prompt list | ground_truth string |
|---|---|
[
{
"content": "Assume that $A$ and $B$ are two $C^{*}$ algebras such that their minimal tensor product is a $Z^{*}$ algebra. Is it true that $A$ or $B$ is a $Z^{*}$ algebra?",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( d \\) be the greatest common divisor of \\( 2^{30^{10}}-2 \\) and \\( 2^{30^{45}}-2 \\). Find the remainder when \\( d \\) is divided by 2013.",
"role": "user"
}
] | 2012 |
[
{
"content": "Determine the number of ordered pairs of nonempty sets \\( \\mathcal{A} \\) and \\( \\mathcal{B} \\) such that \\( \\mathcal{A} \\cup \\mathcal{B} = \\{1,2,\\ldots,10\\}\\), \\( \\mathcal{A} \\cap \\mathcal{B} = \\emptyset\\), the number of elements of \\( \\mathcal{A} \\) is an element of \\( \\m... | 186 |
[
{
"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... | Magnitude: 47.8 m/s, Direction: -13 degrees |
[
{
"content": "Determine the sequence $s(k,n)$, which represents the number of sides of the intersection of a unit-radius regular polygon $P_k$ with $k$ sides and a rotating unit-radius regular polygon $P_n$ with $n \\ge k$ sides, as the angle of rotation $\\theta$ varies from $0$ to $2\\pi$. Provide the sequenc... | 2k |
[
{
"content": "Determine whether the product of two non-normal subgroups of a finite group can have an order that divides the order of the group. Answer with 'Yes' or 'No'.",
"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... | 8.05e-19 |
[
{
"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... | $2,799.91 |
[
{
"content": "Solve the equation: \\[ 4^{x} - 3^{x - \\frac{1}{2}} = 3^{x + \\frac{1}{2}} - 2^{2x - 1} \\]",
"role": "user"
}
] | \dfrac{3}{2} |
[
{
"content": "Let $n > 2$ and $\\sigma^2 > 0$. In a math exam with $n$ participants, the scores $X_1, X_2, \\ldots, X_n$ follow a normal distribution with mean $\\mu_X$ and variance $\\sigma^2$. The average score is given by \\( \\overline{X} = \\frac{1}{n} \\sum_{i=1}^n X_i \\). For each $i = 1, \\ldots, n$, f... | 0 |
[
{
"content": "How many solutions in positive integers are there to the equation \\(x_1 \\cdot x_2 \\cdot x_3 \\cdot x_4 = 2^{20} \\cdot 13^{13}\\)? Assume that \\(x_1, x_2, x_3,\\) and \\(x_4\\) are distinguishable.",
"role": "user"
}
] | 991760 |
[
{
"content": "Calculate the sum of the infinite series: \\[ \\sum_{n=0}^\\infty \\frac{(n+1)^2}{2^n} \\]",
"role": "user"
}
] | 12 |
[
{
"content": "Determine the number of 8-tuples in the set $[4]^8 = \\{1, 2, 3, 4\\}^8$ such that each number from $[4]$ appears exactly twice and no number occupies its corresponding index (e.g., $i$ does not appear in the $i$th position for all $i \\in [4]$). Is this number equal to $8!/16$ minus the number of... | 864 |
[
{
"content": "Calculate the determinant of the matrix $A$ given by:\n$$A = \\begin{bmatrix} a_1b_1+1 &a_1b_2 + 1&\\cdots&a_1b_n + 1\\\\ a_2b_1+1 &a_2b_2 + 1&\\cdots&a_2b_n + 1\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ a_nb_1+1 &a_nb_2 + 1&\\cdots&a_nb_n + 1 \\end{bmatrix}$$",
"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... | E(x, y) = (9y)i + (9x - 6y)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... | <img>/cimages/multimages/16/2-ethoxypentane_structural_formula4976515697038679131.png</img> |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 13.04 |
[
{
"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... | 0.9 |
[
{
"content": "Determine the number of terms, $n$, required in the series expansion of $\\pi = 4\\arctan(1/2) + 4\\arctan(1/3)$ such that the error is less than $10^{-10}$. Provide your answer as an integer.",
"role": "user"
}
] | 15 |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} \\int_{0}^{1} x^n e^{\\sqrt{x}} \\, dx \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the Maclaurin series for the function \\( g(x) = \\int_0^x \\frac{1}{1+t^3} \\, dt \\) and determine its radius of convergence.",
"role": "user"
}
] | 1 |
[
{
"content": "Determine whether the sequence \\(a_n= \\frac{7^n + 6^n -n^{100}}{(7.1)^n-7^n+n^{101}}\\) converges, and if so, find its limit.",
"role": "user"
}
] | 0 |
[
{
"content": "Given the polynomial \\( f(x) = x^4 + ax^3 + 3x^2 + bx + 1 \\), where \\( a, b \\in \\mathbb{R} \\), if \\( f(x) \\geq 0 \\) holds for all \\( x \\in \\mathbb{R} \\), find the maximum value of \\( a^2 + b^2 \\).",
"role": "user"
}
] | 40 |
[
{
"content": "Evaluate \\( \\lim_{x \\to 1^-} \\ln x \\cdot \\ln(1-x) \\) using the change of variable method.",
"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... | Actual Costs Incurred Flexible Budget Variance
Activity level (in units) 5,250 5,250 0
Variable costs
Indirect materials $24,182 $23,782.50 $399.50 U
Utilities $22,356 $22,995 $639 F
Fixed costs
Administration $63,450 $65,500 $2,050 F
Rent $65,317 $63,904 $1,413 U |
[
{
"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... | -$5000 |
[
{
"content": "Determine whether the following expression has a limit as \\( k \\) approaches infinity, given that \\( 2^mf\\notin \\mathbb{Z} \\) for every \\( m\\in\\mathbb{Z} \\): \\[ \\lim_{k\\rightarrow \\infty}\\frac{\\sin^2(2^{k}\\pi f)}{\\sin^2 (2^{k-1}\\pi f)}\\prod_{j=1}^{k} \\cos^2(2^{k-j}\\pi f) \\]"... | 0 |
[
{
"content": "Determine the radius of convergence for the Taylor series of the function \\( f(z) = \\frac{1}{1+e^z} \\) centered at \\( z = 0 \\).",
"role": "user"
}
] | \pi |
[
{
"content": "Let $f:\\mathbb{C} \\rightarrow \\mathbb{C}$ be an analytic function such that $|f(z)| \\in \\mathbb{Z}$ for all $z \\in \\mathbb{C}$. Is $f$ necessarily a constant function?",
"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... | test statistics = 7.2736, degree of freedom = 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... | 10.45% |
[
{
"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... | 303.968 bar |
[
{
"content": "Determine if the function \\[ f(x)= \\begin{cases} x^2\\cos\\left(\\frac{\\pi}{x^2}\\right) &\\quad\\text{if} \\hspace{5mm} x\\in (0,1]\\\\ 0 &\\quad\\text{if} \\hspace{5mm} \\text{x = 0} \\end{cases} \\] is differentiable at the point \\(x = 0\\) using the limit definition of the derivative.",
... | 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... | 0.51515152 |
[
{
"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... | 1.08 |
[
{
"content": "Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 3.$ Find the maximum value of\n\\[(xy + z)(xz + y).\\]",
"role": "user"
}
] | 4 |
[
{
"content": "Find the minimum area of a triangle that can completely contain a unit square.",
"role": "user"
}
] | 2 |
[
{
"content": "Suppose $F(x)$ is a polynomial with real coefficients such that $F(y) \\neq 0$ for all $y \\geq 0$. Determine whether there exists a polynomial $T(x)$ with real coefficients such that $T(x) \\cdot F(x)$ has all its coefficients non-negative.",
"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... | The tangential acceleration is {eq}\frac{4}{\sqrt 5 }{/eq} and the normal acceleration is {eq}\frac{2}{\sqrt 5 }{/eq} |
[
{
"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... | 0.16 m |
[
{
"content": "Let \\(\\omega\\) be a primitive root modulo a prime \\(p\\). Determine the value of the Legendre symbol \\(\\left(\\frac{\\omega}{p}\\right)\\).",
"role": "user"
}
] | -1 |
[
{
"content": "Determine whether the derivative of the function \\( f(x) = \\frac{\\sin x}{|x| + \\cos x} \\) at \\( x = 0 \\) is defined.",
"role": "user"
}
] | 1 |
[
{
"content": "Find the smallest positive integer $n$ such that $n^8 + (n + 1)^8$ is a composite number.",
"role": "user"
}
] | 2 |
[
{
"content": "In triangle \\( \\triangle ABC \\), given that \\( \\overrightarrow{AB} \\cdot \\overrightarrow{AC} = 7 \\) and \\( |\\overrightarrow{AB} - \\overrightarrow{AC}| = 6 \\), determine the maximum possible area of \\( \\triangle ABC \\).",
"role": "user"
}
] | 12 |
[
{
"content": "In a group \\((G, \\cdot)\\) with order \\(n\\), where \\(p\\) is a prime divisor of \\(n\\), consider the set \\(M\\) defined as:\n\\[M=\\left\\{(g_1,g_2,\\dots,g_p)\\in G^p:g_1\\cdot g_2\\cdots g_p=e_G\\right\\}\\]\nDetermine the cardinality of the set \\(M\\).",
"role": "user"
}
] | n^{p-1} |
[
{
"content": "Suppose there are 11 people seated around a circular table. When everyone stands up and randomly chooses a new seat, what is the expected number of pairs that end up swapping places (i.e., person A takes person B's seat and person B takes A's seat)?",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Let $K$ be a number field, $\\mathcal{O}_K$ the ring of integers, and $\\mathcal{O}_K^\\times$ the unit group of $\\mathcal{O}_K$. Using Dirichlet's theorem, we know that $\\mathcal{O}_K^\\times$ is a finitely generated group. Is the quotient group $\\mathcal{O}_K^\\times / (\\mathcal{O}_K^\\times... | Yes |
[
{
"content": "Let $f$ be a real-valued function and $x_0$ a real number. If $f(x)$ does not have a finite limit as $x$ approaches $x_0$, must there exist a sequence $\\{z_n\\}_{n=1}^{\\infty}$ such that $z_n$ converges to $x_0$ and the sequence $\\{f(z_n)\\}_{n=1}^{\\infty}$ does not have a finite limit? Answer... | Yes |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to +\\infty} \\frac{n}{\\ln(n)}(\\sqrt[n]{n}-1) \\]",
"role": "user"
}
] | 1 |
[
{
"content": "Evaluate the integral of the Fourier series over the interval \\(-\\pi, \\pi\\):\n\n$$\\int_{-\\pi}^{\\pi} \\left( \\frac{3}{\\pi}\\sum_{n = 0}^{\\infty} \\frac{\\sin\\left((2n + 1)x\\right)}{2n + 1} \\right) \\, dx$$",
"role": "user"
}
] | 0 |
[
{
"content": "Phillip flips an unfair coin eight times. This coin is twice as likely to come up heads as tails. How many times as likely is Phillip to get exactly three heads than exactly two heads?",
"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... | 49.5 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... | Date Accounts Debit Credit
2009
Dec. 13 Notes Receivable 84,000
_____Sales 84,000
Dec 31 Interest Receivable ($84,000 x 9% x 18 days / 360 days) 378
_____Interest Income 378
Dec 31 Income Summary 378
_____Interest Income 378
2010
Mar. 12 Cash 85,890
_____Interest Receivable 378
_____Interest Income... |
[
{
"content": "Does there exist a metric on a compact, connected Lie group $G$ that induces the group's underlying topology, making $G$ compact and connected with respect to this metric?",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit: \\( \\lim_{x \\rightarrow 0}(1-\\cos x)^{\\tan x} \\)",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the asymptotic behavior of the recurrence relation \\( T(n) = T(n-1) + (n-1)! \\cdot n^k \\) in terms of \\( \\Theta \\) or \\( O \\) notation, where \\( k \\) is usually 0, 1, or 2.",
"role": "user"
}
] | \Theta(n! \cdot n^{k-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... | 5.64e+3 J |
[
{
"content": "Determine the point-wise limit of the sequence of functions \\( \\{f_n(x)\\} \\) defined on the interval \\([0,1]\\) by:\n\n\\[\nf_n(x) = \\begin{cases} 0 & \\text{if } x=0 \\\\ n^2x & \\text{if } x \\in [0,\\frac{1}{n}] \\\\ -n^2x+n^2 & \\text{if } x \\in (\\frac{1}{n},\\frac{2}{n}] \\\\ 0 & \\te... | 0 |
[
{
"content": "Compute the triple integral \\( \\int_D f(x, y, z) \\, dV \\) where \\( D \\) is the region in the first octant \\((0, \\infty)^3\\) between the cone \\( z^2 = (x - 1)^2 + y^2 \\) and the sphere \\( x^2 + y^2 + z^2 = 1 \\). The function \\( f : D \\to \\mathbb{R} \\) is given by \\( f(x, y, z) = z... | \dfrac{2}{75} |
[
{
"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 profit for Refined Oil is -$5,220, and the profit for Top Quality Oil is $61,740 |
[
{
"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... | Use AgNO3 to test for Cl- (precipitate indicates CaCl2), Ca(NO3)2 to test for SO4^2- (precipitate indicates CuSO4). If neither forms a precipitate, it's likely NaNO3. |
[
{
"content": "A sequence of three real numbers forms an arithmetic progression with a first term of 9. If 2 is added to the second term and 20 is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term of the geometric progression... | 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... | \vec{v}=0.8\vec{e_x}+0.6\vec{e_y} |
[
{
"content": "Determine whether the evaluation homomorphism from the affine space $A^n$ over $\\mathbb{C}$ to the set of ring homomorphisms from the polynomial ring $\\mathbb{C}[X_1, \\ldots, X_n]$ to $\\mathbb{C}$ is surjective.",
"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... | 6.5% |
[
{
"content": "A 98 x 98 chessboard is colored alternately black and white. A move consists of selecting a rectangular subset of the squares (with boundaries parallel to the sides of the board) and changing their color. Determine the smallest number of moves required to make all the squares black.",
"role": ... | 98 |
[
{
"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... | Date|Cash|Office Supplies|Accounts Receivables|Equipment|Accounts Payable|Notes Payable|Common Stock|Service Revenue|Rent Expense|Advertising Expense|Dividends|Salaries Expense|Utilities Expense
---|---|---|---|---|---|---|---|---|---|---|---|---|---
May 1|22,350|0|0|0|0|0|22,350|0|0|0|0|0|0
May 2|-894|0|0|0|0|0|0|0|89... |
[
{
"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... | $14,391.67 |
[
{
"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... | 1.476 ft lb |
[
{
"content": "Compute \\( \\lim_{x\\to 0^{+}}x^{x^{x}} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Compute the expected value \\( E[(be^{-X}-a)^+] \\) where \\( X \\sim \\mathcal{N}(0,1) \\), and \\( a > 0 \\) and \\( b < 0 \\) are real numbers.",
"role": "user"
}
] | 0 |
[
{
"content": "Find the value of \\( u(1/2, 1/2) \\) where \\( u(x,t) \\) is the solution to the partial differential equation \\( u_{tt} - u_{xx} = 1 \\) with initial conditions \\( u(x,0) = 0 \\) and \\( u_t(x,0) = 0 \\) for \\( x \\in \\mathbb{R} \\) and \\( t > 0 \\).",
"role": "user"
}
] | \dfrac{1}{8} |
[
{
"content": "In a bipartite graph $G$ with $K$ left nodes and $M$ right nodes, each of the $KM$ potential edges is present independently with probability $p$. As $K$ and $M$ approach infinity, what is the expected number of left nodes connected to exactly one right node?",
"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... | 6006 |
[
{
"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... | $525 |
[
{
"content": "Determine whether the limit \\( \\lim_{n\\rightarrow\\infty}\\sum_{k=1}^n\\ln\\left(\\frac{k(1+\\alpha)}{n-k+1+(n-k)\\alpha}\\right) \\) exists.",
"role": "user"
}
] | 0 |
[
{
"content": "Player A has £1 and Player B has £99. They flip a coin, and the loser pays the other £1. What is the expected number of games they will play before one player is bankrupt?",
"role": "user"
}
] | 99 |
[
{
"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... | 258.36 g |
[
{
"content": "Determine whether the sequence \\( \\frac{1}{\\sum_{i=n}^\\infty i^{-2}} \\) converges as \\( n \\to \\infty \\). If it converges, find the limit.",
"role": "user"
}
] | \infty |
[
{
"content": "What is the value of $\\aleph_1^{\\aleph_0}$?",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Determine the number of positive integers less than or equal to $6 \\cdot 7 \\cdot 8 \\cdot 9$ that satisfy the following system of congruences: \\[ \\begin{align*} m &\\equiv 5 \\pmod{6}, \\\\ m &\\equiv 4 \\pmod{7}, \\\\ m &\\equiv 3 \\pmod{8}, \\\\ m &\\equiv 3 \\pmod{9}. \\end{align*} \\]",
... | 0 |
[
{
"content": "In the two regular tetrahedra \\(A-OBC\\) and \\(D-OBC\\) with coinciding bases, \\(M\\) and \\(N\\) are the centroids of \\(\\triangle ADC\\) and \\(\\triangle BDC\\) respectively. Let \\(\\overrightarrow{OA}=\\boldsymbol{a}, \\overrightarrow{OB}=\\boldsymbol{b}, \\overrightarrow{OC}=\\boldsymbol... | 439 |
[
{
"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... | 2.003e8 m/s |
[
{
"content": "Evaluate the improper integral: $$\\int_{\\mathbb{R}} \\frac{\\sin^{4}(x)}{x^2}dx$$.",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Is it possible to construct an uncountable chain of subsets of the natural numbers?",
"role": "user"
}
] | Yes |
[
{
"content": "Given the function \\( f(x,y) = \\frac{y\\ln(1+x^2 + ay^2) } {x^2 + 2y^2} \\) for \\((x,y)\\neq (0,0)\\), and \\(f(0,0)=0\\), with the additional information that \\( f_y (0,0) = 2 \\), find the value of \\(a\\).",
"role": "user"
}
] | 4 |
[
{
"content": "Calculate the surface area of the solid formed by rotating the curve \\(x^{2/3} + y^{2/3} = 1\\) around the \\(y\\)-axis from \\(y = 0\\) to \\(y = 1\\).",
"role": "user"
}
] | \dfrac{6\pi}{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... | 12.1 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... | 8,844.22 |
[
{
"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... | Variable cost = $0.02 per unit
Fixed cost = $2,500 |
[
{
"content": "An ordered vertical stack of \\( n \\) books is on a desk. Each day, one book is picked uniformly at random from the stack and placed on top. What is the expected number of days for the books to return to their original order?",
"role": "user"
}
] | 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... | 16.65% |
[
{
"content": "In tetrahedron \\(ABCD\\), points \\(M, N, K\\) are located on edges \\(AD, AB, BC\\) respectively, such that \\(AM:MD = 2:3\\), \\(BN:AN = 1:2\\), and \\(BK = KC\\). Determine the ratio in which the plane passing through points \\(M, N, K\\) divides edge \\(CD\\).",
"role": "user"
}
] | 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... | -7\sqrt{240}/240 ft/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... | 1.1 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... | 26.0 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... | $201.95 |
[
{
"content": "Determine the group homomorphisms from the multiplicative group of non-zero complex numbers, \\( \\mathbb{C}^\\ast \\), to the additive group of integers, \\( \\mathbb{Z} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "A function \\(f\\) is defined by \\(f(z) = (4 + i)z^2 + az + b\\) for all complex numbers \\(z\\), where \\(a\\) and \\(b\\) are complex numbers. If \\(f(1)\\) and \\(f(i)\\) are both real, find the smallest possible value of \\(|a| + |b|\\).",
"role": "user"
}
] | \sqrt{2} |
[
{
"content": "Determine the maximum possible number of discontinuities for a function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that has both a left limit and a right limit at each point of \\( \\mathbb{R} \\).",
"role": "user"
}
] | \aleph_0 |
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