prompt list | ground_truth string |
|---|---|
[
{
"content": "Evaluate the definite integral \\( \\int_{-\\infty}^{\\infty} \\frac{\\sin^2(\\pi Rx)}{R(\\pi x)^2}dx \\) for \\( R > 0 \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Given a conical tent with a fixed volume, determine the ratio of its height to the radius of its base that minimizes the amount of canvas required for its construction.",
"role": "user"
}
] | \sqrt{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... | -19.2 m/s |
[
{
"content": "Determine the value of \\(a\\) such that the function \\(f(x) = ax^2 + 1 - \\ln(1+x)\\), defined for \\(x \\in (-1, \\infty)\\), has exactly two extreme points.",
"role": "user"
}
] | (-\infty, -2) |
[
{
"content": "Calculate the degree of the field extension \\(\\mathbb{Q}(\\sqrt{p_1},\\dots,\\sqrt{p_n})\\) over \\(\\mathbb{Q}\\) for distinct prime numbers \\(p_1,\\dots,p_n\\).",
"role": "user"
}
] | 2^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... | $122,550 |
[
{
"content": "A bag contains 100 uniquely labeled marbles. You draw 9 marbles at random from the bag and then return them. This process is repeated 50 times. What is the probability that you have selected less than 70% of all the marbles at least once during these 50 draws?",
"role": "user"
}
] | 0 |
[
{
"content": "Given the quadratic function \\( f(x) = x^2 + b x + 8 \\) where \\( b \\neq 0 \\), which has two distinct real roots \\( x_1 \\) and \\( x_2 \\), and the expressions \\( x_1 + \\frac{1}{x_2} \\) and \\( x_2 + \\frac{1}{x_1} \\) are the roots of another quadratic function \\( g(x) \\) with a leadin... | -8 |
[
{
"content": "Determine the smallest integer \\( n \\) such that the proportion \\( p(n) = \\frac{\\text{number of asymmetric graphs with } n \\text{ nodes}}{\\text{number of graphs with } n \\text{ nodes}} \\) exceeds 0.99.",
"role": "user"
}
] | 15 |
[
{
"content": "Evaluate the integral \\( \\int_C \\frac{z+2}{(z^2-2z+1)^2}dz \\) using the residue theorem, where \\( C \\) is a semi-circle of radius \\( R>1 \\) in the right half-plane connected by a straight line on the imaginary axis that connects the two endpoints of the semi-circle.",
"role": "user"
... | 0 |
[
{
"content": "Let \\( \\alpha \\) be the closed curve along the square with vertices at \\( 1, i, -1, -i \\). Calculate \\( \\frac{1}{2\\pi i}\\int_\\alpha\\frac{dz}{z} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Given positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{12}$, what is the smallest possible value for $x + y$?",
"role": "user"
}
] | 49 |
[
{
"content": "You are on a basketball team, and at the end of every practice, you shoot half-court shots until you make one. Each half-court shot, independent of all other shots, has a 0.1 probability of going in. Your team has 100 practices per season. Estimate the probability that you shoot more than 1111 hal... | 0.12 |
[
{
"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... | Prepaid Expense A/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... | 391.83 m/sec |
[
{
"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... | 4.62e-18 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... | Xerox: 0.304, Qwest: 0.182, Liz Claiborne: 0.514 |
[
{
"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... | 564.57 J |
[
{
"content": "Find the optimal solution \\((x^*, y^*)\\) for the following concave maximization problem using the Karush-Kuhn-Tucker conditions: \\[\\max_{x, y} f(x,y) = 2\\ln x - \\ln (x+y) - x -2y\\] subject to \\(x \\geq 0, y \\geq 0\\).",
"role": "user"
}
] | (1, 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... | [Ni(NH_3)_4]^{2+} + 2OH^- |
[
{
"content": "Find the supremum of the set \\( \\{ \\operatorname{Re}(iz^3+1) : |z|<2 \\} \\).",
"role": "user"
}
] | 9 |
[
{
"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... | Neutrons |
[
{
"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... | T_2 = \dfrac{1}{\dfrac{1}{T_1} - \dfrac{R\ln(0.25)}{E_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... | 1,133 |
[
{
"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... | 5.7 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... | -0.05 |
[
{
"content": "Determine the interval of convergence for the power series:\n\n$$\n1-\\frac{x}{2 \\cdot 2}+\\frac{x^{2}}{3 \\cdot 2^{2}}-\\frac{x^{3}}{4 \\cdot 2^{3}}+\\ldots+(-1)^{n} \\frac{x^{n}}{(n+1) 2^{n}}+\\ldots\n$$",
"role": "user"
}
] | (-2, 2] |
[
{
"content": "If the function \\( f(x) = \\sin(\\ln x) - 2\\cos(\\ln x) \\) is increasing in the interval \\((e^\\lambda, e^\\mu)\\), find the value of \\( \\sqrt{5}\\cos(\\mu-\\lambda) \\).",
"role": "user"
}
] | -\sqrt{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... | 0.187 M/s |
[
{
"content": "Let $(G, \\cdot)$ be a centerless group with $n$ elements, where $n$ is an integer not divisible by $4$. Determine whether there exist distinct elements $a, b \\in G$ such that their centralizers are equal, i.e., $C(a) = C(b)$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Compute the value of $f(2)$ for an entire function $f(z)$ satisfying the inequality $|f(z) - z + 2z^2| \\leq \\arctan(|z|)$ for all $z \\in \\Bbb C$. Use the property of $f$ being entire to find the solution.",
"role": "user"
}
] | -6 |
[
{
"content": "Find all integer solutions to the equation \\( f(g(f(x))) = g(f(g(x))) \\), where \\( f(x) = x^2 + 4x + 3 \\) and \\( g(x) = x^2 + 2x - 1 \\).",
"role": "user"
}
] | -2 |
[
{
"content": "Calculate the arc length of the curve given by the equation \\(e^{2x+2y}=x-y\\) from the point \\((0,0)\\) to the point \\((1,1)\\).",
"role": "user"
}
] | \sqrt{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... | 88.5e-16 C/m^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... | Owning more than 25% of the company or having a board seat and the ability to significantly alter operations or financial policy. |
[
{
"content": "Prove the equality: \\[ 1=(-1)^n\\sum_{k=0}^n\\frac{x_k^n}{\\prod_{\\substack{l=0 \\\\ l \\neq k}}^n(x_l-x_k)} \\]",
"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... | 16.4 Volt |
[
{
"content": "Find the gradient of the function $f(M) = x^T M x$ with respect to the matrix $M$, where $x \\in \\mathbb{R}^n$ and $M \\in \\mathbb{R}^{n \\times m}$. Express your answer in terms of $x$ and $M$. ",
"role": "user"
}
] | xx^T |
[
{
"content": "Let $X$ and $Y$ be integral, proper curves of finite type over an algebraically closed field $k$, and let $f: X \\rightarrow Y$ be a finite étale morphism of degree $n$. For a closed point $P \\in X$ and its image $Q = f(P) \\in Y$, the induced map $f^\\#: O_{Y,Q} \\rightarrow O_{X, P}$ is finite ... | 1 |
[
{
"content": "Determine the number of homomorphisms from the group \\(C_2 \\times C_3\\) to the symmetric group \\(S_4\\).",
"role": "user"
}
] | 18 |
[
{
"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... | 0.659 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... | 13.0 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... | 2.8 |
[
{
"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... | 1.04 m/s |
[
{
"content": "If $f:\\mathbb{R}\\to\\mathbb{R}$ is a differentiable function such that $x^3f'(x)\\to 0$ as $x\\to\\infty,$ determine whether it is true that $f(x)\\to c\\in\\mathbb{R}$ as $x\\to\\infty$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the number of roots of the equation \\( f(x) = \\frac{1}{(x+1)^3} - 3x + \\sin x \\) without using a graphing calculator.",
"role": "user"
}
] | 2 |
[
{
"content": "In parallelogram \\(ABCD\\), \\(\\angle B = 111^\\circ\\) and \\(BC = BD\\). A point \\(H\\) is located on segment \\(BC\\) such that \\(\\angle BHD = 90^\\circ\\). Let \\(M\\) be the midpoint of side \\(AB\\). Determine the measure of angle \\(AMH\\) in degrees.",
"role": "user"
}
] | 132 |
[
{
"content": "Determine whether the following statement is true or false: For a function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$, is it true that $f \\circ f = f$ if and only if $f \\upharpoonright \\text{rng}(f) = \\text{id}_{\\text{rng}(f)}$? Justify your answer.",
"role": "user"
}
] | True |
[
{
"content": "Find the arc length of the astroid defined by the parametric equations \\(x = 2^{3/2}\\cos^3(t)\\) and \\(y = 2^{3/2}\\sin^3(t)\\) for \\(0 \\leq t \\leq \\pi/2\\).",
"role": "user"
}
] | 3\sqrt{2} |
[
{
"content": "Let $S$ be the region enclosed by the curves $C_1: y = ax^2$ and $C_2: y = -ax^2 + 2abx$, where $a$ and $b$ are positive constants. Calculate the ratio of the volume $V_x$ of the solid formed by revolving $S$ around the $x$-axis to the volume $V_y$ of the solid formed by revolving $S$ around the $... | ab |
[
{
"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... | catalyst |
[
{
"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... | $876,970 |
[
{
"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 = (5y^2-4xy)i + (-2x^2+10xy+1)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... | 9.94% |
[
{
"content": "Determine the number of homomorphisms $\\psi: D_4 \\longrightarrow \\mathbb{Z}_4$, where $D_4$ is the dihedral group of order 8.",
"role": "user"
}
] | 4 |
[
{
"content": "Determine the number of permutations in the symmetric group $S_8$ that are conjugate to the permutation $\\tau = (1\\ 3\\ 5)(2\\ 4\\ 6\\ 7)$. Provide your answer as a single integer.",
"role": "user"
}
] | 3360 |
[
{
"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... | PbI2 |
[
{
"content": "Determine the least possible value of \\( M \\) such that there exists a function \\( f: \\{1, 2, \\ldots, 2013\\} \\rightarrow \\{1, 2, \\ldots, M\\} \\) satisfying the condition: if \\( n \\leq 2013 \\) and \\( k < n \\) with \\( k \\mid n \\), then \\( f(k) < f(n) \\).",
"role": "user"
}
... | 11 |
[
{
"content": "Determine whether the sum of factorials of consecutive integers, $1! + 2! + \\ldots + n!$, is divisible by 9 for any integer $n > 8$. Provide a proof or counterexample.",
"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... | 3.01 |
[
{
"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... | 679.83J |
[
{
"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... | 0.17 |
[
{
"content": "Calculate the flux of the vector field \\( \\bar{V} = z\\bar{i} + x\\bar{j} + y\\bar{k} \\) through the surface defined by \\( x^2 + y^2 + z^2 = 1 \\), \\( x = 0 \\), \\( y = 0 \\), \\( z = 0 \\) with \\( x > 0 \\), \\( y > 0 \\), \\( z > 0 \\) using Gauss's theorem.",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the cardinality of the set $G$ of all entire functions $g$ such that $g(x) = x^k$ for all $x \\in (-1, 1)$, where $k$ is an odd integer $\\geq 1$. \\( K \\) is given as an odd integer $\\geq 1$. \\( G \\) is the set of all entire functions $g$ such that $g(x) = x^k$ for all $x \\in (-1, ... | 1 |
[
{
"content": "Evaluate the line integral \\(I=\\oint_\\Gamma \\sin(x^3)\\,dx + 6xy\\,dy\\) where \\(\\Gamma\\) is the counterclockwise triangular path in \\(\\mathbb{R}^2\\) connecting the points (0,0), (2,2), and (0,2).",
"role": "user"
}
] | 16 |
[
{
"content": "Determine the maximum possible rank of an $n \\times n$ real matrix $A$ for which there exists an $n \\times n$ real matrix $B$ such that $AB - BA = A$. Provide examples of matrices $A$ and $B$ that achieve this maximum rank.",
"role": "user"
}
] | n-1 |
[
{
"content": "Evaluate the limit: \\[ A = \\lim_{n \\to \\infty} \\left(\\binom{n}{0} \\binom{n}{1} \\binom{n}{2}\\cdots \\binom{n}{n}\\right)^{1/((n)(n+1))} \\]",
"role": "user"
}
] | \sqrt{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... | H(X)= \frac{1}{k} \sum_{z=1}^k H_z + \log_2(k) |
[
{
"content": "Let $A$ and $B$ be two non-empty subsets of $X = \\{1, 2, \\ldots, 8\\}$ such that $A \\cup B = X$ and $A \\cap B = \\emptyset$. Let $P_A$ be the product of all elements of $A$ and $P_B$ be the product of all elements of $B$. Find the minimum possible value of $P_A + P_B$. \\(\\text{Note: This is ... | 402 |
[
{
"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... | 5.74 m/s |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\rightarrow 0^{+}}\\frac{\\int_{1}^{+\\infty}\\frac{e^{-xy}-1}{y^3}dy}{\\ln(1+x)}$$ as x approaches 0 from the right.",
"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... | Confabulation |
[
{
"content": "Evaluate the limit of the series: $$\\lim_{n\\to\\infty}\\sum_{r=1}^{n}\\frac{1}{(r+2)r!}$$",
"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... | 8.64% |
[
{
"content": "Determine whether the limit \\( \\lim_{(x,y) \\to (0,0)} f(x,y) \\) exists for the function \\( f(x,y) = x\\sin (1/y)+y\\sin (1/x) \\), where the function is defined as 0 when \\( x=0 \\) or \\( y=0 \\). If the limit exists, find its value.",
"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... | 1.32e-3 mol/L 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... | d=v^2/(2*\mu*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... | 0.412 |
[
{
"content": "Determine the rank of the map \\(\\pi: \\mathbb{S}^n \\subseteq \\mathbb{R}^{n+1} \\rightarrow \\mathbb{P}^n \\mathbb{R}\\), which is the restriction of the projection map from \\(\\mathbb{R}^{n+1} \\setminus \\{0\\}\\) to \\(\\mathbb{P}^n \\mathbb{R}\\), at any point on \\(\\mathbb{S}^n\\).",
... | 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... | 61.6 |
[
{
"content": "Evaluate the integral: \\[ \\int^{\\frac{\\pi}{2}}_0 \\frac{\\cos^2x\\,dx}{\\cos^2x+4\\sin^2x} \\]",
"role": "user"
}
] | \dfrac{\pi}{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... | 2.2e3 g/cm^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... | 4.49 m/s |
[
{
"content": "In triangle $ABC$, the angle bisector $AD$ divides side $BC$ in the ratio $BD:DC = 2:1$. Determine the ratio in which the median $CE$ divides the angle bisector $AD$. Express your answer as a ratio.",
"role": "user"
}
] | 3:1 |
[
{
"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... | 6.3 N |
[
{
"content": "Evaluate the integral \\( \\int_C \\tan z \\, dz \\), where \\( C \\) is the positively oriented circle \\( |z|=2 \\).",
"role": "user"
}
] | -4\pi i |
[
{
"content": "Determine the coefficient of \\(\\frac{1}{z}\\) in the series expansion of \\(\\frac{1}{z^4 \\cdot \\sin z}\\).",
"role": "user"
}
] | \dfrac{7}{360} |
[
{
"content": "Which interpretation correctly represents convergence in distribution for a sequence of real-valued random variables $X_1, X_2, \\cdots$ converging to a random variable $X$? Consider the definition: $ \\lim _{n\\to \\infty }F_{n}(x)=F(x) $ for every $x \\in \\mathbb{R}$ at which $F$ is continuous,... | 1 |
[
{
"content": "Let \\( y \\) be the unique positive number such that \\( \\frac{y}{\\log_{2} y} = - \\frac{3}{5} \\). The value of \\( y \\) is of the form \\( e^{-W(z \\ln 2)} \\) for some rational number \\( z \\). What is the value of \\( z \\)?",
"role": "user"
}
] | \dfrac{5}{3} |
[
{
"content": "Let U be a random variable uniformly distributed on the interval [-1,1]. Define V = U^2. Calculate the correlation coefficient, corr(U, V).",
"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... | $62.11 |
[
{
"content": "A cylindrical tank with hemispherical ends is to be constructed to hold a given volume of liquid. If the cost of making the hemispherical ends is twice as much as the cost of making the cylindrical part, determine the ratio of the height to the radius of the cylinder that minimizes the total cost ... | 4 |
[
{
"content": "How many different lines pass through at least two points in this 3-by-3 grid of lattice points shown?\n\n[asy]\nsize(30);\ndot((0,0));\ndot((1,0));\ndot((2,0));\ndot((0,1));\ndot((1,1));\ndot((2,1));\ndot((0,2));\ndot((1,2));\ndot((2,2));\n[/asy]",
"role": "user"
}
] | 20 |
[
{
"content": "Calculate the limit: $$\\lim_{n\\rightarrow\\infty} \\frac{7^{\\sqrt{n}}\\cdot(n/2)!\\cdot(n/2)!}{n!}$$",
"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... | - 2.00e-18 J |
[
{
"content": "Find the values of $p$ for which the improper integral \\( \\int_{0}^{\\infty}\\frac{e^{-x}-1}{x^{p}}dx \\) converges.",
"role": "user"
}
] | (1, 2) |
[
{
"content": "Determine whether there exist groups $G, H, I$ and injective group homomorphisms $h : G \\rightarrow H$ and $i : G \\rightarrow I$ such that $\\mathrm{img}(h)$ and $\\mathrm{img}(i)$ are normal subgroups of $H$ and $I$ respectively, $H/\\mathrm{img}(h) \\cong I/\\mathrm{img}(i)$, but $H$ and $I$ a... | 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... | 18.26% |
[
{
"content": "Compute the radius of convergence of the series \\(\\sum_{n\\ge 0}{\\log((2n)!)x^n}\\) using the Stirling formula.",
"role": "user"
}
] | 1 |
[
{
"content": "Let \\( G \\) be a finite non-abelian group of order 39. Determine the number of subgroups of order 3 in \\( G \\).",
"role": "user"
}
] | 13 |
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