prompt list | ground_truth string |
|---|---|
[
{
"content": "Calculate the surface area of the surface formed by rotating the curve \\(x = \\sqrt{81 - y^2}\\) about the y-axis from \\(y = 0\\) to \\(y = 6\\).",
"role": "user"
}
] | 108\pi |
[
{
"content": "Determine whether the Riemann integral of a function over an interval always equals its Lebesgue integral over the same interval. Provide a justification for your answer.",
"role": "user"
}
] | No |
[
{
"content": "Given two jointly normal random variables $X$ and $Y$ with means of zero, variances $\\sigma^2_{X}$ and $\\sigma^2_{Y}$, and covariance $\\sigma_{XY}$, calculate the expected value of $Z = X \\cdot Y^2$. Find $E(Z)$. \\(\\text{Express your answer in terms of } \\sigma^2_{X}, \\sigma^2_{Y}, \\text{... | 0 |
[
{
"content": "What is the smallest positive integer that satisfies the congruence $4x \\equiv 13 \\pmod{27}$?",
"role": "user"
}
] | 10 |
[
{
"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... | 74 degrees |
[
{
"content": "Evaluate the limit:\n\n$$\\lim_{\\delta \\to 0} \\frac{1}{\\delta} \\int_{-\\infty}^{\\infty} \\varphi \\left(\\frac{x}{\\delta}\\right) g(x)dx$$\n\nwhere \\( \\varphi : \\mathbb{R} \\rightarrow \\mathbb{R} \\) is a continuous function such that \\( \\varphi(x) \\geq 0 \\) for all \\( x \\in \\mat... | 2g(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... | 40/3 |
[
{
"content": "If \\( x \\) and \\( y \\) are of the same sign, find the value of the expression:\n\\[\n\\frac{x^3}{2} \\csc^2 \\left(\\frac{1}{2}\\tan^{-1} \\frac{x}{y}\\right) + \\frac{y^3}{2} \\sec^2 \\left(\\frac{1}{2}\\tan^{-1} \\frac{y}{x} \\right)\n\\]\nChoose the correct option:\n(A) \\((x-y)(x^2+y^2)\\)... | C |
[
{
"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... | corrosive |
[
{
"content": "Find the remainder when \\( P = 2008^{2007} - 2008 \\) is divided by \\( Q = 2008^2 + 2009 \\).",
"role": "user"
}
] | 4032066 |
[
{
"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... | $280,000 |
[
{
"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... | $236,088.00 |
[
{
"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 years |
[
{
"content": "Compute the value of the infinite series \\( \\sum_{n=0}^{\\infty}{\\frac{n(n+2)}{3^n}} \\).",
"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... | 1.3e+2 mg/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... | 3.6% |
[
{
"content": "Each of the numbers $a_1,$ $a_2,$ $\\dots,$ $a_{95}$ is $\\pm 1.$ Find the smallest possible positive value of\n\\[\\sum_{1 \\le i < j \\le 95} a_i a_j.\\]",
"role": "user"
}
] | 13 |
[
{
"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... | $77.32 |
[
{
"content": "Evaluate the integral \\( \\int_0^{90} f(x) \\, dx \\), where the function \\( f(x) \\) is defined as follows:\n\\[\nf(x) = \\begin{cases} \n(x-3k)^2 & \\text{if } 3k-1 \\le x < 3k+1 \\\\\n1 & \\text{if } 3k+1 \\le x < 3k+2 \n\\end{cases}\n\\]\nfor \\( k \\in \\mathbb{Z} \\).",
"role": "user"
... | 50 |
[
{
"content": "Consider all $m \\times n$ matrices where each entry is either $0$ or $1$. Determine the number of such matrices in which the number of $1$s in each row and each column is even.",
"role": "user"
}
] | 2^{(m-1)(n-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... | 1.21 g |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.112 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... | 58.8% |
[
{
"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... | 3359.2 km/h |
[
{
"content": "Determine the number of fields, up to isomorphism, of order 6.",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to \\infty} \\sum \\limits_{k=1}^{n}\\frac{k^2 }{n^3 +n^2 +k}$$ without using integration.",
"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... | $74,387,500.00 |
[
{
"content": "Evaluate the limit: \\( \\lim_{n \\to \\infty} n^2 \\int_{1/(n+1)}^{1/n} \\frac{\\tan^{-1}(nx)}{\\sin^{-1}(nx)} \\, dx \\). Choose the correct answer from the following options: (a) 1, (b) 0, (c) -1, (d) \\frac{1}{2}.",
"role": "user"
}
] | D |
[
{
"content": "Does a set being dense imply that it is not enumerable?",
"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... | 72.8ms^-1 |
[
{
"content": "Determine whether the set \\( M = \\{(x,y,z) \\in \\mathbb{R}^3 : x^2 + y^2 = z^2, z \\neq 0\\} \\) is a submanifold of \\(\\mathbb{R}^3\\). If it is, find its dimension.",
"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... | Common Stock, $60 par: $960,000
Paid-In Capital from Sale of Treasury Stock: $38,000
Paid-In Capital in Excess of Par-Common Stock: $304,000
Total Paid-in-capital: $1,302,000
Retained Earnings: $518,000
Treasury Stock: ($18,000)
Total Stockholder's Equity: $1,802,000 |
[
{
"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... | 60,500 |
[
{
"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... | 3.23e-10 esu |
[
{
"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... | 1/3 L |
[
{
"content": "Let $Y$ be a closed subscheme of a scheme $X$, and let $i: Y \\rightarrow X$ be the inclusion morphism. Is the morphism $i$ quasi-compact? Justify 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... | 22.8 g |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 4.1e-1 m/s^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... | 1.37435 |
[
{
"content": "Let $U$ be the vector subspace of $\\mathbb{R}^5$ generated by $\\{(1,3,-3,-1,-4),(1,4,-1,-2,-2),(2,9,0,-5,-2)\\}$, and let $V$ be the vector subspace of $\\mathbb{R}^5$ generated by $\\{(1,6,2,-2,3),(2,8,-1,-6,-5),(1,3,-1,-5,-6)\\}$. Determine the dimension of the vector space $U \\cap V$. \\( \\... | 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... | $1,810.39 |
[
{
"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... | 2.86 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... | 8.4 years |
[
{
"content": "In a topological vector space $X$, let $K$ and $C$ be closed subsets such that $K \\cap C = \\emptyset$. Does there exist a neighborhood $V$ of $0$ such that $(K+V) \\cap (C+V) = \\emptyset$?",
"role": "user"
}
] | No |
[
{
"content": "In triangle $ABC$, let $P$ and $R$ be the feet of the perpendiculars from $A$ onto the external and internal bisectors of $\\angle ABC$, respectively; and let $Q$ and $S$ be the feet of the perpendiculars from $A$ onto the internal and external bisectors of $\\angle ACB$, respectively. Given that ... | 84 |
[
{
"content": "Let $f:\\mathbb{R}\\rightarrow \\mathbb{R}$ be a continuous function satisfying $|f(x)-f(y)|\\le L|x-y|$ for some constant $L<1$. Does the equation $f(x)=x$ have a solution?",
"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... | 26684 |
[
{
"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... | 378 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... | 6.23% |
[
{
"content": "Determine whether the statement \\( \\frac{f(x)}{g(x)} = \\frac{a+bx}{c+dx} + O(x^2) \\) is true, given that \\( f(x) = a + bx + O(x^2) \\) and \\( g(x) = c + dx + O(x^2) \\).",
"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... | 21.33*10^6 min |
[
{
"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... | 200 MeV |
[
{
"content": "A park is in the shape of a regular hexagon $2$ km on a side. Starting at a corner, Alice walks along the perimeter of the park for a distance of $5$ km. How many kilometers is she from her starting point?\n$\\textbf{(A)}\\ \\sqrt{13}\\qquad \\textbf{(B)}\\ \\sqrt{14}\\qquad \\textbf{(C)}\\ \\sqrt... | \sqrt{13} |
[
{
"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... | Prior Year: 0.2
Current Year: 0.17 |
[
{
"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... | 1.96e-6 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... | 4.56x10^{-3}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... | 10.5 imes 10^{-3} m^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... | $50.50 |
[
{
"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... | 7.82% |
[
{
"content": "Given two simple curves $M, N \\subset \\mathbb{R}^n$ with the same finite length $L[M] = L[N] < \\infty$, determine whether there exists a smooth bijection $F: M \\to N$ such that for every parametrized curve $\\gamma$ in $M$, the length condition $L[\\gamma] = L[F \\circ \\gamma]$ holds.",
"... | 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... | 635,525.00 |
[
{
"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... | 2.49 km/s |
[
{
"content": "In triangle \\( \\triangle ABC \\), the interior angles \\( \\angle A, \\angle B, \\angle C \\) satisfy the relationship \\( \\angle A = 3 \\angle B = 9 \\angle C \\). Calculate the value of the expression:\n\\[ \\cos A \\cdot \\cos B + \\cos B \\cdot \\cos C + \\cos C \\cdot \\cos A. \\]",
"r... | -\dfrac{1}{4} |
[
{
"content": "Calculate the limit: $$ \\lim _{(x,y,z)\\to (0,0,0) } \\frac{x^3+y^3+z^3}{x^2+y^2+z^2} $$ without using spherical coordinates.",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | +2 and +3 |
[
{
"content": "Evaluate the integral \\( \\int_{|z|=2} \\frac{1}{z^2-1} \\, dz \\) where \\( C \\) is the circle \\(|z|=2\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Find the degree of the intersection divisor of a homogeneous polynomial $F$ of degree $m$ with a curve $Y$ of degree $n$ in the projective plane.",
"role": "user"
}
] | mn |
[
{
"content": "Determine whether the following statement is true or false: For a topological space $X$ and a sequence of subsets $A_n \\subset X$ for all $n \\in \\mathbb{N}$, is it true that $\\bigcup\\limits_{n \\in \\mathbb{N}} \\overline{A_n} = \\overline{ \\bigcup\\limits_{n \\in \\mathbb{N}}A_n}$?",
"r... | False |
[
{
"content": "Evaluate the sum \\(\\sum_{d\\mid2020}{\\sigma(d)}\\), where \\(\\sigma(n)\\) is the sum of all divisors of \\(n\\).",
"role": "user"
}
] | 7931 |
[
{
"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... | 173130.5745 |
[
{
"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... | 11.85 yr |
[
{
"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... | 7.62e-12 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... | 2.85e-8 cm |
[
{
"content": "Find the best order 2 least squares approximation to the data points \\((0,0), (0.25,1), (0.5,0), (0.75,-1)\\) using the basis functions \\(1\\) and \\(\\cos(2\\pi t)\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{\\infty} \\frac{1}{x^6 + 1} \\,\\mathrm{d}x \\) using methods from real analysis, without employing complex analysis techniques.",
"role": "user"
}
] | \dfrac{\pi}{3} |
[
{
"content": "Find the range of values for \\( \\phi \\) that satisfy the inequality:\n\\[ -1.96 \\leq \\frac{6.4-10\\phi}{\\sqrt{1-\\phi^2}} \\leq 1.96 \\]",
"role": "user"
}
] | [0.467, 0.766] |
[
{
"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... | 405.69 nm/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... | 608K |
[
{
"content": "Solve for \\(x\\) in the equation: \\[ \\tan^{-1}(x+1) + \\tan^{-1}x + \\tan^{-1}(x-1) = \\tan^{-1}3. \\]",
"role": "user"
}
] | 5 - \sqrt{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... | 0.25 mol |
[
{
"content": "Determine the values of $x$ for which the series \\( \\sum_{n=1}^{\\infty} \\left(\\frac{x(x+n)}{n}\\right)^n \\) converges.",
"role": "user"
}
] | (-1, 1) |
[
{
"content": "Evaluate the limit: \\( \\lim_{h\\rightarrow0^-}\\frac{e^{-1/|h|}}{h^2} \\)",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.076 rad/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... | 2263.696 watts |
[
{
"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. Depolarization: Sodium ions move into the cell, making the membrane potential less negative.
2. Threshold: The membrane potential reaches a critical level, triggering an action potential.
3. Repolarization: Potassium ions move out of the cell, restoring the negative membrane potential.
4. Refractory Period: Sodium c... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 1} \\left( \\frac{2017}{1-x^{2017}} - \\frac{2018}{1-x^{2018}} \\right) \\]",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "Determine whether a continuous function $f:[a,b] \\to \\mathbb{R}$ that satisfies $\\int_a^b f(x)g(x)dx = 0$ for all continuous functions $g:[a,b] \\to \\mathbb{R}$ with $g(a) = g(b) = 0$ must necessarily be identically zero.",
"role": "user"
}
] | 0 |
[
{
"content": "Is the one-point compactification $Y = X \\cup \\{\\infty\\}$ of a topological space $X$ compact? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "In a circle with center $O$ and perpendicular diameters $AC$ and $BD$, a string $AE$ is drawn such that $E$ lies on arc $BC$. The line $AE$ intersects $BD$ at $M$ such that $BM = MO$. Calculate the length of $ON$, where $N$ is the intersection point of $ED$ and $AC$, given that $OA = 12$. Provide ... | 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... | 12.641 |
[
{
"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... | 67.97 ft/s |
[
{
"content": "Calculate the expected value of \\(\\sum_{i=1}^{n}(X_i - \\bar{X_n})^2\\) for independent random variables \\(X_1, X_2, \\ldots, X_n\\) with mean \\(\\mu\\) and variance \\(\\sigma^2\\), where the variables are not necessarily identically distributed.",
"role": "user"
}
] | (n-1)\sigma^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... | 3.84 m/s |
[
{
"content": "Find the value of the Lagrange multiplier for the optimization problem: \\( \\begin{array}{ll} \\text{maximize} & x^2+y^2 \\\\ \\text{subject to} & y^2=9-\\frac{9}{25}x^2 \\end{array} \\)",
"role": "user"
}
] | \dfrac{25}{9} |
[
{
"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... | 3.86 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... | 0.13 m |
[
{
"content": "Compute the induced group homomorphism \\((g_n)_* : \\pi_1(S^1,1) \\to \\pi_1(S^1,1)\\) for the map \\(g_n: S^1 \\to S^1\\) defined by \\(g_n(z) = z^{2n}\\), with the identification of \\(\\pi_1(S^1,1)\\) with \\(\\mathbb{Z}\\).",
"role": "user"
}
] | 2n |
[
{
"content": "For a bounded linear operator on a Hilbert space, does the range of the operator restricted to the orthogonal complement of its kernel have the same dimension as the orthogonal complement of its kernel? Answer with 'Yes' or 'No'.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\(P(x) = x^4 + ax^3 + bx^2 + cx + d\\) be a polynomial where \\(a, b, c, d\\) are integers. Given that \\(P(x)\\) has remainders 24, -6, 4, -6, and 24 when divided by \\(x-2012\\), \\(x-2013\\), \\(x-2014\\), \\(x-2015\\), and \\(x-2016\\) respectively, find the remainder when \\(P(x)\\) is d... | 274 |
[
{
"content": "Find the sum of all possible values of \\(\\theta\\) in the interval \\((0, 2\\pi)\\) such that the quadratic equation \\(x^2+(2-\\tan\\theta)x-(1+\\tan\\theta)=0\\) has two integral roots.",
"role": "user"
}
] | 4\pi |
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