prompt list | ground_truth string |
|---|---|
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | The partial pressure of N2O4 is approximately 199832.67 Pa, and the partial pressure of NO2 is approximately 167.33 Pa. |
[
{
"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... | 1799 lb |
[
{
"content": "Given the homographic function \\( y = \\frac{ax+b}{cx+d} \\), derive the expression \\( ad-bc \\) using an alternative method to the one provided. Show your steps clearly to arrive at the condition \\( \\boxed{ad-bc} \\).",
"role": "user"
}
] | ad - bc |
[
{
"content": "Determine whether a real skew-symmetric matrix with purely imaginary eigenvalues can be diagonalized over the complex numbers.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine which is larger: $e^\\pi$ or $\\pi^e$. Provide your reasoning without using a calculator.",
"role": "user"
}
] | e^\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... | 6.219 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... | 383000 km |
[
{
"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... | SnSO4 |
[
{
"content": "Find the smallest possible value of the moment generating function $m_X(2)$ for a random variable $X$ with an expected value of $0$ and a moment generating function $m_X(1) = z \\geq 1$. Provide your answer in terms of $z$.",
"role": "user"
}
] | 2z^2 - 1 |
[
{
"content": "For a parametric curve defined by \\(x(t) = \\cos(t)\\) and \\(y(t) = \\cos(2^k t)\\), where \\(k\\) is a natural number, determine the order of the polynomial obtained by eliminating the parameter \\(t\\).",
"role": "user"
}
] | 2^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... | 1-methoxy-3-methyl butane |
[
{
"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 redox reaction is given by the following balanced molecular equation:
{eq}6FeSO_4 + 3Br_2 -> 2Fe_2(SO_4)_3 + 2FeBr_3{/eq}
The net ionic equation is:
{eq}6Fe^{2+} + 3Br_2 -> 6Fe^{3+} + 6Br^{-}{/eq} |
[
{
"content": "Evaluate the sum: $$\\displaystyle\\sum\\limits_{n=1}^{\\infty}\\frac{(-1)^n \\sin(n)}{n}$$",
"role": "user"
}
] | -\dfrac{1}{2} |
[
{
"content": "Consider the sequence defined by:\n\n\\[\n\\begin{aligned}\nx_{1} & =19, \\\\\nx_{2} & =95, \\\\\nx_{n+2} & =\\operatorname{lcm}\\left(x_{n+1}, x_{n}\\right)+x_{n},\n\\end{aligned}\n\\]\n\nfor \\(n > 1\\). Find the greatest common divisor of \\(x_{1995}\\) and \\(x_{1996}\\).",
"role": "user"
... | 19 |
[
{
"content": "Given a list of numbers where every number except one appears twice, find the number that appears only once. For example, in the list $2, 3, 4, 3, 2, 1, 1$, identify the number that does not have a pair.",
"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... | 2.85 rad/s |
[
{
"content": "Determine whether the operator \\(S:C[0,1]\\rightarrow C[0,1]\\) defined by \\((Sx)(t)=\\int^1_t x(s)ds, \\ \\ \\forall t\\in[0,1]\\) is compact.",
"role": "user"
}
] | Yes |
[
{
"content": "Calculate the average value of \\((a-b)^2+(b-c)^2+(c-d)^2+(d-e)^2+(e-f)^2+(f-g)^2\\) over all possible permutations \\((a, b, c, d, e, f, g)\\) of the numbers \\(1, 2, 3, 11, 12, 13, 14\\). Choose the correct answer from the following options:\n\n(A) 398\n(B) 400\n(C) 396\n(D) 392\n(E) 394",
"... | D |
[
{
"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... | 1620 seconds |
[
{
"content": "Find the product abc for positive integers a, b, and c that satisfy the following system of equations:\n\n1. \\(-10(c-2ab)+\\frac{b-5}{a}=52\\)\n2. \\(a-b+c=53\\)\n3. \\(a(b+7)=54\\)",
"role": "user"
}
] | 2013 |
[
{
"content": "Evaluate the expression \\( l+m \\), where \\( l = \\lim_{x\\rightarrow 0}\\int_{0}^{x}\\frac{(1+\\cos t)^2}{x}dt \\) and \\( m = \\lim_{x\\rightarrow \\infty}\\int_{0}^{x}\\frac{(1+\\cos t)^2}{x}dt \\).",
"role": "user"
}
] | \dfrac{11}{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... | 65.18 quarters |
[
{
"content": "For the function \\( k(x) = 4\\sqrt{x} + \\frac{2}{\\sqrt{x}} \\) on the interval \\([\\frac{1}{4}, 1]\\), find the value \\( c \\) that satisfies the Mean Value Theorem.",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Determine whether the following statement is true or false: Every $2 \\times 2$ matrix over the complex numbers $\\Bbb C$ is a square of some matrix.",
"role": "user"
}
] | B |
[
{
"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... | 15.0 m/s |
[
{
"content": "Determine the cardinality of the set \\( S = \\mathbb{N} \\times A \\), where \\( A \\) is a set with infinitely many elements and \\( \\mathbb{N} \\) is the set of natural numbers.",
"role": "user"
}
] | |A| |
[
{
"content": "Using the Extended Euclidean algorithm, find the inverse of the polynomial $x^4$ modulo $x^5+x^2+1$. Verify that the inverse is $x^4+x^2+1$ and identify any errors in the calculation if a different result is obtained.",
"role": "user"
}
] | x^3 + x + 1 |
[
{
"content": "Let $\\{a_n\\}$ be a sequence such that $0\\le a_n\\lt 1$. If the series $\\sum_{n=0}^{\\infty} \\frac{a_n}{1+a_n}$ converges, does it follow that the series $\\sum_{n=0}^{\\infty} a_n$ converges? Answer 'yes' or 'no' and justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A jar has $10$ red candies and $10$ blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n,$ where $m$ and $n$ are relatively prime positive integer... | 441 |
[
{
"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... | 107.1 nm |
[
{
"content": "Let $B$ be an orthogonal matrix with $\\det(B) < 0$. Determine the range of possible values for $\\det(B-I)$.",
"role": "user"
}
] | (-\infty, 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... | 16670.47 |
[
{
"content": "An insurer finds that the time in hours it takes to process a loss amount $x$ has a uniform distribution on the interval $(\\sqrt{x}, 2\\sqrt{x})$. Given that the loss distribution has a probability density function $f(x) = \\frac{1}{x^2}$ for $x > 1$, find the expected time to process a claim of ... | 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... | 3.2e5 m |
[
{
"content": "Given the polynomial \\( f(x) = a(x-1)(x-2)\\dots(x-101) \\), where \\(1, 2, \\dots, 101\\) are its roots, find the sum of the roots of its derivative \\( f'(x) \\).",
"role": "user"
}
] | 5100 |
[
{
"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... | 56.9 percent |
[
{
"content": "Determine the order of growth for the recurrence relation given by:\n\n\\[ T(n) = n + \\sum\\limits_{k=1}^{n-1} [T(n-k) + T(k)] \\]\n\nwith the initial condition \\( T(1) = 1 \\). What is the order of \\( T(n) \\) in big O notation?",
"role": "user"
}
] | O(3^n) |
[
{
"content": "Evaluate the limit: \\[ \\lim_{m \\to \\infty} m \\cdot E \\left[ \\log\\left( \\frac{N+\\frac{1}{2}}{m+1} \\right) \\right] \\] where \\( N \\) is a Poisson random variable with mean \\( m \\).",
"role": "user"
}
] | -1 |
[
{
"content": "Let $f$ be an entire function on $\\mathbb{C}$ such that $|f(z)|\\le 100 \\ln |z|$ for $|z|\\ge 2$ and $f(i)=2i$. Determine the value of $f(1)$. \\( \\boxed{} \\)",
"role": "user"
}
] | 2i |
[
{
"content": "Find the coefficient of \\(x^{26}\\) in the expansion of \\((x^2+x^3+x^4+x^5+x^6)^8\\).",
"role": "user"
}
] | 13140 |
[
{
"content": "Evaluate the limit: $$\\ell =\\lim_{x\\to 0}\\frac{\\sin^{200}(x)}{x^{199}\\sin(4x)}.$$ Determine if this limit exists, and if so, find its value.",
"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... | 2328K |
[
{
"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... | /cimages/multimages/16/z-5-pentadecen-16-diol4280967040600664354.jpg |
[
{
"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... | -9.976 MPa |
[
{
"content": "Determine the sum of all real numbers $x$ satisfying\n\\[(x^2-4x+2)^{x^2-5x+2} = 1.\\]",
"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... | 21.40 m/s |
[
{
"content": "Let $A$ be a $10 \\times 10$ matrix with complex entries, where all eigenvalues are non-negative real numbers and at least one eigenvalue is strictly positive. Determine which of the following conditions can be satisfied by some matrix $B$:\n\nA. $AB - BA = B$\nB. $AB - BA = A$\nC. $AB + BA = A$\n... | ACD |
[
{
"content": "Determine whether the following statement is true or false: If $a^n$ is an eigenvalue of $M^n$, then $a$ is an eigenvalue of $M$.",
"role": "user"
}
] | B |
[
{
"content": "Let $U_1$ and $U_2$ be uniformly distributed random variables on the unit interval. If $U_1$ and $U_2$ are correlated, determine the maximum possible value of $E[U_1 U_2]$. Express your answer as a single value.",
"role": "user"
}
] | \dfrac{1}{3} |
[
{
"content": "In the metric space $(X, d)$, determine if the set $B = \\{z \\in X: d(x,z) > d(y,z)\\}$ is open.",
"role": "user"
}
] | Yes |
[
{
"content": "Suppose that X is the sum of m independent Bernoulli random variables that are not necessarily identically distributed, and thus it has a Poisson binomial distribution. Determine whether mX is also a Poisson binomial random variable.",
"role": "user"
}
] | No |
[
{
"content": "Calculate the average value of the sum \\( S = x_1x_2 + x_2x_3 + \\cdots + x_9x_{10} + x_{10}x_1 \\), where \\( x_1, x_2, \\ldots, x_{10} \\) are all possible permutations of the numbers \\( 1, 2, 3, \\ldots, 10 \\).",
"role": "user"
}
] | \dfrac{880}{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... | 472500 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... | $6,262 |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\pi} \\lfloor \\pi^2 \\cos^3 x \\rfloor \\sin x \\, dx \\), where \\( \\lfloor x \\rfloor \\) denotes the floor function.",
"role": "user"
}
] | -1 |
[
{
"content": "Determine whether the following statement is true or false: If a commutative ring with $1$ does not have a unit of additive order $2$, then it does not have any element of additive order $2$. Justify your answer.",
"role": "user"
}
] | False |
[
{
"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... | 370 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... | idea screening |
[
{
"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 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... | Oxygen-15 |
[
{
"content": "Determine whether there can exist an associative, but non-commutative binary operation that has an identity element and inverses for all elements.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the definite integral \\( \\int_{0}^{2\\pi} \\frac{dx}{5-3\\cos x} \\).",
"role": "user"
}
] | \dfrac{\pi}{2} |
[
{
"content": "Given $A, B, C$ are $n \\times n$ matrices, where $B$ and $C$ are nonsingular, and $b$ is an $n$-vector, solve for the vector $x$ using LU factorization, where:\n\\[ x = C^{-1}(3A-I)(B^{-1}-A)b \\]",
"role": "user"
}
] | C^{-1}(3A - I)(B^{-1} - A)b |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\to\\infty}\\frac{\\{x\\}^{2n}-1}{\\{x\\}^{2n}+1} \\) for \\( 0 < \\{x\\} < 1 \\), where \\( \\{x\\} \\) represents the fractional part of \\( x \\).",
"role": "user"
}
] | -1 |
[
{
"content": "Suppose we have the set $[0,1]$ equipped with the Lebesgue measure. A measurable function $f$ is said to be non-decreasing on a full-measure set if there exists a set $E$ with $\\mu(E)=1$ such that for all $y,z\\in E$, if $z\\leq y$, then $f(z)\\leq f(y)$. If $f$ is not non-decreasing on a full-me... | Yes |
[
{
"content": "Calculate the length of the curve defined by the equations $z^2 = 2x$ and $3y = xz$ from the point $(0, 0, 0)$ to the point $(\\frac{1}{2}, \\frac{1}{6}, 1)$. Provide your answer as a single numerical value.",
"role": "user"
}
] | \dfrac{7}{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... | 1.02e-7 N/C |
[
{
"content": "Let the sequence \\(a_{0}, a_{1}, a_{2}, \\cdots\\) satisfy:\n\\[ a_{0} = a_{1} = 11, \\quad a_{m+n} = \\frac{1}{2}(a_{2m} + a_{2n}) - (m-n)^{2}, \\quad m, n \\geq 0. \\]\n\nFind \\(a_{45}\\).",
"role": "user"
}
] | 1991 |
[
{
"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... | 5.89% |
[
{
"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... | 3.33 * 10^-10 M |
[
{
"content": "The decimal $0.76$ is equal to the fraction $\\frac{4b+19}{6b+11}$, where $b$ is a positive integer. What is the value of $b$?",
"role": "user"
}
] | 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... | 339 K |
[
{
"content": "Determine if the function \\( f(x) \\) defined as \\( f(x)=2|x|+1 \\) for \\( x \\) rational and \\( f(x)=0 \\) for \\( x \\) irrational is discontinuous on a set of positive measure on the interval \\([0, 5]\\).",
"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... | 1.53 mm |
[
{
"content": "Let\n\\[z = \\frac{-\\sqrt{3} + i}{2}.\\]Compute $z^6.$",
"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... | 20 g |
[
{
"content": "Determine whether the following statement is true or false: If $u$ and $v$ are any two vectors in $\\mathbb{R}^n$ and $\\mathcal{B}$ is any basis for $\\mathbb{R}^n$, then $[u \\cdot v]_{\\mathcal{B}} = [u]_{\\mathcal{B}} \\cdot [v]_{\\mathcal{B}}$. Justify your answer.",
"role": "user"
}
] | False |
[
{
"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... | 10.3 m/s^2 |
[
{
"content": "Find all positive integer solutions \\((n, m)\\) to the equation \\(n^5 + n^4 = 7^m - 1\\).",
"role": "user"
}
] | (2, 2) |
[
{
"content": "Calculate the limit: \\[ \\lim_{x\\to 0} \\frac{(\\sin(x)-\\tan(x))^2}{(e^x-1-\\ln(x+1))^3} \\]",
"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... | 414.35 ft |
[
{
"content": "Calculate the scalar surface integral of the function \\( f(x, y, z) = yz \\) over the half cylinder \\( S \\), defined by the equation \\( x^2 + z^2 = a^2 \\) with \\( z \\leq 0 \\) and \\( 0 \\leq y \\leq h \\). Verify if the result is \\( -a^2h^2 \\).",
"role": "user"
}
] | -a^2 h^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... | 3.6e-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... | 20 degrees Celcius |
[
{
"content": "Find the limit function $f(x)$ of the sequence $f_n(x)$ defined on $\\mathbb{R}$, where $f_n(x) = \\int_{0}^{1}f_{n-1}(t)\\sin(x-t)dt$ for $n \\geq 2$ and $f_1(x) = \\cos x$. Assume the sequence converges uniformly to a function $f(x)$. Determine $f(x)$ by analyzing the integral transformation.",
... | 0 |
[
{
"content": "If \\( \\alpha \\), \\( \\beta \\), and \\( \\gamma \\) are the roots of the polynomial \\( x^3 - x - 1 = 0 \\), find the value of \\( \\frac{1+\\alpha}{1-\\alpha} + \\frac{1+\\beta}{1-\\beta} + \\frac{1+\\gamma}{1-\\gamma} \\).",
"role": "user"
}
] | -7 |
[
{
"content": "Determine the positive integers \\(k\\) for which the series \\(\\sum_{n=1}^{\\infty}\\frac{(n!)^2}{(kn)!}\\) is convergent.",
"role": "user"
}
] | k \geq 2 |
[
{
"content": "Find the minimum value of the function \\( f(x, y) = \\frac{2015(x+y)}{\\sqrt{2015 x^{2} + 2015 y^{2}}} \\) and determine all pairs \\((x, y)\\) where this minimum value is achieved.",
"role": "user"
}
] | -\sqrt{4030} |
[
{
"content": "Determine the condition under which a holomorphic function $f(z)$ defined in the open disk $D(0,1)$ can be represented as $f(z) = zg(z)$, where $g(z)$ is also holomorphic.",
"role": "user"
}
] | f(0) = 0 |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^{x} \\frac{1}{\\sqrt{2\\pi}} e^{-\\left(\\frac{t^2}{2} + 2t + 2\\right)} \\, dt \\) for \\(-\\infty < x < \\infty\\).",
"role": "user"
}
] | \Phi(x + 2) |
[
{
"content": "Determine whether a nonsingular symmetric matrix that is not positive semidefinite can be factorized into the product of a matrix and its transpose, where the factor matrix is allowed to have complex entries.",
"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... | 242 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... | 974600 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... | 11.08% |
[
{
"content": "Given that the series \\( \\sum_{n=3}^{\\infty}a_n x^{n} \\) has a radius of convergence \\( R \\), determine the radius of convergence for the series \\( \\sum_{n=3}^{\\infty}a_n x^{n^2} \\).",
"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... | 3.6e-14 N |
[
{
"content": "In the context of a matrix $A$ as a linear transformation with elements denoted by $a^{i}_{j}$, how should the inverse of $A$ be represented? Specifically, if a bilinear form is expressed as $a_{ij}$, should the inverse be denoted as $a^{ij}$ or $a^{ji}$?",
"role": "user"
}
] | a^{ij} |
[
{
"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 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... | 65 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... | 1.68 s |
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