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... | 8.75% |
[
{
"content": "Does there exist an infinite \\(\\sigma\\)-algebra that contains only countably many members?",
"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... | Prairie Corp.
Stockholders' Equity Section
As of December 31, 2014
ParticularsAmount ($)
Stockholder's equity:
Common stock216,000
Add: Paid-in capital in excess of cost, Treasury stock900
Add: Paid-in capital in excess of stated value, Preferred stock4,000
Add: Preferred stock62,000
Total paid-in capital (A)282,900
Retained earnings:
Appropriated5,500
Add: Unappropriated26,250
Total retained earnings (B)31,750
Treasury stock (C)3,600
Total stockholder's equity (D = A + B - C)311,050 |
[
{
"content": "Calculate the limit: $$\\lim_{x\\to+\\infty}\\left(\\frac{x^2 -x +1}{x^2}\\right)^{\\frac{-3x^3}{2x^2-1}}$$ without using de l'Hôpital's rule.",
"role": "user"
}
] | e^{\frac{3}{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... | -2758.92 m/s^2 |
[
{
"content": "Determine whether the differential equation \\( \\frac{\\text{d}y}{\\text{d}x}=-y+\\sin(x^2) \\) has a unique and well-defined solution for \\( y \\in [0,\\infty) \\) given any initial condition \\( y(0) \\in \\mathbb{R} \\).",
"role": "user"
}
] | Yes |
[
{
"content": "Determine whether the following statement is true or false: If $p$ is an irreducible polynomial in $n$ indeterminates, then the ideal $(p)$ generated by it is prime.",
"role": "user"
}
] | A |
[
{
"content": "Calculate the surface area of the portion of the cylinder \\( x^2+y^2 = 2x \\) that lies inside the cone \\( z^2 = x^2 + y^2 \\).",
"role": "user"
}
] | 16 |
[
{
"content": "Consider a sequence of operators \\(\\{A_n\\}_{n=1}^{\\infty}\\) where each \\(A_n: \\ell_1 \\to \\ell_1\\) and \\(\\|A_n\\|_1 \\leq 1\\). Given that the unit ball \\(\\{T : \\ell_1 \\to \\ell_1 \\text{ such that } \\|T\\|_1 \\leq 1\\}\\) is compact in the weak operator topology, does the sequence... | Yes |
[
{
"content": "A toy consists of 9 differently sized rings and a tapering pole. The rings can be stacked on the pole in any order until the top of the pole is reached, at which point they must be in decreasing order of size. How many different configurations of the rings can be created without reaching the top o... | 623530 |
[
{
"content": "Consider the space $H^2(\\mathbb{D})$. Let $||\\cdot||_X$ be a norm on $H^2(\\mathbb{D})$ such that the point evaluations at $z = 0$ and $z = 1/2$ are continuous. If $H^2(\\mathbb{D})$ is complete under $||\\cdot||_X$, does it follow that $||\\cdot||_X$ is equivalent to $||\\cdot||_H$? Provide a j... | Yes |
[
{
"content": "Let \\( T \\) be a \\( 4 \\times 4 \\) real matrix such that \\( T^4 = 0 \\). Define \\( k_i = \\dim \\ker T^i \\) for \\( 1 \\leq i \\leq 4 \\). Determine which of the following sequences \\( k_1 \\leq k_2 \\leq k_3 \\leq k_4 \\) is not possible:\n\n(i) \\( 3 \\leq 4 \\leq 4 \\leq 4 \\)\n(ii) \\(... | ii |
[
{
"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... | 30280.90 |
[
{
"content": "Determine if the following statement is true: There exists a unique holomorphic function $f$ on the open disc $D(0,3/2)$ in the complex plane with center $0$ and radius $3/2$, satisfying the conditions: $|f(z)|=1/2$ when $|z|=1$, $z=0$, $z=1/2$, and $z=-1/2$ are the only simple zeros of $f$ in $D(... | True |
[
{
"content": "Determine whether there exists a non-Abelian group $G$ such that the map $x \\mapsto x^n$ is a homomorphism for some positive integer $n$. Provide a justification for your answer.",
"role": "user"
}
] | Yes |
[
{
"content": "Determine the values of \\( \\alpha > 0 \\) for which the improper integral \\( \\int_{2}^{\\infty} \\frac{\\mathrm{d}x}{x^{2} \\ln^{\\alpha}(x)} \\) is convergent.",
"role": "user"
}
] | \alpha > 0 |
[
{
"content": "Find all values of \\(a\\) for which the image of the function \\(y=\\frac{\\sqrt{a}-2\\cos x+1}{\\sin^2x+a+2\\sqrt{a}+1}\\) contains the interval \\([2, 3]\\).",
"role": "user"
}
] | 0 |
[
{
"content": "Calculate the limit \\(\\lim_{n\\to\\infty}(2-\\sqrt[n]{2})^n\\).",
"role": "user"
}
] | \dfrac{1}{2} |
[
{
"content": "Let $n$ be a positive integer and $x_1, x_2, \\ldots, x_n$ be independent and identically distributed $\\mathop{Bernoulli}(\\frac{1}{2})$ random variables. Using Hoeffding's inequality, determine an upper bound for the probability that the sum $S_n = x_1 + x_2 + \\cdots + x_n$ exceeds $\\frac{n}{2... | \dfrac{1}{n^2} |
[
{
"content": "Aerith and Bob take turns picking a nonnegative integer, each time changing exactly one digit from the other's last number. The first person to pick a number that they picked before loses. If Aerith goes first, and both play optimally, who will win the game?",
"role": "user"
}
] | Bob |
[
{
"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... | Trial Balance
Debit
Credit
Cash
$4,000
Accounts receivables
$8,800
Supplies
$300
Trucks
$132,000
Building
$48,000
Accounts payable
$4,000
Note payable
$4,000
Common stock
$72,000
Dividends
$5,400
Retained earnings
$50,000
Service revenue
$80,000
Insurance expense
$600
Supplies expense
$400
Salary
$7,000
Fuel expense
$3,000
Utility expense
$500
Total
$210,000
$210,000 |
[
{
"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... | A |
[
{
"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... | 855.16 yd |
[
{
"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... | 7.84 m/s |
[
{
"content": "In a circular arrangement, determine the number of ways to arrange $n$ men and $m$ women such that no two women are adjacent, given that $n < m$. Provide your answer as a single number.",
"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... | 2.16e2 m/s |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 3.19 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... | $289,000 |
[
{
"content": "Consider a matrix $A$ of size ${n\\choose 3} \\times {n\\choose 2}$, where each row corresponds to a triplet $\\{i,j,k\\}$ and each column corresponds to a pair $\\{s,t\\}$. The entry $A_{\\{i,j,k\\},\\{s,t\\}}$ is defined to be 1 if $\\{s,t\\} \\subseteq \\{i,j,k\\}$ and 0 otherwise. For $n \\ge ... | Yes |
[
{
"content": "Find the limit of the sequence as \\( n \\to \\infty \\):\n\\[ \\sum_{i=1}^n \\frac{\\cos(i^2)}{n^2+i^2} \\]",
"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... | 104 s |
[
{
"content": "Evaluate the integral \\( \\int_0^{\\infty}\\frac{\\arctan^2(x)}{x^2}\\,dx \\).",
"role": "user"
}
] | \pi \ln 2 |
[
{
"content": "Consider a Markov chain $Z_k$ with 6 states, structured as a pyramid where each node has a probability of $1/2$ to move to a neighboring node. Starting from state $1$ with $P(Z_0=1)=1$, calculate the expected time $E[T_B]$ to first reach one of the bottom states, which are $3$, $4$, or $5$. Provid... | 4 |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.44e-14 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... | 6.93e-15 J |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 1.5 g/mL |
[
{
"content": "Find the minimum value of \\(|z| + |z - e^{ix}| + |z - 5e^{ix}| + |z - 7e^{ix}| + |z - 32e^{ix}|\\) where \\(z\\) is a complex number.",
"role": "user"
}
] | 38 |
[
{
"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... | 2.54558e-11 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... | $31,657.14 |
[
{
"content": "Given the continuous-time state-space representation:\n\\[ \\frac{d}{dt}x(t)=Ax(t)+Bu(t), \\quad y(t)=Cx(t), \\quad t \\in \\mathbb{R}^+, \\]\nwith matrices:\n\\[ A =\\begin{bmatrix}-4&-5&0&5\\\\-4&-4&0&4\\\\0&0&-2&0\\\\-4&-5&0&5\\end{bmatrix}, \\quad B=\\begin{bmatrix}2\\\\0\\\\0\\\\1\\end{bmatri... | E |
[
{
"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... | frequency |
[
{
"content": "An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon?",
"role": "user"
}
] | 567 |
[
{
"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... | 343.60 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... | 82.423 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... | Cl2, S2(2-), SCl(-), Ar2(2+) |
[
{
"content": "Provide an example of a bounded sequence that is not convergent, but where the differences of consecutive terms converge to zero.",
"role": "user"
}
] | a_n = \sin(\sqrt{n}) |
[
{
"content": "Determine the number of continuous functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) that satisfy the functional equation \\( f(f(x)) + f(x) + x = 0 \\).",
"role": "user"
}
] | 0 |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n\\rightarrow\\infty}\\frac{1}{n\\sqrt{n}}\\sum^n_{k=1}\\sqrt{k} \\]",
"role": "user"
}
] | \dfrac{2}{3} |
[
{
"content": "Find the minimum value of the expression \\( \\sin \\alpha + \\sin \\beta + \\sin \\gamma \\), where \\( \\alpha, \\beta, \\gamma \\in \\mathbb{R} \\) and \\( \\alpha + \\beta + \\gamma = \\pi \\). Choose the correct option: (a) positive, (b) negative, (c) 0, (d) -3.",
"role": "user"
}
] | B |
[
{
"content": "A hexagon is drawn with its vertices at $$(0,0),(1,0),(2,1),(2,2),(1,2), \\text{ and } (0,1),$$ and all of its diagonals are also drawn, as shown below. The diagonals cut the hexagon into $24$ regions of various shapes and sizes. These $24$ regions are shown in pink and yellow below. If the smalle... | 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... | 107.5 m/s |
[
{
"content": "Given the cubic equation \\(3x^3+8x^2-1=0\\), if \\(a\\), \\(b\\), and \\(y\\) are its roots, find the value of the expression \\((b+\\frac{1}{y})(y+\\frac{1}{a})(a+\\frac{1}{b})\\).",
"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... | 5/18 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... | 419J |
[
{
"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... | 30.4^o east of north |
[
{
"content": "Let $X$ and $Y$ be independent random variables with respective moment generating functions $M_X(t) = \\frac{(8+e^t)^2}{81}$ and $M_Y(t) = \\frac{(1+3e^t)^3}{64}$ for $-\\infty<t<\\infty$. Find $P(X+Y = 1)$. \\(\\boxed{?}\\)",
"role": "user"
}
] | \dfrac{37}{324} |
[
{
"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... | $15.2 million |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | $53,233 |
[
{
"content": "Find the convolution of two Heaviside step functions, one centered at \\( \\frac{1}{2} \\) and the other at \\( -\\frac{1}{2} \\).",
"role": "user"
}
] | t \, H(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... | 725 psia |
[
{
"content": "Given the matrices \\(A = \\begin{bmatrix}0 & 1 & 0 & 0 & 1\\\\ 0 &0& 1& 1& 0\\\\ 1& 0& 0& 0& 0\\\\ 1& 0& 0& 0& 1\\\\ 0& 1& 0& 1& 0\\end{bmatrix}\\) and \\(M=A+A^2 = \\begin{bmatrix}0& 2& 1& 2& 1\\\\ 2& 0& 1& 1& 1\\\\ 1& 1& 0& 0& 1\\\\ 1& 2& 0& 1& 2\\\\ 1& 1& 1& 2& 1\\end{bmatrix}\\), determine th... | 3 |
[
{
"content": "Suppose $f$ is a real-valued continuous non-constant function defined on all of $\\mathbb{R}$. Let $A = \\text{image} f$. Suppose there exists $L > 0$ such that for every half-open interval $I \\subseteq \\mathbb{R}$ with $|I| = L$, $\\text{image} f|_{I} = A$. Must $f$ be periodic? Provide a justi... | 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... | 2.42 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... | 782 nm |
[
{
"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... | 1.4e+3 kg/m^3 |
[
{
"content": "Find the smallest positive integer $x$ that satisfies the following system of congruences: \\[ 77x \\equiv 1 \\pmod{144} \\] \\[ 67x \\equiv 3 \\pmod{77} \\]",
"role": "user"
}
] | 4997 |
[
{
"content": "Find the volume of the region bounded by the paraboloid \\( z = x^2 + y^2 \\) and the plane \\( z = 1 - x - y \\).",
"role": "user"
}
] | \dfrac{9\pi}{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... | 3.68e6 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... | 125/9 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... | 900,000 |
[
{
"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... | 2.8 |
[
{
"content": "Calculate the expectation \\( \\mathbb{E}[e^{t}W(e^{-2t})] \\), where \\( W(t) \\) is a Brownian motion.",
"role": "user"
}
] | 0 |
[
{
"content": "If $A$ is a closed subset of a topological space $(X, \\mathscr{T})$, and both $A$ and its complement $X \\setminus A$ are compact, does it follow that $X$ is compact?",
"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... | 2.248e-3 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... | -20.5 J |
[
{
"content": "Evaluate the double integral \\( \\int \\int_R |xy| \\, dx \\, dy \\) where \\( R \\) is the region enclosed by the curves \\( x^2 + 4y^2 \\geq 1 \\) and \\( x^2 + y^2 \\leq 1 \\).",
"role": "user"
}
] | \dfrac{3}{8} |
[
{
"content": "Are two matrices with identical characteristic and minimal polynomials but different ranks similar?",
"role": "user"
}
] | No |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to 0} \\frac{x\\sin(nx)}{1-\\frac{\\sin x}{x}} \\]",
"role": "user"
}
] | 6n |
[
{
"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... | 8.069e18 |
[
{
"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... | $44.2 |
[
{
"content": "Calculate the conditional expectation $E(\\xi | \\xi^2)$ where $\\xi$ is a random variable uniformly distributed on the interval $[-1, 1]$. Provide your answer as a function of $\\xi^2$. \\( \\boxed{} \\)",
"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... | $29,233 |
[
{
"content": "Given the following Cayley table, identify the symmetry group that corresponds to this table.\n\\[\n\\begin{array}{c|c|c|c|c}\n\\hline\n\\cdot & e & a & b & c \\\\\n\\hline\ne & e & a & b & c \\\\\n\\hline\na & a & e & c & b \\\\\n\\hline\nb & b & c & a & e \\\\\n\\hline\nc & c & b & e & a \\\\\n\... | C_4 |
[
{
"content": "Find the minimum value of\n\\[\\frac{x^2}{y - 1} + \\frac{y^2}{x - 1}\\]for real numbers $x > 1$ and $y > 1.$",
"role": "user"
}
] | 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... | 229 lines/mm |
[
{
"content": "In a game involving a $2008 \\times 2008$ matrix, Alan and Barbara take turns filling the matrix with real numbers, starting with Alan. The game ends when the matrix is completely filled. Alan wins if the determinant of the matrix is nonzero, while Barbara wins if it is zero. Determine which playe... | Barbara |
[
{
"content": "Let $F$ be the vector field $F=4xz\\,i+xz^2\\,j+x^7y^3\\,k$. Let $S$ be the portion of the paraboloid $z=x^2+y^2$ below $z=9$, along with the disk $x^2+y^2\\leq9$ in the plane $z=9$, all with outward orientation. Compute the surface integral $\\iint_S F \\cdot dS$. \\(\\text{Use the Divergence The... | 972\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... | 42 mph |
[
{
"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... | 17.5 m/s |
[
{
"content": "A conversation between User and Assistant. The user asks a question, and the Assistant solves it. The assistant first thinks about the reasoning process in the mind and then provides the user with the answer. The reasoning process and answer are enclosed within <think> </think> and <answer> </answ... | 42675.10 |
[
{
"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... | dividends payable |
[
{
"content": "Determine the intersection of the fields $\\mathbb{Q}(\\pi^2)$ and $\\mathbb{Q}(\\pi^2-\\pi)$. Provide a proof for your answer.",
"role": "user"
}
] | \mathbb{Q} |
[
{
"content": "The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term? Type the letter of the correct option.\n\nA. $-\\sqrt{3}$\n\nB. $-\\frac{2\\sqrt{3}}{3}$\n\nC. $-\\frac{\\sqrt{3}}{3}$\n\nD. $\\sqrt{3}$\n\nE. $3$",
"role": "user"
}
] | B |
[
{
"content": "Consider two sequences of real numbers: an arithmetic sequence \\((a_{n})_{n \\in \\mathbb{N}}\\) and a non-constant geometric sequence \\((g_{n})_{n \\in \\mathbb{N}}\\). It is given that \\(a_{1}=g_{1} \\neq 0\\), \\(a_{2}=g_{2}\\), and \\(a_{10}=g_{3}\\). Determine if, for each positive integer... | Yes |
[
{
"content": "Determine the values of the complex number $x$ for which the power series \\( P(x) = \\sum_{n=0}^{\\infty} \\frac{n!}{n^n} \\cdot x^n \\) converges.",
"role": "user"
}
] | |x| < e |
[
{
"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... | 8.16 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... | 1.6*10^-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... | 6.777 m/s |
[
{
"content": "You have $n^2$ distinct balls, each of which can be colored in $n^2$ different ways. You randomly select $n^b$ balls without replacement. Determine the asymptotic probability that at least $n^a$ balls have the same color, where $0 < a < b \\leq 2$. Provide your answer in terms of $n$. ",
"role... | 0 |
[
{
"content": "Let $W_t$ be a standard Brownian motion and define the process $$X_t = X_0 e^{t(1-\\sigma^2/2) + \\sigma W_t}$$ where $\\sigma$ has an exponential distribution given by $$ P(\\sigma \\leq x) = 1-e^{-x}$$ for $x \\geq 0$. Determine the probability $$P\\left(\\lim_{t \\to \\infty}X_0 e^{t(1-\\sigma^... | e^{-\sqrt{2}} |
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