prompt list | ground_truth string |
|---|---|
[
{
"content": "Find the natural number \\( x = 9^n - 1 \\) such that \\( x \\) has exactly three distinct prime divisors, one of which is 11. Determine the value of \\( x \\).",
"role": "user"
}
] | 59048 |
[
{
"content": "If $H$ is a finite-dimensional Hilbert space, are the norm topology, strong operator topology, weak operator topology, σ-strong topology, and σ-weak topology on $B(H)$ all equivalent?",
"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... | 710 |
[
{
"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... | 0.27 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... | 0.029 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 77.72 |
[
{
"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... | 0.32 degrees |
[
{
"content": "Determine whether complete regularity implies regularity in topological spaces.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the dual one-form field corresponding to the vector field $V = x\\, \\frac{\\partial}{\\partial z} + y\\, \\frac{\\partial}{\\partial x}$ in $(x, y, z) \\in \\mathbb{R}^3$. \\( \\text{Express your answer as a one-form.} \\)",
"role": "user"
}
] | y \, dx + x \, dz |
[
{
"content": "Given a $(0,1)$-matrix $B$ with $n$ columns and $n^3$ rows, where each row contains exactly four $1$'s, determine the minimum number of columns that must be deleted so that each row contains fewer than four $1$'s.",
"role": "user"
}
] | n - 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... | 3291.3 s |
[
{
"content": "Find the value of \\( \\dfrac{d z^*}{dz} \\), where \\( z^* \\) denotes the complex conjugate of \\( z \\) and \\( z \\in \\mathbb{C} \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Determine whether the expression \\( \\frac{1^\\alpha + 2^\\alpha + \\cdots + n^\\alpha}{n^{\\alpha+1}} \\) diverges to infinity as \\( n \\to \\infty \\) for \\( \\alpha \\in (-\\infty, -1) \\).",
"role": "user"
}
] | \infty |
[
{
"content": "In a universe $U$ shaped like $\\mathbb{R}^3$, points are randomly distributed in a Poisson fashion with density parameter $\\lambda$, such that in any measurable set $S$ of volume $V$, the expected number of points in $S$ is $V$. Consider a random line $L$ that intersects the origin. What is the ... | 0 |
[
{
"content": "Determine if metrics $d_1$ and $d_2$ on a space $X$ are equivalent given that for any sequence $\\{x_n\\}_{n=1}^\\infty \\subset X$ and point $x_0 \\in X$, the condition \\( \\lim_{n \\to \\infty}d_1(x_n,x_0)=0 \\iff \\lim_{n \\to \\infty}d_2(x_n,x_0)=0 \\) holds. Equivalence implies that they ind... | 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... | 14.52% |
[
{
"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... | 6.36 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... | LOS GATOS CORPORATION
Balance sheet
at December 31,2016
Assets
Current Assets
Cash
$39,000
Accounts receivable, net of allowance for uncollectible accounts of $24,000
$74,000
Inventories
$74,000
Total Current Assets
$187,000
Investments
Bond sinking fund
$39,000
Notes receivables
$39,000
Total Investments
$78,000
Property, plant and equipment: Machinery
$228,000
Less: Accumulated depreciation
$89,000
Net Property, plant and equipment
$139,000
Intangible assets
Franchise
$49,000
Total Asset
$453,000
Liabilities and shareholder's equity
Current Liabilities
Accounts Payable
$88,000
Interest payable
$24,000
Notes payable
$69,000
Total Current liabilities
$181,000
Long term liabilities
Bonds payable
$129,000
Shareholder's equity
Common stock, no par value: $200,000 shares authorized, shares issued and outstanding$89,000
Retained Earnings
$54,000
Total shareholder's equity
$143,000
Total liabilities and shareholder's equity
$453,000 |
[
{
"content": "Determine the value of \\( k \\) for the function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfies the equation:\n\n\\[ f(x+y) = e^{y}f(x) + e^{x}f(y) + k, \\quad \\forall x,y \\in \\mathbb{R}. \\]",
"role": "user"
}
] | 0 |
[
{
"content": "Solve the equation \\( t \\log t = 10 \\) for \\( t \\).",
"role": "user"
}
] | 10 |
[
{
"content": "Consider a projection valued measure $E$ from the complex plane into the projections on a Hilbert space $\\mathcal{H}$, with compact support $K \\subseteq \\mathbb{C}$. Determine the spectrum of the normal operator $\\int_K z \\, dE(z)$ in $\\mathcal{B}(\\mathcal{H})$. \\( \\mathcal{B}(\\mathcal{H... | 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... | 17.28 cm |
[
{
"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... | 12.90% |
[
{
"content": "Consider a commutative ring $A$ without a multiplicative identity. Let $r \\in A$ be a nonzero element such that $r$ is contained in every proper prime ideal of $A$. Can we conclude that $r = 0$? Justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Determine the values of \\(\\alpha\\) for which the integral \\(I = \\int\\limits_0^{+\\infty} \\frac {\\ln^{\\alpha} x}{x} \\sin x \\, \\text{d} x\\) converges uniformly on the interval \\([0, 1]\\).",
"role": "user"
}
] | [0, 1] |
[
{
"content": "Evaluate the limit \\( \\lim_{x\\to\\infty} \\frac{1}{x \\sqrt{x}} \\int_{0}^{x} \\ln \\left( \\frac{t^3 + 2}{t^2 + 1} \\right) dt \\) using L'Hopital's Rule. Justify the application of the rule and find the value of the limit.",
"role": "user"
}
] | 0 |
[
{
"content": "Given that the determinant of the 3x3 matrix \\( A \\) is \\( \\det(A) = -4 \\), where:\n\n\\[ A = \\begin{vmatrix} a_1 & a_2 & a_3 \\\\ b_1 & b_2 & b_3 \\\\ c_1 & c_2 & c_3 \\end{vmatrix} \\] \n\nFind the determinant of the matrix \\( B \\):\n\n\\[ B = \\begin{vmatrix} 2a_3 & 2a_2 & 2a_1 \\\\ b_3... | 8 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.08e-2 m |
[
{
"content": "Determine whether a real analytic function is equal to its Taylor series at the endpoints of the interval of convergence, assuming the series converges at the endpoint.",
"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... | 588 pm |
[
{
"content": "Evaluate the integral \\( \\int_1^\\infty \\frac{1}{x^4 \\sqrt{x^2 + 3} } \\, dx \\) using a suitable substitution.",
"role": "user"
}
] | \dfrac{4}{27} |
[
{
"content": "On a genus 1 smooth projective curve $C$ over $\\mathbb{C}$, if $p$ and $q$ are two points and there exists a rational function $f$ on $C$ such that the divisor of $f$ is $(f) = p - q$, must $p$ equal $q$? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Evaluate the limit \\( \\lim_{n\\rightarrow \\infty}\\int _{a}^b f(t)\\sin(nt)\\,\\mathrm dt \\) where \\( f \\) is a continuous function on \\([a, b]\\).",
"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... | $807,000 |
[
{
"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... | -$1,350,000 |
[
{
"content": "Let \\( f(x) \\) be defined as follows:\n\\[\nf(x) = \\begin{cases} e^{-1/x^2}, & x \\neq 0 \\\\ 0, & x = 0 \\end{cases}\n\\]\nCalculate \\( f'''(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... | 11.063 g |
[
{
"content": "Consider a compact connected Riemannian 2-manifold $M$ with positive curvature and diameter 1. Determine the supremum of the average distance between two randomly chosen points on $M$, distributed according to the volume form on $M$. Assume $M$ is diffeomorphic to a 2-sphere. What is this supremum... | \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... | Work in Process Inventory - Job 1: $2,361
Work in Process Inventory - Job 2: $1,736
Work in Process Inventory - Job 3: $1,676
Factory Overhead: $1,137
Factory Wages Payable: $5,862
Payroll Taxes Payable: $1,048 |
[
{
"content": "Given a parabola \\( C: y^{2}=2px(p>0) \\) and points \\( A(a, 0) \\) and \\( A'(-a, 0) \\) where \\( a > 0 \\), a line passing through point \\( A' \\) intersects the parabola at points \\( P \\) and \\( Q \\). Find the sum of the slopes of lines \\( AP \\) and \\( AQ \\).",
"role": "user"
... | 0 |
[
{
"content": "Find the integer value of $x$ such that \\( \\sqrt{x+\\frac{1}{2}\\sqrt{2011}} - \\sqrt{x-\\frac{1}{2}\\sqrt{2011}} = y \\), where \\( x, y \\in \\mathbb{Z} \\).",
"role": "user"
}
] | 503 |
[
{
"content": "Given a set of $n$ vectors in $n$-dimensional space, where all angles between any two vectors are equal, determine if the cosine of the angle between any two vectors is $-\\frac{1}{n}$. Answer with True or False.",
"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... | $50,120.50 |
[
{
"content": "Calculate the expected value of \\(\\frac{r}{n}\\), where \\(r\\) is the sum of the first \\(k\\) sorted frequencies in an array \\(\\mathbf{q}\\). The array \\(\\mathbf{q}\\) is constructed by sampling an array \\(\\mathbf{p}\\) of size \\(m\\) with elements uniformly distributed in \\([0,1]\\), ... | \dfrac{k}{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... | 9000 N |
[
{
"content": "Let $k \\in \\mathbb{N}$ and let $f : \\mathbb{R}^d \\rightarrow [0,\\infty)$ be a lower semicontinuous function. Determine whether there exists an increasing sequence of $k$-times continuously differentiable functions $(g_n)_n \\subset C^k(\\mathbb{R}^d ; [0,\\infty))$ that converges pointwise to... | Yes |
[
{
"content": "Let \\( A \\) and \\( B \\) be subsets of a set \\( M \\) and define \\( S_0 := \\{A,B\\} \\). For \\( i \\ge 0 \\), define \\( S_{i+1} \\) inductively as the collection of subsets of \\( M \\) taking the form \\( C \\cup D \\), \\( C \\cap D \\), or \\( M - C \\) where \\( C,D \\in S_i \\). Deter... | 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... | 0.824 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\sum_{k=0}^n \\frac{\\binom{n}{k}}{k2^n+n} \\]",
"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... | $114,379.80 |
[
{
"content": "Evaluate the limit \\( \\lim\\limits_{x \\to +\\infty} \\frac{\\cos(2x^{-\\frac{1}{2}})-1}{\\frac{\\pi}{2}-\\arctan(3x)} \\) without using integrals, derivatives, or L'Hopital's rule.",
"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... | $61,981.87 |
[
{
"content": "Find the limit of the expression \\( \\frac{\\ln(\\frac{\\cos\\alpha}{\\cos\\beta})}{\\beta-\\alpha} \\) as \\( \\beta \\) approaches \\( \\alpha \\), given that \\( 0<\\alpha<\\beta<\\frac{\\pi}{2} \\).",
"role": "user"
}
] | \tan \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... | 56.75 Days |
[
{
"content": "Evaluate the expression \\[ P = \\dfrac{a^2}{2a^2+bc} + \\dfrac{b^2}{2b^2+ca} + \\dfrac{c^2}{2c^2+ab} \\] given that \\( a + b + c = 0 \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Calculate the limit: $$\\lim_{x\\to0}\\frac{1}{x}\\left(\\sqrt[3]{\\frac{1-\\sqrt{1-x}}{\\sqrt{1+x}-1}}-1\\right)$$",
"role": "user"
}
] | \dfrac{1}{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... | $155,000 |
[
{
"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... | 29580 bottles |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 8.6 |
[
{
"content": "Determine whether there exists a real number \\(a\\) such that \\[ \\int_{-a}^a \\frac{x^4}{e^x+1}dx = -\\frac{32}{5}. \\]",
"role": "user"
}
] | -2 |
[
{
"content": "If $M$ is a free $R$-module of finite rank $n$ over a principal ideal domain $R$, do proper submodules of $M$ have strictly less rank than $M$? Answer yes or no and justify your answer.",
"role": "user"
}
] | No |
[
{
"content": "Suppose $G_1$ and $G_2$ are groups with orders $|G_1| = 2p$ and $|G_2| = 2q$, where $p$ and $q$ are distinct prime numbers. Consider a non-trivial homomorphism $\\phi: G_1 \\rightarrow G_2$. Determine the order of the kernel, $|\\ker(\\phi)|$. \\(\\boxed{\\text{Your answer here}}\\)",
"role": ... | p |
[
{
"content": "Given that \\( N \\) is the number of triples \\((x, y, z)\\) of positive integers such that \\( x < y < z \\) and \\( x \\cdot y \\cdot z = 2^2 \\cdot 3^2 \\cdot 5^2 \\cdot 7^2 \\cdot 11^2 \\cdot 13^2 \\cdot 17^2 \\cdot 19^2 \\), find the remainder when \\( N \\) is divided by 100. Choose the cor... | C |
[
{
"content": "Calculate \\( \\lfloor z \\rfloor \\) where \\( z = \\frac{ \\left\\{ \\sqrt{3} \\right\\}^2 - 2 \\left\\{ \\sqrt{2} \\right\\}^2 }{ \\left\\{ \\sqrt{3} \\right\\} - 2 \\left\\{ \\sqrt{2} \\right\\} } \\).",
"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... | 85.67 |
[
{
"content": "Evaluate the limit: $$\\lim_{x \\to 0} \\frac{x^6(1-\\cos(x^6))}{(x-\\sin(x))^6}.$$",
"role": "user"
}
] | 23328 |
[
{
"content": "Evaluate the limit using the Dominated Convergence Theorem: $$\\lim\\limits_{n\\rightarrow \\infty}\\int_{0}^{\\pi}\\left(\\sin\\left(\\frac{x}{n}\\right)\\right)^{n}dx$$",
"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... | -$5,000 |
[
{
"content": "Calculate the functional derivative of the functional \\( J(y) = \\int_{x_0}^{x_1} \\left( y^6(x) + 3 y^2(x) \\right) \\, \\mathrm{d}x \\).",
"role": "user"
}
] | 6y^5(x) + 6y(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... | 0.0045 m^3 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.171 m/s^2 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 2.5 hours and 2.3 hours |
[
{
"content": "Given the series:\n\\[ a = 1 + \\frac{x^{3}}{3!} + \\frac{x^{6}}{6!} + \\cdots \\]\n\\[ b = \\frac{x}{1!} + \\frac{x^{4}}{4!} + \\frac{x^{7}}{7!} + \\cdots \\]\n\\[ c = \\frac{x^{2}}{2!} + \\frac{x^{5}}{5!} + \\frac{x^{8}}{8!} + \\cdots \\]\nLet \\( u = a^2 + b^2 + c^2 \\). Find the value of \\( u... | \dfrac{5}{3} |
[
{
"content": "Determine whether there exists a real solution to the equation \\(A^2 + B^2 + C^2 = \\frac{(A+ iB )(A-iB) + (A-i B) (A+iB)}{2} + C^2\\), where \\(A\\), \\(B\\), and \\(C\\) are arbitrary \\(n \\times n\\) matrices and \\(i\\) is the imaginary unit.",
"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... | 8.73e6 m/s |
[
{
"content": "Does every maximum independent set in a tree contain a leaf? Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.09 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... | 9.87e-4 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... | 0.0697 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... | 881.51 kg |
[
{
"content": "Evaluate the limit: $$\\lim_{n \\to \\infty} \\frac{\\left[(n+1)(n+2)\\cdots(2n)\\right]^{\\frac{1}{n}}}{n}$$",
"role": "user"
}
] | \dfrac{4}{e} |
[
{
"content": "Let $L$ be a nonperfect Lie algebra (i.e., $L \\ne [L, L]$) which is not nilpotent. Determine whether there exists an ideal $M$ of $L$ such that the intersection of the derived subalgebra of $L$ and the center of $M$ is nonzero.",
"role": "user"
}
] | No |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | For a generic compound A2B3 where A is the cation A3+ and B is the anion B2-, the ionic equation for its dissolution is A2B3(s) <=> 2A3+ + 3B2-. For the ionic compound AB3, where A is the cation A3+ and B is the anion B-, the ionic equation for its dissolution is AB3(s) <=> A3+ + 3B- |
[
{
"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... | 1.732e-7 T |
[
{
"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... | Date accounts & explanation debit ($) credit ($)
Dec 3 Notes receivable 36,000
Account receivable 36,000
(To record notes receivable)
Dec 31 Interest receivable (36,000 * 6% * 28/ 360) 168
Interest revenue 168
(To record adjusting entry)
Dec 31 Interest revenue 168
Income summary 168
(To record closing entry)
Mar 2 Cash 36,540
Notes receivable 36,000
Interest receivable 168
Interest revenue (36,000 * 6% * 62/ 360) 372
(To record amount received) |
[
{
"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... | 0.866c |
[
{
"content": "Given a $3 \\times 3$ matrix $A$ with rational entries such that $A^3 = 2A$ and $A$ is not the zero matrix, calculate the trace of $A^2$, denoted as $\\text{tr}(A^2)$. It is known that the determinant of $A$, $\\det(A)$, is zero.",
"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... | s(4) = \int_{0}^{4} v(\tau) d\tau + 5 |
[
{
"content": "Calculate \\(3^{P^5} \\mod P\\) where \\(P = 174449211009120179071170527\\).",
"role": "user"
}
] | 3 |
[
{
"content": "Determine the cardinality of the conjugacy class of the permutation $\\sigma = (1,3,5)(2,9)(4,12)(6,11)(7,10,8)$ in the symmetric group $S_{12}$. The permutations in the conjugacy class are of the form $(abc)(de)(fg)(hi)(jkl)$ with $a, \\ldots, l \\in \\{1,2,3,\\ldots,12\\}$.",
"role": "user"
... | 554400 |
[
{
"content": "Provide an example of a simple equation with rational coefficients whose graph has no rational coordinates.",
"role": "user"
}
] | x^2 + y^2 = 3 |
[
{
"content": "Compute the contour integral:\n\\[ \\oint_{|z-1-i| = 5/4} \\frac{\\log(z)}{(z-1)^2} \\, dz. \\]",
"role": "user"
}
] | 2\pi i |
[
{
"content": "Find the remainder when $70!$ is divided by $125$ using the Chinese Remainder Theorem.",
"role": "user"
}
] | 0 |
[
{
"content": "Let $U = Z(x^2 - y)$ in $k[x,y]$, where $k$ is a field. Show that $I(U) = (x^2 - y)$. Recall that $Z(I) = \\{(x,y) \\in k^2 \\mid \\forall f \\in I, f(x,y) = 0\\}$ and $I(U) = \\{f \\in k[x,y] \\mid \\forall (x,y) \\in U, f(x,y) = 0\\}$. Prove that $f \\in I(U)$ if and only if $f(x, x^2) = 0$ for ... | (x^2 - y) |
[
{
"content": "For any integer \\( n \\) where \\( n \\geq 2 \\), the positive numbers \\( a \\) and \\( b \\) satisfy the equations \\( a^n = a + 1 \\) and \\( b^{2n} = b + 3a \\). Determine the correct relationship between \\( a \\) and \\( b \\):\n\n(A) \\( a > b > 1 \\) \n(B) \\( b > a > 1 \\) \n(C) \\( a ... | A |
[
{
"content": "Determine the power to which the root \\( x_0 \\) of the equation \\( x^{11} + x^{7} + x^{3} = 1 \\) must be raised to obtain the value \\( x_0^{4} + x_0^{3} - 1 \\).",
"role": "user"
}
] | 15 |
[
{
"content": "Determine the cardinality of the set $A$, where $A$ consists of all sequences $a = (a_n)_{n \\ge 1}$ of natural numbers satisfying the following conditions for all $n \\ge 1$: (a) $a_n < a_{n+1}$ (strictly increasing), and (b) the limit $\\lim_{n\\to \\infty} \\frac{\\#\\{j \\mid a_j \\le n\\}}{n}... | 2^{\aleph_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... | $2,895 |
[
{
"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... | Cl2(l) + 2Br-(aq) -> 2Cl-(aq) + Br2(l) |
[
{
"content": "Let the sequence \\( \\{x_n\\} \\) be defined by \\( x_1 \\in \\{5, 7\\} \\) and, for \\( k \\ge 1, x_{k+1} \\in \\{5^{x_k} , 7^{x_k} \\} \\). Determine the sum of all possible values for the last two digits of \\( x_{2012} \\).",
"role": "user"
}
] | 75 |
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