prompt list | ground_truth string |
|---|---|
[
{
"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... | Tl-209 |
[
{
"content": "Given that the function \\( f(x) \\) satisfies the condition that for all natural \\( n > 1 \\), there exists a prime divisor \\( p \\) of \\( n \\) such that\n\\[ f(n) = f\\left(\\frac{n}{p}\\right) - f(p) \\]\nand it is known that \\( f(1001) = 1 \\), determine the value of \\( f(1002) \\).",
... | 1 |
[
{
"content": "Let $p$ be an integer with $p \\geq 2$ and define a sequence by \\(x_n = \\prod_{j=0}^{n-1} \\frac{n + jp + 1}{n + jp}\\) for each positive integer $n$. Assuming \\(\\lim_{n \\to \\infty} x_n\\) exists, evaluate it.",
"role": "user"
}
] | (1 + p)^{1/p} |
[
{
"content": "Consider the equation \\((\\log_2 x)^2 - 4\\log_2 x - (m^2 + 2m + 13) = 0\\). Let the real roots of the equation be \\(x_1\\) and \\(x_2\\) such that \\(x_1 < x_2\\). Find the sum of the maximum value of \\(x_1\\) and the minimum value of \\(x_2\\).",
"role": "user"
}
] | \dfrac{257}{4} |
[
{
"content": "Compute \\(\\lim_{n \\rightarrow \\infty} \\left(1 + \\frac{i}{n^2 + in}\\right)^n\\).",
"role": "user"
}
] | 1 |
[
{
"content": "What is the proportion of prime numbers \\(p\\) such that the Legendre symbols \\(\\left(\\frac{2}{p}\\right) = 1\\) and \\(\\left(\\frac{-5}{p}\\right) = -1\\)?",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Find the value of \\( k \\in \\mathbb{R} \\) such that \\[ \\lim_{x \\to 0} \\frac{1 - \\cos(kx)}{x^2} = \\int_0^\\infty xe^{-x} \\, \\mathrm{d}x. \\]",
"role": "user"
}
] | \pm \sqrt{2} |
[
{
"content": "Consider a convex octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ where no three diagonals are concurrent. A \"button\" is defined as the intersection of any two diagonals. A \"sub quadrilateral\" is a convex quadrilateral formed by any four vertices of the octagon. Determine the smallest number $n$ such that ... | 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... | $307,468.50 |
[
{
"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 mol |
[
{
"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... | 1.46 m |
[
{
"content": "Derive an expression for the derivative of the step function, \\( \\theta'(x) \\), without using the Fourier transform.",
"role": "user"
}
] | \theta'(x) = \delta(x) |
[
{
"content": "Find the interval of convergence of the series \\(\\sum_{n=1}^{\\infty} \\frac{(-1)^n x^n}{4^n \\ln(n)}.\\)",
"role": "user"
}
] | (-4, 4] |
[
{
"content": "Determine the number of automorphisms of the symmetric group $S_6$. Provide your answer as a single integer.",
"role": "user"
}
] | 1440 |
[
{
"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... | 0.67 m s^{-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... | 11.075 W |
[
{
"content": "Does the theorem that every ideal $I$ dense in a $C^*$-algebra $A$ admits an approximate unit imply that every $C^*$-algebra has an approximate unit?",
"role": "user"
}
] | Yes |
[
{
"content": "In triangle \\( \\triangle ABC \\), point \\( D \\) is on side \\( AC \\). Given that \\( \\overline{AB} = \\overline{DC} = 1 \\), \\( \\angle DBC = 30^{\\circ} \\), and \\( \\angle ABD = 90^{\\circ} \\), find the length of \\( \\overline{AD} \\).",
"role": "user"
}
] | \sqrt[3]{2} |
[
{
"content": "Is the group of $3 \\times 3$ invertible matrices with coefficients in $\\Bbb Z_3$ isomorphic to the symmetric group $S_4$?",
"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... | 32.74 g/mol |
[
{
"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... | 113.3 kJ |
[
{
"content": "Find all functions \\( f: \\Re \\rightarrow \\Re \\) such that \\[ x^{2} \\cdot f(x) + f(1-x) = 2x - x^{4} \\]",
"role": "user"
}
] | 1 - x^2 |
[
{
"content": "We have a triangle $\\triangle ABC$ such that $AB = AC = 8$ and $BC = 10.$ What is the length of the median $AM$?",
"role": "user"
}
] | \sqrt{39} |
[
{
"content": "Compute the exterior derivative \\(d\\omega\\) of the 1-form \\(\\omega = \\frac{x^1dx^2 - x^2dx^1}{(x^1)^2+(x^2)^2}\\) for \\((x^1,x^2) \\neq 0\\).",
"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... | 6.99978 |
[
{
"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... | 0.0677 A |
[
{
"content": "Given that $A, B, C \\in \\mathbb{R}$ and $\\cos(A-B) + \\cos(B-C) + \\cos(C-A) = -\\frac{3}{2}$, evaluate the expression \\( \\frac{\\sum \\cos^3(\\theta+A)}{\\prod\\cos(\\theta+A)} \\), where $\\theta \\in \\mathbb{R}$. Provide your answer in simplest form.",
"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... | The two Lewis structures for C4H10 are:
1. Straight-chained butane: CH3-CH2-CH2-CH3
2. Y-chained isobutane (methylpropane): CH(CH3)3 |
[
{
"content": "Evaluate the contour integral \\( \\oint_C f(z) \\, dz \\) where \\( f(z) = \\pi e^{\\pi \\cdot \\overline{z}} \\) and \\( C \\) is the square contour with vertices at \\( 0 \\), \\( 1 \\), \\( 1+i \\), and \\( i \\).",
"role": "user"
}
] | 4(e^{\pi} - 1) |
[
{
"content": "Circles $C_1$ and $C_2$ have radii $4$ and $9$, respectively, and intersect at distinct points $A$ and $B$. A point $P$ lies on the line through $A$ and $B$, outside circle $C_2$, and is $6$ units away from the center of $C_1$. If $PQ$ is tangent to circle $C_2$ at point $Q$, find the length of $P... | 2\sqrt{5} |
[
{
"content": "Suppose $g$ is a holomorphic function on $\\mathbb{C} \\backslash L$, where $L = \\{1, 2, \\dots\\}$, and is bounded. If $g(0) = 2$, find all possible functions $g$. Provide your answer in the form of a function or expression.",
"role": "user"
}
] | 2 |
[
{
"content": "Determine whether it is possible to construct a tournament with 27 participants such that every player is a pseudo-winner, using the concept of Steiner triple systems.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether for all natural numbers \\( n > 5 \\), there exist integers \\( a \\) and \\( b \\) such that \\( a^2 + b^2 \\) is not a prime number and \\( n = a + b \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Given a weighted graph $G=(V,E)$ with some edges having negative weights, and a designated source node $s$, assume the graph contains no negative cycles. For every node $v \\in V$, there exists a minimum weight path from $s$ to $v$ that uses at most $k$ edges. Determine if there is an algorithm th... | 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... | 1.9 s |
[
{
"content": "Given two independent random variables, X and Y, with probability density functions:\n\nFor X: \\( f_X(x) = 4x^3 \\) for \\( 0 < x < 1 \\), and 0 otherwise.\n\nFor Y: \\( f_Y(y) = \\frac{y}{2} \\) for \\( 0 < y < 2 \\), and 0 otherwise.\n\nCalculate the probability \\( P(Y > 4X) \\).",
"role":... | \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... | 101 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... | 1.2318 |
[
{
"content": "Consider a matroid $M$ of rank 3 with three bases $E_1, E_2, E_3$. Let $e_{i,j}$ denote the $i$-th element of base $E_j$. Determine if there always exists a permutation $s: \\{1,2,3\\} \\to \\{1,2,3\\}$ such that $e_{s(1),1}, e_{s(2),2}, e_{s(3),3}$ forms a base of $M$. Provide a proof or countere... | 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... | 15.222 pF |
[
{
"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... | 1.203 nm |
[
{
"content": "Determine whether the function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) defined by\n\n\\[ f(x) = \\begin{cases} \n x^2\\sin\\left(\\frac{1}{x}\\right) & \\text{for } x \\neq 0 \\\\\n 0 & \\text{for } x = 0\n \\end{cases} \\]\n\nhas a derivative at every point \\( x \\) in \\( \\mathbb{R... | 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... | 9.09% |
[
{
"content": "Evaluate the integral \\( \\int_{C} \\frac{\\cosh(4z)}{(z-4)^3} \\, dz \\), where \\( C \\) consists of the contour \\( |z|=6 \\) oriented counterclockwise and \\( |z-3|=2 \\) oriented clockwise.",
"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... | messenger RNA (mRNA) |
[
{
"content": "Evaluate the expression \\( \\sum _ {k=1}^{n}{\\frac{k}{n}\\dbinom{n}{k}t^k(1-t)^{n-k}} \\) using the binomial theorem and differentiation.",
"role": "user"
}
] | t |
[
{
"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... | 2/3 this quantity |
[
{
"content": "Given the sequence \\(\\{a_{n}\\}\\) defined by\n\\[ a_{n} = \\left\\lfloor (2+\\sqrt{5})^{n} + \\frac{1}{2^{n}} \\right\\rfloor \\quad \\text{for} \\quad n \\in \\mathbb{Z}_{+}, \\]\nwhere \\(\\lfloor x \\rfloor\\) denotes the greatest integer less than or equal to \\(x\\). Let \\(C\\) be a real ... | \dfrac{1}{288} |
[
{
"content": "Determine the asymptotic complexity of finding the value of $x$ using the extended version of the Chinese Remainder Theorem, given that $x$ is known modulo all primes less than $p$. Provide your answer in terms of big O notation.",
"role": "user"
}
] | O(p^2) |
[
{
"content": "Calculate the residue of the function \\( f(z) = \\frac{1}{z^2 \\sin z} \\) at \\( z = 0 \\).",
"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... | 5238.55 |
[
{
"content": "In a group of $n$ children of different heights, they randomly form a circle. Each child considers themselves \"tall\" if they are taller than both their immediate neighbors. Define the random variable $\\xi_i$ to be 1 if the $i$th child considers themselves tall, and 0 otherwise. Find the expecte... | \dfrac{n}{3} |
[
{
"content": "Calculate \\( S = [\\sqrt{1}] + [\\sqrt{2}] + \\cdots + [\\sqrt{1988}] \\) and find \\( [\\sqrt{S}] \\).",
"role": "user"
}
] | 241 |
[
{
"content": "Let $T$ be a linear bounded operator on a Banach space $X$. Suppose that $(a_n)$ and $(b_n)$ are two increasing sequences of positive integers such that $b_n > a_n$ for any $n$ and such that $\\sup_{n}(b_n - a_n)<\\infty$. Furthermore, assume that $\\sup_{n}\\|T^{b_n}\\|<\\infty$. Does it necessar... | 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... | $46.68 |
[
{
"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... | 205665.13 |
[
{
"content": "Let \\( y = \\frac{x}{1+x} \\), where \\( x = {{\\omega^{2009}}^{2009}}^{\\ldots \\text{up to } 2009 \\text{ times}} \\) and \\( \\omega \\) is a complex cube root of unity. Determine the value of \\( y \\).\\n\\n(a) \\( \\omega \\); \\quad (b) \\( -\\omega \\); \\quad (c) \\( \\omega^2 \\); \\qua... | 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... | 2.1e+06 V |
[
{
"content": "Determine all integer pairs $(a, b)$ such that the expression \\( \\frac{\\sqrt{2} + \\sqrt{a}}{\\sqrt{3} + \\sqrt{b}} \\) is a rational number.",
"role": "user"
}
] | (3, 2) |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\infty} e^{-x} \\sum_{k=0}^{\\infty} \\frac{(-1)^k 2017^{2k} x^{2k}}{(2k)!^2} \\, dx \\).",
"role": "user"
}
] | \cos(2017) |
[
{
"content": "Given the joint density function of two random variables \\(X\\) and \\(Y\\):\n\\[\nf_{X,Y}(x,y)=\\begin{cases}0.0009x(1+y), & \\text{if } (x,y)\\in \\Omega \\\\\n0, & \\text{else}\\end{cases}\n\\]\nand the marginal distributions:\n\\[\nf_X(x)=0.0036(21x-2x^2-x\\sqrt{x}), \\quad \\text{and} \\quad... | 0.42 |
[
{
"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... | 3.54 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... | 73,800 |
[
{
"content": "Evaluate whether the integral \\( \\int_{\\Omega}(\\partial_x^2-\\partial_y^2)u\\phi \\, d\\omega = 0 \\) is zero, given that \\( \\phi \\in C_0^{\\infty}(\\Omega) \\) with \\( \\Omega = (0,1)\\times(0,1) \\) and \\( u \\in L_2(\\Omega) \\) is defined by \\( u(x,y) = 1 \\) for \\( x>y \\) and \\( ... | 0 |
[
{
"content": "A circle is inscribed in an angle with vertex \\( O \\) and touches its sides at points \\( A \\) and \\( B \\). The ray \\( OX \\) intersects this circle at points \\( C \\) and \\( D \\), with \\( OC = CD = 1 \\). If \\( M \\) is the intersection point of the ray \\( OX \\) and the segment \\( A... | \dfrac{4}{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... | 5.786e7 N/C |
[
{
"content": "Consider a projective variety $X$ with a hypersurface section $Z$ and its complement $U = X \\setminus Z$, which is an open affine subscheme of $X$. Let $i: U \\hookrightarrow X$ be the open embedding. Given a coherent sheaf $M$ on $U$, the quasi-coherent sheaf $i_*M$ on $X$ has a filtration by co... | 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... | k=Ae^(-Ea/RT) |
[
{
"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... | 1st Quarter
Beginning Cash Balance: 36,000
Cash Receipts: 260,000
Total Cash Available: 296,000
Cash Disbursements: 316,000
Remaining Cash Balance: -20,000
Loan Proceeds: 30,000
Loan Payment: 0
Interest Payment: 0
Ending Cash Balance: 10,000
2nd Quarter
Beginning Cash Balance: 10,000
Cash Receipts: 410,000
Total Cash ... |
[
{
"content": "The base $ABCD$ of a tetrahedron $P-ABCD$ is a convex quadrilateral with diagonals $AC$ and $BD$ intersecting at $O$. Given that the area of $\\triangle AOB$ is 36, the area of $\\triangle COD$ is 64, and the height of the tetrahedron is 9, find the minimum volume of the tetrahedron.",
"role":... | 588 |
[
{
"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... | 5e9 N |
[
{
"content": "Evaluate whether the following limit holds: $$\\lim_{n\\to \\infty} \\binom{2n}{n}(1-p)^{n}p^{n}=0$$ for $p\\in[0,1]$ with $p\\ne 1/2$. Provide a justification for your answer.",
"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... | 9.455 |
[
{
"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... | 5.92 mT |
[
{
"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... | 5X/36 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... | 160/3L |
[
{
"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-methylcyclohexene |
[
{
"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... | x(t)=0.02cos(14t) |
[
{
"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... | 2705 |
[
{
"content": "Evaluate the Lebesgue integral \\( \\int_{[0, 1]} f \\, d\\mu \\), where the function \\( f(x) \\) is defined as follows:\n\n\\[ f(x)= \\begin{cases} \\sin \\pi x,& \\text{if } x \\in [0,1/2] -C\\\\ \\cos \\pi x , &\\text{if } x\\in (1/2, 1]-C\\\\ x^2, & \\text{if } x \\in C \\end{cases} \\]\n\nHe... | 0 |
[
{
"content": "Can a function with the property $f'(0) = f''(0)$ be a characteristic function of a random variable?",
"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... | 0.09 |
[
{
"content": "Determine whether the following statement is true or false: If \\( G \\subset \\mathrm{GL}(n,\\mathbb{R}) \\) is a subgroup, then its topological closure \\( \\overline{G} \\) is a topologically closed subgroup of \\( \\mathrm{GL}(n,\\mathbb{R}) \\).",
"role": "user"
}
] | A |
[
{
"content": "For a group of 20 people, using a randomized greedy algorithm, how many iterations are required to ensure that each person meets every other person if each iteration allows for 4 new meetings?",
"role": "user"
}
] | 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... | 54.3 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... | AgCl(s) + 2NH_3(aq) -> [Ag(NH_3)_2]^+ (aq) + Cl^- (aq) |
[
{
"content": "Find the norm of the linear functional \\( f \\) defined on \\( \\mathscr{C}[-1,1] \\) by \\[ f(x) = \\int_{-1}^0 x(t)\\, dt - \\int_0^1 x(t)\\, dt \\]",
"role": "user"
}
] | 2 |
[
{
"content": "In the prism $ABC A_1 B_1 C_1$, vertices $A$ and $B$ are located on the axis of a cylinder, while the remaining vertices lie on the lateral surface of the cylinder. Determine the dihedral angle formed with edge $AB$ in this prism.",
"role": "user"
}
] | 60^\circ |
[
{
"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... | 13.94% |
[
{
"content": "In quadrilateral $ABCD$, the angles are given as $\\angle DAB=60^{\\circ}$, $\\angle ABC=90^{\\circ}$, and $\\angle BCD=120^{\\circ}$. The diagonals $AC$ and $BD$ intersect at point $M$. If $MB=1$ and $MD=2$, determine the area of quadrilateral $ABCD$. Express your answer in simplest form.",
"... | \dfrac{9}{2} |
[
{
"content": "Assume that $F$ is a field such that $GL_n(F)$ is a divisible group. Is it true that the multiplicative group $F^*$ is also divisible?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the Galois group of the quadratic extension $\\mathbb{Q}(\\sqrt{p})$, where $p$ is a prime number not congruent to 1 modulo 4.",
"role": "user"
}
] | \mathbb{Z}/2\mathbb{Z} |
[
{
"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... | -1.8 |
[
{
"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... | 2.24e-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... | 9h1 |
[
{
"content": "Let $H$ be a real Hilbert space, and let $B : H \\times H \\to \\mathbb{R}$ be a bilinear and symmetric form. Suppose there exists a constant $C$ such that for all $x \\in H$, $|B(x,x)| \\le C \\|x\\|^2$. Must $B$ be continuous? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine if there exists any group other than $S_3$ that contains elements of order 2 and 3, such that the product of these elements also has order 2.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty} \\frac{1^{n}+2^{n}+\\ldots+n^{n}}{n^{n+1}}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Find the radius of convergence for the power series \\( g(z) = \\sum_{n=1}^{\\infty} (n!)z^n \\), where \\( z \\) is a complex variable.",
"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... | y=2sin(1000t) + sin(1001t) + sin(999t) |
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