",
"role": "user"
}
] | 1 |
[
{
"content": "The sequence \\( \\{a_n\\} \\) of real numbers satisfies the recursive relation \\( n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n \\) for every positive integer \\( n \\), where \\( a_0 = a_1 = 1 \\). Calculate the sum \\( \\frac{a_0}{a_1} + \\frac{a_1}{a_2} + \\cdots + \\frac{a_{2008}}{a_{2009}} \\).",
... | 2019045 |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\to \\infty}\\frac{\\ln(a^n+b^n+c^n)}{\\sqrt{n^2+n+1}} \\) where \\( 0<a<b<c \\).",
"role": "user"
}
] | \ln c |
[
{
"content": "Consider normed linear spaces X and Y, and a finite-rank continuous linear transformation T from a subspace W of X to Y. Is it possible to extend T to a continuous linear transformation from X to Y that maps X onto the range of T?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether the following statement is true or false: If a linear algebraic group defined over \\(\\mathbb{R}\\) has a faithful irreducible representation, then it is reductive.",
"role": "user"
}
] | True |
[
{
"content": "In a population of 10,000 birds consisting of 8 equally sized species, determine the minimum number of birds you need to catch to have an 80% probability of capturing at least one bird from each species.",
"role": "user"
}
] | 28 |
[
{
"content": "Calculate the set $II = \\{xy \\mid x, y \\in I\\}$ for the ideal $I = (2, 1+\\sqrt{-5})$ in the ring $\\mathbb{Z}[\\sqrt{-5}]$. Provide the process of calculation and the resulting set.",
"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... | 2MnO4- + 5H2O2 + 6H+ -> 2Mn2+ + 5O2 + 8H2O |
[
{
"content": "Is it possible to divide a circle into an infinite number of non-congruent quadrilaterals?",
"role": "user"
}
] | Yes |
[
{
"content": "Is the estimator \\( \\hat{F}(x_0)=\\frac{1}{n}\\sum_{i=1}^nI\\{X_i\\leq x_0\\} \\) a consistent estimator for \\( F(x_0) \\), where \\( X_1,X_2,\\ldots,X_n \\) are i.i.d. random variables with common CDF \\( F \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Consider the heat equation $u_t = u_{xx}$ on the domain $[0, \\pi] \\times [0, L]$ with boundary conditions $u(0,t) = u(\\pi,t) = 0$ for $t \\in [0, L]$ and initial condition $u(x,0) = \\phi(x)$ for $x \\in [0, \\pi]$. If $u(x,L) = f(x)$, determine which of the following statements is true for a s... | A |
[
{
"content": "Let $M$ be a connected, non-compact, non-orientable topological manifold of dimension $n$. Is the top singular cohomology group $H^n(M, \\mathbb{Z})$ zero?",
"role": "user"
}
] | 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... | 0.853L |
[
{
"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... | 3.12e-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... | Present Operating Conditions:
Sales: $70,000
Variable Expenses: $40,000
Contribution Margin: $30,000
Fixed Expenses: $46,000
Net Operating Income: -$16,000
After Proposed Changes:
Sales: $78,750
Variable Expenses: $50,000
Contribution Margin: $28,750
Fixed Expenses: $46,000
Net Operating Income: -$17,250 |
[
{
"content": "Find the integrating factor of the form \\( \\mu(x,y) = x^m y^n \\) that makes the first-order ODE \\((2y^2-3xy)dx + (4xy-3x^2)dy = 0\\) exact.",
"role": "user"
}
] | x y^2 |
[
{
"content": "Determine whether there exists an integer \\( n \\) such that \\( \\sqrt{p+n} + \\sqrt{n} \\) is an integer prime \\( p \\). Choose the correct option:\n\n(A) None exist \n(B) Only one exists \n(C) More than one exist, but finitely many \n(D) Infinitely many exist",
"role": "user"
}
] | D |
[
{
"content": "Let $a,$ $b,$ and $c$ be real numbers such that $a^2 + b^2 + c^2 = 1.$ Find the minimum value of\n\\[ab + ac + bc.\\]",
"role": "user"
}
] | -\frac{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... | $49,782.80 |
[
{
"content": "Determine the infimum of the set \\( A = \\{ |\\sin(n)| : n \\in \\mathbb{N} \\} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the surface integral \\(\\iint _S F\\cdot n \\space dA\\) using the divergence theorem, where \\( F = [xy, yz, zx]\\) and \\(S\\) is the surface of the cone defined by \\(x^2 + y^2 \\le 4z^2\\) with \\(0 \\le z \\le 2\\).",
"role": "user"
}
] | 16\pi |
[
{
"content": "Consider a finite extension $L/\\mathbb{Q}$ that does not lie in $\\mathbb{R}$. Determine the degree of the extension $L/ (L \\cap \\mathbb{R})$. Provide your answer as a single integer or expression.",
"role": "user"
}
] | 2 |
[
{
"content": "Find the number of cubic centimeters in the volume of the cylinder formed by rotating a square with side length 14 centimeters about its vertical line of symmetry. Express your answer in terms of $\\pi$.",
"role": "user"
}
] | 686\pi |
[
{
"content": "Find the limit of \\( \\frac{(n!)^{1/n}}{n} \\) as \\( n \\to \\infty \\).",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"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... | 13.9 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... | $77,000 |
[
{
"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... | 6.12e9 Joules |
[
{
"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... | $2032 |
[
{
"content": "Given the polynomial \\((1-z)^{b_{1}}(1-z^{2})^{b_{2}}(1-z^{3})^{b_{3}} \\cdots(1-z^{32})^{b_{32}}\\), where \\( b_{i} \\) are positive integers for \\( i = 1, 2, \\ldots, 32 \\), and this polynomial has the property that after expanding it and removing the terms with \\( z \\) raised to powers hi... | 2^{27} - 2^{11} |
[
{
"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... | $333,066.67 |
[
{
"content": "Evaluate the integral \\( \\int_{\\mathbb{R}} (F(x+b)-F(x+a)) \\, dx \\) for any real numbers \\(a\\) and \\(b\\), where \\(F\\) is a distribution function.",
"role": "user"
}
] | b - a |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty}\\frac{x^2}{1+x^6} \\, dx \\) using a suitable substitution.",
"role": "user"
}
] | \dfrac{\pi}{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... | kR^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... | 3.45e4 |
[
{
"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... | $3.05 per share |
[
{
"content": "In the context of an algebraically closed field of characteristic 0 and finitely dimensional semisimple Lie algebras, is the decomposition of a representation into a direct sum of irreducible subrepresentations unique up to ordering?",
"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... | 0.291 kg |
[
{
"content": "Suppose over the integers \\( \\mathbb{Z} \\), we have an irreducible polynomial \\( p(x) \\). Does \\( p(x) \\) represent a prime number for some integer values of \\( x \\)? Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"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... | E(k)=E_0+((\hbar^2 k^2)/(2m^*)) |
[
{
"content": "Evaluate the integral: \\[ \\int_{0}^\\infty \\frac{\\sin^3(2x)}{x} \\, dx \\]",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Calculate the surface area of the portion of the cone defined by $x^2 + y^2 = z^2$ that lies inside the sphere $x^2 + y^2 + z^2 = 6z$. What is the surface area of this ice cream cone?",
"role": "user"
}
] | 9\sqrt{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... | -4/3 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... | 1169.58 g |
[
{
"content": "Calculate the volume of the solid bounded by the planes $z = 0$, $x = 1$, $x = 3$, $y = 1$, $y = 2$, and the surface $z = x^2y^2$ using triple integration in Cartesian coordinates. Evaluate the integral:\n\\[ \\int_1^3 \\int_1^2 \\int_0^{x^2y^2} dz \\, dy \\, dx \\]",
"role": "user"
}
] | \dfrac{182}{9} |
[
{
"content": "Find an equivalent form for \\( f(x)=\\sum_{n=0}^{\\infty} e^{-x\\sqrt{n}} \\) as \\( x \\rightarrow 0^+ \\).",
"role": "user"
}
] | \dfrac{2}{x^2} |
[
{
"content": "How many ways are there to put 4 indistinguishable balls into 2 distinguishable boxes?",
"role": "user"
}
] | 5 |
[
{
"content": "Let $X_1, X_2, \\ldots, X_n$ be independent and identically distributed random variables with probability density function $f_X(x) = \\frac{1}{x^2}$ for $x \\geq 1$ and $f_X(x) = 0$ otherwise. Find the expected value $\\mathbb{E}[Y]$, where $Y = \\sum_{i=1}^{n} \\log X_i$. Provide your answer in t... | 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... | $5,055.57 |
[
{
"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... | Molecular Clock |
[
{
"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... | 2,183,333 |
[
{
"content": "Determine if the function \\( g(x) = \\sqrt{|x|} \\) belongs to the Sobolev space \\( W^{1,p}((-1, 1)) \\) for \\( p \\in [1, \\infty) \\).",
"role": "user"
}
] | [1, 2) |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0} \\left(a^x + b^x - c^x\\right)^{\\frac{1}{x}}$$ where \\(a, b, c\\) are positive real numbers.",
"role": "user"
}
] | \dfrac{ab}{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... | 9.0 |
[
{
"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... | 216540 km/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... | Comprehensive Income: $224,280 |
[
{
"content": "Let $a, b, c$ be the roots of the polynomial equation $x^3 - (k+1)x^2 + kx + 12 = 0$, where $k$ is a real number. Given that $(a-2)^3 + (b-2)^3 + (c-2)^3 = -18$, determine the value of $k$. Express your answer as a single number.",
"role": "user"
}
] | 5 |
[
{
"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... | 1.87e6 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... | -0.0255 cm |
[
{
"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... | $67.20 million |
[
{
"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... | 0.002 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... | 3.16 |
[
{
"content": "Determine the possible values of the integral \\( \\frac{1}{2\\pi i}\\int_{\\gamma}\\frac{2i}{z^2+1}dz \\), where \\( \\gamma \\) is a closed continuously differentiable path in the upper half-plane that does not pass through \\( i \\).",
"role": "user"
}
] | \mathbb{Z} |
[
{
"content": "Evaluate the limit \\( \\lim_{x \\to 0} \\frac{\\int_{0}^{x}t\\sin t \\,\\text{d}t}{x-\\sin x} \\) using L'Hopital's rule.",
"role": "user"
}
] | 2 |
[
{
"content": "Calculate the flux of the vector field \\( \\mathbf{F}(x, y, z) = \\langle e^{z^2}, 2y + \\sin(x^2z), 4z + \\sqrt{x^2 + 9y^2} \\rangle \\) through the surface \\( S \\), where \\( S \\) is defined by the region \\( x^2 + y^2 \\leq z \\leq 8 - x^2 - y^2 \\).",
"role": "user"
}
] | 96\pi |
[
{
"content": "The infinite sequence $S=\\{s_1,s_2,s_3,\\ldots\\}$ is defined by $s_1=7$ and $s_n=7^{s_{n-1}}$ for each integer $n>1$. What is the remainder when $s_{100}$ is divided by $5$?",
"role": "user"
}
] | 3 |
[
{
"content": "Find the remainder when \\(12^{75}\\) is divided by 85 using the Chinese Remainder Theorem.",
"role": "user"
}
] | 23 |
[
{
"content": "Compute the integral \\( \\int_0^\\infty \\frac{1}{(1+x^{2015})(1+x^2)} \\, dx \\).",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Let $T: V \\rightarrow V$ be a linear operator on a vector space $V$. If $T$ is compact and $T^k$ has no eigenvectors for some positive integer $k$, does it follow that $T$ has no eigenvectors? Answer yes or no and justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the minimum value of the function \\( y = \\tan x + \\frac{\\tan x}{\\sin(2x - 90^\\circ)} \\) for \\( 45^\\circ < x < 90^\\circ \\).",
"role": "user"
}
] | 3\sqrt{3} |
[
{
"content": "Given an open subset $\\Omega\\subseteq {\\mathbb R}^n$ and a smooth injective embedding $i: \\Omega\\to {\\mathbb R}^N$ where $N\\geq n$, determine whether there exists a smooth mapping $f: A\\to \\Omega$ such that $A\\supseteq i(\\Omega)$, $A$ is open in ${\\mathbb R}^N$, and $f|_{i(\\Omega)} = ... | Yes |
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