prompt list | ground_truth string |
|---|---|
[
{
"content": "Find all compact self-adjoint operators $T: H \\to H$ such that $T^{2017} = 0$. Provide a description or characterization of these operators.",
"role": "user"
}
] | 0 |
[
{
"content": "On a blackboard, the numbers $1, 2, 3, \\ldots, 2000$ are written. You can repeatedly erase two numbers $a$ and $b$ (where $a \\geq b$) and write the three numbers $a - b$, $\\sqrt{ab}$, and $\\sqrt{ab}$ (i.e., $\\sqrt{ab}$ is written twice). After performing this operation 8000 times, there will ... | 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... | Adjusted Balance: $8,550 |
[
{
"content": "Determine if there exists a sequence such that, when the described process is applied, the resulting sequence $X$ repeats itself.",
"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... | 70.9246 amu |
[
{
"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... | $21,580 |
[
{
"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... | 1130 g |
[
{
"content": "Does the uniform continuity of a function $f$ on $\\mathbb{R}$ imply the existence of a constant $C$ such that the supremum of $|f(y) - f(x) - C|x - y||$ over all $(x, y) \\in \\mathbb{R} \\times \\mathbb{R}$ is finite?",
"role": "user"
}
] | No |
[
{
"content": "Determine whether the following statement is true or false: If $f(z) = u(z) + iv(z)$ is an analytic function, then both $u(z)$ and $v(z)$ are analytic functions.",
"role": "user"
}
] | B |
[
{
"content": "In a study of product lifespan, consider the hypothesis test:\n\n\\[H_0: \\mu = 2\\]\n\\[H_1: \\mu < 2\\]\n\nwhere \\(\\mu\\) is the expected lifetime of the product, assumed to follow an exponential distribution with parameter \\(\\lambda\\). The probability density function is:\n\n\\[f_X(x) = \\... | 1.085 |
[
{
"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... | 0.0672 M |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\to \\infty}\\left(\\sum_{k=n+1}^{2n}\\left(2(2k)^{\\frac{1}{2k}}-k^{\\frac{1}{k}}\\right)-n\\right) \\]",
"role": "user"
}
] | (\ln 2)^2 |
[
{
"content": "For a finitely generated module $M$ over a principal ideal domain $R$, is it true that $M$ having free rank 0 is equivalent to $M$ being a torsion module? Justify your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Let \\( x_1, x_2, \\ldots, x_{42} \\) be real numbers such that \\( 5x_{i+1} - x_i - 3x_ix_{i+1} = 1 \\) for each \\( 1 \\le i \\le 42 \\), with the condition \\( x_1 = x_{43} \\). Determine the product of all possible values of \\( x_1 + x_2 + \\cdots + x_{42} \\).",
"role": "user"
}
] | 588 |
[
{
"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... | Sales
225,600
Sales discounts
2,270
Sales Returns and Allowances
13,500
Income and Expense Summary
209,830
To close the revenue and contra-revenue accounts.
Income and Expense Summary
73,400
Cost of Goods Sold
73,400
To close the Cost of Goods Sold.
Income and Expense Summary
88,500
Sales salaries expense
32,600
Rent... |
[
{
"content": "A point is randomly selected on the segment [3, 8]. Let this point be denoted as \\( k \\). Determine the probability that the roots of the quadratic equation \\((k^{2}-2k-3)x^{2}+(3k-5)x+2=0\\) satisfy the condition \\( x_{1} \\leq 2x_{2} \\).",
"role": "user"
}
] | \dfrac{4}{15} |
[
{
"content": "Let $A$ and $B$ be two $n \\times n$ complex matrices that satisfy $AB - BA = 0$. Determine whether $A$ and $B$ have a common eigenvector.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether a finite automaton can be constructed for the language $L_1 = (P \\cup \\{b\\})^*$, where $P = \\{a^p \\mid p \\text{ is a prime number}\\}$. If not, explain why the regular expression might need to be simplified to $b^*$. Provide a justification for your answer.",
"role": "u... | b^* |
[
{
"content": "Let \\( \\psi: \\mathbb{C}[X] \\to \\mathbb{R} \\) be a homomorphism such that \\( \\psi(X) = 2 \\). Determine the ideal generated by \\( \\ker \\psi \\).",
"role": "user"
}
] | (X - 2) |
[
{
"content": "Calculate the determinant of the matrix \\( \\begin{vmatrix} n & 0 & 0 & \\ldots & 0 & 0 \\\\\n0 & 0 & 2 & \\ldots& 0 & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n0 & 0 & 0 & \\ldots & 0 &n-1 \\\\ \n0 &1 & 0 &\\ldots & 0 & 0 \\end{vmatrix} \\).",
"role": "user"
... | (-1)^n n! |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to0}\\frac{(1+x\\cdot2^x)^{\\frac{1}{x^2}}}{(1+x\\cdot3^x)^{\\frac{1}{x^2}}}$$ as \\(x\\) approaches 0.",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?",
"role": "user"
}
] | 8 |
[
{
"content": "Calculate the integral \\( \\int_D{f^*\\omega } \\) where \\( f:D \\rightarrow \\mathbb{R}^3 \\) is given by \\( f(u,v)=(u,v,u^2+v^2+1) \\) with \\( D=[0,1]\\times [0,1] \\), and \\( \\omega=y\\,\\mathrm{d}y\\wedge \\mathrm{d}z+xz\\,\\mathrm{d}x\\wedge \\mathrm{d}z \\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Calculate the maximum value of \\[ \\frac{bc}{(b + c)^3(a^2 + 1)} + \\frac{ca}{(c + a)^3(b^2 + 1)} + \\frac{ab}{(a + b)^3(c^2 + 1)} \\] where \\(a, b, c\\) are positive real numbers satisfying \\(abc = 1\\).",
"role": "user"
}
] | \dfrac{3}{16} |
[
{
"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... | 4:1 |
[
{
"content": "Compute the integral \\( \\int_{2-i\\infty}^{2+i\\infty} \\frac{1}{t} \\, \\mathrm{d}t \\).",
"role": "user"
}
] | i\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... | 4796J |
[
{
"content": "Given two 20-sided dice, X and Y, calculate the expected value of X given that X is greater than Y, denoted as E(X | X > Y).",
"role": "user"
}
] | 14 |
[
{
"content": "Let $f,g:|K|\\to|L|$ be continuous maps such that the sets $f^{-1}(\\text{star}(v,L))\\cap g^{-1}(\\text{star}(v,L))$ with $v$ a vertex of $L$ form an open covering of $|K|$. Determine an integer $m$ and a simplicial map $s:|K^m|\\to|L|$ that simplicially approximates both $f$ and $g$. Provide the... | 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... | 33809 rpm |
[
{
"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... | 0.005 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... | 12.49 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... | $49,600 |
[
{
"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... | government intervention or regulation |
[
{
"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... | 240 ATM x l |
[
{
"content": "Let $A$ be a $3 \\times 3$ matrix with eigenvalues $1$, $2$, and $3$. Find $\\det(B)$ where $B = A^2 + A^T$. \\(A^T\\) denotes the transpose of matrix \\(A\\).",
"role": "user"
}
] | 144 |
[
{
"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... | $86.83 |
[
{
"content": "Minimize the function \\((x-3)^2+(y-1)^2\\) subject to the constraints:\n\\[ 2x+y \\leq 2 \\]\n\\[ x^2 + 2y = 3 \\]\n\\[ x, y \\geq 0 \\]",
"role": "user"
}
] | 25 - 10\sqrt{3} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | Sticks & Stones Builders Mart
Balance Sheet
For Oct. 31, 2014
| Particulars | Amount ($) | Percent |
|---|---|---|
| **Assets:** | | |
| Current assets: | | |
| Cash | 45,260 | 4.14% |
| Accounts receivable | 267,580 | 24.46% |
| Merchandise inventory | 213,200 | 19.49% |
| Prepaid expenses | 13,400 | 1.22% |
| Suppl... |
[
{
"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... | 18000 Vm |
[
{
"content": "In an arithmetic progression $a, b, c, d, e$, the sum $a + b + c + d + e$ is a cube of an integer, and the sum $b + c + d$ is a square of an integer. What is the least possible number of digits for the natural number $c$?",
"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... | -$366,000 |
[
{
"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... | input |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 26.78 m/s |
[
{
"content": "Let $X$ be a random variable with the moment generating function $$M_X(t) = \\frac{6}{\\pi^2}\\sum_{n\\geq 1}\\frac{e^{\\frac{t^2}{2n}}}{n^2}.$$ Determine the probability that $X$ is a rational number, $P(X \\in \\mathbb{Q})$. Choose the correct answer: \\(A\\) 0, \\(B\\) \\(\\frac{1}{4}\\), \\(C\... | A |
[
{
"content": "Consider the following algorithm:\n\nStep 0. Set \\( n = m \\).\n\nStep 1. If \\( n \\) is even, divide \\( n \\) by two. If \\( n \\) is odd, increase \\( n \\) by one.\n\nStep 2. If \\( n > 1 \\), go to Step 1. If \\( n = 1 \\), end the algorithm. \n\nHow many natural numbers \\( m \\) exist suc... | 610 |
[
{
"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... | Accounts Debit Credit
Cash 11,700
Accounts Receivable 14,000
Equipment 12,500
Allowance for Doubtful Accounts 3,300
Vopat, Capital 34,900 |
[
{
"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... | 1.23 |
[
{
"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... | It should pay back $137,467.5 on January 3, and $2,467.5 is interest. |
[
{
"content": "Evaluate the triple integral \\( \\iiint_R \\left(1-\\frac{y}{\\sqrt{x^2+y^2}}\\right)\\,dx\\,dy\\,dz \\), where \\( R \\) is the region bounded by \\( z = x^2 + y^2 \\) and \\( z = 1 - x^2 - y^2 \\).",
"role": "user"
}
] | \dfrac{\pi}{4} |
[
{
"content": "Determine the number of solutions to the equation \\((\\tan x)\\sin^2(2x) = \\cos x\\) within the interval \\(-2\\pi \\leq x \\leq 2\\pi\\).",
"role": "user"
}
] | 4 |
[
{
"content": "Find the real value of \\(x\\) that satisfies the equation: \\[2^{144} - \\frac{1}{2^{144}} = 8^x - 8^{-x}\\]",
"role": "user"
}
] | 48 |
[
{
"content": "A cylindrical fuel tank with hemispherical ends must hold 300 liters of jet fuel. The side wall of the cylinder costs $0.1 per cm² to manufacture, while the hemispherical ends cost $0.3 per cm². Determine the optimal radius of the tank that minimizes the total manufacturing cost.",
"role": "us... | 21.7 |
[
{
"content": "Find the function \\( f(x) \\) that satisfies the functional equation \\( f\\left(\\frac{x}{y}\\right) = \\frac{f(x)}{f(y)} \\) for \\( f(y) \\neq 0 \\) and the condition \\( f'(1) = 2 \\).",
"role": "user"
}
] | x^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... | $22,265.29 |
[
{
"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... | 78 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... | 38300g |
[
{
"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 |
[
{
"content": "Find the maximum value of $x+y$ given that $x$ and $y$ are positive real numbers satisfying the equation \\[x^3+y^3+(x+y)^3+36xy=3456.\\]",
"role": "user"
}
] | 12 |
[
{
"content": "Let \\( f(x) = \\int_0^x \\sum_{i=0}^{n-1} \\frac{(x-t)^i}{i!} \\, dt. \\) Find the nth derivative \\( f^{(n)}(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... | 24.74 miles per hour |
[
{
"content": "Find the limit: $$\\lim_{n \\to\\infty} \\dfrac{\\ln(\\ln(\\frac{n}{n-1}))}{\\ln(n)}$$",
"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... | 20000 miles or 32000 kilometers |
[
{
"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... | 440 N |
[
{
"content": "Evaluate the integral \\( \\int_0^1 \\frac{x^{10}-1}{\\log (x)}\\,dx \\).",
"role": "user"
}
] | \ln 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... | 4.2 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... | 1516.22 |
[
{
"content": "Determine the values of \\(\\alpha\\) for which the improper integral \\(\\iint_{\\Bbb R^2}\\frac{dxdy}{(1+x^2+xy+y^2)^\\alpha}\\) converges.",
"role": "user"
}
] | \alpha > 1 |
[
{
"content": "Given the complex number \\( z \\) such that \\( \\frac{z - \\mathrm{i}}{z - 1} \\) is a purely imaginary number, determine the maximum value of \\( |z| \\).",
"role": "user"
}
] | \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... | 108 days |
[
{
"content": "Let \\( f(z) = \\frac{az+b}{cz+d} \\) for \\( a, b, c, d \\in \\mathbb{C} \\). Given that \\( f(1) = i, f(2) = i^2, \\) and \\( f(3) = i^3 \\), find the real part of \\( f(4) \\) expressed as \\( \\frac{m}{n} \\) for relatively prime positive integers \\( m \\) and \\( n \\). Compute \\( m^2 + n^2... | 34 |
[
{
"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... | $122.38 million |
[
{
"content": "Calculate the sum of the series \\(\\sum_{n=1}^{\\infty} \\left(\\cos\\frac{a}{n^3}\\right)^{n^3}\\), where \\(a\\) is a constant.",
"role": "user"
}
] | \infty |
[
{
"content": "Determine the smallest closed set containing $C^1[-1, 1]$ in the space $({\\scr{B}}[-1, 1],||\\cdot||_\\infty)$. Provide your answer as a set description.",
"role": "user"
}
] | C[-1, 1] |
[
{
"content": "Compute the limit: $$\\lim\\limits_{x \\to 0}\\frac{\\int_0^{x^2}\\cos t^2dt}{(\\sin x)^2}$$.",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the Laplacian \\( \\Delta u = u_{xx} + u_{yy} + u_{zz} = \\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} + \\frac{\\partial^2 u}{\\partial z^2} \\) for the function \\( u(x,y,z) = \\frac{1}{\\sqrt{x^2 + y^2 + z^2}} \\) on \\( \\mathbb{R}^3 \\setminus \\{0\\} \... | 0 |
[
{
"content": "Find the most efficient sequence of digits to enter on a keycode doorlock to cover all possible 4-digit passcodes, given that the lock accepts any 4 consecutive digits as a passcode.",
"role": "user"
}
] | 10003 |
[
{
"content": "Determine the number of 3x4 matrices with non-negative integer entries such that each row sums to 3 and no column consists entirely of zeros.",
"role": "user"
}
] | 4380 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 14.11 m/s |
[
{
"content": "Determine all polynomials $P(x)$ with real coefficients such that $P(x)$ is a rational number if and only if $x$ is a rational number.",
"role": "user"
}
] | P(x) = ax + b |
[
{
"content": "Let $\\mathbb{R}^A$ be the vector space of functions from a set $A$ to $\\mathbb{R}$. Suppose $x_1, \\ldots, x_n \\in \\mathbb{R}^A$ are linearly independent. Determine whether there always exist $\\alpha_1, \\ldots, \\alpha_n \\in A$ such that the vectors $$(x_1(\\alpha_1), \\ldots, x_1(\\alpha_n... | Yes |
[
{
"content": "Let \\( f \\) be an analytic function on the closed unit disk \\( \\{z \\in \\mathbb{C} \\mid |z| \\leq 1\\} \\) such that \\( |f(z)| \\leq 1 \\) for all \\( z \\) in the disk. Determine which of the following is not a possible value of \\( (e^{f})^{''}(0) \\):\n\n(a) \\( 2 \\)\n(b) \\( 6 \\)\n(c)... | 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... | {\vec{v}_{ave.} = \dfrac{2v_o(v_1 + v_2)}{2v_o + v_1 + v_2} \hat{s}} |
[
{
"content": "What is the algorithm with the lowest time complexity for finding the nullspace of a singular $n \\times n$ square matrix $A$? For example, using Singular Value Decomposition (SVD) has a time complexity of $O(n^3)$. Provide the algorithm and its time complexity.",
"role": "user"
}
] | O(n^\omega) |
[
{
"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... | 47.5 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... | $2 million |
[
{
"content": "Calculate the Lebesgue integral of the function \\(x^2\\) over the interval \\((0,1]\\).",
"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... | nurture |
[
{
"content": "Let A be a positive definite matrix. If the inverse of A is expressed as \\( \\text{inverse}(A) = \\frac{1}{\\det(A)} B \\), is the matrix B still positive definite?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine if the set \\(B=\\{(x,y)\\in\\mathbb{R}^2:0<x+y\\leq 2, 0<x^2-y^2\\leq x-y\\}\\) is a Borel set.",
"role": "user"
}
] | Yes |
[
{
"content": "In an equilateral triangle ABC, three rods of lengths 3, 4, and 5 units are placed such that they intersect at a common point O, with the other ends at vertices A, B, and C, respectively. Given that BO = 3 units and CO = 4 units, find the angle ∠BOC.",
"role": "user"
}
] | 150^\circ |
[
{
"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... | 5.2 km/h |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0^+} \\sqrt{x} (\\ln(x))^{2015} \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Given the probability density function \\( f(x;\\lambda) = 1 - \\frac{2}{3}\\lambda + \\lambda\\sqrt{x} \\) for \\( 0 \\le x \\le 1 \\) and 0 otherwise, find the maximum likelihood estimate of the parameter \\( \\lambda \\) based on two independent observations \\( x_1 = \\frac{1}{4} \\) and \\( x... | -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... | 2 atoms and 1 lone pair |
[
{
"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... | 5296.33 |
[
{
"content": "Determine the radius of convergence of the power series \\( \\sum_{n=1}^\\infty b_n z^{n^3} \\), where \\( b_1=\\frac{1}{2} \\) and \\( b_n=4^{-n}b_{n-1} \\) for \\( n \\geq 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... | Fe(NO_3)_3(aq) + 3KI(aq) rightleftharpoons3K(NO_3)(aq) + FeI_3(aq) |
[
{
"content": "Use the divergence theorem to evaluate the surface integral \\( \\iint_S \\mathbf{F} \\cdot d\\mathbf{S} \\), where \\( \\mathbf{F}(x,y,z) = x^3 \\hat{\\imath} + y^3 \\hat{\\jmath} + z^3 \\hat{k} \\) and \\( S \\) is the surface of the solid bounded by the cylinder \\( x^2 + y^2 = 1 \\) and the pl... | 11\pi |
[
{
"content": "Determine all continuous functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) such that \\( f(x) = f(\\sin(x)) \\) for all \\( x \\in \\mathbb{R} \\).",
"role": "user"
}
] | f(x) = c |
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