prompt list | ground_truth string |
|---|---|
[
{
"content": "Given that \\((\\sin^{-1}a)^2 + (\\cos^{-1}b)^2 + (\\sec^{-1}c)^2 + (\\csc^{-1}d)^2 = \\frac{5\\pi^2}{2}\\), find the value of \\((\\sin^{-1}a)^2 - (\\cos^{-1}b)^2 + (\\sec^{-1}c)^2 - (\\csc^{-1}d)^2\\). Assume \\(a\\), \\(b\\), \\(c\\), and \\(d\\) are real numbers within the appropriate domains ... | 0 |
[
{
"content": "Determine the maximum step size for the forward Euler method such that the solution of the system of ODEs \\( a' = -a - 2b \\) and \\( b' = 2a - b \\) with initial conditions \\( a(0)=1 \\) and \\( b(0)=-1 \\) goes to zero. Use the eigenvalues of the matrix \\( \\begin{pmatrix} -1-h & -2-2h \\\\ 2... | \dfrac{2}{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... | m1u1 + m2u2 = m1v1 + m2v2 |
[
{
"content": "Evaluate the sum \\(\\sum_{n=2}^\\infty \\frac{1}{n(n-1)(n+1)}\\) using the integral representation of the zeta function.",
"role": "user"
}
] | \dfrac{1}{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... | 1.77 * 10^(-12) 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... | 9.31e6 m/s |
[
{
"content": "Given that $f(0)=3$ and $f(x)\\times f'(-x)=f(-x)\\times f'(x)$ for all $x$, find the value of $f(x)\\times f(-x)$. Provide your answer as a single value.",
"role": "user"
}
] | 9 |
[
{
"content": "Consider a positive definite, symmetric $n \\times n$ matrix $G$ with integer entries. The matrix $G$ can be diagonalized as $G = Q^T D Q$, where $Q$ is an orthogonal matrix and $D$ is a diagonal matrix with eigenvalues of $G$. Determine whether the matrix $Q$ must also have integer entries.",
... | No |
[
{
"content": "What is the sum of $101_2$ and $1011_2$? Express your answer in base 2.",
"role": "user"
}
] | 10000_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... | 10.605 g/mL |
[
{
"content": "Calculate the limit: \\[ \\lim_{x \\to 0} \\frac{ \\sqrt{x+\\sqrt{x+\\sqrt{x}}} }{ 2\\sqrt{x}+4\\sqrt[8]{x}} \\]",
"role": "user"
}
] | \dfrac{1}{4} |
[
{
"content": "Find all non-empty sets of real numbers $A$ such that for any real numbers $a$ and $b$, if $a+b \\in A$, then $ab \\in A$. These sets are called complete sets.",
"role": "user"
}
] | \mathbb{R} |
[
{
"content": "Determine whether the limit \\( \\lim_{n \\to \\infty} \\frac{p_n}{n} \\) exists, where \\( p_n \\) is the largest prime number less than or equal to \\( n \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Determine the positive values of $P$ for which the series \\( \\sum_{n=1}^{\\infty}\\frac{(n!)^3}{(Pn)!} \\) converges.",
"role": "user"
}
] | P \geq 3 |
[
{
"content": "Is it always possible to select 11 numbers from any permutation of the sequence $1, 2, 3, \\ldots, 101$ such that the selected numbers form either an increasing or decreasing sequence?",
"role": "user"
}
] | 11 |
[
{
"content": "What is the name of the function defined by \\( f(x) = \\sum\\limits_{k=0}^\\infty {x \\choose k} \\)?",
"role": "user"
}
] | 2^x |
[
{
"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... | Third-person limited |
[
{
"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... | 1e-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... | 4.5 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... | 3.88e5 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... | 3.51e+3 grams |
[
{
"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... | 74.6 g |
[
{
"content": "Find the range \\([a, b]\\) of the expression \\(\\sum_{n=0}^\\infty\\frac{\\cos nx}{3^n}\\). Calculate the value of \\(b-a\\).",
"role": "user"
}
] | \dfrac{3}{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... | 111 x 10^3 Pa |
[
{
"content": "Suppose that \\( f(x)=\\frac{2x+1}{x-2} \\) and that \\( y=g(x) \\) is a linear function. If \\( f^{-1}(g(2))=7 \\) and \\( g^{-1}(f(1))=\\frac{4}{5} \\), find the \\( x \\)-intercept of \\( y=g(x) \\).",
"role": "user"
}
] | \dfrac{7}{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... | $348,161 |
[
{
"content": "Let $K = \\mathbb{C}(x_1, \\ldots, x_n)$ and let $G$ be a countable group of automorphisms of $K$, where $G \\cong \\mathbb{Z}$. The field of $G$-invariants, $K^G$, is an extension of $\\mathbb{C}$ with transcendence degree $d$. For a generic $n$-tuple $x \\in \\mathbb{C}^n$, consider the Zariski ... | n - d |
[
{
"content": "Determine the minimum value of \\( p \\) for which the series \\( \\sum_{n=0}^\\infty \\frac{\\sqrt{n+1}}{n^{p+1}+n+1} \\) converges.",
"role": "user"
}
] | \dfrac{1}{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... | $53.93 |
[
{
"content": "Does every primitive digraph have a directed cycle? A digraph is a directed graph, and a directed cycle is a directed circuit where the only repeated vertices are the first and last vertices. A digraph is primitive if its adjacency matrix is primitive, meaning there exists a positive integer \\( k... | 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... | 0.594 |
[
{
"content": "The function $f(x) = -3x^2 + 36x - 7,$ defined for all real numbers, does not have an inverse. However, when its domain is restricted to $x \\in [c,\\infty),$ the function does become invertible, for certain values of $c.$ What is the smallest such value of $c$?",
"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... | -2.64e-17 J |
[
{
"content": "Find [the decimal form of] the largest prime divisor of $100111011_6$.",
"role": "user"
}
] | 181 |
[
{
"content": "For integers \\(m, n > 0\\) and \\(x > 1\\), determine the conditions on \\(m\\) and \\(n\\) such that \\((x+m) \\mid (x-1)^n\\).",
"role": "user"
}
] | n \geq 2 |
[
{
"content": "Given positive numbers \\(x, y, z\\) such that \\(x^2 + y^2 + z^2 = 1\\), find the minimum value of the expression:\n\\[ S = \\frac{xy}{z} + \\frac{yz}{x} + \\frac{zx}{y}. \\]",
"role": "user"
}
] | \sqrt{3} |
[
{
"content": "Two lines pass through point \\(N\\) and are tangent to a circle with center \\(O\\). Point \\(A\\) is on one line, and point \\(B\\) is on the other such that \\(OA = OB\\) and \\(OA > ON\\). Given \\(NA = a\\), \\(NB = b\\), and \\(OA = c\\) where \\(a \\neq b\\), find the length of \\(ON\\).",
... | \sqrt{c^2 + ab} |
[
{
"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... | -0.190 m/s |
[
{
"content": "Let \\( f(x) = x^{3} + 3x^{2} + 6x + 14 \\) be a real-valued function. Given that \\( f(a) = 1 \\) and \\( f(b) = 19 \\) for real numbers \\(a\\) and \\(b\\), find the value of \\((a + b)^{2}\\).",
"role": "user"
}
] | 4 |
[
{
"content": "Let $f(m,1) = f(1,n) = 1$ for $m \\geq 1, n \\geq 1,$ and let $f(m,n) = f(m-1,n) + f(m,n-1) + f(m-1,n-1)$ for $m > 1$ and $n > 1.$ Also, let\n\n$$S(k) = \\sum_{a+b=k} f(a,b), \\text{ for } a \\geq 1, b \\geq 1.$$Note: The summation notation means to sum over all positive integers $a,b$ such that ... | 2 |
[
{
"content": "Determine all positive integers \\( n \\) such that the quotient \\( \\frac{Q_n(x,y,z)}{P_n(x,y,z)} \\) is a polynomial with rational coefficients, where:\n\\[P_n(x,y,z)=(x-y)^{2n}(y-z)^{2n}+(y-z)^{2n}(z-x)^{2n}+(z-x)^{2n}(x-y)^{2n}\\]\nand\n\\[Q_n(x,y,z)=[(x-y)^{2n}+(y-z)^{2n}+(z-x)^{2n}]^{2n}.\\... | 1 |
[
{
"content": "What is $333_4-344_5$ when expressed in base 10?",
"role": "user"
}
] | -36 |
[
{
"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... | $120,240 |
[
{
"content": "Determine if there exist values of $n$ such that the number of primes of the form $1 + 6k$ exceeds the number of primes of the form $5 + 6k$ for $k \\leq n$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Given that $f(x)=ax^2+bx+c$ and $g(x)=bx^2+cx+a$ have a common root, and $a+b+c=0$, determine the value of \\(\\frac{a^3+b^3+c^3}{abc}\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Evaluate the limit using polar coordinates: $$\\lim\\limits_{(x,y)\\to(1,1)}\\dfrac{\\log\\left(\\dfrac{1}{\\sqrt{(x-1)^2+(y-1)^2}}\\right)}{\\dfrac{1}{\\sqrt{(x-1)^2+(y-1)^2}}}$$",
"role": "user"
}
] | 0 |
[
{
"content": "Consider a rectangle tiled with dominos. Is it always possible to transform this tiling into any other domino tiling of the same rectangle by repeatedly performing the following move: pick two dominos that share a long edge and rotate them in place by 90 degrees?",
"role": "user"
}
] | Yes |
[
{
"content": "Given the moment generating function of a random variable X:\n\\[ M_X(t) = \\frac{1}{(1-2500t)^4} \\]\ncalculate the standard deviation of the claim size for this class of accidents.",
"role": "user"
}
] | 5000 |
[
{
"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... | 757.8 mm |
[
{
"content": "Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Given that t... | 405 |
[
{
"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... | 0.95 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... | 5.7 meters directed at about 72-degrees north of east |
[
{
"content": "Find the degree of the splitting field of the polynomial \\(x^{616} - 1\\) over the field \\(\\mathbb{F}_{2}\\).",
"role": "user"
}
] | 30 |
[
{
"content": "Determine whether every square matrix over \\(\\mathbb{R}\\) or \\(\\mathbb{C}\\) that lacks an LU decomposition will admit an LU decomposition when a scalar multiple of the identity matrix is added to it.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine if there exists an integer value of \\( m \\) such that both expressions \\((m - 2)^3 + 1\\) and \\(m^2 - 4m + 5\\) are perfect cubes.",
"role": "user"
}
] | 2 |
[
{
"content": "Let \\( a_1, a_2, a_3, \\ldots \\) be a sequence of positive real numbers such that:\n\n(i) For all positive integers \\( m, n \\), we have \\( a_{mn} = a_m a_n \\).\n\n(ii) There exists a positive real number \\( B \\) such that for all positive integers \\( m, n \\) with \\( m < n \\), we have \... | 0 |
[
{
"content": "Determine the number of solutions to the equation \\(x^4 - 1 = e^x\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Solve the convolution equation \\((y * \\sin)(t) = t^2\\) using Laplace transforms and verify that the solution is \\(y(t) = t^2 + 1\\).",
"role": "user"
}
] | y(t) = t^2 + 2 |
[
{
"content": "Does a topological space $X$ being both locally connected and second-countable guarantee the existence of a countable basis consisting of connected sets?",
"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... | 8000 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... | Cash, Checking Account Deposits, Savings Account Deposits, Money Market Deposits, Certificates of Deposit (CDs), Traveler's Checks, Credit, Debit Cards, Cryptocurrencies |
[
{
"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... | 1.35 |
[
{
"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... | $1,482,893.75 |
[
{
"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... | 200,000 |
[
{
"content": "Determine the number of distinct $4 \\times 4$ matrices with entries in \\( \\{0,1\\} \\) such that each row is non-zero and distinct, and the matrix has a real rank of 3 or 4.",
"role": "user"
}
] | 32760 |
[
{
"content": "Evaluate the infimum \\( I = \\inf_{f \\in \\mathcal{F}} \\left( \\sup_{0 \\leq t \\leq 1} \\left| 1 - f(t) \\right| + \\int_{0}^{1} \\left| 1 - f(t) \\right| dt \\right) \\), where \\( \\mathcal{F} \\) is the vector space of all continuous functions \\( f: [0,1] \\rightarrow \\mathbb{R} \\) such ... | 1 |
[
{
"content": "Evaluate the integral \\( \\int_0^\\infty Ai(y) \\, dy \\), where the Airy function \\( Ai(y) \\) is given by the integral representation \\( Ai(y) = \\frac{1}{\\pi} \\int_0^\\infty \\cos(\\alpha y + \\frac{\\alpha^3}{3}) \\, d\\alpha \\).",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "Let $X$ be a topological space and $U$ be a proper dense open subset of $X$. Determine which of the following statements is correct:\n\n(A) If $X$ is connected, then $U$ is connected.\n(B) If $X$ is compact, then $U$ is compact.\n(C) If $X \\setminus U$ is compact, then $X$ is compact.\n(D) If $X$... | D |
[
{
"content": "Let \\( A \\) be the sum of all non-negative integers \\( n \\) satisfying\n\\[ \\left\\lfloor \\frac{n}{27} \\right\\rfloor = \\left\\lfloor \\frac{n}{28} \\right\\rfloor. \\]\nDetermine \\( A \\).",
"role": "user"
}
] | 95004 |
[
{
"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... | 3.42e-5 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... | 200\sqrt{10} seconds |
[
{
"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... | 11.96% |
[
{
"content": "Verify the identity \\( \\arcsin x + \\arcsin y = \\arcsin\\left(x \\sqrt{1-y^2} + y \\sqrt{1-x^2}\\right) \\) by differentiating both sides with respect to \\( x \\) and evaluating at \\( x = 0 \\). Show that the derivatives are equal or identify any discrepancies.",
"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... | 133.70 |
[
{
"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... | 0.564 m |
[
{
"content": "In a category with zero morphisms, kernels, and cokernels, consider a kernel morphism $m$ and a cokernel morphism $e$ such that the composition $e \\circ m$ is defined. If $e \\circ m$ has an image, is this image always a normal monomorphism in the category?",
"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... | 3.54e-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... | 3.12 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... | 0.0172 N |
[
{
"content": "Determine whether the generalized Laguerre function $L^{\\alpha}_n(x)$ can be defined for $\\alpha = -1$ with all $n \\geq 0$ and $x > 0$.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether the following statement is true or false: For a number field $K$ and a discrete valuation $v$ on $K$, the completion of the localization of the ring of integers $\\mathcal{O}_{K}$ at $v$, denoted $\\widehat{\\mathcal{O}_{K,v}}$, is isomorphic to the inverse limit of the sequence ... | 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... | 76,377 |
[
{
"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... | 1.18e-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... | $336.66 Million |
[
{
"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... | 10.383 m/s |
[
{
"content": "Calculate the limit: \\[ \\lim_{{x \\downarrow 0}} \\left(\\frac{1}{\\sin x} - \\frac{1}{x}\\right)^x. \\]",
"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... | Preferred shares' dividend per share is $1 for each year while the common shares' dividend per share is $0.10 in Year 1, $0.50 in Year 2, and $0.90 in Year 3. |
[
{
"content": "Suppose we have three non-negative random variables $X, Y, Z$, which may be dependent and satisfy $Z \\geq \\min\\{X, Y\\}$. Given that there exists a constant $A$ such that $E(X) \\geq A$ and $E(Y) \\geq A$, determine if it is true that $E(Z) \\geq A$.",
"role": "user"
}
] | No |
[
{
"content": "Let $V$ be a finite-dimensional vector space over the field $\\mathbb{F}_q$ with $q$ elements, and let $U$ be a subspace of $V$. Determine the number of subspaces $W$ of $V$ such that $W \\cap U = \\{0\\}$ and $V = W \\oplus U$. Provide your answer as a single integer.",
"role": "user"
}
] | q^{k(n - 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... | 6.74 mm |
[
{
"content": "Let $(a_n)$ be a real sequence such that $a_n > 0$ for $n > 0$ and \\( \\lim\\limits_{n \\to \\infty} \\frac{a_n a_{n+2}}{a_{n+1}^2} = L^2 \\). Given that $L \\ge 0$, find \\( \\lim\\limits_{n \\to \\infty} \\sqrt[n^2]{a_n} \\) in terms of $L$. \\( \\)",
"role": "user"
}
] | 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... | 1.2R |
[
{
"content": "Find the equation of the circle whose center is at \\(a + i\\) (where \\(a\\) is a real number) and that intersects the circles \\(|z| = 1\\) and \\(|z - 1| = 4\\) orthogonally. Choose the correct option:\n\n(a) \\(|z-7+i| = 7\\)\n(b) \\(|z-2+i| = 7\\)\n(c) \\(|z+7-i| = 7\\)\n(d) \\(|z+2+i| = 7\\)... | C |
[
{
"content": "Evaluate the limit:\n\\[\n\\lim_{n \\to \\infty} \\frac{\\sin(1) + 2\\sin\\left(\\frac{1}{2}\\right) + 3\\sin\\left(\\frac{1}{3}\\right) + \\dots + n\\sin\\left(\\frac{1}{n}\\right)}{n}\n\\]",
"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... | 30.77% |
[
{
"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... | 1089/8 |
[
{
"content": "Determine the asymptotic behavior of the expected number of flips required for the number of heads to exceed the number of tails in a sequence of fair coin flips.",
"role": "user"
}
] | \infty |
[
{
"content": "Evaluate the integral \\(\\int_{|z|=3}\\frac{1}{z^2-1}dz\\) using the Cauchy integral formula.",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the volume of the region defined by the inequality \\( \\{(x,y,z) \\in \\mathbb{R}^3 \\mid 2\\cdot \\max(\\lvert x\\rvert,\\lvert y\\rvert)^2+z^2\\leq 4\\} \\).",
"role": "user"
}
] | \dfrac{64}{3} |
[
{
"content": "Determine the value of \\( \\lim_{x \\to 0^+} f(x) \\) for a non-zero function \\( f: [0,1] \\rightarrow [-1,1] \\) that satisfies \\( f(2x) = 3f(x) \\) for all \\( x \\in \\left[0, \\frac{1}{2}\\right] \\).",
"role": "user"
}
] | 0 |
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