prompt list | ground_truth string |
|---|---|
[
{
"content": "Suppose $u \\in \\mathcal{S}'(\\mathbb{R}^n)$, where $\\mathcal{S}'(\\mathbb{R}^n)$ denotes the space of tempered distributions. Determine whether there exists an integer $m \\ge 0$ and a constant $C > 0$ such that for all Schwartz functions $\\phi \\in \\mathcal{S}(\\mathbb{R}^n)$, the inequality... | Yes |
[
{
"content": "Given that \\( \\int_{0}^{9} f(x) \\, dx = 4 \\), find the value of \\( \\int_{0}^{3} x f(x^2) \\, dx \\).",
"role": "user"
}
] | 2 |
[
{
"content": "Determine the radius of the largest circle that can be inscribed on an ellipsoid with semi-axes lengths of \\( a > b > c \\).",
"role": "user"
}
] | b |
[
{
"content": "Given that $f(x) + 3x^2 = 2f(1-x)$ and $\\lim _{x\\to 1}f(x) =7$, find $\\lim _{x\\to 0} f(x)$. \\(\\text{The correct answer is } \\lim_{x\\to 0} f(x)=14.\\)",
"role": "user"
}
] | 14 |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\sum_{i=1}^n \\frac{1}{\\sqrt i}$$",
"role": "user"
}
] | \infty |
[
{
"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 26,000
Common Stock 10,000
Additional Paid-in Capital 16,000 |
[
{
"content": "Use the Divergence Theorem to evaluate \\( \\iint_S \\mathbf{G} \\cdot d\\mathbf{S} \\), where \\( \\mathbf{G}(x, y, z) = \\langle y^2z, x^3-z^2, xz+y^3 \\rangle \\) and \\( S \\) is the boundary of the solid enclosed by the paraboloids \\( z = 2x^2+2y^2 \\) and \\( z = 10 - x^2 - y^2 \\), oriente... | 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... | 11.11 |
[
{
"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... | 444000 |
[
{
"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... | bacterial nucleic acid |
[
{
"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... | 0.1 mol/L |
[
{
"content": "If $(x + y)^2 = 1$ and $xy = -4$, what is the value of $x^2 + y^2$?",
"role": "user"
}
] | None |
[
{
"content": "What is the sum of the two values which will make the following expression undefined? $$\\frac{x-3}{2x^2-8x+7}$$",
"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... | 32,000 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... | 92.125 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... | 0.548 earth years |
[
{
"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. 6:
Salaries Payable - $1,000
Salaries Expense - $600
Cash - $1,600
Dec. 8:
Cash - $2,000
Accounts Receivable - $2,000
Dec. 10:
Accounts Receivable - $6,800
Sales - $6,800
Cost of goods sold - $4,200
Inventory - $4,200
Dec. 13:
Inventory - $9,000
Accounts Payable - $9,000
Dec. 15:
Supplies - $1,600
Cash - $1,600
De... |
[
{
"content": "Given that $P \\left( \\left| \\frac{S_n}{n} - p \\right| \\geq \\epsilon \\right) = e^{-\\frac{n\\epsilon^2}{4}}$, determine the probability that there exists a number $n_0$ such that the event $A_n = \\left\\{ \\left| \\frac{S_n}{n} - p \\right| \\leq \\epsilon \\right\\}$ occurs for every $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... | 0.00248 m^3/sec |
[
{
"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... | 15,464.28571 |
[
{
"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... | $3,455.37 |
[
{
"content": "For which values of \\(a\\) is the matrix \\(\\begin{pmatrix} 4 & 0 & 0 \\\\ 4 & 4 & a \\\\ 4 & 4 & 4 \\end{pmatrix}\\) diagonalizable over \\(\\mathbb{R}\\)?",
"role": "user"
}
] | a > 0 |
[
{
"content": "Determine whether there exist two integer polynomials $P(x)$ and $Q(x)$ that touch at a point where both the x-coordinate and y-coordinate are irrational.",
"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... | 270.6 mL |
[
{
"content": "Let $K/\\mathbb{Q}$ be a number field, and suppose a prime $p \\in \\mathbb{Z}$ factors in $\\mathcal{O}_K$ as $\\prod_{i=1}^r \\mathfrak{p}_i^{e_i}$. Given the identity $[K:\\mathbb{Q}]=\\sum_{i=1}^r e_if_i$, determine if it is true that for each $i$, the local extension degree satisfies $[K_{\\m... | Yes |
[
{
"content": "Compute the limit of the sequence: $$ \\lim_{n\\to\\infty} \\left(\\frac{3n^2-n+1}{2n^2+n+1}\\right)^{\\large \\frac{n^2}{1-n}} $$",
"role": "user"
}
] | 0 |
[
{
"content": "Find the minimum value of the expression \\( a + b + \\frac{1}{ab} \\) for positive real numbers \\( a \\) and \\( b \\) such that \\( a^2 + b^2 = 1 \\).",
"role": "user"
}
] | 2 + \sqrt{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... | 87 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... | -119.1463 N.m |
[
{
"content": "In a Hausdorff locally compact space $X$, determine whether every neighborhood of a point $x$ contains a compact neighborhood of $x$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( f \\) be a function that is continuous on \\( \\mathbb{R} \\) and differentiable for all \\( x \\neq 0 \\). Given that \\( \\lim_{x \\rightarrow 0} f'(x) = 1 \\), prove that \\( f'(0) \\) exists and is equal to 1.",
"role": "user"
}
] | 1 |
[
{
"content": "Six distinct positive integers \\( a, b, c, d, e, f \\) are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has 10 prime numbers while Jill claims that she has 9 prime numbers among the sums. Determine who has the correct claim.",
"role": "user"
}
... | Jill |
[
{
"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... | 94.12 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 66.4 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... | \theta = {\cos ^{ - 1}}\left( {\dfrac{{g{T^2}}}{{4{\pi ^2}L}}} \right) |
[
{
"content": "Determine whether the supremum of the expression \\( \\min_{n\\ge 1} \\frac{1-x}{(1-xy)x^{n-1}(1-y^n)} \\) over all \\( x, y \\in (0,1) \\) is finite or infinite.",
"role": "user"
}
] | \infty |
[
{
"content": "If two random variables have identical distributions and the same mathematical operation is performed on both, will the resulting random variables also have identical distributions?",
"role": "user"
}
] | Yes |
[
{
"content": "Simplify \\( \\frac{2^{2017}+1}{3 \\cdot 2^{2017}} \\) to \\( \\frac{n}{m} \\) where \\( n \\) and \\( m \\) are coprime. Find the remainder when \\( m+n \\) is divided by 1000.",
"role": "user"
}
] | 763 |
[
{
"content": "The circles whose equations are $x^2 + y^2 - 4x + 2y - 11 = 0$ and $x^2 + y^2 - 14x + 12y + 60 = 0$ intersect in the points $A$ and $B.$ Compute the slope of $\\overline{AB}.$",
"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... | 36.23 kJ/kg |
[
{
"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... | 7.58% |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{\\left( x^2 \\cos^2 x - \\sin^2 x \\right)\\left( x^3 - \\sin^3 x \\right)}{\\left( e^x + e^{-x} - x^2 - 2 \\right)^2 \\sin x} \\]",
"role": "user"
}
] | -48 |
[
{
"content": "Provide an example of a complete metric space that is not compact.",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "A student must choose a program of four courses from a list of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?",
"role": "user"
}
] | 9 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to\\infty} p(x)^{\\frac{1}{\\ln x}} \\] where $p(x)$ is a polynomial of degree $n$ with leading coefficient $a_n \\neq 0$.",
"role": "user"
}
] | e^n |
[
{
"content": "In $\\triangle{ABC}$, $\\angle ABC=120^\\circ,AB=3$ and $BC=4$. If perpendiculars constructed to $\\overline{AB}$ at $A$ and to $\\overline{BC}$ at $C$ meet at $D$, then $CD=$\n$\\text{(A) } 3\\quad \\text{(B) } \\frac{8}{\\sqrt{3}}\\quad \\text{(C) } 5\\quad \\text{(D) } \\frac{11}{2}\\quad \\tex... | \frac{10}{\sqrt{3}} |
[
{
"content": "Evaluate the limit \\( \\lim_{n \\to \\infty} \\int_{0}^{\\frac{\\pi}{2}} e^{-n \\sin x} \\, dx \\) using the Dominated Convergence Theorem. Determine if the integral evaluates to 0.",
"role": "user"
}
] | 0 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.98c |
[
{
"content": "Find the maximum value of the expression \\(3x^2 + 2\\sqrt{2}xy\\) given the constraint \\(x^4 + y^4 = 1\\), where both \\(x\\) and \\(y\\) are positive.",
"role": "user"
}
] | 2\sqrt{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... | 0.42 m |
[
{
"content": "Calculate the limit: $$\\lim_{n \\to \\infty} \\int^{n}_{-n} \\left(\\cos\\left(\\frac{x}{n}\\right)\\right)^{n^{2}}\\,dx .$$ Use the inequality $$r^{k} \\leq \\exp(-k(1-r)),$$ where $0\\leq r \\leq 1$ and $k\\in \\mathbb{N}$, to determine the value of the limit.",
"role": "user"
}
] | \sqrt{2\pi} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $53.72 |
[
{
"content": "Let \\( a, b, \\) and \\( c \\) be positive integers such that \\( a^{2} = 2b^{3} = 3c^{5} \\). Determine the minimum possible number of factors of \\( abc \\), including 1 and \\( abc \\).",
"role": "user"
}
] | 77 |
[
{
"content": "Evaluate the limit: $$\\lim_{p\\to\\infty}\\frac{\\int_{0}^{p}x^{m+1}e^{x}dx}{\\int_{0}^{p}x^{m}e^{x}dx},$$ where \\(m\\in \\mathbb{N}\\).",
"role": "user"
}
] | \infty |
[
{
"content": "What is the probability of selecting $r \\leq n$ vectors $a_i \\in \\mathbb{R}^n$ such that they are linearly independent?",
"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... | The total revenue will decrease by 0.01%. |
[
{
"content": "Find the least possible degree of the polynomial \\( f(x) \\) in the expression \\( \\frac{d^{100}}{dx^{100}}\\left(\\frac{p(x)}{x^3-x}\\right) = \\frac{f(x)}{g(x)} \\), where \\( p(x) \\) is a polynomial that does not have the factor \\( x^3-x \\).",
"role": "user"
}
] | 200 |
[
{
"content": "Alice and Bob are on a 1 km long closed and dark train. They start at random positions and can only see their own and each other's information on their iPhone 133, which includes their facing direction, total walking distance, and whether they are at the front or end of the train. They can discuss... | 1 |
[
{
"content": "Consider the exact sequence of modules: \\(0 \\rightarrow A \\rightarrow B \\rightarrow C \\rightarrow 0\\), with maps \\(f: A \\rightarrow B\\) and \\(g: B \\rightarrow C\\). Let \\(Q\\) and \\(P\\) be submodules of \\(B\\). Determine whether the following statement is true: If \\(g(Q) = g(P)\\) ... | False |
[
{
"content": "Find the smallest integer \\( n \\) such that any subset of \\( n \\) elements from the set \\( M = \\{1, 2, \\ldots, 100\\} \\) contains four elements that are pairwise coprime.",
"role": "user"
}
] | 75 |
[
{
"content": "Evaluate the integral of $w^{-2}$ over the triangle with vertices $(0, 0)$, $(1, 0)$, and $(0, 1)$ using Cauchy's Integral Formula.",
"role": "user"
}
] | 0 |
[
{
"content": "Given that \\(a > b > c > 0\\) and \\(7a + 8b = 15c + 24\\sqrt[3]{abc}\\), find the maximum value of the expression:\n\\[ \\frac{a-b}{c} + \\frac{b-c}{a} + \\frac{c-a}{b} \\]",
"role": "user"
}
] | \dfrac{32}{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... | 7608 V/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... | 2.06 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... | 0.206 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... | $600,753.17 |
[
{
"content": "Determine whether the limit of the sequence of sets \\(A_n = \\left\\{ \\frac{k}{n}, 0 \\le k \\le n \\right\\}\\) as \\(n\\) approaches infinity is equal to the interval \\([0,1]\\).",
"role": "user"
}
] | [0,1] |
[
{
"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... | 29310344.83 |
[
{
"content": "Let \\( \\alpha \\) be a continuous function on \\([0,T] \\times [0,1]\\) where \\( T > 0 \\) is a fixed positive number. Consider the subspace of \\( H^1(0,1) \\):\n\\[ V(t) = \\left\\{ v \\in L^2(0,1) \\mid \\partial_x v + \\alpha(t,x)v \\in L^2(0,1), \\ t \\in [0,T] \\right\\} \\]\nIs \\( V(t) ... | Yes |
[
{
"content": "A point moves on a circle with a radius of 2 meters at an angular velocity of \\(\\frac{1}{2} \\text{ rad}\\cdot s^{-1}\\), starting at an angle of \\(\\frac{\\pi}{4}\\). Determine the time \\(t\\) within the interval \\([0, \\frac{\\pi}{2}]\\) when the cosine of the angle between the tangent at \... | \dfrac{\pi}{6} |
[
{
"content": "Evaluate the integral \\( \\int _{0}^{2\\pi} e^{it} e^{( e^{-it})} \\, dt \\) using the method of Taylor series expansion.",
"role": "user"
}
] | 2\pi |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 11.35 g/cm^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. Flexible labor market with less regulation on hiring and firing
2. Less generous unemployment benefits, thus greater incentive to work
3. Lower taxes and benefits create higher incentive to work since individuals keep more of their paychecks |
[
{
"content": "Consider a sequence of real numbers \\(\\{a_n\\mid n \\ge 1\\}\\) such that the series of absolute values \\(\\sum_{n=1}^\\infty |a_n|\\) diverges. If the power series \\(\\sum_{n=0}^\\infty a_nx^n\\) has a radius of convergence \\(R\\), can \\(R\\) be greater than 1?",
"role": "user"
}
] | No |
[
{
"content": "Determine whether the quotient field of $R[a]$ is equal to $R(a)$, given the definitions: $R[x]$ is the polynomial ring with coefficients from $R$, $R[a] = \\{f(a): f \\in R[x]\\}$ for $a \\in S$, and $R(a)$ is the smallest field containing $a$ and $R$. Assume $S$ is a field.",
"role": "user"
... | Yes |
[
{
"content": "Simplify the expression \\( \\int_0^{\\infty}kx^{k-1}[1-F(x)]\\,dx \\) for a random variable \\( X \\) with density function \\( f \\) and distribution function \\( F \\).",
"role": "user"
}
] | E[X^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... | 0.17% |
[
{
"content": "In triangle \\( \\triangle ABC \\), the interior angles \\( A \\), \\( B \\), and \\( C \\) form an arithmetic sequence. Additionally, the reciprocals of the side lengths \\( a \\), \\( b \\), and \\( c \\) also form an arithmetic sequence. Determine the measures of angles \\( A \\), \\( B \\), an... | 60^\circ |
[
{
"content": "Determine the smallest positive integer \\( N \\) such that for any initial arrangement of \\( 2n \\) balls, each colored with one of \\( n \\) colors (two balls per color), Phoe Wa Lone can always sort the balls into \\( n \\) boxes, each containing two balls of the same color, using at most \\( ... | 2n |
[
{
"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... | 2.1 m |
[
{
"content": "Evaluate the double integral: $$\\int_0^\\pi \\int_0^\\pi|\\cos(x+y)|\\,dx\\,dy$$",
"role": "user"
}
] | 2\pi |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3 kg |
[
{
"content": "The square of 15 is 225. The square of what other number is 225?",
"role": "user"
}
] | -15 |
[
{
"content": "Evaluate the contour integral \\( \\oint_{c(2,1)} \\frac{z^{7}+1}{z^{2}(z^{4}+1)}\\,\\mathrm{d}z \\), where the contour \\( c(2,1) \\) is the circle centered at 2 with radius 1, parametrized by \\( t \\rightarrow 2 + e^{it} \\).",
"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... | 4.5 L |
[
{
"content": "Simplify the expression \\(7^{101} + 8^{101}\\) modulo \\(25\\) using modular arithmetic and the Lifting The Exponent (LTE) Lemma.",
"role": "user"
}
] | 15 |
[
{
"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... | 20 cents |
[
{
"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... | 0.11 m |
[
{
"content": "Determine the degree of the differential equation for the family of circles that touch the lines $y^2 - x^2 = 0$ and lie in the first and second quadrants, with centers on the $y$-axis.",
"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... | 1.11 |
[
{
"content": "Let $H_{1}$ and $H_{2}$ be finite-dimensional Hilbert spaces with dimensions $n_{1}$ and $n_{2}$, respectively. Consider the linear operator $T: H_{1} \\to H_{2}$. Suppose for any orthonormal bases $\\{ e_{i} \\}$ for $H_{1}$ and $\\{ f_{j} \\}$ for $H_{2}$, the inequality \\(|\\langle f_{j}, Te_{... | Yes |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to \\infty}\\frac{1}{x}\\int_0^x|\\sin(t)|dt$$",
"role": "user"
}
] | \dfrac{2}{\pi} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.028 s |
[
{
"content": "Find the value of \\( f^{(2016)}(0) \\) for the function \\( f(x) = \\sin(x^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... | 2.06 |
[
{
"content": "Find the general formula for the sequence $a_n$ defined by $a_0 = 3$ and $a_n = 2 + a_0 a_1 a_2 \\ldots a_{n-1}$ for $n \\geq 1$. Express your answer in terms of $n$.",
"role": "user"
}
] | 2^{2^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... | C3H4F2 |
[
{
"content": "Evaluate the expression \\( \\tan^{-1}\\left(\\frac{3 \\sin 2\\alpha}{5 + 3 \\cos 2\\alpha}\\right) + \\tan^{-1}\\left(\\frac{1}{4} \\tan \\alpha\\right) \\) for \\( \\frac{-\\pi}{2} < \\alpha < \\frac{\\pi}{2} \\).",
"role": "user"
}
] | \alpha |
[
{
"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... | +187011.98 N/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... | 1.12e-20 J |
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