prompt list | ground_truth string |
|---|---|
[
{
"content": "Let \\( a \\) and \\( b \\) be any real numbers. Consider the equation\n\\[ x^2 + |a| x + b = 0 \\]\nwhose set of solutions in the complex numbers \\(\\mathbf{C}\\) is \\( A \\). Define the set\n\\[ B = \\{ n \\mid n = |A|, a, b \\in \\mathbf{R} \\} \\]\nFind the set \\( B \\).",
"role": "user... | \{1, 2\} |
[
{
"content": "Determine the number of solutions of the equation \\( a - z - e^{-z} = 0 \\) in the half-plane \\( \\{ z \\in \\mathbb{C} : \\text{Re}(z) > 0 \\} \\), where \\( a \\in \\mathbb{R} \\) and \\( a > 1 \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Given the sine function $y = \\sin x$ with an arc length $L$ from $0$ to $2\\pi s$, where $s \\in \\mathbb{Q}$ and $s > 1$, the period is changed from $2\\pi$ to $\\frac{4\\pi}{3}$. Determine the new amplitude $|A|$ of the sine function such that the arc length from $0$ to $2\\pi s$ remains $L$. U... | \dfrac{2}{3} |
[
{
"content": "Compute the Lebesgue measure of the set \\(A = \\{(x,y) \\in \\mathbb{R}^2 : xy = 1\\}\\).",
"role": "user"
}
] | 0 |
[
{
"content": "How many fractions in the form $\\frac{n}{99}$, with $0<n<99$, are in lowest terms?",
"role": "user"
}
] | 60 |
[
{
"content": "Find \\( \\lim_{n\\to\\infty}\\left(\\frac{\\binom{3n}{n}}{\\binom{2n}{n}}\\right)^{\\frac{1}{n}} \\).",
"role": "user"
}
] | \dfrac{27}{16} |
[
{
"content": "Determine whether every automorphism of a group that fixes all maximal subgroups is the identity automorphism.",
"role": "user"
}
] | No |
[
{
"content": "Determine whether the vector field \\( F(x,y) = y^2e^{xy}\\vec{i} + (1+xy)e^{xy}\\vec{j} \\) is conservative. If it is, find a function \\( f \\) such that \\( F = \\nabla f \\).",
"role": "user"
}
] | f(x, y) = y e^{xy} |
[
{
"content": "Consider a monotonic decreasing function $f(x)$ defined on the interval $[0,+\\infty)$. Calculate the error committed when computing the definite integral $\\int_{0}^{+\\infty}f(x)\\,\\mathrm{d}x$ using the Taylor expansion of $f(x)$ around $x=0$, truncated at a certain order $n$. What is the erro... | \infty |
[
{
"content": "Find the probability that two independent continuous random variables, $Y_1$ and $Y_2$, take on the same value.",
"role": "user"
}
] | 0 |
[
{
"content": "Let $X_1, X_2, \\ldots, X_n$ be independent and identically distributed (i.i.d.) random variables with the probability distribution: $P(X_i=0)=\\frac{1}{18}$, $P(X_i=1)=\\frac{8}{9}$, and $P(X_i=2)=\\frac{1}{18}$. Define $T_n = 100(X_1 + \\cdots + X_n)$. Find the limit of $P(T_n > 97n)$ as $n$ app... | 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... | 0.34 mm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 63,000 |
[
{
"content": "Calculate the volume of the region of the ellipsoid \\( \\frac{x^2}{49} + y^2 + z^2 = 1 \\) that lies outside the sphere \\( x^2 + y^2 + z^2 = 1 \\).",
"role": "user"
}
] | 8\pi |
[
{
"content": "If the image of a non-trivial group morphism $f$ is commutative, does it imply that the domain group is also commutative? 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... | $18,902.33 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Jan. 1, 2011
Investment in Exchange Corp. Assets 252,000
Investment in Goodwill 128,000
Bank 380,000
Dec. 31, 2011
Investment in Exchange Corp. 84,000
Income from investment in Exchange Corp. 84,000
Dec. 31, 2011
Bank 12,000
Investment in Exchange Corp. 12,000 |
[
{
"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... | 0.2655 mm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.47 |
[
{
"content": "Evaluate the limit as \\( n \\to \\infty \\):\n\\[ \\lim_{n \\to \\infty} \\sqrt[n]{n^{(4n)}+(4n)^n}\\left[\\left(2+\\frac{1}{n^2}\\right)^{18}-\\left(4+\\frac{4}{n^2}\\right)^9\\right] \\]",
"role": "user"
}
] | 589824 |
[
{
"content": "Find the norm of the linear operator \\( S: \\ell^1 \\rightarrow \\ell^2 \\) defined by \\( Sx = \\left( \\frac{1}{2^k} x_k \\right)_{k=1}^{\\infty} \\), where \\( \\ell^1 \\) and \\( \\ell^2 \\) are the sequence spaces of absolutely summable and square-summable sequences, respectively.",
"rol... | \dfrac{1}{2} |
[
{
"content": "Let $f:[0,\\infty) \\rightarrow[0,\\infty]$ be a continuous function on $[0,\\infty]$ and differentiable on $(0,\\infty)$. Given that $f(x)=\\int_{0}^{x}\\sqrt{f(t)}dt$, determine the range of possible values of $f(6)$. \\( \\text{Provide your answer as an interval or set of values.} \\)",
"ro... | [0, 9] |
[
{
"content": "The fraction $\\frac{a}{a+27}$, where $a$ is a positive integer, equals $0.865$. What is the value of $a$?",
"role": "user"
}
] | 173 |
[
{
"content": "Given a 2x2 matrix \\( A = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\) with a non-zero determinant (\\(\\operatorname{det}(A) \\neq 0\\)), determine if there exist two non-singular matrices \\( B \\) and \\( C \\) such that the product \\( BAC \\) equals \\( \\begin{pmatrix} a & -b \\\\ -... | Yes |
[
{
"content": "Consider a connected, undirected simple graph where all edges have weights from the set \\( \\{1, 10, 100, 1000\\} \\). Is it possible to find the minimum spanning tree (MST) of this graph in \\( O(m+n) \\) time, where \\( m \\) is the number of edges and \\( n \\) is the number of vertices?",
... | Yes |
[
{
"content": "Determine the points in the domain of the function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) defined by\n\\[\nf(x) =\n\\begin{cases}\nx + 1, & \\text{if } x \\in \\mathbb{Q} \\\\\n2, & \\text{if } x \\in \\mathbb{R} - \\mathbb{Q}\n\\end{cases}\n\\]\nwhere the function is continuous and where it is ... | 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... | Speed = 544.14 mph, Heading = 340.77 degrees |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.13 g of NaHCO3 and 7.92 g of Na2CO3 |
[
{
"content": "Determine if the function \\( f(x) = \\frac{x}{2} + \\frac{a}{2x} \\) is a contraction mapping on the domain \\((1, \\infty)\\) for \\( a \\in (2, 4) \\). If a fixed point exists, find it.",
"role": "user"
}
] | \sqrt{a} |
[
{
"content": "Given a non-unit fraction \\( \\frac{a}{b} \\) in lowest terms, where \\( 0 < \\frac{a}{b} < 1 \\), and \\( \\frac{1}{n} \\) is the largest unit fraction less than \\( \\frac{a}{b} \\), let \\( \\frac{a'}{b'} = \\frac{a}{b} - \\frac{1}{n} \\). Is it true that \\( 0 < a' < a \\)? Assume \\( a, b, n... | Yes |
[
{
"content": "In the adjoining figure, $CD$ is the diameter of a semicircle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semicircle, and $B$ is the point of intersection (distinct from $E$) of line segment $AE$ with the semicircle. If length $AB$ equals length $OD$,... | 15^\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... | 4000/3 W |
[
{
"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... | $1,706.16 |
[
{
"content": "In the vector space $V$ of polynomials with degree at most 2 over the real numbers, a bilinear form $\\psi: V \\times V \\rightarrow \\mathbb{R}$ is defined by \\( (p, q) \\mapsto \\int_{0}^{1}p(x)q(x)dx \\). Given a linear form \\( \\phi: p \\mapsto p(0) \\), find a polynomial \\( q \\in V \\) su... | 30x^2 - 36x + 9 |
[
{
"content": "Find the minimum value of the function \\( f(x) = \\tan^2 x - 4 \\tan x - 8 \\cot x + 4 \\cot^2 x + 5 \\) on the interval \\( \\left( \\frac{\\pi}{2}, \\pi \\right) \\).",
"role": "user"
}
] | 9 + 8\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... | 0.32 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... | $82,857 |
[
{
"content": "Evaluate the constant \\( \\alpha \\) such that \\( \\iiint_{g(S)}(2x+y-2z)\\,dxdydz = \\alpha \\iiint_S z\\,dxdydz \\), where the transformation \\( g: \\mathbb{R}^3 \\to \\mathbb{R}^3 \\) is defined by \\( g(x,y,z) = (3x+4z, 2x-3z, x+3y) \\) and \\( S = \\{(x,y,z) \\in \\mathbb{R}^3 : 0 \\leq x ... | 255 |
[
{
"content": "Use the orbit-stabilizer theorem to find the order of the group of automorphisms of \\( \\mathbb{Z}_6 \\). Consider the action of the symmetric group \\( S_6 \\) on \\( \\mathbb{Z}_6 \\) defined by \\( \\sigma([a]) = [\\sigma(a)] \\) for \\( \\sigma \\in S_6 \\) and \\( a \\in \\{1, \\ldots, 6\\} ... | 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... | Right of Peaceful Assembly |
[
{
"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... | 270 nm |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Division E: Sales $365,000, Variable Expenses $182,500, Contribution Margin $182,500, Fixed Expenses $290,000, Net Profit $(107,500); Division C: Sales $670,000, Variable Expenses $194,300, Contribution Margin $475,700, Fixed Expenses $327,000, Net Profit $148,700; Division W: Sales $540,000, Variable Expenses $205,200... |
[
{
"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... | [Ar] 3d^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... | 0.36 kg |
[
{
"content": "Calculate \\( \\lim\\limits_{n \\to \\infty} \\frac{n^k}{k^n} \\) for \\( k > 1 \\).",
"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... | 298.7 m/s^2 |
[
{
"content": "Find the function $u(x)$ that minimizes the functional $F[u(\ncdot)] = \\frac{1}{2}\\int_0^1 f(u'(x)) \\, dx$, where $f: \\mathbb{R} \\to \\mathbb{R}$ is a strictly convex $C^2$ function. The function $u(x)$ must belong to the set $\\mathscr{A} = \\{ u: [0,1] \\to \\mathbb{R} \\mid u \\in C^1, u(0... | u(x) = A + (B - A)x |
[
{
"content": "In the sequence \\(\\{a_{n}\\}\\), it is given that \\(a_{1}=1\\) and \\(a_{n+1}>a_{n}\\). The sequence satisfies the equation \\(a_{n+1}^{2}+a_{n}^{2}+1=2(a_{n+1}+a_{n}+2a_{n+1}a_{n})\\). Determine \\(\\lim \\limits_{n \\rightarrow \\infty} \\frac{S_{n}}{n a_{n}}\\).",
"role": "user"
}
] | 0 |
[
{
"content": "In a topological space, does every closed set have a point on its boundary? Answer yes or no.",
"role": "user"
}
] | No |
[
{
"content": "Suppose $f$ is a linear function defined on a Banach space $E$ such that $f(x)=0$ for all $x$ with $\\|x\\|<1$. Determine if the operator norm of $f$ is 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... | 0.17 M |
[
{
"content": "Describe the group of homomorphisms $f: C_m \\rightarrow A$, where $C_m$ is the cyclic group of order $m$ and $A$ is a finite abelian group, with group composition defined by $(f * g)(x) = f(x) + g(x)$. What is this group?",
"role": "user"
}
] | A[m] |
[
{
"content": "Calculate the probability that $X_1 + X_2 < 2$ and $X_1 - X_2 > 0$, given that $(X_1, X_2)$ follows a bivariate normal distribution with mean $\\mu = \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}$ and covariance matrix $\\Sigma = \\begin{pmatrix} 3 & -1\\\\ -1 & 3 \\end{pmatrix}$. Note that the events ... | 0.1116 |
[
{
"content": "Evaluate the improper integral: $$\\int_{0}^{1}\\ln(e^{1/x}+1)\\ dx$$.",
"role": "user"
}
] | \infty |
[
{
"content": "Let $W_{1}$ and $W_{2}$ be subspaces of a finite-dimensional vector space $V$ with $\\dim(W_{1}) = \\dim(W_{2})$. Determine the necessary and sufficient conditions on $W_{1}$ and $W_{2}$ such that $\\dim(W_{1}+W_{2}) = \\dim(W_{1})$. Express your answer in terms of the relationship between $W_1$ a... | W_1 = W_2 |
[
{
"content": "Let $R$ be a local (not Noetherian) domain satisfying the ascending chain condition (a.c.c.) on radical ideals and the descending chain condition (d.c.c.) on prime ideals. Is the Krull dimension of $R$ finite?",
"role": "user"
}
] | Yes |
[
{
"content": "In triangle $ABC$, let $D$ be a point on side $BC$. Points $P$ and $Q$ are distinct points on segment $AD$. Define $K = BP \\cap AC$, $L = CP \\cap AB$, $E = BQ \\cap AC$, and $F = CQ \\cap AB$. If $KL \\parallel EF$, determine all possible values of the ratio $\\frac{BD}{DC}$.",
"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... | 8.89% |
[
{
"content": "Determine whether there exists a right-angled triangle with integer sides such that its perimeter $P$ and area $A$ satisfy either $P = A \\cdot k$ or $k \\cdot P = A$, where $k$, $P$, and $A$ are positive integers.",
"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... | $20.19 |
[
{
"content": "Find the expected value \\( E[\\sin(B_t)] \\), where \\( B_t \\) is a standard Brownian motion.",
"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... | 2.987 A |
[
{
"content": "Calculate the work done by the vector field \\( \\vec{F}(x,y) = xy^2 \\hat{i} + yx^2 \\hat{j} \\) along the path from \\((0,0)\\) to \\((1,0)\\) and then to \\((1,1)\\).",
"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... | Short term debt: $10,000,000
Long term Debt: $17,740,444
Common equity: $54,000,000
Total capital: $81,740,444 |
[
{
"content": "Is there a Lebesgue measurable function \\( F: \\mathbb{R}^2 \\to \\mathbb{R} \\) such that for every Borel function \\( f: \\mathbb{R} \\to \\mathbb{R} \\), there exists some \\( y_f \\) such that \\( f(x) = F(x, y_f) \\) for all \\( x \\)?",
"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... | The structure of the product formed during inhibition is shown in Figure 2 of the response. |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to\\infty} \\int_0^{2 \\pi} \\frac{1}{x+\\sin^n x+ \\cos^n x} \\, dx \\]",
"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... | 7 tablets |
[
{
"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... | 11.88 |
[
{
"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... | 2143 J and 5357 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... | 8.2568 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... | 310 cm/s^2 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to0^-}\\frac{\\sqrt{x^4\\cos^2{x}+2x^2\\sin^2{2x}-x^4}}{x^2} \\]",
"role": "user"
}
] | 2\sqrt{2} |
[
{
"content": "Given any real numbers \\(a\\), \\(b\\), and \\(c\\), define an operation ※ with the following properties:\n1. \\(a ※ (b ※ c) = (a ※ b) \\cdot c\\),\n2. \\(a ※ a = 1\\),\n\nwhere \" \\(\\cdot\\) \" denotes multiplication.\n\nIf the solution to the equation \\(2016 ※ (6 ※ x) = 100\\) is \\(x = \\fr... | 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... | 7.3*10^6 m/s |
[
{
"content": "In a round-robin chess tournament with 12 players, each player creates 12 lists. The first list contains only themselves, and the $(k+1)$-th list contains the players from the $k$-th list and those who were defeated by any of the players in the $k$-th list. It is known that the 12th list of each p... | 54 |
[
{
"content": "Determine whether every natural number can be extended to form a prime number by appending digits. Specifically, for any natural number \\( n \\), does there exist a prime number \\( p \\) such that \\( n \\) is a truncation of \\( p \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( f(x) \\) be a \\(2\\pi\\)-periodic function defined on \\([0, 2\\pi)\\) as follows:\n\\[\nf(x) = \\begin{cases} \n1, & x \\in [0, \\frac{7\\pi}{16}) \\\\\n5, & x \\in [\\frac{7\\pi}{16}, \\frac{11\\pi}{16}) \\\\\n4, & x \\in [\\frac{11\\pi}{16}, 2\\pi)\n\\end{cases}\n\\]\nFind the Fourier ... | 3 |
[
{
"content": "Determine whether the following statement is equivalent to saying that $S$ is finitely generated as an $R$-algebra: \"For some $n \\in \\mathbb{N}$, there exists a surjective ring homomorphism from $R[t_1,\\dots,t_n]$ to $S$.\"",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the cardinality of the set of sequences of rational numbers that converge to 1.",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "In triangle \\( \\triangle ABC \\), \\(\\cos A\\), \\(\\sin A\\), and \\(\\tan B\\) form a geometric sequence with a common ratio of \\(\\frac{3}{4}\\). Determine the value of \\(\\cot C\\).",
"role": "user"
}
] | -\dfrac{53}{96} |
[
{
"content": "Compute the integral \\( \\oint_{|z|=2} \\frac{d z}{1+z+z^{2}+z^{3}} \\) using the residue theorem.",
"role": "user"
}
] | 0 |
[
{
"content": "Given a circle with 800 points labeled in sequence clockwise as \\(1, 2, \\ldots, 800\\), dividing the circle into 800 arcs, initially, one point is painted red. Subsequently, additional points are painted red according to the following rule: if the \\(k\\)-th point is already red, the next point ... | 25 |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to \\infty} \\left[\\left(\\Gamma(x+2)\\right)^{\\frac{1}{x+2}} - \\left(\\Gamma(x+1)\\right)^{\\frac{1}{x+1}}\\right]$$",
"role": "user"
}
] | \dfrac{1}{e} |
[
{
"content": "Evaluate the limit \\(\\lim_{n\\to\\infty}\\frac{\\prod_{k=1}^n(2k-1)}{(2n)^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... | $98.99 |
[
{
"content": "Given that $f(x)$ and $xf(x)$ are particular solutions of the differential equation \\(y''+R(x)y'+S(x)y=0\\), find the solution of the differential equation \\(y''+R(x)y'+S(x)y=f(x)\\) in terms of $f(x)$. Assume a solution of the form $(ax^2+bx+c)f(x)$ and determine the suitable value of $a$.",
... | \dfrac{1}{2} |
[
{
"content": "Determine whether the following statement is true or false: If \\( \\lim \\limits_{n\\to \\infty}\\ (a_{2n}-a_n)=0 \\), then \\( \\lim \\limits_{n\\to \\infty}\\ a_n \\) exists.",
"role": "user"
}
] | False |
[
{
"content": "The line $y = \\frac{1}{2} x + 4$ is paramterized by the form\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} -7 \\\\ s \\end{pmatrix} + t \\begin{pmatrix} l \\\\ -5 \\end{pmatrix}.\\]Enter the ordered pair $(s,l).$",
"role": "user"
}
] | \left( \frac{1}{2}, -10 \right) |
[
{
"content": "Let $k$ be an infinite field and $f(X,Y) \\in k[X,Y]$ be a polynomial with two variables. If $f(X,Y)$ vanishes on all points $(x,y) \\in k^2$, i.e., \\(f(x,y) = 0, \\forall (x,y) \\in k^2\\), can we conclude that $f(X,Y) = 0$? In other words, is $k[X,Y]$ isomorphic to the ring of polynomial functi... | Yes |
[
{
"content": "Can both of the numbers 739 and 741 be beautiful, given that a number \\( a \\) is defined as beautiful if for any integer \\( x \\), \\( g(x) = g(a - x) \\), where \\( g \\) is a function defined on the integers such that \\( g(x) \\neq x \\) for every integer \\( x \\)?",
"role": "user"
}
... | Yes |
[
{
"content": "Can two distinct piecewise differentiable simple closed curves in \\(\\mathbb{R}^2\\) intersect in an infinite set of points if they are not identical over any range of the parameter \\(t\\)? Answer yes or no.",
"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... | 0.0028 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... | 263 molecules/cm^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... | $187,295 |
[
{
"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... | John would prefer equal pay for equal work or strong unions over minimum wage laws, because minimum wage laws may create job losses in the economy and immigrants tend to have less job security. |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\frac{n^{n}}{e^{n^{3/2}}}.$$",
"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... | 0.340 |
[
{
"content": "Determine the character of the point \\((x,y)\\) in the topological space \\(X = (0,1) \\times (0,1)\\) with the anti-lexicographic order topology.",
"role": "user"
}
] | \aleph_0 |
[
{
"content": "Evaluate the double integral \\( \\iint_R (x^2+y^2)^{50}\\,dx\\,dy \\), where \\( R \\) is the region enclosed by the circle \\( x^2+y^2=1 \\).",
"role": "user"
}
] | \dfrac{\pi}{51} |
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