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... | 9.87347 KW |
[
{
"content": "Calculate the surface area of the portion of the plane \\(2x + y + 2z = 16\\) that lies within the region bounded by \\(x = 0\\), \\(y = 0\\), and \\(x^2 + y^2 = 64\\).",
"role": "user"
}
] | 24\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... | 117.69yen |
[
{
"content": "Provide an example of a sequence of functions \\( f_{n} \\in L_{1}(\\mathbb{R}) \\cap L_{2}(\\mathbb{R}) \\) for \\( n=1,2,3,\\ldots \\) such that \\( \\frac{||f_n||_1}{||f_n||_2} \\rightarrow 0 \\).",
"role": "user"
}
] | f_n(x) = n \cdot \chi_{[0, 1/n]}(x) |
[
{
"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... | 297 m |
[
{
"content": "Let the function \\( f(x) \\) be defined as \\[ f(x)= \\begin{cases} e^x & x\\text{ is rational} \\\\ e^{(1-x)} & x\\text{ is irrational} \\end{cases} \\] for \\( x \\) in \\((0,1)\\). Determine the one point in \\((0,1)\\) at which \\( f \\) is continuous.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Let \\( x_{1} \\) and \\( x_{2} \\) be the largest roots of the polynomials \\( f(x) = 1 - x - 4x^{2} + x^{4} \\) and \\( g(x) = 16 - 8x - 16x^{2} + x^{4} \\), respectively. Calculate \\( \\frac{x_{1}}{x_{2}} \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine whether \\(3^{1/2}\\) is an element of the field \\(\\mathbb{Q}(3^{1/3}+3^{1/2})\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Solve the differential equation \\(\\frac{d}{dx}(2y y') = (y')^2\\) using the substitution \\(y' = z\\), subject to the boundary conditions \\(y(0) = 0\\) and \\(y(1) = 0\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Given the sequence \\( \\{a_n\\} \\) defined by \\( a_1 = 1 \\) and \\( a_{n+1} = (-1)^n \\left( \\frac{1}{2} \\right) \\left( |a_n| + \\frac{2}{|a_n|} \\right) \\) for \\( n \\ge 1 \\), find the value of \\( \\sup \\{a_n\\} \\).",
"role": "user"
}
] | \dfrac{17}{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... | 1097.74 |
[
{
"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... | 320 m |
[
{
"content": "Calculate the area enclosed by the curve $g(x)$, the lines $x=-3$ and $x=5$, and the $x$-axis, where $g(x)$ is the inverse function of $f(x)=x^3+3x+1$. Use techniques of definite integrals to find the area.",
"role": "user"
}
] | \dfrac{9}{2} |
[
{
"content": "Calculate the divergence of the vector field \\( \\frac{\\mathbf{r}}{r^2} \\) using the product rule of divergence.",
"role": "user"
}
] | \dfrac{1}{r^2} |
[
{
"content": "Determine the number of ways to express the number 2160 as a product of factors that are relatively prime to each other.",
"role": "user"
}
] | 5 |
[
{
"content": "Given \\( x, y \\in \\left[-\\frac{\\pi}{4}, \\frac{\\pi}{4}\\right], a \\in \\mathbb{R} \\), and the equations:\n\\[\n\\left\\{\n\\begin{array}{l}\nx^{3} + \\sin x - 2a = 0, \\\\\n4y^{3} + \\sin y \\cdot \\cos y + a = 0\n\\end{array}\n\\right.\n\\]\nfind \\(\\cos(x + 2y)\\).",
"role": "user"
... | 1 |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to0^-}\\frac{\\int_{x}^{0}\\cos(t^2)\\,dt}{x^3}$$",
"role": "user"
}
] | -\infty |
[
{
"content": "Let $F$ be an ordered field and $A$ a convex subring of $F$, which is a valuation ring and thus a local ring with maximal ideal $m_A$. For any subfield $C$ of $F$ contained in $A$ and maximal with respect to these conditions, the residue field $F_A = A / m_A$ is algebraic over the image $C_A$ of $... | 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... | 4.40 m |
[
{
"content": "Evaluate the integral \\( \\int_0^\\pi \\sin(\\sin(x))\\sin(x)\\, dx \\).",
"role": "user"
}
] | \pi J_1(1) |
[
{
"content": "In the derived category $D^b(X)$ of constructible sheaves on a reasonable space $X$, consider an open inclusion $j: U \\to X$ and the closed complement $i: Y \\to X$. Let $M, N \\in D^b(X)$ and $f \\in \\text{Hom}_{D^b(X)}(M, N)$. If $i^*f = 0$ and $j^*f = 0$, is it true that $f = 0$? Justify your... | 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... | 4.7586% |
[
{
"content": "Let $f$ be a bounded complex-valued function defined on a locally compact group $G$, which is right uniformly continuous. Is it always true that the left translation of $f$ by $g$, defined as $(L_gf)(x) = f(g^{-1}x)$, is also right uniformly continuous?",
"role": "user"
}
] | Yes |
[
{
"content": "Given a matrix $A$ of size $6\\times6$ that satisfies the equation $A^2 + 12I = 7A$, and knowing that the rank of $A-3I$ is $1$, determine the rank of $A-4I$. Provide your answer as a single integer.",
"role": "user"
}
] | 5 |
[
{
"content": "Given non-zero complex numbers \\( x \\) and \\( y \\) satisfying\n\\[ y^{2}(x^{2}-xy+y^{2})+x^{3}(x-y)=0, \\]\nevaluate the expression\n\\[ \\sum_{m=0}^{29} \\sum_{n=0}^{29} x^{18mn} y^{-18mn}. \\]",
"role": "user"
}
] | 180 |
[
{
"content": "Find the derivative of the solution to the differential equation \\( \\ddot{x} = \\dot{x}^2 + x^3 \\) with initial conditions \\( x(0) = 0 \\) and \\( \\dot{x}(0) = A \\), with respect to \\( A \\) at \\( A = 0 \\).",
"role": "user"
}
] | t |
[
{
"content": "Evaluate \\( \\lim_{t\\to \\infty}\\int_0^1 \\frac{e^{-tx}}{1+x^2}\\ dx \\) using the Dominated Convergence Theorem (DCT).",
"role": "user"
}
] | 0 |
[
{
"content": "Let G be an infinite group. Suppose that the commutator subgroup [G,G] of G and the abelianization of G are finitely generated. Does this imply that G is finitely generated? Provide a justification for your answer.",
"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... | 400 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... | 45280 ounces |
[
{
"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... | 1.1 m/s |
[
{
"content": "Determine the values of the parameter \\( a \\) for which the equation \\[ 3^{x^{2}+6 a x+9 a^{2}} = a x^{2}+6 a^{2} x+9 a^{3}+a^{2}-4 a+4 \\] has exactly one solution.",
"role": "user"
}
] | 1 |
[
{
"content": "Given the nonlinear minimization model with a single variable:\n\n$$\\min_{x \\in [0,h]} \\ z\\left(x\\right) := a \\, x^b + c \\, \\left(h-x\\right)^d$$\n\nwhere the parameters satisfy $0 \\le b, d \\le 1$ and $a, c > 0$, and the derivatives are given by:\n\n$$\\frac{dz}{dx}=abx^{b-1}-cd{\\left(h... | Yes |
[
{
"content": "Evaluate the line integral \\( \\int_C \\sin(y) \\, dx + x \\cos(y) \\, dy \\) where \\( C \\) is the ellipse defined by the equation \\( x^2 + xy + y^2 = 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... | behavioral therapist, behavior modification |
[
{
"content": "Let $X_1, X_2, \\ldots, X_n$ be random variables, and define $Y_k = \\sum_{i=1}^{k} X_i$ for $k = 1, 2, \\ldots, n$. Suppose that $Y_1, Y_2, \\ldots, Y_n$ are jointly Gaussian. Are the random variables $X_1, X_2, \\ldots, X_n$ jointly Gaussian?",
"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... | x = \sqrt{\dfrac{2mgh}{k}} |
[
{
"content": "Given a group $G$ and a Sylow $p$-subgroup $P$ of $G$, determine whether the normalizer $N(P)$ of $P$ is a $p$-group.",
"role": "user"
}
] | No |
[
{
"content": "Determine whether there are any simple groups, other than $PSL(2,p)$, that have cyclic odd Sylow subgroups.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the equation of the hyperbola obtained by rotating the graph of \\( f(x) = \\frac{1}{x} \\) by an angle of \\( \\frac{\\pi}{4} \\) using the rotation matrix \\( M_- \\).",
"role": "user"
}
] | x^2 - y^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... | 655,500 |
[
{
"content": "For any positive integer $n$, the value of $n!$ is the product of the first $n$ positive integers. For example, $4! = 4\\cdot 3\\cdot 2\\cdot 1 =24$. What is the greatest common divisor of $5!$ and $7!$ ?",
"role": "user"
}
] | 120 |
[
{
"content": "Determine the value of $k$ such that the equation $(1+x^2)e^x-k=0$ has exactly one real root.",
"role": "user"
}
] | \dfrac{2}{e} |
[
{
"content": "Given that \\( \\frac{a}{b+c+d} + \\frac{b}{a+c+d} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c} = 1 \\), find the value of \\( \\frac{a^2}{b+c+d} + \\frac{b^2}{a+c+d} + \\frac{c^2}{a+b+d} + \\frac{d^2}{a+b+c} \\).",
"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... | -0.6667 |
[
{
"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... | 229.63 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... | 294.34 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... | 1.09e3 mmHg |
[
{
"content": "Determine whether the function \\( f(x) = \\begin{cases} 0 & \\text{if $x=0$ or $x = 1/n$ for some $n \\in \\mathbb{N}$,} \\\\ 1 & \\text{otherwise.} \\end{cases} \\) is Riemann-Stieltjes integrable on the interval \\([0,1]\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Let $f: \\mathbb{N}^{+} \\to \\mathbb{N}^{+}$ be a function such that for any $x \\neq y$, if a prime $p$ divides $f(x) + f(y)$, then $p$ also divides $x + y$. Determine the value of $f(1)$. ",
"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... | - 2 \hbar \hat X e^{i \hat{X^2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 10615 nm |
[
{
"content": "Find the number of irreducible monic polynomials of degree 6 in the polynomial ring \\( \\mathbb{Z}/(2)[x] \\).",
"role": "user"
}
] | 9 |
[
{
"content": "Find all functions \\( f: \\mathbb{Q}_+ \\to \\mathbb{Q}_+ \\) that satisfy the following conditions:\n\n1. \\( f(x+1) = f(x) + 1 \\) for all \\( x \\in \\mathbb{Q}_+ \\).\n2. \\( f(x^2) = f(x)^2 \\) for all \\( x \\in \\mathbb{Q}_+ \\).",
"role": "user"
}
] | f(x) = x |
[
{
"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... | 2543 K |
[
{
"content": "Let $n \\geq 2$ be an integer and consider the vector space $V=F^{2n}$ with the standard basis $e_1,\\ldots,e_{2n}$. The second exterior power $\\Lambda^2(V)$ contains the element $$\\omega=e_1 \\wedge e_2 + e_3 \\wedge e_4 + \\cdots + e_{2n-1} \\wedge e_{2n}.$$ Since $\\Lambda^{2n}(V)$ is spanned... | n! |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3.5% |
[
{
"content": "Calculate the double integral \\( \\int\\int_R xe^y \\, dA \\), where \\( R \\) is the region inside the circle \\( x^2 + y^2 = 1 \\) and outside the circle \\( x^2 + y^2 = 2y \\).",
"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... | 0.00236 moles |
[
{
"content": "Suppose \\( \\det(A) = 0 \\). Is there always a non-zero matrix \\( B \\) such that \\( AB = BA = 0 \\), where \\( 0 \\) denotes the zero matrix?",
"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... | v1f = +4/sqrt(5) m/s; v2f = -8/sqrt(5) m/s |
[
{
"content": "Determine whether every square matrix $A$ can be expressed as $A = B_1B_2 = B_2B_1$ for some matrices $B_1$ and $B_2$. Provide a justification for your answer.",
"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... | 375.0 pm |
[
{
"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... | 3.38 x 10^6 N/C |
[
{
"content": "Let\n\\[f(x) = \\sqrt{x(50 - x)} + \\sqrt{x(2 - x)}\\]for $0 \\le x \\le 2.$ Let $M$ be the maximum value of $f(x),$ and let $x = x_0$ be the point where the maximum value is attained. Enter the ordered pair $(x_0,M).$",
"role": "user"
}
] | \left( \frac{25}{13}, 10 \right) |
[
{
"content": "Find the greatest common divisor of the numbers \\(2^{71} - 2, 3^{71} - 3, \\ldots, 100^{71} - 100\\).",
"role": "user"
}
] | 4686 |
[
{
"content": "Determine the value of \\(a\\) for which the function \\(f(x) = x^3 + 3x^2 + ax + 3\\) is an open map from \\(\\mathbb{R}\\) to \\(\\mathbb{R}\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Find the supremum of the set \\(\\mathbb{A} = \\left\\{ ab - \\left(\\frac{a^2}{2} + \\frac{b^4}{4}\\right) \\mid a, b \\in \\mathbb{R}^{+} \\right\\}.\"",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Determine the number of non-isomorphic connected unicyclic graphs on 6 vertices.",
"role": "user"
}
] | 13 |
[
{
"content": "Let $(E, \\mathcal{A}, \\mu)$ be a finite measure space and $\\{f_n\\}$ be a sequence of integrable functions such that for all $\\epsilon \\in (0, \\frac{\\sqrt{2}}{2}]$, \\[ \\lim_{n \\to \\infty} \\mu(\\{t : \\sup_{k \\geq n} |f_k(t) - f(t)| > \\epsilon\\}) = 0. \\] Determine if it follows that... | 0 |
[
{
"content": "Find the radius of convergence of the series \\( \\sum_{n=0}^{\\infty} \\left(\\frac{3}{2\\pi}\\right)^n (\\arctan n)^n x^n \\).",
"role": "user"
}
] | \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... | 0.174 L |
[
{
"content": "Solve the equation for \\( x \\): \\[ \\frac{\\pi}{4} = \\frac{e^x - e^{-x}}{2} + \\arctan(x+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... | 0.23 |
[
{
"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... | 24 N |
[
{
"content": "Let $u_n, u \\in L^{\\infty}(E)$ and suppose $u_n \\rightarrow u$ almost everywhere. Determine whether the following equality holds: \\[ \\sup_{m \\geq n} \\, \\esssup_E |u_n - u_m| = \\esssup_E \\, \\sup_{m \\geq n} |u_n - u_m|. \\]",
"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... | 11 km/s |
[
{
"content": "In a metric space $X$, consider two complete metric spaces $X'$ and $Y$, with isometries $i : X \\rightarrow X'$ and $f : X \\rightarrow Y$. Suppose a sequence $(i(x_n))_n$ in $i[X]$ converges to some $p \\in \\overline{i[X]} = X'$. This implies $(f(x_n))_n$ converges to some $q \\in Y$. If anothe... | Yes |
[
{
"content": "Find the gradient of the function \\( F(A) = X^T \\cdot A \\cdot Y \\) with respect to the matrix \\( A \\), where \\( X \\) and \\( Y \\) are \\( m \\times 1 \\) vectors and \\( A \\) is an \\( m \\times m \\) matrix.",
"role": "user"
}
] | X Y^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... | 1.10 s^{-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... | <img>/cimages/multimages/16/diag_22789359681696238395.jpg</img> |
[
{
"content": "Evaluate the integral \\( \\int_A y^{-1/2}\\sin x \\,dm^2 \\), where \\( A=\\{(x,y)\\in[0,1]^2:x+y\\not \\in \\mathbb Q \\;\\&\\;xy \\not \\in \\mathbb Q\\} \\) and \\( m^2 \\) is the Lebesgue measure on \\( \\mathbb{R}^2 \\).",
"role": "user"
}
] | 2(1 - \cos 1) |
[
{
"content": "Provide an example of a countable set $S$ such that its set of limit points $S'$ is uncountable.",
"role": "user"
}
] | \mathbb{Q} |
[
{
"content": "Suppose you throw a fair die 1995 times. What is the probability that the total sum of the outcomes is divisible by 3?",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "For a complex number $z$ with $\\Re(z) \\ge 0$ and a real number $x > 1$, find the number of solutions to the equation $z + e^{-z} - x = 0$ subject to the conditions $0 < x - 1 < |z| < x + 1$ and $-1 < \\Im(z) < 1$. Provide the number of solutions.",
"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... | 3.8m |
[
{
"content": "Determine the interior of the set of even numbers in the topological space \\((\\mathbb{N}, \\text{Cofin})\\), where \\(\\text{Cofin}\\) is the topology on \\(\\mathbb{N}\\) consisting of the cofinite subsets of \\(\\mathbb{N}\\).",
"role": "user"
}
] | \emptyset |
[
{
"content": "Let $X$ be the ordinal space $[0,\\Omega)$, with the order topology, where $\\Omega$ is the first uncountable ordinal. Let $f\\colon X \\rightarrow \\mathbb{R}$ be a continuous, real-valued function on $X$. Determine whether $f$ is eventually constant, i.e., whether there exists a countable ordina... | 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... | 4446 mL |
[
{
"content": "Evaluate the integral \\( \\int\\limits_{B^n}\\Delta f(x)dx \\), where \\( B^n=\\{x\\in\\mathbb{R}^n:|x|<1\\} \\) and \\( f(x)=(|x|^2-1)^{20}\\sin(x_1+\\cdots+x_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... | 10^{-19} s |
[
{
"content": "Determine how the prime number 2 splits in the splitting field of the polynomial \\(X^3 - 3X + 1 = 0\\).",
"role": "user"
}
] | (2) |
[
{
"content": "Find the limit of \\( x^{f(x)} - f(x) \\) as \\( x \\to 0^+ \\), where \\( f(x) = x^{x^x} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 22.82 |
[
{
"content": "In triangle \\( \\triangle ABC \\), the ratio \\( (\\sin B + \\sin C) : (\\sin C + \\sin A) : (\\sin A + \\sin B) = 4 : 5 : 6 \\). Determine the degree measure of the largest angle in the triangle. Choose from the following options:\n(A) \\(72^{\\circ}\\)\n(B) \\(84^{\\circ}\\)\n(C) \\(1.5^{\\circ... | D |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to\\infty}\\frac{\\sum_{r=1} ^x re^\\frac{r}{x}}{x^2}$$",
"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... | 9439.5 m |
[
{
"content": "Determine if the point $(-1,0,0)$ lies on the curve defined by the parametric equations $x(t)=\\text{Re}(t^{i\\pi})$, $y(t)=\\text{Im}(t^{i\\pi})$, $z(t)=0$ for $t \\in [0,1]$. Consider the condition $t^{i\\pi}=-1$ and the restriction on $t$. Is the point included in the curve?",
"role": "user... | Yes |
[
{
"content": "Determine whether $T^p - T$ is the greatest common divisor of the set \\( \\{(T+u)^n - (T+u) : u \\in \\mathbb{F}_p\\} \\) in the polynomial ring \\( \\mathbb{F}_p[T] \\), given that \\( n > 1 \\), \\( p \\) is an odd prime, \\( p-1 \\mid n-1 \\), and \\( p^k - 1 \\nmid n-1 \\) for any \\( k > 1 \... | T^p - T |
[
{
"content": "Find the limit of the sequence \\((U_n)\\) defined by the recursion:\n\\[ x_1 = \\sqrt[2022]{\\frac{2022}{2021}} \\]\n\\[ x_{n+1} = \\frac{n+2}{3n+11}\\left(\\sqrt{x_n} + \\sqrt[3]{7 + x_n}\\right), \\quad \\forall n \\in \\mathbb{N^*} \\]",
"role": "user"
}
] | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.