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... | -71493.41 |
[
{
"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... | 1.36 pF |
[
{
"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... | 150.5 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... | $3,500.00 |
[
{
"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... | 150 V |
[
{
"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... | -1.136 |
[
{
"content": "Determine the order of the quotient group \\((G \\times G) / \\mathrm{diag}(G)\\) for a finite group \\(G\\).",
"role": "user"
}
] | |G| |
[
{
"content": "Find the volume of the region \\( F = \\{(x,y,z) \\in \\mathbb{R}^{3} \\mid 0 \\leq x \\leq 1, y^{2} + z^{2} \\leq x^{2}(1-x)^{2} \\} \\).",
"role": "user"
}
] | \dfrac{\pi}{30} |
[
{
"content": "During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^3,z^5,z^7,\\ldots,z^{2013}$ in that order; on Sunday, he begins at $1$ and delivers milk to houses located at $z^2,z^4,z^6,\\ldots,z^{2012}$ in that order. Eli alwa... | \frac{1005}{1006} |
[
{
"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... | 193.33% |
[
{
"content": "Let the set \\( T \\) consist of integers between 1 and \\( 2^{30} \\) whose binary representations contain exactly two 1s. If one number is randomly selected from the set \\( T \\), what is the probability that it is divisible by 9?",
"role": "user"
}
] | \dfrac{5}{29} |
[
{
"content": "Given a bounded Lipschitz domain $\\Omega \\subset \\mathbb{R}^n$ and a function $v \\in H^1_0(\\Omega)$ with $|v(x)| \\leq 1$ almost everywhere in $\\Omega$, determine if there exists a sequence $(v_j)_{j \\in \\mathbb{N}} \\subset \\mathcal{C}^\\infty_0(\\Omega)$ such that $|v_j(x)| \\leq 1$ alm... | Yes |
[
{
"content": "Consider the graph of the function $f(x) = x^3 - 3x$. How many squares can be formed by selecting four distinct points from this function's graph, given the following conditions: The vertices of the square, labeled $M$, $N$, $P$, and $Q$ in clockwise order, satisfy that $M$ and $P$ lie between the... | 1 |
[
{
"content": "Determine the minimum number of moves required to make an entire \\( m \\times n \\) checkerboard monochrome by selecting rectangles and inverting the colors of all squares within them.",
"role": "user"
}
] | \left\lfloor \dfrac{m}{2} \right\rfloor + \left\lfloor \dfrac{n}{2} \right\rfloor |
[
{
"content": "Evaluate the partial derivative \\( \\frac{\\partial f(t,v)}{\\partial v} \\) for the function \\( f(t, v) = 25(\\cos t - v + t/5) \\) at the point \\( t = 0, v = 1 \\).",
"role": "user"
}
] | -25 |
[
{
"content": "Provide an example of a separable metric space that is not complete.",
"role": "user"
}
] | \mathbb{Q} |
[
{
"content": "The circles \\(\\omega_{1}\\) and \\(\\omega_{2}\\) intersect at points \\(A\\) and \\(B\\). A circle centered at point \\(O\\) encompasses \\(\\omega_{1}\\) and \\(\\omega_{2}\\), touching them at points \\(C\\) and \\(D\\) respectively. Given that points \\(A\\), \\(C\\), and \\(D\\) are colline... | 90^\circ |
[
{
"content": "How many sequences of positive integers contain all positive integers?",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "Evaluate the limit: $$\\lim_{x\\to0^-} (x^{-x^x+1})$$",
"role": "user"
}
] | 1 |
[
{
"content": "Find the absolute value of the difference between the two possible values of \\( A \\) in the solution of the matrix equation:\n\\[\n\\left[\\begin{matrix}2A+1&-5\\\\-4&A\\end{matrix}\\right]^{-1}\\cdot \\left[\\begin{matrix}A-5&B\\\\2A-2&C\\end{matrix}\\right]= \\left[\\begin{matrix}14&D\\\\E&F\\... | D |
[
{
"content": "Find the limit of \\( y\\sin(1/x) \\) as \\((x,y) \\to (0,0)\\).",
"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... | 309 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... | 1.14 kg/m^3 |
[
{
"content": "Find the number of solutions in non-negative integers to the equation \\(x_1 + x_2 + x_3 + x_4 + x_5 = 9\\) where \\(x_1, x_2, x_3, x_4, x_5 \\neq 1\\). Use generating functions to determine the coefficient of \\(x^9\\) in the expression \\((1 + x^2 + x^3 + x^4 + \\dots)^5\\).",
"role": "user"... | 185 |
[
{
"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... | GDP = TNI + T + D + NFFI, or Gross Domestic Product = Total National Income + Sales Taxes + Depreciation + Net Foreign Factor Income |
[
{
"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 kJ |
[
{
"content": "In $\\triangle{ABC}$, the coordinates of point $A$ are $(1,2)$. The line $y=x$ is the perpendicular bisector of $AB$, and the line $x-2y+1=0$ is the angle bisector of $\\angle{BCA}$. The equation of line $BC$ is $ax+by-5=0$. Find the value of $a+b$. \\(\\text{Express your answer as a single number... | 2 |
[
{
"content": "Evaluate the limit \\( \\lim\\limits_{x\\to a}\\left(\\frac{x^x-a^x}{x^a-a^a}\\right) \\) using L'Hospital's rule.",
"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... | 309.21 mins |
[
{
"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... | 6.27e-21 J |
[
{
"content": "What is the maximum number of knights that can be placed on a 2013 x 2013 chessboard such that no two knights attack each other?",
"role": "user"
}
] | 2026085 |
[
{
"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... | Q = 1,440 units, Z = 4,320 units |
[
{
"content": "If a group $G$ satisfies the maximal condition, meaning every subgroup of $G$ is finitely generated, does it follow that $G$ itself is finitely generated?",
"role": "user"
}
] | Yes |
[
{
"content": "If the system of equations\n\n\\begin{align*}\n2x-y&=a,\\\\\n3y-6x &=b.\n\\end{align*}has a solution, find $\\frac{a}{b},$ assuming $b \\neq 0.$",
"role": "user"
}
] | -\frac{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... | 7.61% |
[
{
"content": "Evaluate the expression \\(\\frac{q}{1+q^2}+\\frac{q^2}{1+q^4}+\\frac{q^3}{1+q^6}\\), where \\(q\\) is a complex number such that \\(q^7=1\\) and \\(q\\neq 1\\).",
"role": "user"
}
] | -2 |
[
{
"content": "Determine the number of integral solutions for the equation \\(a + b + c + d = 40\\) such that the greatest common factor (GCF) of \\(a, b, c,\\) and \\(d\\) is 3, and none of the variables can be zero.",
"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... | 1.8% |
[
{
"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... | $7,850 |
[
{
"content": "Prove using the greedy algorithm that the chromatic number \\(\\chi(G)\\) of a finite graph \\(G\\) satisfies \\(\\chi(G) \\leq D(G) + 1\\), where \\(D(G)\\) is the maximum degree of a vertex in \\(G\\).",
"role": "user"
}
] | \chi(G) \leq D(G) + 1 |
[
{
"content": "Consider the function \\(f(x)=\\int_x^{2x}\\frac{\\sin t}{t}dt\\). Determine the value of \\(x\\) on the interval \\((0,\\frac{3\\pi}{2})\\) at which this function attains a local maximum.",
"role": "user"
}
] | \dfrac{\pi}{3} |
[
{
"content": "Let $(X, d)$ be a compact metric space and $f: X \\to X$ be a continuous function. If the diameter of $f(X)$ is less than the diameter of $X$, can $f$ be surjective? Provide a justification for your answer.",
"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... | 16.4 mm |
[
{
"content": "Suppose that $f$ is a function defined on the set of natural numbers such that \\[ f(1)+ 2^2f(2)+ 3^2f(3)+\\ldots+n^2f(n) = n^3f(n) \\] for all positive integers $n$. Given that $f(1)= 2013$, find the value of $f(2013)$. \\( \\boxed{} \\)",
"role": "user"
}
] | \dfrac{1}{2013} |
[
{
"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... | 1/3600 |
[
{
"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... | 0.63 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... | 30 ms^-1 |
[
{
"content": "Determine whether the function \\( f(x) = \\frac{1}{\\sqrt{x(1-x)}} \\) is Lebesgue-integrable on the interval \\((0,1)\\), and if so, compute its integral.",
"role": "user"
}
] | \pi |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 0.537 rev |
[
{
"content": "A square has sides of length 2. Set $\\cal S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $\\cal S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$.",
"role": "... | 86 |
[
{
"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... | V=K\sqrt{\frac{T}{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... | 30442 J |
[
{
"content": "Let $V$ be a finite-dimensional vector space over the complex numbers $\\mathbb{C}$, and let $T: V \\rightarrow V$ be a linear transformation with a minimal polynomial of $x^{13}(x-1)$. Determine the minimal polynomial of $T^4$.",
"role": "user"
}
] | x^4(x - 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... | y(t) = R(1 - cos(sqrt(k)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... | $0.054 |
[
{
"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... | 13169.38 |
[
{
"content": "Compute the line integral of the vector field \\( F = \\langle 6y, -6x \\rangle \\) over the circle \\( x^2 + y^2 = 4 \\), oriented clockwise.",
"role": "user"
}
] | 48\pi |
[
{
"content": "Evaluate the integral \\( \\int_0^1 A(x) \\, dx \\), where the function \\( A: [0,1] \\rightarrow \\mathbb{R} \\) is defined by \\( A(x) = 1 \\) if \\( x \\) is an algebraic number and \\( A(x) = 0 \\) otherwise.",
"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... | 519.615 km/h |
[
{
"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... | 491 m/s^2 |
[
{
"content": "Find the magnitude \\( |z| \\) of the complex number \\( z \\) that satisfies the equation \\( 3z^6 + 2iz^5 - 2z - 3i = 0 \\).",
"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... | 64.07 g/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... | 406K |
[
{
"content": "Evaluate the integral \\(\\int_{|z|=1} \\frac{dz}{z^4+16z^2+4}\\) using the residue theorem.",
"role": "user"
}
] | 0 |
[
{
"content": "Simplify the ordinal expression \\(((\\omega + 1) + (\\omega + 1))^\\omega\\) and express it in Cantor Normal Form.",
"role": "user"
}
] | \omega^{\omega} |
[
{
"content": "Apply the linear multistep method given by the equation \\( x_{n+2} - 2x_{n+1} + x_{n} = h(f_{n+1} - f_{n}) \\) to solve the initial value problem \\( x'(t) = -x(t) \\) with the initial condition \\( x(0) = 1 \\) and starting values \\( x_0 = x_1 = 1 \\). Express the solution in exact form.",
... | 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... | 1876 s |
[
{
"content": "Determine whether there exists a family of open, bounded intervals of rational numbers (intervals of the form \\(]x,y[\\) where \\(x,y \\in \\mathbf{Q}\\)) whose union is the set of all rational numbers, \\(\\mathbf{Q}\\).",
"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... | 8.4% |
[
{
"content": "Find the sum of the roots of $\\tan^2x-9\\tan x+1=0$ that are between $x=0$ and $x=2\\pi$ radians.",
"role": "user"
}
] | 3 \pi |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\left( \\frac{4n^2+5n-6}{4n^2+3n-10}\\right)^{3-4n}$$",
"role": "user"
}
] | e^{-2} |
[
{
"content": "Calculate the value of \\( \\frac{a_{20}}{a_{20}+b_{20}} \\) for the following recursive system:\n\n\\[\n\\begin{cases}\na_{n+1} = -2a_n - 4b_n \\\\\nb_{n+1} = 4a_n + 6b_n \\\\\na_0 = 1, \\quad b_0 = 0\n\\end{cases}\n\\]",
"role": "user"
}
] | -39 |
[
{
"content": "Given a finite field \\( K \\) with \\( |K| = q \\) and an irreducible polynomial \\( f \\) with a root \\( \\alpha \\) and degree \\( n \\), determine if \\( |K(\\alpha)| = q^n \\). Justify your answer.",
"role": "user"
}
] | q^n |
[
{
"content": "In triangle $\\triangle ABC$, $\\angle A = 60^{\\circ}$ and $AB > AC$. The circumcenter is point $O$, and the altitudes $BE$ and $CF$ intersect at the orthocenter $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, such that $BM = CN$. Determine the value of $\\frac{MH + NH}{OH}$."... | \sqrt{3} |
[
{
"content": "Evaluate the limit: $$\\lim_{n\\to\\infty}\\left(\\frac{1}{n}\\sum_{i=0}^{n} \\sqrt{\\frac{i}{n}}\\right)$$",
"role": "user"
}
] | \dfrac{2}{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... | 21.41 s |
[
{
"content": "Find the limit \\( \\lim_{n \\to \\infty} \\left( \\sqrt[10]{n^{10} + 8n^9} - n \\right) \\) using the standard limits \\( \\lim_{n \\to \\infty} \\frac{e^{\\alpha_n}-1}{\\alpha_n} = 1 \\) and \\( \\lim_{n \\to \\infty} \\frac{\\ln(1+\\beta_n)}{\\beta_n} = 1 \\) where \\( \\alpha_n, \\beta_n \\to ... | \dfrac{4}{5} |
[
{
"content": "Find the coefficient of \\(x^5\\) in the expansion of the generating function \\((1+x+x^2+x^3)^4\\).",
"role": "user"
}
] | 40 |
[
{
"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... | 45.5 J |
[
{
"content": "Determine the cardinality of the set \\( \\prod_{n\\in\\mathbb{N}} (n+1) \\).",
"role": "user"
}
] | 2^{\aleph_0} |
[
{
"content": "The trapezoid shown has a height of length $12\\text{ cm},$ a base of length $16\\text{ cm},$ and an area of $162\\text{ cm}^2.$ What is the perimeter of the trapezoid? [asy]\ndraw((0,0)--(0,12)--(11,12)--(16,0)--cycle);\nlabel(\"12 cm\",(-1,6),W);\ndraw((0.8,0)--(0.8,0.8)--(0,.8));\ndraw((0,11.2)... | 52 |
[
{
"content": "In the tetrahedron ABCD, let M and N be the centroids of the triangular faces ABD and BCD, respectively. Given that the length of edge AC is a, determine the length of the segment MN.",
"role": "user"
}
] | \dfrac{a}{3} |
[
{
"content": "Evaluate the complex integral \\( \\int_C z^3\\exp{\\left(\\dfrac{-1}{z^2}\\right)} dz \\) where \\( C \\) is the contour \\( |z|=5 \\). Use the Residue Theorem to find the value of the integral.",
"role": "user"
}
] | \pi i |
[
{
"content": "Geoff and Trevor each roll a fair six-sided die. What is the probability that the product of the numbers they roll is even?",
"role": "user"
}
] | \dfrac34 |
[
{
"content": "Consider the Hilbert space \\(\\mathcal H=L^2(\\mathbb R_{+})\\) and the operator \\(Af(x)=-f''(x)\\) with domain \\(D(A)=\\{f\\in H_2(\\mathbb R_{+})\\;\\;| \\;\\;f'(0)+\\beta f(0)=0\\}\\) for some \\(\\beta \\in \\mathbb R\\). Determine the conditions on \\(\\beta\\) for which the operator \\(A\... | \beta > 0 |
[
{
"content": "Determine if there exists a vector $y \\in \\ell^\\infty$ with $||y||_\\infty = 1$ such that for the operator $S:(\\ell^\\infty,||.||_\\infty) \\rightarrow (c_0,||.||_\\infty)$ defined as $S(y)=(y_1,\\frac{1}{2}y_2,...,\\frac{1}{n}y_n,...)$, we have $||S(y)||_\\infty = ||S||$. Provide a justificat... | 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... | 25.13 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... | Preferred stock - par value and Additional paid-in capital- preferred stock |
[
{
"content": "Calculate \\( \\lim_{n \\to \\infty} \\mathbb{P}(X + n = Y) \\) where \\( X \\sim \\mathrm{Poisson}(n) \\) and \\( Y \\sim \\mathrm{Poisson}(2n) \\) are independent random variables.",
"role": "user"
}
] | 0 |
[
{
"content": "Is it possible to construct a surjective homomorphism from the multiplicative group of 12th roots of unity to the group U(18), where U(18) is the multiplicative group modulo 18?",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether it is possible to construct an injection from the power set of the rational numbers to the power set of the natural numbers.",
"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... | 4.48 |
[
{
"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... | $3.13 |
[
{
"content": "Determine the constant $c_0$ in the power series representation of the shifted exponential function $y = e^{-a(gx-b)^{2}}$. The power series is given by:\n\n$$e^{-a(gx-b)^{2}} = c_{0}\\sum_{k=1}^{\\infty}\\frac{(-2ag)^{k}}{(2k)!!}(gx-b)^{k}$$\n\nwhere the coefficients $c_{2k}$ are defined by the r... | 1 |
[
{
"content": "Determine whether the function $h_a(t) = f(p + ta)$ has a local minimum at $t = 0$ for $p = (0,0)$ and $f(x,y) = x^2 - 2xy + 3y^2$, for any non-zero vector $a = (a_1, a_2) \\in \\mathbb{R}^2$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Find the positive root of\n\\[x^3 - 3x^2 - x - \\sqrt{2} = 0.\\]",
"role": "user"
}
] | 2 + \sqrt{2} |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 12 g |
[
{
"content": "Given the functional equation \\( f(f(x))[1+f(x)] = -f(x) \\), find the value of \\( f(3) \\).",
"role": "user"
}
] | -\dfrac{3}{4} |
[
{
"content": "Find the radius of convergence of the power series \\( \\sum_{k=0}^{\\infty } n^{k^{2}} \\).",
"role": "user"
}
] | 1 |
[
{
"content": "Prove that if \\( \\int_0^x f(t)\\, dt = f(x) \\) for all \\( x \\), then \\( f(x) \\) is identically zero.",
"role": "user"
}
] | 0 |
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