prompt list | ground_truth string |
|---|---|
[
{
"content": "Is it possible to find an inner product on a finite-dimensional vector space such that a given set of linearly independent vectors becomes orthogonal with respect to that inner product?",
"role": "user"
}
] | Yes |
[
{
"content": "Given that the positive real numbers \\( x \\) and \\( y \\) satisfy the equation \\( x^3 + y^3 = (4x - 5y)y \\), find the maximum value of \\( y \\).",
"role": "user"
}
] | \dfrac{1}{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... | 0.46 cm |
[
{
"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... | $48646.71 |
[
{
"content": "Evaluate the integral \\( \\int _0^6 2\\pi \\left(\\sqrt{81-y^2}\\right)\\sqrt{1+\\frac{y^2}{81-y^2}}\\,dy \\).",
"role": "user"
}
] | 108\pi |
[
{
"content": "Suppose $X$ and $Y$ are independent random variables with the same geometric distribution, where $\\mathbb{P}(X=k)=pq^{k-1}$ for $k \\geq 1$ and $q=1-p$. Determine the distribution of $\\mathbb{P}(X=k \\mid X+Y=n+1)$, where $n$ is a positive integer.",
"role": "user"
}
] | \dfrac{1}{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... | 107 kg |
[
{
"content": "Provide an example of an element \\( a \\in K((T)) \\) that is algebraic over \\( K(T) \\) but not an element of \\( K(T) \\), for some field \\( K \\).",
"role": "user"
}
] | \sqrt{1 + T} |
[
{
"content": "Calculate the sum of the infinite series: \\[ S = \\frac{1}{2} + \\frac{3}{2^2} + \\frac{5}{2^3} + \\frac{7}{2^4} + \\cdots \\]",
"role": "user"
}
] | 3 |
[
{
"content": "Determine the values of $r$ for which the series \\( \\sum\\limits_{n=1}^{\\infty} \\frac{1}{n^{r+1}}\\cos(nx) \\) converges uniformly for all real numbers $x$.",
"role": "user"
}
] | r > 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... | 8.93 m/s |
[
{
"content": "Consider a sequence of functions $(g_n)$ such that each function has infinitely many sign changes. Can the sequence $(g_n)$ uniformly converge to a function $g$ that has only a finite number of sign changes? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "When the graph of a certain function $f(x)$ is shifted $2$ units to the right and stretched vertically by a factor of $2$ (meaning that all $y$-coordinates are doubled), the resulting figure is identical to the original graph.\n\nGiven that $f(0)=0.1$, what is $f(10)$?",
"role": "user"
}
] | 3.2 |
[
{
"content": "Given a non-bipartite, undirected, connected graph $G(V,E)$, a new graph $G'(V', E')$ is constructed as follows: $V' = V \\times V = \\{(v_i,v_j) \\mid v_i, v_j \\in V\\}$ and $E' = \\{((v_i,v_j),(v_k,v_l)) \\mid (v_i,v_k),(v_j,v_l) \\in E\\}$. Is $G'$ guaranteed to be connected?",
"role": "us... | Yes |
[
{
"content": "Let $u \\in H^1(U)$ with $Du = 0$ and $Tu = 0$. Given that $u$ is constant on each connected component of $U$ and $u \\in H^1(U) \\cap C(\\overline{U})$, can you conclude that $u = 0$?",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit of the infinite series: $$ \\lim_{n \\to \\infty} \\sum_{k=0}^n \\left( \\sum_{j=0}^k \\binom{k}{j} \\left(-\\frac{1}{3}\\right)^j \\right). $$",
"role": "user"
}
] | 3 |
[
{
"content": "Determine if the function \\( g(t) = \\frac{1-\\exp(-t)-t\\exp(-t)}{t^2} \\) for \\( t > 0 \\) is a probability density function.",
"role": "user"
}
] | Yes |
[
{
"content": "A line is expressed in the form\n\\[\\begin{pmatrix} -2 \\\\ -5 \\end{pmatrix} \\cdot \\left( \\begin{pmatrix} x \\\\ y \\end{pmatrix} - \\begin{pmatrix} 1 \\\\ 11 \\end{pmatrix} \\right) = 0.\\]The equation of the line can be expressed in the form $y = mx + b.$ Enter the ordered pair $(m,b).$",
... | \left( -\frac{2}{5}, \frac{57}{5} \right) |
[
{
"content": "Calculate the limit: $$\\lim_{x\\to 1 }\\left( \\frac{x-1-x\\ln(x) }{x(x-1)^{2}}\\right)$$ without using L'Hopital's rule.",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "Determine whether the following statement is true or false: If $f(x) \\leq a \\leq g(x)$ and $f(x) \\leq b \\leq g(x)$ for every $x \\in \\mathbb{R}$, and for all $\\epsilon > 0$ there exists $x$ such that $g(x) - f(x) < \\epsilon$, then $a = b$. Suppose $f, g$ are real-valued functions and $a, b$... | True |
[
{
"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... | 1409.17 kg |
[
{
"content": "Determine whether $2^{1093} - 2$ is divisible by $1093^2$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the value of A in the infinite power tower: \\( A = \\sqrt{2}^{\\sqrt{2}^{\\sqrt{2}^{\\cdots}}} \\).",
"role": "user"
}
] | 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... | 8.85% |
[
{
"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... | -30.25 |
[
{
"content": "How many ways can a committee of 17 politicians be formed from indistinguishable Democrats, Republicans, and Independents, such that each party has at least two members and no party has a majority of the committee members?",
"role": "user"
}
] | 33 |
[
{
"content": "Does there exist a twice differentiable function that approximates a threshold such that \\( f(x) = 0 \\) for \\( x \\leq a \\), \\( f(x) = L \\) for \\( x \\geq a + \\epsilon \\), and \\( f(x) \\) takes values between these two for \\( x \\in (a, a+\\epsilon) \\), where \\( \\epsilon \\) is a sma... | 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... | 4.225 J |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty} \\int_0^1 \\sqrt[n]{x} e^x \\,dx.$$",
"role": "user"
}
] | e - 1 |
[
{
"content": "Evaluate the surface integral \\( \\iint_S x\\;dy \\times dz + y \\; dz \\times dx + z \\; dx \\times dy \\) over the part of the plane \\( x+y+z=2 \\) in the first octant, where the normal vector \\( n \\) satisfies \\( n \\cdot (0,1,0) \\geq 0 \\).",
"role": "user"
}
] | 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... | 109344 |
[
{
"content": "Determine the conditions on \\(x_0\\) and \\(y_0\\) such that the differential equation \\( y' = x^4 \\ln y \\) with the initial condition \\( y(x_0) = y_0 \\) has a unique solution.",
"role": "user"
}
] | y_0 > 0 |
[
{
"content": "Determine the values of the parameter $a$ for which the inequality \\(|x^2 - ax + 1| < 3(x^2 + x + 1)\\) holds for every real number $x$. Provide your answer in terms of $a$.",
"role": "user"
}
] | (-5, 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... | 451 billions of solar masses |
[
{
"content": "Find the minimum value of the expression \\( A = \\frac{(3xy + x^{2}) \\sqrt{3xy + x - 3y} + (3xy + y^{2}) \\sqrt{3xy + y - 3x}}{x^{2} y + y^{2} x} \\) for \\( x, y \\in [1,3] \\).",
"role": "user"
}
] | 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... | 6.6 mL |
[
{
"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... | Net Income: 150,000 |
[
{
"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... | 6.69 * 10^-4 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... | 15.04% |
[
{
"content": "The equation \\[\\frac{x^2}{36} + \\frac{(y+5)^2}{16} = 0\\]describes a degenerate ellipse, because the right-hand side is $0$ instead of $1$ (as in the standard form for an ellipse). Of all the points on the graph of this equation, what is the largest possible $y$-coordinate?",
"role": "user"... | -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... | 9.8 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... | 128.124 |
[
{
"content": "Determine whether the following statement is true: If $X$ and $Y$ are identically distributed random variables, then $E(XY) = E(X^2) = E(Y^2)$. Assume $X$ and $Y$ are not necessarily independent.",
"role": "user"
}
] | False |
[
{
"content": "Let $X_1, X_2, \\ldots, X_n$ be independent uniform random variables over the interval $[0,1]$. Given a positive integer $k$ such that $1 \\leq k < n$, determine the expected number of $X_i$ that are greater than the $k$th largest value among $X_1, X_2, \\ldots, X_n$. Provide your answer as a sing... | k - 1 |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} \\sum_{k=1}^{n} \\frac{(n+1)^k}{n^{k+1}} \\) using the concept of Riemann sums.",
"role": "user"
}
] | e - 1 |
[
{
"content": "Evaluate the double integral \\(\\int_R \\exp\\left({\\frac{x^2}{25}+\\frac{y^2}{16}}\\right)\\,\\mathrm dA\\) over the region bounded by the ellipses \\(\\frac{x^2}{25}+\\frac{y^2}{16}=1\\) and \\(\\frac{x^2}{25}+\\frac{y^2}{16}=4.\\)",
"role": "user"
}
] | 20\pi(e^4 - e) |
[
{
"content": "Twelve congruent disks are placed on a circle $C$ of radius 1 in such a way that the twelve disks cover $C$, no two of the disks overlap, and so that each of the twelve disks is tangent to its two neighbors. The resulting arrangement of disks is shown in the figure below. The sum of the areas of t... | 135 |
[
{
"content": "Determine the radius of convergence of the series \\( \\sum_{k = 0}^{\\infty} 3^{k^2}x^{k^2} \\).",
"role": "user"
}
] | \dfrac{1}{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... | -30.1kJ |
[
{
"content": "How many different isosceles triangles have integer side lengths and perimeter 23?",
"role": "user"
}
] | 6 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | <4/3, -2/3> ms^-1 |
[
{
"content": "Evaluate the limit: $$\\lim\\limits_{x\\to0}\\ \\left(\\frac {\\tan x} {x}\\right)^\\frac {1} {x^2}\\ .$$",
"role": "user"
}
] | e^{\frac{1}{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... | 6907.65 rev |
[
{
"content": "Find all positive integer solutions \\((a, b, x, y)\\) to the equation \\( a^x - b^y = 2 \\) given that \\( x \\geq 2 \\) and \\( y \\geq 2 \\).",
"role": "user"
}
] | (3, 5, 3, 2) |
[
{
"content": "Evaluate the double integral over the first quadrant: \\[ \\int_0^\\infty \\int_0^\\infty \\frac{\\exp(-x-y)}{(x+y)}\\,dx\\,dy. \\]",
"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... | 906430 |
[
{
"content": "Evaluate the limit of the improper integral: $$\\lim_{n\\to\\infty}\\int_0^{\\pi/2} \\frac{\\sin^n(x)}{1+x^2} \\, dx$$",
"role": "user"
}
] | 0 |
[
{
"content": "Provide an example of a continuous function \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that is closed but not open.",
"role": "user"
}
] | f(x) = 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... | 110.53 J |
[
{
"content": "Given non-zero vectors $\\overrightarrow{A B}$ and $\\overrightarrow{A C}$ satisfying $\\left(\\frac{\\overrightarrow{A B}}{|\\overrightarrow{A B}|} + \\frac{\\overrightarrow{A C}}{|\\overrightarrow{A C}|}\\right) \\cdot \\overrightarrow{B C} = 0$ and $\\frac{\\overrightarrow{A B}}{|\\overrightarr... | D |
[
{
"content": "Determine the number of roots of the function \\( h(x) = x - e^{-Bx^2}\\sin(x) \\) over the real numbers, where \\( B > 0 \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\int_{0}^{n} \\left(1 + \\frac{x}{2n}\\right)^n e^{-x} \\, dx \\]",
"role": "user"
}
] | 2 |
[
{
"content": "Does the flow of a smooth vector field on a closed smooth manifold exist for all time $t \\in \\mathbb{R}$?",
"role": "user"
}
] | Yes |
[
{
"content": "Find the total length of the curve defined by the parametric equations \\( \\gamma(t) = (t - \\cos(t), 1 + \\sin(t)) \\) for \\( t \\in (-\\pi, \\pi) \\). Use a suitable substitution to evaluate the integral.",
"role": "user"
}
] | 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... | Year|Interest paid|Interest expense|Amortization amount|Unamortized amount|Carrying amount
---|---|---|---|---|---
Year 0|-|-|-|$425,594|$2,325,594
Year 1|$47,500|$33,314|$14,186|$411,408|$2,311,408
Year 2|$47,500|$33,314|$14,186|$397,222|$2,297,222 |
[
{
"content": "Determine whether the set of zeroes of the polynomial \\( p(x) = x^4 + 4 \\) in \\( \\mathbb{C} \\) forms a group under multiplication. Answer true or false.",
"role": "user"
}
] | False |
[
{
"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... | 51.1 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... | 4.28 |
[
{
"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... | 161.3 W |
[
{
"content": "Determine whether there exists an algebraic solution for the equation \\(x^2 + e^x = e^{2x}\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Find the smallest positive integer \\( n \\) such that \\( n, n+1, \\) and \\( n+2 \\) each have exactly four positive factors.",
"role": "user"
}
] | 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... | 2.10e-13 m |
[
{
"content": "Compute the limit: \\[ \\lim_{n\\to \\infty} \\int_0^\\infty \\int_0^\\infty \\frac{n}{x} \\sin\\left( \\frac{x}{ny} \\right) e^{-\\frac{x}{y} - y}\\ dx\\ dy. \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Find all functions \\( f: [0, 1] \\rightarrow [0, 1] \\) that satisfy the following conditions: \\( f(0) = 0 \\), \\( f(1) = 1 \\), and \\( f(x+y) - f(x) = f(x) - f(x-y) \\) for any \\( x, y \\geq 0 \\) such that \\( (x-y), (x+y) \\in [0, 1] \\).",
"role": "user"
}
] | f(x) = x |
[
{
"content": "Find the determinant of an invertible $6 \\times 6$ matrix $A$ such that $A^4 + 2A = 0$. What is the value of $\\det(A)$?",
"role": "user"
}
] | 4 |
[
{
"content": "Find the number of ordered quadruples $(a,b,c,d)$ of real numbers such that\n\\[\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}^{-1} = \\renewcommand{\\arraystretch}{1.5} \\begin{pmatrix} \\frac{1}{a} & \\frac{1}{b} \\\\ \\frac{1}{c} & \\frac{1}{d} \\end{pmatrix} \\renewcommand{\\arraystretch}{1}... | 0 |
[
{
"content": "Determine the value of \\( h \\) such that the matrix \\( \\begin{pmatrix} h & h+1 & 0 \\\\ 2h & 1 & h-1 \\\\ 2h-2 & 0 & 1 \\end{pmatrix} \\) can be diagonalized by an orthogonal matrix.",
"role": "user"
}
] | 1 |
[
{
"content": "Let $M$ be a diagonalizable $n \\times n$ matrix. If the set of matrices commuting with $M$ is not simultaneously diagonalizable, does this imply that $M$ has repeated eigenvalues? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Compute the residue of the function \\( \\frac{z}{e^z - 1} \\) at its poles.",
"role": "user"
}
] | 2\pi i k |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $105.127 |
[
{
"content": "Find the volume of the region bounded by the planes $4x+2y+4z=6$, $y=x$, $x=0$, and $z=0$.",
"role": "user"
}
] | \dfrac{3}{4} |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to 0} \\int_0^1 \\cos\\left(\\frac{1}{xt}\\right)\\, dt$$",
"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... | 21.69% |
[
{
"content": "Find all polynomials \\( P(x) \\) with integer coefficients such that the equation \\( P(x)=0 \\) has no positive integer solutions and for all positive integers \\( n \\), \\( \\phi(n) \\mid \\phi(P(n)) \\). Assume \\( P(n) > 0 \\) for all positive integers \\( n \\).",
"role": "user"
}
] | P(x) = x^k |
[
{
"content": "Given the system of equations with positive real numbers $x, y, z$:\n\\begin{align*}\n\\frac{x}{y} + \\frac{y}{x} + \\frac{x}{z} + \\frac{z}{x} + \\frac{y}{z} + \\frac{z}{y} &= 2018 \\\\\n\\sqrt{\\frac{x}{y}} + \\sqrt{\\frac{y}{z}} + \\sqrt{\\frac{z}{x}} &= 17,\n\\end{align*}\nfind the value of $\... | 43 |
[
{
"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... | c > a > d > b |
[
{
"content": "Consider a topological space $X$ of infinite cardinality that is homeomorphic to $X \\times X$. Determine which of the following statements is true:\n\n(a) $X$ is not connected\n(b) $X$ is not compact\n(c) $X$ is not homeomorphic to a subset of $\\mathbb{R}$\n(d) None of the above",
"role": "u... | D |
[
{
"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... | 3.84 m/s |
[
{
"content": "Let $R$ be a ring (unital, not necessarily commutative), and let $M$ be a finitely presented left $R$-module. Suppose $m_1, \\ldots, m_n \\in M$ generate $M$, determining a surjection $f: R^n \\to M$. Is the kernel of $f$ finitely generated?",
"role": "user"
}
] | Yes |
[
{
"content": "Derive the formula for the sum of the first \\( n \\) terms of the series: \\( 1^3 + 3^3 + 5^3 + \\ldots + (2n-1)^3 \\).",
"role": "user"
}
] | n^2(2n^2 - 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... | 141.90 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... | 9.37 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... | \overrightarrow{\mu_l} = -\frac{e}{2m} \overrightarrow{l} |
[
{
"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... | 85 m/s |
[
{
"content": "Suppose $M$ is a random variable following a Poisson distribution with parameter $\\lambda$. For each of the $M$ objects, you decide to keep it with probability $p$. Let $N$ be the total number of objects you keep. Given that the expectation of $N$ is $\\lambda p$, find the variance of $N$. Assume... | \lambda p |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 1} \\left( \\frac{x}{\\ln x} - \\frac{1}{x \\ln x} \\right) \\] without using L'Hospital's Rule.",
"role": "user"
}
] | 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... | 3.53 atm |
[
{
"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... | 19600 dynecm^{-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... | $167.92 |
[
{
"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... | Thor Company: 4.021
Gunnar Company: 9.91 |
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