inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the cardinality of the quotient set of the equivalence relation defined on the set $\mathbb{Z}^{\mathbb{N}}$ of all infinite sequences of integers, where $f \equiv g$ if and only if $((f(2n) = g(2n)) \wedge (f(n) \cdot g(n) > 0 \vee f(n) = 0 = g(n)))$ for $f, g \in \mathbb{Z}^{\mathbb{N}}$. What is the power of this quotient set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Jill and Kelly are doing spring cleaning. Jill takes an exponentially distributed time with a mean of 30 minutes to clean the kitchen, and Kelly takes an exponentially distributed time with a mean of 40 minutes to clean the bathroom. The first one to finish will then rake leaves, which takes an exponentially distributed time with a mean of 60 minutes. If the second person finishes their task, they will join the first person, and the raking will be completed at twice the original rate. What is the expected time until all chores are completed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{408}{7}}
Suppose the prime numbers $p$ and $q$ satisfy $q^2+3p=197p^2+q$.Write $\frac{p}{q}$ as $l+\frac{m}{n}$, where $l,m,n$ are positive integers , $m<n$ and $GCD(m,n)=1$. Find the maximum value of $l+m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
32}
How many nonempty subsets of ${1, 2, \dots, 15}$ are there such that the sum of the squares of each subset is a multiple of $5$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\( 7583\)}
In a finite abelian group \( G \), two subgroups \( H \) and \( K \) both have an index of 3 in \( G \). Determine the index of the intersection \( H \cap K \) in \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
You have 100 chips and are offered a game with a winning probability of \( p = 0.66 \) and a payout of 1:1. Determine the optimal number of chips to bet in this game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Given a multiset of symbols \((A, A, B, B, B, C, D, D)\), find the nth permutation of the symbols when sorted in alphabetical order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{BDCADABB}
Find the smallest number that rounds to 0.7 after rounding to the nearest thousandth, then to the nearest hundredth, and finally to the nearest tenth. Provide your answer as a decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6445}
Determine the non-negative integer value(s) of $a$ for which the following limit equals $\frac{1}{4}$:
\[\lim_{x \to 1 }\left\{ \frac{-ax+\sin(x-1)+a}{x+\sin(x-1)-1} \right\}^{\frac{1-x}{1-\sqrt{x}}}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $f(z) = u + iv$ be an entire function. If the Jacobian matrix \( \begin{bmatrix} u_{x}(a) & u_{y}(a) \\ v_{x}(a) & v_{y}(a) \\ \end{bmatrix} \) is symmetric for all complex numbers $a$, determine the form of $f(z)$. What is $f(z)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(z) = az + b}
The New York Public Library requires patrons to choose a 4-digit Personal Identification Number (PIN) to access its online system.  (Leading zeros are allowed.)  The PIN is not allowed to contain either of the following two forbidden patterns:\n* A digit that is repeated 3 or more times in a row.  For example, 0001 and 5555 are not PINs, but 0010 is a PIN.   \n* A pair of digits that is duplicated.  For example, 1212 and 6363 are not PINs, but 1221 and 6633 are PINs.  \nHow many distinct possible PINs are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
9720
}
In a town of 9,000 people, 20 people have heard a rumor by 8 am. The spread of the rumor follows the logistic equation \(\frac{dP}{dt}=kP\left(1-\frac{P}{C}\right)\). By 12 noon, 500 people have heard the rumor. Determine how long it takes for half the population to have heard the rumor. Provide your answer in hours, rounded to one decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5}
Determine the period of the real-valued function $f(x)$ that satisfies the equation $f(x) + f(x + 4) = f(x + 2) + f(x + 6)$ for all $x \in \mathbb{R}$ using elementary algebraic manipulations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the flux of the vector field $$\mathbf{F}=(-y,x,z)$$ across the surface S that is part of the paraboloid $$z=x^{2}+y^{2}$$ that lies between the planes $z=2$ and $z=5$, using cylindrical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21π/2}
Given the function $f(x) = \frac {|x|+x}{2}+1$, find the range of $x$ that satisfies the inequality $f(1-x^2) > f(2x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, \sqrt{2}-1)}
Let \(a > 0\) be a real number, and consider the sequence \(\{x_n\}\) defined by \(x_1 = 1\) and \(ax_n = x_1 + x_2 + \cdots + x_{n+1}\) for all \(n \geq 1\). Determine the minimum value of \(a\) such that \(x_n > 0\) for every \(n \geq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the volume of the solid obtained by revolving the area under the first arc of the cycloid, given by the parametric equations \(x(t) = a(t - \sin t)\) and \(y(t) = a(1 - \cos t)\), around the \(y\)-axis. Use the formula \( V = \pi \int [f(x)]^2 \, dx \) to find the volume.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6\pi^3 a^3}
Evaluate the integral \( \int_m^{n}\frac{1}{\frac{b}{x -\frac{1}{2b}}-\lfloor{b/x}\rfloor} \, dx \) where \( b, m, n, \) and \( x \) are natural numbers such that \( m < n < b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - m}
Determine the minimum number of dimensions $n$ for the system described by the ordinary differential equation $\dot{\vec{x}} = A\vec{x}$, where the first coordinate of the solution vector $\vec{x}(t)$ is given by $x_1(t) = 5t^2\cos(3t) + 2\sin(3t) - e^{2t}\sin(3t) + 4t$, and $A$ is an $n \times n$ matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a deck of cards numbered from 1 to 200, calculate the expected number of cycles formed when performing a card-flipping process. What is the expected number of cycles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H_{200}}
Find the ideal $J$ of $\mathbb{Z}[x]$ such that the quotient ring $\mathbb{Z}[x]/J$ is isomorphic to $\mathbb{Z}/3\mathbb{Z}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, x)}
Determine the constraint on the parameter $a$ such that the function \( f(x) = \frac{x^2 + 2x + a}{x^2 + 4x + 3a} \) is surjective.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
Find all integer triples \((x, y, z)\) with each integer greater than 1 such that:
\[ x \mid yz - 1, \quad y \mid zx - 1, \quad z \mid xy - 1. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 3, 5)}
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 local extremums $A(-1, 2)$ and $B(1, -2)$, $N$ lies to the right of $B$, $Q$ lies to the left of $A$, and the square has its center at the origin $O(0, 0)$. Determine the number of such squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $S$ be a set of 2020 distinct points in the plane. Define \( M = \{ P : P \text{ is the midpoint of } XY \text{ for some distinct points } X, Y \text{ in } S \} \). Determine the minimum possible number of points in the set \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4037}
In a soccer game, how many ways can the first 12 goals be scored such that the difference in scores between the two teams never exceeds 2 at any point during the game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{972}
Find the last four digits of the number \(2^{1965}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1232}
Given two independent normally distributed random variables, X and Y, with means \( E[X] = 1 \) and \( E[Y] = 10^{-3} \), and variances \( \text{Var}(X) = (0.12)^2 \) and \( \text{Var}(Y) = (0.05 \times 10^{-3})^2 \), find the variance of the random variable \( Z = \frac{X}{Y} \). Assume that X and Y are independent. What is \( \text{Var}(Z) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16900}
Calculate the limit \( \lim_{n \rightarrow \infty} \sqrt[n]{\binom{n^2k}{n}} \) without using Stirling's approximation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Express the function \( g(x) \) in terms of \( f(x) \) given the equation \( f(x) = g(x) \cdot g(-x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(x) = \sqrt{f(x)}}
Calculate the determinant of the matrix \( A \) defined as:
\[
A = \begin{pmatrix} 3 & 2 & 2 & . & . & . & 2 \\
2 & 3 & 2 & . & . & . & 2 \\
2 & 2 & 3 & . & . & . & 2 \\
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
2 & 2 & 2 & . & . & . & 3 \end{pmatrix}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the range of values of $a$ for which the function \( f(x) = \frac{(x-1)}{x(x+a)} \) has no stationary points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -1]}
Evaluate the integral \(\int_{-\infty}^{\infty}\frac{\sin(cx) - \sin(dx)}{x^2} \mathrm{d}x\) where \(c\) and \(d\) are positive, non-zero constants, using a suitable indented contour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the dimension of the linear span of the powers of an $n \times n$ matrix $A$ up to its $k$-th power, given that $A$ has rank $n$ and satisfies the Cayley–Hamilton theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min\{k + 1, n\}}
Determine the parameters of the binary restriction of the Hamming code $Ham(3,4)$, which is originally a $[21,18,3]$ code over $\mathbb{F}_4$. What are the parameters for $Ham(3,4)\mid_{F_2}=Ham(3,4)\cap F_2^{21}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[21, 15, 3]}
Evaluate the expression \( S = \frac{A^3-B^3}{B^3}+\frac{A^4+B^4}{B^4} \) where \( A=\int_0^1(1-x^{2023})^{\frac{1}{2022}}dx \) and \( B=\int_0^1(1-x^{2022})^{\frac{1}{2023}}dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the smallest connected graph that is regular but not vertex-transitive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the number of distinct $7 \times 7$ symmetric matrices where each row contains all integers from $1$ to $7$ exactly once, and both the main and secondary diagonals also contain all integers from $1$ to $7$ exactly once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit \(\lim_{x\to0^+}\int_x^1\frac{f(t)}{t^3}\,\mathrm{d}t\), where \(f:[0,1]\) is a continuous function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the flux integral \(\int_S (\nabla \times \vec{F}) \cdot \vec{n} \, dS\) where \(\vec{F} = (y,z,x^2y^2)\) and \(S\) is the surface given by \(z = x^2 + y^2\) with \(0 \leq z \leq 4\), oriented so that \(\vec{n}\) points downwards.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Given that a scalar random variable \( y = z + v \), where the probability density function (pdf) of \( v \) is \( p_{v}(t) = \frac{t}{4} \) on the interval \([0,4]\), and the pdf of \( z \) is \( p_{z}(t) = t \) on the interval \([0,2]\), both densities are zero elsewhere. If a single measurement value \( y = 3 \) is available, compute the maximum a posteriori (MAP) estimate of \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Calculate the derivative of \( \frac{1}{2} \log(\det(X+X^T)) \) with respect to \( X \), given that the derivative of \( \log(\det(X)) \) with respect to \( X \) is \( X^{-1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(X + X^T)^{-1}}
设复数 $z=(\omega+2)^{3}(\omega-3)^{2}$ 。求当 $\omega$取遍所有模长为 1 的复数时, $z$ 的模长的最大值。原始答案为$m^{\frac{p}{q}}$的形式，其中$p,q$互质，请给出$m+p+q$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In triangle \( \triangle ABC \), the sides are given as \( AB = n \), \( AC = n-1 \), and \( BC = n+1 \), where \( n \) is a natural number. The angles opposite these sides are \( A \), \( 2A \), and \( \pi - 3A \) respectively. Determine the value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
It is known that $\cos 157^o = a$, where $a$ is given. Calculate $1^o$ in terms of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$1^{\circ} = \dfrac{1}{157} \cdot \arccos(a)$
}
Determine the cardinality of the set of functions that are either multiplicative or completely multiplicative, defined on the set of natural numbers and taking natural numbers as their values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Given the joint probability density function \(f(x,y,z) = kxyz^2\) for \(0 < x < 1\), \(0 < y < 1\), and \(0 < z < 2\), find the probability \(P(Z > X + Y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{20}}
There are 1001 numbers arranged in a circle. For any two neighboring numbers \(x\) and \(y\), the inequalities \(|x-y| \geqslant 4\) and \(x+y \geqslant 6\) are satisfied. Determine the minimum possible sum of all the numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3009}
Find the limit of the sequence \( a_n = \frac{n!(0.5)^{n-1}}{\sqrt{n}\{[(n-1)/2]!\}^2} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\dfrac{2}{\pi}}}
Determine the number of sets of four primes \((p, q, r, s)\) such that their sum is a prime number, \(p^2 + qs\) and \(p^2 + qr\) are both perfect squares, and all conditions are met simultaneously.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Task 2. Lёnya has cards with digits from 1 to 7. How many ways are there to glue them into two three-digit numbers (one card will not be used) so that their product is divisible by 81, and their sum is divisible by 9?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["36"]}
For an ordered 10-tuple of nonnegative integers \( {a}_{1},{a}_{2},\ldots ,{a}_{10} \) ,we denote
 \( f\left( {{a}_{1},{a}_{2},\ldots ,{a}_{10}}\right)  = \left( {\mathop{\prod }\limits_{{i = 1}}^{{10}}\left( \begin{matrix} {20} - \left( {{a}_{1} + {a}_{2} + \cdots  + {a}_{i - 1}}\right) \\  {a}_{i} \end{matrix}\right) }\right)  \cdot  \left( {\mathop{\sum }\limits_{{i = 1}}^{{10}}\left( \begin{matrix} {18} + i \\  {19} \end{matrix}\right) {a}_{i}}\right) . \)
 When \( i = 1 \) ,we take \( {a}_{1} + {a}_{2} + \cdots  + {a}_{i - 1} \) to be 0 . Let \( N \) be the average of \( f\left( {{a}_{1},{a}_{2},\ldots ,{a}_{10}}\right) \) over all 10-tuples of
 nonnegative integers \( {a}_{1},{a}_{2},\ldots ,{a}_{10} \) satisfying
 \( {a}_{1} + {a}_{2} + \cdots  + {a}_{10} = {20}. \)
 Compute the number of positive integer divisors of \( N \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
462
}
Note that there are exactly three ways to write the integer 4 as a sum of positive odd integers where the order of the summands matters:
$1 + 1 + 1 + 1 = 4,$
$1 + 3 = 4,$
$3 + 1 = 4.$
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the summands matters. Find the remainder when $f(2008)$ is divided by 100.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$71$

}
Evaluate the double integral \( \iint (x^2 \cdot y + y \cdot \sin(x^8)) \, dx \, dy \) over the region \( D = \{(x,y) : x^2 + y^2 \leq 2, y > 0\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8\sqrt{2}}{15}}
Given that \(a, b, c > 0\) satisfy the equation \((8a - 7b - 3c)^2 = 6(a^2 - b^2 - c^2)\), find the ratio \(a:b:c\) in terms of constants \(b_1\) and \(c_1\) such that \(b = ab_1\) and \(c = ac_1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8:7:3}
Determine the number of integer solutions \((i, j)\) to the equation:
\[3i^2 + 2j^2 = 77 \cdot 6^{2012}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Evaluate the surface integral of the vector field $\mathbf{F}(x, y, z) = \left(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\right)$ over a sphere $S$ of radius $R$ with outward orientation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\pi R}
In $\triangle ABC$, $AB$, $BC$, and $CA$ have lengths $3$, $4$, and $5$, respectively. Let the incircle, circle $I$, of $\triangle ABC$ touch $AB$, $BC$, and $CA$ at $C'$, $A'$, and $B'$, respectively. Construct three circles, $A''$, $B''$, and $C''$, externally tangent to the other two and circles $A''$, $B''$, and $C''$ are internally tangent to the circle $I$ at $A'$, $B'$, and $C'$, respectively. Let circles $A''$, $B''$, $C''$, and $I$ have radii $a$, $b$, $c$, and $r$, respectively. If $\frac{r}{a}+\frac{r}{b}+\frac{r}{c}=\frac{m}{n}$ where $m$ and $n$ are positive integers, find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$m + n = 23$}
Determine the order of convergence of the sequence \(\{y_n\}_{n=0}^{\infty }\) defined by \(y_{n+1}=3y_n-2 {y_n}^3\) with \(y_0=1/2\), given that the limit of the sequence is \(1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit \( \lim_{x\rightarrow 0} \frac{\sin x^2}{ \ln ( \cos x^2 \cos x + \sin x^2 \sin x)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Calculate the number of open subsets of the real line, denoted as \( o(\mathbb{R}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
In a game, Ann and Bob alternately toss a fair coin. In each round, they flip the coin $2n$ times. Let $X_n$ be the number of heads in these $2n$ flips during the first $n$ rounds. Define $r_m$ as the probability that there exists an $n$ for which $X_n = m$. Using a renewal theorem, find $\lim\limits_{m\to\infty} r_m$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_0^1 \frac{1+x^4}{1+x^6} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Evaluate the limit: $$\lim_{\alpha\rightarrow 0} \int_{0}^1 \frac{(1 - x)^\alpha}{\ln x} dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\infty}
Determine the upper bound of the function $g(x)$ that satisfies the differential equation $g'(x) = \frac{1}{x^{2} + (g(x))^{2}}$ with the initial condition $g(1) = 0$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the minimum value of \( k \in \mathbb{R}^{+} \) such that the inequality
\[
\frac{a^3}{(b-c)^2} + \frac{b^3}{(c-a)^2} + \frac{c^3}{(a-b)^2} \geq k (a+b+c)
\]
holds for all distinct positive real numbers \( a, b, c \) that do not form the sides of a triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit: $$\lim_{n \to \infty} n \int_0^\pi (\sin x - \cos x)^n dx$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the number of ordered pairs \((\alpha_1, \alpha_2)\) of permutations in the symmetric group \(S_n\) such that \(\alpha_1 \circ \alpha_2 = \alpha_2 \circ \alpha_1\), where \(\alpha_1, \alpha_2 \in S_n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n! \cdot p(n)}
Find all prime numbers $p$ such that $4p^4 + 1$ is also a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Compute the integral of the differential form \( \omega = \dfrac{x \, dy \wedge dz + y \, dz \wedge dx + z\, dx \wedge dy}{(x^2+y^2+z^2)^{3/2}} \) over the unit sphere \( S^2 \) using the parametrization \( f(u, v) = (\sin u \cos v, \sin u \sin v, \cos u) \), where \( (u, v) \in [0, \pi] \times [0, 2\pi] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
How many distinct arrangements of the letters in the word "MATHEMATICS" exist if both 'T's appear before both 'A's, both 'A's appear before both 'M's, or both 'M's appear before the 'E'?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2835360}
Determine the value of \(\aleph_1^{\aleph_0}\) in set theory.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the largest square-free positive integer $n$ such that the ring of integers of the number field $\mathbb{Q}(\sqrt{-n})$ is a unique factorization domain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{163}
Seven students count from $1$ to $1000$ as follows:
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1, 3, 4, 6, 7, 9, \cdots, 997, 999, 1000.$
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
Finally, George says the only number that no one else says.
What number does George say?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
365
}
Find the limit of the following nested square root expression as \( n \) approaches infinity:

\[ \lim_{n \to \infty}\left( 1 + \sqrt{2 + \sqrt{3+ \ldots + \sqrt {n + \sqrt{n+1}}}}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the limit \(\lim_{n\rightarrow\infty} \int^{1}_{0}\frac{x^n}{\ln(1+x^n)}dx.\) Use the inequality \(\ln(1+t) \leq t\) to assist in your evaluation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}