prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to0} \\frac{e^{\\frac{1}{2}x^2}\\cdot\\frac{2}{3}x^{-1/3}}{\\frac{d}{dx}\\left(\\int^{\\sqrt[3]{x^2}}_0e^{\\frac{1}{2}x^2}dx-x^{2/3}\\right)}$$",
"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... | 2500 |
[
{
"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... | $1,050 |
[
{
"content": "Determine the number of Hahn-Banach extensions of a non-zero continuous linear functional $f:(c_{00},\\|\\cdot\\|_1) \\to \\mathbb{C}$ to the space $(\\ell^1,\\|\\cdot\\|_1)$. Provide your answer as a single number.",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the triple integral \\( \\iiint_V xyz \\, dxdydz, \\) where \\( V \\) is the volume in \\( \\mathbb{R}^3 \\) defined by the inequalities \\( x^2+y^2+z^2\\leq1, \\quad x\\geq0, \\quad y\\geq0, \\quad z\\geq0. \\) Use spherical coordinates to perform the integration.",
"role": "user"
... | \dfrac{1}{48} |
[
{
"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... | 1049 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... | 0.0220 g |
[
{
"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... | Handling materials: $2.50
Inspecting product: $600.00
Processing purchase orders: $160.00
Paying supplies: $360.00
Insuring the factory: $7.00
Designing packaging: $47,000.00 |
[
{
"content": "Using the method of Lagrange multipliers, determine the minimum value of the function \\( f(x,y,z) = x^2y^2z^2 \\) subject to the constraint \\( x^2 + y^2 + z^2 = r^2 \\). Given the critical point \\((\\pm \\frac{r}{\\sqrt{3}}, \\pm \\frac{r}{\\sqrt{3}}, \\pm \\frac{r}{\\sqrt{3}})\\), identify the... | 0 |
[
{
"content": "Find the Laurent series of the function \\( g(z) = \\frac{\\cos({\\frac{1}{iz^3}})}{z^3} \\) at \\( z_0=0 \\) and determine its residue.",
"role": "user"
}
] | 0 |
[
{
"content": "Determine the interval of convergence for the series \\(1+\\frac{x}{2}+\\frac{2! x^2}{3^2}+\\frac{3!x^3}{4^3}+\\ldots\\).",
"role": "user"
}
] | (-e, e) |
[
{
"content": "Derive the infinite sum that converges to \\( \\tan\\theta \\) using the telescoping property:\n\n\\[\\sum_{n=1}^\\infty \\frac{\\tan\\dfrac{\\theta}{2^n}}{\\cos\\dfrac{\\theta}{2^{n-1}}}\\]\n\nShow that this series simplifies to \\( \\tan \\theta \\) by using tangent and trigonometric identities.... | \tan \theta |
[
{
"content": "Let $(X, \\mathcal{A})$ be a measurable space and let $\\mu$ be a signed measure on $\\mathcal{A}$. If $E, F \\in \\mathcal{A}$ and $F \\subset E$, does $\\mu(E)=0$ imply $\\mu(F)=0$? Answer yes or no and justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Given that \\( z_{1} \\) and \\( z_{2} \\) are complex numbers such that \\( \\left|z_{1}\\right|=3 \\), \\( \\left|z_{2}\\right|=5 \\), and \\( \\left|z_{1}+z_{2}\\right|=7 \\), find the value of \\( \\arg \\left(\\frac{z_{2}}{z_{1}}\\right)^{3} \\).",
"role": "user"
}
] | \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... | 2.6e+22 molecules |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{2\\pi} e^{\\cos \\theta} \\cos( \\sin \\theta) \\, d\\theta \\). To do this, define \\( I(\\alpha) = \\int_{0}^{2\\pi} e^{\\alpha \\cos \\theta} \\cos(\\alpha \\sin \\theta) \\, d\\theta \\) for \\( \\alpha \\in \\mathbb{R} \\). Calculate \\( I(0) \\) and use i... | 2\pi |
[
{
"content": "Determine the equilibrium solutions for the first order differential equation: \\[ \\frac{dy}{dx}=y^2-r\\sqrt{x}, ~~ x\\geqslant 0 \\]",
"role": "user"
}
] | y = 0 |
[
{
"content": "Benjamin is trying to break a combination lock. This particular type of lock has 5 digits from 0 to 9, and Benjamin just happens to know that there can be no repeated digits in the code for this type of lock. How many valid codes are there?",
"role": "user"
}
] | 30,\!240 |
[
{
"content": "Determine if there exist real numbers $x$ and $y$, where either $x$ or $y$ is not an integer, such that both $xy$ and $x - y$ are integers.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether there exists an $n \\times n$ matrix $M$ such that $M^2 = -I_n$ for even $n$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find a lower bound for the expression \\(\\frac{1 - \\cos(2x^{2})}{2x^{6}}\\) using the inequality \\(x^2y^2 \\le \\dfrac{(x^2+y^2)^2}{4}\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the residue of the function \\( \\frac{e^{z + \\frac{1}{z}}}{1-z^2} \\) at \\( z=1 \\).",
"role": "user"
}
] | -\dfrac{e^2}{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... | 5176 km/h |
[
{
"content": "Let $B_1, B_2 \\subset \\mathbb{R}^3$ be two topological balls with non-empty interiors in the usual Euclidean topology. Determine whether the connected components of the intersection $B_1 \\cap B_2$ are also topological balls.",
"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... | undervalued |
[
{
"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... | 70 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... | -0.365 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... | Cash Budget:
Budgeted Cash Receipts:
March | April
------- | --------
Cash Sales ($450,000*35%) | 157,500 | 182,000 ($520,000*35%)
Collections of Account Receivable | 285,000 | 328,900
Budgeted Cash Payments:
Particular | March | April
----------|-------|-------
Manufacturing Costs | $174,500 | $305,000
Selling & Administrative Expenses | $37,000 | $37,500
Capital Additions | $250,000 | -
Income Taxes | - | $40,000
Total Cash Payments | $461,500 | $382,500
Cash Increase/(Decrease) | ($19,000) | $128,400
Beginning Cash Balance | $45,000 | $26,000
Ending Cash Balace | $26,000 | $154,400
Minimum Cash Balance Required | $20,000 | $20,000
Excess/(Deficiency) | $6,000 | $134,400 |
[
{
"content": "Find the shortest distance between the hyperboloid $H = \\{(x,y,z) \\in \\mathbb{R}^3: x^2 + y^2 - z^2 + 4 = 0\\}$ and the point $P = (2,4,0)$. Express your answer as a single numerical value.",
"role": "user"
}
] | \sqrt{14} |
[
{
"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... | 47.78e4 C |
[
{
"content": "Find an irreducible polynomial of degree 4 in \\(\\mathbb{Z}_2[x]\\).",
"role": "user"
}
] | x^4 + x + 1 |
[
{
"content": "Evaluate \\( \\lim_{n\\to\\infty}\\int^2_0\\dfrac{dx}{1+x^{2n}} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether there exists a number divisible by 2020 in which each digit from 0 to 9 appears an equal number of times.",
"role": "user"
}
] | Yes |
[
{
"content": "In triangle $ABC$, $AB = AC = 2AD$. Let $H$ be the intersection of $BD$ and $AE$, where $AE$ is perpendicular to $BD$. Given that the ratio of the areas of $\\triangle ABH$ to $\\triangle ADH$ is $8:1$, determine the ratio $BH:HD$. \\( \\text{Express your answer as a simplified ratio.} \\)",
"... | 8:1 |
[
{
"content": "Find the volume of the solid $E$ bounded by the surfaces $z = 0$, $z = y$, and $y = 1 - x^2$. The region $E$ is defined as $$E = \\{(x,y,z) \\mid -1 \\leq x \\leq 1, \\ 0 \\leq y \\leq 1-x^2, \\ 0 \\leq z \\leq y \\}.$$",
"role": "user"
}
] | \dfrac{8}{15} |
[
{
"content": "Let $\\{a_n\\} _{n\\in\\mathbb{N}} $ be a sequence of real numbers such that $\\sum_{i=0}^{\\infty} a_i \\in\\mathbb{R} ^{+} $. Determine if it is true that \\( \\lim_{k\\rightarrow\\infty} \\sum_{i=k} ^{\\infty} a_i = \\lim_{n\\rightarrow\\infty} a_n =0, \\) given that the series $\\sum_i a_i$ is... | True |
[
{
"content": "Let $f$ and $g$ be two polynomials such that $f(0)=-24$, $g(0)=30$, and for all $x \\in \\mathbb{R}$, $f(g(x))=g(f(x))$. Find the values of $f(3)$ and $g(6)$. Express your answer as an ordered pair $(f(3), g(6))$. ",
"role": "user"
}
] | (-21, 36) |
[
{
"content": "Let G be the Galois group of a field with nine elements over its subfield with three elements. Determine the number of orbits for the action of G on the field with 9 elements.",
"role": "user"
}
] | 6 |
[
{
"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... | $41,244.40 |
[
{
"content": "Evaluate the limit \\( \\lim\\limits_{n\\to\\infty}\\frac{\\sum_{i=1}^n i^r}{n^{r+2}} \\) for \\( r>0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Let $a < b$ be positive real numbers. Determine if there exist $x, y \\in (a, b)$ such that \\( \\left(\\frac{a+b}{2}\\right)^{x+y} = a^x b^y \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Let $f: \\mathbb{R}^2 \\to \\mathbb{R}$ be a differentiable function with $f(0, -2) = 3$ and $\\nabla f(0, -2) = (-6, 2)$. Define $g(t) = \\|(t, t^2, f(3t, t-2))\\|$. Find $g'(0)$. Provide your answer as a single value.",
"role": "user"
}
] | -16 |
[
{
"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... | 2e-11 Volt |
[
{
"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 mean square distance can be calculated using the integral: <r^2> = \frac{\int_V[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2]\rho(x,y,z)\,dV}{\int_V\rho(x,y,z)\,dV} |
[
{
"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... | -476.2 J/K |
[
{
"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... | 2.2 m/s |
[
{
"content": "Suppose \\(A\\) is an \\(n \\times n\\) matrix with complex entries, and its characteristic polynomial is identical to its minimal polynomial, given by \\(p_{A}(t) = m_{A}(t) = (t - \\lambda)^{n}\\). Determine the dimension of the eigenspace corresponding to the eigenvalue \\(\\lambda\\).",
"r... | 1 |
[
{
"content": "Consider three sets x, y, and z, where x ∈ y, y ∈ z, and z ∈ x. Determine if this configuration violates the Foundation axiom.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether every vector in the set $[-1,+1]^d$ can be expressed as a convex combination of vectors in the set $\\{-1,+1\\}^d$. Provide a proof or counterexample.",
"role": "user"
}
] | Yes |
[
{
"content": "If $n$ is an integer, $1 \\leq n \\leq 2010,$ how many fractions $\\frac{n^2}{2010}$ yield repeating decimals?",
"role": "user"
}
] | 2000 |
[
{
"content": "Prove that the quotient space $\\mathbb{R} \\text{P}^{n} / \\mathbb{R} \\text{P}^{n-1}$ is homeomorphic to the sphere $S^n$.",
"role": "user"
}
] | S^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... | 9 sigma bonds and 2 pi bonds |
[
{
"content": "Determine the values of \\( \\alpha \\in \\mathbb{R} \\) for which the integral \\( \\int_0^1 \\int_0^1 \\frac{1}{|y-x|^\\alpha} \\, dx \\, dy \\) is convergent.",
"role": "user"
}
] | \alpha < 1 |
[
{
"content": "Determine whether the following statement is true or false: For any holomorphic function $f$ on a bounded connected open subset of $\\mathbb{C}$ with smooth boundary, there exists a constant $K$ such that $|f'(z)| \\leq K/\\rho(z)$, where $\\rho(z)$ is the distance from $z$ to the boundary.",
... | B |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{1} \\frac{r^3}{ \\sqrt {1-r^2}}dr \\).",
"role": "user"
}
] | \dfrac{2}{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... | Pele Company's Balance Sheet
Before Action | After Stock Dividend | After Stock Split
---|---|---|
Stockholders' equity: | | |
Paid-in capital | $488,400 | $561,660 | $488,400
Retained earnings | $404,500 | $331,240 | $404,500
Total stockholders' equity | $892,900 | $892,900 | $892,900
Outstanding shares | 81,400 | 85,470 | 162,800 |
[
{
"content": "Determine whether every Eulerian bipartite graph contains an even number of edges.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty} \\left(\\frac{4n^2+2n-3}{3n^2-n-2}\\right)^{\\large \\frac{n^2}{2-n}}$$",
"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... | 3.104 keV |
[
{
"content": "Given that \\( x^2 + y^2 + z^2 = 1 \\) for \\( x, y, z \\in \\mathbb{R} \\), find the maximum value of the expression \\( x^3 + y^3 + z^3 - 3xyz \\).",
"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... | 2H_2S + 4Ag + O_2 \to 2Ag_2S + 2H_2O |
[
{
"content": "Calculate the volume of the solid formed by revolving the region defined by \\((x-1)^2 + z^2 \\leq 1\\) around the Z-axis. Use either a triple integral or a double integral to find the solution.",
"role": "user"
}
] | 2\pi^2 |
[
{
"content": "Determine the value of $x$ that ensures the convergence of the infinite series $$\\sum_{n=0}^{\\infty}\\frac{n^{1000}}{\\sqrt{n!}}x^n.$$",
"role": "user"
}
] | (-\infty, \infty) |
[
{
"content": "Determine the values of \\( x \\) for which the series \\( \\sum_{n=0}^{\\infty} \\left(\\frac{n}{2n+1}\\right)^{2n-1}x^n \\) converges.",
"role": "user"
}
] | (-4, 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... | 5793 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... | $96.10 |
[
{
"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... | 125700 |
[
{
"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... | 2800 |
[
{
"content": "Given that \\(\\lim\\limits_{x \\to 2} \\frac{f(x)-f(2)}{x-2}=\\frac{-1}{3}\\) and \\(y=f(\\sqrt{|x|+3})\\), find \\(\\frac{d}{dx}y(-1)\\).",
"role": "user"
}
] | \dfrac{1}{12} |
[
{
"content": "Consider a polynomial \\( P_n(x) \\) of degree \\( n > 0 \\). Determine if the equation \\( P_n(x) = e^{ax} \\), where \\( a \\neq 0 \\), can have at most \\( n+1 \\) real solutions. Justify your answer.",
"role": "user"
}
] | n+1 |
[
{
"content": "Evaluate the integral \\( \\iiint_K e^z \\, dx\\, dy\\, dz \\) where \\( K \\) is the solid ball defined by \\( K = \\{x^2 + y^2 + z^2 \\leq 1\\} \\).",
"role": "user"
}
] | \dfrac{4\pi}{e} |
[
{
"content": "Evaluate the surface integral \\(\\iint_S (\\nabla \\times \\vec{F}) \\cdot \\vec{n} \\, dS\\) for the vector field \\(\\vec{F} = y\\hat{i} + (x - 2xz)\\hat{j} - xy\\hat{k}\\), where the surface \\(S\\) is the upper half of the sphere with equation \\(x^2 + y^2 + z^2 = a^2\\). Given that \\(\\nabl... | 0 |
[
{
"content": "Does the differentiability of a function over an interval imply the continuity of its derivative over the same interval?",
"role": "user"
}
] | No |
[
{
"content": "Given the solutions \\(\\{z_1, z_2, z_3, z_4, z_5\\}\\) to the equation \\(z^5 = 2i\\) as follows:\n\n\\[z_1 = \\sqrt[5]{2}\\text{cis}\\left(\\frac{\\pi}{10}\\right)\\]\n\\[z_2 = \\sqrt[5]{2}\\text{cis}\\left(\\frac{\\pi}{2}\\right)\\]\n\\[z_3 = \\sqrt[5]{2}\\text{cis}\\left(\\frac{9\\pi}{10}\\rig... | 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... | 0.46 |
[
{
"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... | 107.91 meters per second |
[
{
"content": "Find the limit of the function as \\( x \\) approaches infinity: \\[ \\lim_{x \\to \\infty} x \\left[ \\frac{1}{e} - \\left( \\frac{x}{x+1} \\right)^x \\right] \\]",
"role": "user"
}
] | -\dfrac{1}{2e} |
[
{
"content": "Calculate the residue of the function \\( f(z) = \\frac{1}{z^2 \\sin(z)} \\) at \\( z = 0 \\), given that it is a pole of order 3.",
"role": "user"
}
] | \dfrac{1}{6} |
[
{
"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... | $1,087.42 |
[
{
"content": "Calculate the total work done in moving a particle in a force field given by \\( F=(y^2-x^2)i+(2-x+y)j \\) along the curve \\( y=x^3 \\), from \\((-1,-1)\\) to \\((1,1)\\).",
"role": "user"
}
] | \dfrac{76}{21} |
[
{
"content": "In triangle ABC, where \\( \\angle A = 120^\\circ \\), the lines AD, BE, and CF are the angle bisectors of \\( \\angle A \\), \\( \\angle B \\), and \\( \\angle C \\), respectively. Determine the measure of \\( \\angle FDE \\).",
"role": "user"
}
] | 90^\circ |
[
{
"content": "If $a$ is a constant such that $9x^2 + 24x + a$ is the square of a binomial, then what is $a$?",
"role": "user"
}
] | 16 |
[
{
"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... | 6519.2 N |
[
{
"content": "Determine whether the following statement is true or false: A complex function $f(z)$ is differentiable at $z_0 = x_0 + y_0$ if and only if the Cauchy-Riemann equations hold at $(x_0, y_0)$. Provide a justification for your answer.",
"role": "user"
}
] | B |
[
{
"content": "What is the smallest value of $a$ such that a Möbius strip can be constructed using a rectangular piece of paper with dimensions $[0,1]\\times[0,a]$, where $a > 1$?",
"role": "user"
}
] | \sqrt{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... | 0.45 |
[
{
"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 m/s |
[
{
"content": "Does there exist a function $f \\in C\\{M_n\\}$, where $C\\{M_n\\}$ is a quasi-analytic class of infinitely differentiable functions with inequalities $|f^{(n)}(x)| \\leq \\beta_f B_f^n M_n$ for all $x \\in \\mathbb{R}$ and $n = 0, 1, 2, \\ldots$, having infinitely many zeros within a bounded inte... | No |
[
{
"content": "What is the minimum value of the expression $x^2+y^2+2x-4y+8$ for real $x$ and $y$?",
"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... | 87.88 kJ |
[
{
"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... | 137.5% |
[
{
"content": "Determine which of the following expressions can represent the first fundamental form of a surface:\n\n1. \\( ds^2 = du^2 + 4 dudv + dv^2 \\)\n2. \\( ds^2 = du^2 + 4 dudv + 4 dv^2 \\)\n3. \\( ds^2 = du^2 - 4 dudv + 6 dv^2 \\)\n4. \\( ds^2 = du^2 + 4 dudv - 2 dv^2 \\)\n\nSelect the correct expressi... | 3 |
[
{
"content": "Does the condition that local rings $R$ and $S$ have the same residue field imply that $S$ dominates $R$? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "For an affine variety $V$ in $\nabla^{n}$ with a coordinate ring that is a unique factorization domain (UFD), determine if it is true that any closed subvariety of $V$ with codimension 1 can be defined by a single equation.",
"role": "user"
}
] | True |
[
{
"content": "Can a normal subgroup of a nonabelian finite group be nonabelian?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to +\\infty} \\frac{\\int_{0}^{x} t|\\sin t| \\, dt}{x^{2}}.$$",
"role": "user"
}
] | \dfrac{1}{\pi} |
[
{
"content": "Evaluate the limit of the sequence \\(a_n = \\left( \\left(1 + \\frac{1}{n}\\right) \\left(1 + \\frac{2}{n}\\right) \\cdots \\left(1 + \\frac{n}{n}\\right) \\right)^{\\frac{1}{n}}\\) as \\(n\\) approaches infinity.",
"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... | 29.7 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... | $2,006.42 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} \\left( \\frac{\\sqrt{x^2+2x-3}}{x+2} \\right)^{3-2x} \\]",
"role": "user"
}
] | e^2 |
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