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... | f = \frac{1}{2\pi}\sqrt{\frac{PE}{I}} |
[
{
"content": "Evaluate the integral \\( \\int_{-\\pi/4}^{\\pi/4} \\frac{\\sec^2x}{1+e^x} \\, dx \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Compute the 2-adic expansion of \\( \\frac{2}{3} \\).",
"role": "user"
}
] | \overline{10}110 |
[
{
"content": "Find the area of the region $R$ enclosed by the curve parameterized by $g(t) = (t^4 - t^2, t^6 - t^2)$ for $0 \\leq t \\leq 1$ using Green's Theorem.",
"role": "user"
}
] | \dfrac{1}{60} |
[
{
"content": "Calculate the limit \\[ l = \\lim_{x\\to 0}\\frac{e^{\\sin x}-e^{\\tan x}}{e^{\\sin 2x}-e^{\\tan 2x}} \\]",
"role": "user"
}
] | \dfrac{1}{8} |
[
{
"content": "Determine whether the following statement is true or false: If \\(b_n\\) is a positive sequence and \\(\\lim_{n\\to\\infty}\\frac{b_{n+1}}{b_n}=k\\), then \\(\\lim_{n\\to\\infty} b_n^{\\frac{1}{n}}=k\\).",
"role": "user"
}
] | 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... | 3.697 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... | 1.98 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... | 15.73% |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0}\\frac{\\ln(\\tan(x)+1)-\\sin(x)}{x\\sin(x)} \\] without using L'Hôpital's rule.",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "Let $U$ be a finite set, $D$ another set, and $F = \\{(X_\\lambda, Y_\\lambda)\\}_{\\lambda \\in \\Lambda}$ a finite family of pairs of subsets of $U$. Define $I \\vDash F$ if, for all $(X, Y) \\in F$ and every $s, t \\in I \\subseteq D^U$ (where $D^U$ denotes the set of functions from $U$ to $D$)... | 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... | 0.99 m |
[
{
"content": "Given the functions \\( F(x) = \\int_{1}^{x}{f(t)}\\,dt \\), where \\( f(t) = \\int_{1}^{t^2}{\\frac{\\sqrt{9+u^4}}{u}}\\,du \\), find the value of the second derivative \\( F''(2) \\) using the Fundamental Theorem of Calculus.",
"role": "user"
}
] | \sqrt{265} |
[
{
"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... | $1,446.78 |
[
{
"content": "Find the minimum value of $z + d$ where $z$ and $d$ are 3-digit positive integers such that $24\\text{gcd}(z, d) = \\text{lcm}(z, d)$. ",
"role": "user"
}
] | 374 |
[
{
"content": "Determine whether the group of rational numbers $\\mathbb{Q}$ can be expressed as the direct product of two proper subgroups. Answer with '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... | 7.43 ft/sec |
[
{
"content": "Evaluate the contour integral of \\( h(z) = \\frac{z}{(z+3)(z-2)} \\) over the circle \\( |z| = 4 \\).",
"role": "user"
}
] | 2\pi i |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{(x^2+2x+2)^3}} \\, dx \\).",
"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... | 62.32% of arsenic(III) oxide and 37.68% of arsenic(V) oxide |
[
{
"content": "Calculate the volume integral of the function \\( \\frac{1}{r} \\) over the region outside a sphere with radius \\( R \\) centered at the point \\((a, b, c)\\), where \\( r \\) is the distance from the center of the sphere.",
"role": "user"
}
] | \infty |
[
{
"content": "Let \\( f:\\mathbb{R} \\to \\mathbb{R} \\) be a differentiable function defined by \\( f(x) = x + \\int_0^1 (xy + x^2)f(y) \\, dy \\). Find the value of \\( \\int_0^1 f(x) \\, dx \\).",
"role": "user"
}
] | \dfrac{36}{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... | x(t)=0.0909cos(16.5t) |
[
{
"content": "Through the point \\( P(11,0) \\), draw a line with an angle of inclination \\( \\frac{\\pi}{4} \\) intersecting the parabola \\( y^{2}=4x \\) at points \\( R \\) and \\( Q \\). Then, draw a line parallel to \\( RQ \\) intersecting the arc of the parabola \\( \\overparen{ROQ} \\) at points \\( M \... | 32 |
[
{
"content": "In a complete countably seminormed space $X$, consider a sequence of linear continuous mappings $T_k: X \\rightarrow \\mathbb{R}$ such that for each $x \\in X$, the sequence $(T_k(x))_{k \\in \\mathbb{N}}$ is bounded. By the Banach-Steinhaus theorem for Fréchet spaces, the family $(T_k)_{k \\in \\... | 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... | Dec 31, 2016
Current liabilities
Bonds Payable: $110,000
Interest Payable: $49,500
Long-term liabilities:
Bonds Payable: $220,000
Dec 31, 2017
Current liabilities
Bonds Payable: $110,000
Interest Payable: $33,000
Long-term liabilities:
Bonds Payable: $110,000
Dec 31, 2018
Current liabilities
Bonds Payable: $110,000
Interest Payable: $16,500
Long-term liabilities:
Bonds Payable: $0 |
[
{
"content": "Find the remainder when $2^{1990}$ is divided by $1990$. Express your answer as a single integer.",
"role": "user"
}
] | 1024 |
[
{
"content": "Find the expected value of the second largest number when three numbers are chosen randomly from the interval (0, 1).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine whether there exist integers \\(a\\), \\(b\\), \\(c\\), and \\(d\\) such that \\(a^2 + b^2 + 2d^2 = c^2\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Let $A$ be a skew-symmetric $n \\times n$ matrix with coefficients in $\\mathbb{R}$ or $\\mathbb{C}$. Consider the $2n \\times 2n$ matrix:\n\n$$M = \\begin{pmatrix}0&A\\\\-A&0 \\end{pmatrix}.$$ \n\nDetermine if the signature of $M$ is zero, meaning the number of positive eigenvalues of $M$ is equa... | 0 |
[
{
"content": "Simplify the series: $ \\frac{1}{2^1}+\\frac{3}{2^2}+\\frac{5}{2^3}+\\frac{7}{2^4}+\\cdots $.",
"role": "user"
}
] | 3 |
[
{
"content": "For the expansion of \\((1+0.2)^{1000}\\) using the binomial theorem, the terms are given by \\(A_k = \\binom{1000}{k}(0.2)^k\\) for \\(k = 0, 1, 2, \\ldots, 1000\\). Determine the value of \\(k\\) for which \\(A_k\\) is the largest.",
"role": "user"
}
] | 166 |
[
{
"content": "A particular convex polygon with seven sides has exactly one right angle. How many diagonals does this seven-sided polygon have?",
"role": "user"
}
] | 14 |
[
{
"content": "Let $A$ be a positive definite matrix with a unique square root $B$, and let $A^{-1}$ have a unique square root $D$. Determine whether $B^{-1} = D$. In other words, is the square root of the inverse of a matrix equal to the inverse of its square root?",
"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... | 0.1416 M^-1 s^-1 |
[
{
"content": "Simplify $-3(1+4i)+i(-2-i)$.",
"role": "user"
}
] | -2-14i |
[
{
"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... | $553,010 |
[
{
"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... | 93.132 N |
[
{
"content": "Let \\( \\Omega \\subseteq \\mathbb{R}^{n} \\) be an open set and let \\( f, g: \\Omega \\to \\mathbb{R} \\) be \\( C^{k} \\) functions, where \\( k \\ge 0 \\). Suppose \\( \\int_{\\Omega} f(x)g(x) \\, dx = 0 \\) for every \\( g \\) with compact support. Does it follow that \\( f \\equiv 0 \\) on ... | Yes |
[
{
"content": "Given the equation $a^2 = 2b^3 = 3c^5$, where $a, b, c$ are natural numbers, find the smallest possible value of the product $abc$.",
"role": "user"
}
] | 746496 |
[
{
"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... | 3.63% |
[
{
"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... | 1899.61 |
[
{
"content": "Let $X$ be a complex smooth projective variety with trivial topological Euler characteristic $\\chi_{\\text{top}}(X)=0$. Assume that $D$ is a smooth irreducible divisor in the linear system $|K_X|$ of the canonical divisor $K_X$ of $X$. Determine if $\\chi_{\\text{top}}(D)=0$.",
"role": "user"... | 0 |
[
{
"content": "Let $f(z)$ be an analytic function on $\\mathbb{C}$, where $f$ is not identically zero. For each $\\varepsilon>0$, define $U_\\varepsilon = \\{ z \\in \\mathbb{C} : |f(z)| < \\varepsilon \\}$. Does $m(U_\\varepsilon) \\to 0$ as $\\varepsilon \\to 0$, where $m$ is the Lebesgue measure on $\\mathbb{... | No |
[
{
"content": "Square $ABCD$ has side lengths of 13 units. Point $E$ lies in the interior of the square such that $AE = 5$ units and $BE = 12$ units. What is the distance from $E$ to side $AD$?",
"role": "user"
}
] | 1\frac{12}{13} |
[
{
"content": "Let $C$ be a convex compact subset of a finite-dimensional normed vector space, and let $f: C \\to \\mathbb{R}$ be a strictly convex and uniformly continuous function. Is it true that $f$ has exactly one minimizer?",
"role": "user"
}
] | Yes |
[
{
"content": "An image is divided into two regions: one white and one black. A reading from a point in the white region follows a normal distribution with mean $\\mu_w = 4$ and variance $\\sigma_w^2 = 4$. A reading from a point in the black region follows a normal distribution with mean $\\mu_b = 6$ and varianc... | 4.8 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1,411.27 |
[
{
"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... | 1082.31 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... | Cash: 6850
Supplies: 1530
Prepaid Insurance: 1330
Total Current Assets: 9710
Equipment: 48330
Less: Accumulated Depreciation: 4330
Equipment (net): 44000
Trademarks: 1280
Total Assets: 54990
Accounts Payable: 10330
Salaries and Wages Payable: 830
Unearned Service Revenue: 2330
Total Current Liabilities: 13490
Bonds Payable: 9330
Total Liabilities: 22820
Common Stock: 10330
Retained Earnings: 21840
Total Stockholder's Equity: 32170
Total Liabilities and Stockholder's Equity: 54990 |
[
{
"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... | 1.61e-2 m |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\rightarrow 0} \\left( \\frac{(\\cos{x})^2}{\\sin{(x^2)}}-\\frac{1}{x^2} \\right)$$",
"role": "user"
}
] | -1 |
[
{
"content": "Determine the cardinality of the set \\( S = \\{ (x, y, z) \\mid x^2 + y^2 + z^2 = 2^{2018}, xyz \\in \\mathbb{Z} \\} \\).",
"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... | $95080 |
[
{
"content": "Find the vector to which the vector \\( v = (2, -1, 3) \\) is rotated after a 120-degree rotation around the axis \\( x = y = z \\).",
"role": "user"
}
] | (3, 2, -1) |
[
{
"content": "The infinite series \\( \\sum_{k=0}^{\\infty} \\frac{2^{k}}{5^{2^{k}}+1} \\) can be expressed as a fraction \\( \\frac{p}{q} \\) where \\( p \\) and \\( q \\) are relatively prime positive integers. Calculate \\( p+q \\).",
"role": "user"
}
] | 5 |
[
{
"content": "Evaluate the integral \\( \\int_0^{2\\pi} e^{e^{i\\theta}} \\, d\\theta \\) using complex analysis techniques.",
"role": "user"
}
] | 2\pi |
[
{
"content": "Two spheres with radii \\( R \\) and \\( r \\) (where \\( R > r \\)) are packed inside a cylindrical pipe with a diameter of 100 units. The spheres touch each other and the walls of the pipe. Derive an expression for the distance \\( x \\) between the centers of the spheres along the axis of the p... | 2\sqrt{Rr} |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to 0} (e^x + x)^{1/x} \\) using L'Hopital's rule.",
"role": "user"
}
] | e^2 |
[
{
"content": "Determine whether the extension \\( \\mathbb{K}(X) \\supset \\mathbb{K}(u) \\) is algebraic, where \\( u = \\frac{X^3}{X+1} \\), and find the degree \\([\\mathbb{K}(X):\\mathbb{K}(u)]\\).",
"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... | 33.4m |
[
{
"content": "Determine the dimension of the Clifford algebra $Cl_{2,1}$.",
"role": "user"
}
] | 8 |
[
{
"content": "Let \\( f_n \\in L^1 \\) such that for all \\( g \\in L^1 \\), \\( \\|f_n * g - g\\|_1 \\rightarrow 0 \\). Does this imply that \\( \\int f_n \\rightarrow 1 \\)? Provide a proof or a counterexample.",
"role": "user"
}
] | Yes |
[
{
"content": "Given the function \\( f(x) = \\int_0^x f(t) \\, \\mathrm{d}t \\), find the value of \\( f(\\ln(5)) \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Consider a graph $G$ with $n!$ vertices, where each vertex is labeled by a distinct permutation of the numbers $1, 2, \\ldots, n$. Two vertices are connected by an edge if and only if the label of one vertex can be obtained by swapping two adjacent numbers in the label of the other vertex. Calcula... | n + 9 |
[
{
"content": "Determine the number of positive integer solutions for the equation \\(x! = 3y! + 15z!\\), where \\(x\\), \\(y\\), and \\(z\\) are distinct integers.",
"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... | 0.0015 N/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... | National saving = GDP - Consumption - Government expenditure |
[
{
"content": "Find all real numbers $k$ such that $x^4+kx^3+x^2+4kx+16=0$ is true for exactly one real number $x = r$. Enter all the possible values of $k,$ separated by commas.",
"role": "user"
}
] | \frac{9}{4}, -\frac{9}{4} |
[
{
"content": "Let $\\Omega$ be a bounded domain in $\\mathbb{R}^n$. Consider the space $L^{\\infty}(\\Omega)=\\{f: \\Omega\\to\\mathbb{R} \\text{ measurable}; ||f||_{L^{\\infty}(\\Omega)}<\\infty\\}$. For a function $f \\in L^{\\infty}(\\Omega)$, does there exist a bounded function $\\tilde{f}$, equivalent to $... | Yes |
[
{
"content": "In a graph with $n$ vertices, if the condition $d(u)+d(v) \\geq n-1$ holds for every pair of distinct vertices $u,v$, does it imply that all but one vertex must have a degree of at least $\\frac{n-1}{2}$?",
"role": "user"
}
] | Yes |
[
{
"content": "Given the circle \\(x^2 + y^2 + 6x - 24y + 72 = 0\\) and the hyperbola \\(x^2 - y^2 + 6x + 16y - 46 = 0\\), find the sum of the distances from the four intersection points to the point \\((-3, 2)\\).",
"role": "user"
}
] | 40 |
[
{
"content": "Evaluate the integral \\( \\int_A \\frac{dz}{4z^2+1} \\), where \\( A = \\{ z \\in \\mathbb{C} \\mid |z-i|=1 \\} \\), in the negative direction.",
"role": "user"
}
] | -\dfrac{\pi}{2} |
[
{
"content": "The teacher wrote the quadratic polynomial \\(x^{2} + 10x + 20\\) on the board. Each student, in turn, either increased or decreased by one either the coefficient of \\( x \\) or the constant term, but not both at the same time. As a result, the quadratic polynomial \\(x^{2} + 20x + 10\\) ended up... | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3.6 V |
[
{
"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... | <= 2.413 g/mL |
[
{
"content": "Find the value of $a$ such that the solution to the differential equation \\(9y''-12y'+4y=0\\) with initial conditions \\(y(0)=a>0\\) and \\(y'(0)=-1\\) changes from positive to negative. The solution is given by \\(y=ae^{\\frac{2}{3}t}+(-1-\\frac {2}{3}a)te^{\\frac{2}{3}t}\\). Determine the value... | \dfrac{3}{2} |
[
{
"content": "To transmit a positive integer less than 1000, the Networked Number Node offers two options.\n\nOption 1. Pay $\\$$d to send each digit d. Therefore, 987 would cost $\\$$9 + $\\$$8 + $\\$$7 = $\\$$24 to transmit.\n\nOption 2. Encode integer into binary (base 2) first, and then pay $\\$$d to send e... | 503 |
[
{
"content": "Is the Volterra-type integral operator \\( T(f)(x) = \\int_0^x f(t) \\, dt \\) compact when considered as an operator from \\( L^2([0,1]) \\) to \\( L^2([0,1]) \\)?",
"role": "user"
}
] | Yes |
[
{
"content": "In a convex decagon, no three diagonals intersect at a single point inside the decagon. Determine the number of line segments formed by the intersections of the diagonals.",
"role": "user"
}
] | 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... | 0.244097 |
[
{
"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... | 9.7% |
[
{
"content": "The prime numbers are added in order starting with $2$: $2$, $2 + 3$, $2 + 3 + 5$, and so on. How many of the first 12 such sums are also prime?",
"role": "user"
}
] | 5 |
[
{
"content": "Let $G$ be a finite group. Define a simple graph $X(G)$, called the commuting graph, where the vertices are the elements of $G$ and two distinct vertices $x, y$ are connected by an edge if and only if $xy = yx$. If $c$ is the chromatic number of $X(G)$, does $G$ contain an abelian subgroup of orde... | 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... | 9.12% |
[
{
"content": "Determine which number is greater: \\[ A=\\frac{2.00\\ldots04}{1.00\\ldots04^2+2.00\\ldots04}, \\text{ or } B=\\frac{2.00\\ldots02}{1.00\\ldots02^2+2.00\\ldots02}, \\] where each of the numbers above contains 1998 zeros.",
"role": "user"
}
] | B |
[
{
"content": "Determine the degree of the splitting field of the polynomial \\( f(x) = x^3 + 5x + 5 \\) over the finite field \\( \\mathbb{Z}_3 \\). What is \\([K : \\mathbb{Z}_3]\\), where \\(K\\) is the splitting field?",
"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... | 1254.92 kJ/mol |
[
{
"content": "Let \\( a_{1}=1, a_{2}=2 \\) and for all \\( n \\geq 2 \\), \\[ a_{n+1}=\\frac{2 n}{n+1} a_{n}-\\frac{n-1}{n+1} a_{n-1}. \\] It is known that \\( a_{n} > 2 + \\frac{2009}{2010} \\) for all \\( n \\geq m \\), where \\( m \\) is a positive integer. Find the least value of \\( m \\).",
"role": "u... | 4021 |
[
{
"content": "Consider the function $f(x)$ defined as the definite integral:\n\n$$f(x) = \\frac{1}{2}\\int_{0}^{x}(x-t)^2g(t) dt$$\n\nwhere $g(x)$ is a function such that $g(1)=6$. Find the third derivative, $f'''(1)$. Express your answer as a single number.",
"role": "user"
}
] | 6 |
[
{
"content": "Determine whether it is true that for any Pythagorean triple where the hypotenuse is an odd prime number, the hypotenuse can be expressed in the form \\(4n + 1\\) for some positive integer \\(n\\).",
"role": "user"
}
] | 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... | 5.397e-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... | $61,445.67 |
[
{
"content": "Determine whether the following statement is true or false: Suppose that $A_1,A_2,\\ldots ,A_m$ are $m$ linearly independent real $n\\times n$ matrices such that $A_iA_j=0$ for all $i\\neq j$. Then $m\\le n$. Provide a proof or a counterexample.",
"role": "user"
}
] | True |
[
{
"content": "Find the differential of the integral \\( \\int_0^t B_s \\, ds \\) with respect to time \\( t \\), where \\( B_s \\) is the standard Brownian motion at time \\( s \\).",
"role": "user"
}
] | B_t \, dt |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\rightarrow 0}\\frac{e^{x^2}-\\ln(1+x^2)-1}{\\cos2x+2x\\sin x-1} \\) using Maclaurin's series expansion.",
"role": "user"
}
] | 3 |
[
{
"content": "Determine whether the identity matrix \\(I\\) is an extreme point of the set \\(S = \\{A \\in M_{2\\times2} : \\|A\\|_1 \\leq 1\\}\\).",
"role": "user"
}
] | Yes |
[
{
"content": "Let $A$ be a set with $n$ elements. Let $F$ be the set of all functions from $A$ to $A$, and let $R$ be the relation on $F$ defined by $fRg$ if and only if $f(A)$ and $g(A)$ have the same number of elements. Determine the number of equivalence classes of the relation $R$. \\( \\text{Express your a... | n |
[
{
"content": "Calculate the flow of the vector field \\( F(x, y, z) = (0, 0, z) \\) through the portion of the plane \\( x + y + z = 1 \\) that is located in the first octant.",
"role": "user"
}
] | \dfrac{1}{6} |
[
{
"content": "Evaluate the series: \\[ \\sum_{n=2}^\\infty \\frac{\\prod_{k=1}^{n-1} (2k-1) }{2^n n!} = \\frac{1}{8} + \\frac{1}{16} + \\frac{5}{128} + \\frac{7}{256} + \\ldots \\]",
"role": "user"
}
] | \dfrac{1}{2} |
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