inclusionAI/Ring-lite-rl-data huggingface dataset:
设S={1,2,⋯,100}，求最小的正整数n，使得S的每一个n 元子集都含有4个两两互质的数．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Determine the maximal depth of an ideal in the polynomial ring $k[x,y]$ over an arbitrary field $k$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the conditions on the function $a(t)$ such that the stochastic process $X(t)$, defined by the differential equation \( dX = a(t)Xdt + b(t)XdW(t) \), is a $P$-martingale, where $b(t)$ is non-zero almost surely and $W(t)$ is a Wiener process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a(t) = 0}
The circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point $P$ is on $C_1$, and the tangent line at $P$ passes through $O$. Let the ray $OP$ intersect with $C_2$ at $Q$, and let the tangent line of $C_1$ passing through $A$ intersect with the line $OP$ at $R$. If the radius of $C_2$ is 9 and $PQ = QR$, find the length of the segment $OP$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A princess is equally likely to sleep on any number of mattresses from six to twelve each night. On half of the nights, a pea is placed under the lowest mattress. She never sleeps well if a pea is under six mattresses. If the pea is under seven mattresses, she sleeps well 10% of the time; under eight mattresses, 20% of the time; and so on, up to 60% of the time under twelve mattresses. One morning, she reports having slept incredibly well. What is the expected number of mattresses she slept on that night?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{121}{13}}
Let $AP_1, AP_2, AP_3$ be three arithmetic progressions with common terms $m, n, r$, respectively. If $AP_1 \cap AP_2 = AP(7, 15)$ and $AP_2 \cap AP_3 = AP(a, d)$, find the value of $a + d$. \( \text{Calculate } a + d. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Given an arithmetic-geometric sequence $\{ a_{n} \}$ that satisfies $a\_1 + a\_3 = 10$, $a\_2 + a\_4 = 5$, find the maximum value of the product $a\_1 a\_2 \ldots a\_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Determine the values of \( p > 0 \) for which the integral \( \int_{-1}^{0} x^{-p} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
Six numbers are written in a row. It is known that among them there is one unit and any three consecutive numbers have the same arithmetic mean. Find the maximum value of the geometric mean of any three consecutive numbers in this row if the arithmetic mean of all 6 numbers is A.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[3]{\frac{(3A - 1)^2}{4}}}
Calculate the limit \(\displaystyle\lim_{n\to\infty}\sqrt[n]{|1-z^n|}\) for \(z\in\mathbb{C}\) with \(|z|\ne 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max\{1, |z|\}}
The length of an open interval $\left(a,b\right)$ is defined as $b-a$. Through estimation, the zero point of the function $f(x)=\frac{1}{{2^x}}-{x^{\frac{1}{3}}}$ belongs to the open interval ____ $($just need to write one interval that meets the condition and does not exceed $\frac{1}{6}$ in length).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{1}{3}, \frac{1}{2}\right)}
Given the function f(x) = x^2 + 2x (x > 0), and the recursive definition f1(x) = f(x), f(n+1)(x) = f(f(n)(x)), n ∈ N*, find the maximum value of f2019(x) on the interval [1, 2]. Express your answer as a single mathematical expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{2^{2019}} - 1}
Let \( l^2= \{ (x_1,x_2,x_3,\dots):x_n \in \mathbb{R} \text{ for all } n \in \mathbb{N} \text{ and } \sum_{n=1}^{\infty} x_n^2 < \infty \}. \) For a sequence \((x_1,x_2,x_3,\dots) \in l^2,\) define \[\lvert \lvert (x_1,x_2,x_3,\dots)\rvert\rvert_2= \bigg( \sum_{n=1}^{\infty} x_n^2\bigg)^{\frac{1}{2}}.\] Consider the subspace \(M=\{ (x_1,x_2,x_3,\dots) \in l^2 :\sum_{n=1}^{\infty} \frac{x_n}{4^n}=0 \}.\) Let \(M^{\perp}\) denote the orthogonal complement of \(M\) in the Hilbert space \((l^2, \lvert \lvert \cdot \rvert \rvert_2).\) Consider \(\big(1,\frac{1}{2},\frac{1}{3},\frac{1}{4}, \dots \big) \in l^2.\) If the orthogonal projection of \(\big(1,\frac{1}{2},\frac{1}{3},\frac{1}{4}, \dots \big)\) onto \(M^{\perp}\) is given by \(\alpha\big( \sum_{n=1}^{\infty} \frac{1}{n4^n}\big)\big(\frac{1}{4},\frac{1}{4^2},\frac{1}{4^3}, \dots \big)\) for some \(\alpha \in \mathbb{R},\) find the value of \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Points $K,A,L,C,I,T,E$ are such that triangles $CAT$ and $ELK$ are equilateral, share a center $I,$ and points $E,L,K$ lie on sides $CA, AT, TC$ respectively. If the area of triangle $CAT$ is double the area of triangle $ELK$ and $CI = 2,$ compute the minimum possible value of $CK.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}-1}
The dispatcher Xiao Zhang from Factory A of a certain brand is responsible for transferring goods among five large supermarkets within the inner ring of the Second Ring Road. These five supermarkets are arranged in the order of A, B, C, D, and E within the ring-shaped area. Currently, the supermarkets have 15, 7, 11, 3, and 14 pieces of Product A, respectively. Now, to make the number of Product A pieces the same in each supermarket, how many pieces should Xiao Zhang transfer from each supermarket to its adjacent ones to minimize the total number of transfers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the angle between the vector \( \vec{a} \) and the vector \( \hat{i} - 2\hat{j} + 2\hat{k} \). The vector \( \vec{a} \) is parallel to the line of intersection of the plane determined by the vectors \( \hat{i} \) and \( \hat{i} + \hat{j} \), and the plane determined by the vectors \( \hat{i} - \hat{j} \) and \( \hat{i} + \hat{k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Evaluate the sum: $$ \sum_{k=1}^{+\infty}\frac{1}{k^{3}+k}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and  }x^3+bx+a \text{ have at least a common real zero .}\}$
How many elements are there in $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
有多少个正整数 $n$ 满足 $\sin (\sqrt{2})+\sin (2 \sqrt{2})+\cdots+\sin (n \sqrt{2})>2$? 填写满足条件的正整数个数。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the sum: $$\cos\left(\frac{2\pi}{2021}\right)+\cos\left(\frac{4\pi}{2021}\right)+\cdots+\cos\left(\frac{2020\pi}{2021}\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Find the value of the limit superior of the sequence \(|\sin(n)|^n\) as \(n\) approaches infinity, where \(n\) is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Whole numbers whose decimal representation reads the same from left to right as from right to left are called symmetrical. For example, the number 5134315 is symmetrical, while 5134415 is not. How many seven-digit symmetrical numbers exist such that adding 1100 to them leaves them unchanged as symmetrical numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{810}
Find \(g(2022)\) if for any real numbers \(x\) and \(y\) the following equation holds:
$$
g(x-y)=2022(g(x)+g(y))-2021 x y .
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2043231}
There are 16 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). Everyone said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $x_1=\sqrt{10}$ and $y_1=\sqrt3$. For all $n\geq 2$, let
$$
\begin{align*}x_n&=x_{n-1}\sqrt{77}+15y_{n-1}\\y_n&=5x_{n-1}+y_{n-1}\sqrt{77}\end{align*}
$$
Find $x_5^6+2x_5^4-9x_5^4y_5^2-12x_5^2y_5^2+27x_5^2y_5^4+18y_5^4-27y_5^6.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4608}
Determine the minimum possible real number \( L \) such that the area of the intersection of the set \( I = \{(x,y) \mid x > (\frac{y^4}{9} + 2015)^{1/4}\} \) and the disk \( x^2 + y^2 \leq r^2 \) is less than \( Lr^2 \) for all \( r > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Determine a sequence of real numbers $(\alpha_n)_n$ such that the process $(Y_n)_{n \geq 0}$, defined by $Y_n := S_n^3 + \alpha_n \cdot S_n$, is a martingale with respect to the natural filtration $(F_n)_{n \geq 0}$ of a simple symmetric random walk $(S_n)$ on $\mathbb{Z}$. Provide the sequence $(\alpha_n)_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha_n = -3n}
Using the parametric equations for the folium of Descartes, \(x = \frac{3t}{1+t^3}\) and \(y= \frac{3t^2}{1+t^3}\), calculate the condition for a vertical tangent at the point \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t \to \infty}
Calculate the inner product \( \langle xx^T, A^TA \rangle \) given the definition \( \langle A, B \rangle = \sum_{i=1}^n\sum_{j=1}^n a_{ij}b_{ij} \). Express your answer in terms of \( \|Ax\|^2_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|Ax\|_2^2}
An archer has a probability of hitting a target of 0.4 with each shot. Determine the minimum number of attempts required for the archer to hit the target at least 80 times with a probability of 0.9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{223}
Let $X \sim \mathsf{Gamma}(2,3)$ and define $Y = 2X$. Find the probability density function (pdf) of $Y$ and evaluate it at $y = 13.5$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0395}
Let $p = 10007$ be a prime number. Determine the number of $2 \times 2$ matrices $X$ with elements from $\mathbb{Z}_p$ such that $X^2 \equiv I \pmod{p}$, where $I$ is the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100150058}
Evaluate the limit of the integral: $$\lim_{n \rightarrow \infty}\int_{0}^{\pi/4}{\frac {\sin^2(\frac{nx}{2})}{n\,\sin^2(\frac{x}{2})} }\,dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
For a difference triangle of order 8, where the total number of integers is 36, determine the closest possible total to 36 that can be achieved with either no repeated values or no missing values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Find the density of the $xy$ coordinates in a 3-dimensional unit ball, given that $Z = 0$. Express your answer as a function of $x$ and $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{\pi}}
Calculate the area of the domain D bounded by the loop of the curve \( (x+y)^5 = x^2y^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{1260}}
Let \( a_1, a_2, \ldots, a_9 \) be nine real numbers, not necessarily distinct, with an average of \( m \). Let \( A \) denote the number of triples \( 1 \le i < j < k \le 9 \) for which \( a_i + a_j + a_k \ge 3m \). Determine the minimum possible value of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Find the minimum value of $a + b$ given that $13 \mid (a + 11b)$ and $11 \mid (a + 13b)$, where $a$ and $b$ are positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Let $\phi: c_0 \to \mathbb{R}$ be a linear functional defined by $\phi(x_n) = \sum_{n=1}^\infty (-2)^{1-n} x_n$, where $c_0$ is the space of sequences that converge to 0. Determine the norm of $\phi$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Suppose you have $n$ identically and symmetrically distributed normal random variables. Determine the most negative common correlation coefficient possible between any two of them.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{n-1}}
Evaluate the following limit of an integral: $$ \lim_{n\to\infty}\int_A\frac{\ln(n+y^3)-\ln n}{\sin(x/n)}d\lambda_2 $$ where $A=\{(x,y)\in\mathbb{R}^2_+:1<xy<4,1<y/x<4\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{225}{8}}
Given the matrix \( A = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{pmatrix} \), find the value of \( E_3(A) \) such that the characteristic polynomial \( P_A(t) = t^4 - 4t^3 + 3t^2 - E_3(A)t - 1 \) is satisfied.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Evaluate the expression \( \frac{\partial f'}{\partial f} \) given that \( f: \mathbb{R} \to \mathbb{R} \) and \( f' = \frac{df}{dt} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: \[ \lim_{n \to \infty} n \int_0^\pi \{ -x \}^n \, dx \] where \( \{ x \} \) denotes the fractional part of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
How many distinct configurations exist for placing 1 to 8 red balls in each of 8 boxes such that the difference in the number of balls between any two adjacent boxes is at most 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11814}
Find the volume of the region of points $(x,y,z)$ such that
\[\left(x^{2}+y^{2}+z^{2}+8\right)^{2}\le 36\left(x^{2}+y^{2}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$ 6\pi^{2} $.}
A fair coin is tossed repeatedly, and the outcomes are recorded as a sequence of 'H' (heads) and 'T' (tails). Let $X$ be the number of tosses required for the sequence 'HTH' to appear for the first time. Calculate the expected value $\mathbb{E}(X)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the points where the partial derivatives of the function \( f(x,y) = |x+y|\sin|x+y| \) are discontinuous.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Compute the conditional expectation \( E[\max(X,Y) | \min(X,Y)] \) when \( X \) and \( Y \) are two independent identically distributed random variables with an exponential distribution, where \( X,Y \sim \exp(1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min(X,Y) + 1}
A linear stationary dynamic system is described by the equation \(3 Y^{\prime}(t) + Y(t) = 4 X^{\prime}(t) + X(t)\). The system receives a stationary random function \(X(t)\) with a correlation function \(\mathbf{k}_{x}(\tau) = 6 e^{-2|\tau|}\). Determine the variance of the random function \(Y(t)\) at the output of the system in the steady-state regime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate \( \lim_{n \to \infty} U_n \) where \( U_n = \int_{n^2+n+1}^{n^2+1} \frac{\tan^{-1}x}{x^{0.5}} \, dx \) without performing the integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Evaluate the limit: \[ \lim_{n\rightarrow\infty}e^{\ln(1-\sin{\frac{1}{n}})^n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
An investor needs to cover a liability of $1000 in two years. The investor can invest in a one-year pure discount bond and a three-year pure discount bond, both with a face value of $100. Assume the yield to maturity is 10%. How many units of the three-year pure discount bond must the investor purchase to ensure the liability is covered, even if the yield to maturity changes in the second year? Provide your answer as a decimal number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.5}
Compute the number of ways there are to completely fill a $3\times 15$ rectangle with non-overlapping $1\times 3$ rectangles
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
189
}
Evaluate the outward flux \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) where \( \mathbf{F}(x,y,z) = \frac{1}{(x^2+y^2+z^2)^{3/2}}\langle x,y,z\rangle \) and \( S \) is the outward oriented surface given by \( 9x^2+4y^2+16z^2=144 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Evaluate the definite integral \( \int_0^3 x \, d[x] \), where \([x]\) is the greatest integer less than or equal to \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the limit: $$\lim_{n\to\infty} \frac{1}{\sqrt{n}}\int_{1/{\sqrt{n}}}^{1}\frac{\ln(1+x)}{x^3}\,dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given a positive integer \( n \), let \( \mathrm{rad}\; n \) be the product of \( n \)'s distinct prime factors and \( \mathrm{spf}\; n \) be the sum of \( n \)'s distinct prime factors. For any \( k \ge 1 \), let \( a_{n,k} \) be defined as \[ a_{n,k} = \begin{cases} 1 & \frac{\mathrm{rad}(n)}{\mathrm{rad}(n+k)} > 1 \\ 0 &\text{ otherwise} \end{cases} \] Find the limiting value of \[ \lim_{x \to \infty}\dfrac{1}{x}\sum_{n \le x}a_{n,k} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the radii of circles drawn around the vertices of an equilateral triangle such that the circles do not intersect each other or the opposite side of the triangle, and together they cover the largest possible portion of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a}{2}}
Let \( M \) be the smallest positive multiple of 2012 that has 2012 divisors.
 Suppose \( M \) can be written as \( {\Pi }_{k = 1}^{n}{p}_{k}^{{a}_{k}} \) where the \( {p}_{k} \) ’s are distinct primes and the \( {a}_{k} \) ’s are positive integers.
 Find \( {\sum }_{k = 1}^{n}\left( {{p}_{k} + {a}_{k}}\right) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
Find the smallest possible value of \(x^2 + y^2\) given that \(x\) and \(y\) are positive integers such that \(x(x+1)\) divides \(y(y+1)\), but neither \(x\) nor \(x+1\) divides either \(y\) or \(y+1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{596}
Determine the number of positive integers less than $10^8$ whose digits sum to 24.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1708575}
Calculate the surface area of the portion of the cylinder defined by $x^2 + y^2 = 8y$ that is located inside the sphere $x^2 + y^2 + z^2 = 64$. Provide your answer in terms of the appropriate units.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Compute the number of ways to arrange the numbers $1, 2, 3, 4, 5, 6,$ and $7$ around a circle such that the product of every pair of adjacent numbers on the circle is at most $20.$ (Rotations and reflections count as different arrangements.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
56
}
Evaluate the integral \( \int_\Gamma (z-y)\,dx-(x+z)\,dy-(x+y)\,dz \) using Stokes' Theorem, where \( \Gamma \) is the intersection of the sphere \( x^2+y^2+z^2=4 \) with the plane \( z=y \). Assume \( \Gamma \) is oriented anticlockwise when viewed from the positive \( z \)-axis. Use \( \nabla \times F = (0, 2, 0) \) to find the result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\sqrt{2}\pi}
Determine the condition under which the expression \( n \binom{2n}{n} = \frac{(2n)!}{n!n!} \) is divisible by 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \neq 1}
Let $a_0 = 2$, $a_1 = 5$, and $a_2 = 8$, and for $n>2$ define $a_n$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$. Find $a_{2018}\cdot a_{2020}\cdot a_{2022}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
112
}
Find the 2002th term of a binary sequence defined as follows: Start with $S_1 = 0$. For $k \ge 2$, obtain $S_k$ from $S_{k-1}$ by replacing every 0 with 01 and every 1 with 001. The sequence satisfies $S_k = S_{k-1}S_{k-2}S_{k-1}$ for all $k \ge 3$. What is the 2002th term of this sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
What is the greatest power of 2 that is a factor of $10^{1002}-4^{501}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{1005}$}
Determine the smallest cardinality of a compact subset of a topological space that does not satisfy a given property.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
Evaluate the integral \( \frac{1}{2\pi i}\oint_{|z|=R} \frac{z^2}{z-2} e^{\frac{1}{z}} \, dz \) using the residue theorem, where \( R \) is a sufficiently large number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{2}}
What is the smallest infinite ordinal that is not order isomorphic to a reordering of the natural numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega_1}
Determine the interval of convergence for the series \( \sum_{n=1}^{\infty} \frac{n!x^n}{1\cdot 3\cdot 5\cdots (2n-1)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 2)}
The moisture content of freshly cut grass is $70\%$, while the moisture content of hay is $16\%. How much grass needs to be cut to obtain 1 ton of hay?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2800 \, \text{kg}}
Determine if there exists a constant \( r \) such that with probability one, \( \limsup_{t \to \infty} \frac{M_t - m_t}{\sqrt{t \log \log t}} = r \), where \( M_t = \max\{B_s : 0 \le s \le t\} \) and \( m_t = \min\{B_s : 0 \le s \le t\} \) for a standard Brownian motion \( B_t \). If such an \( r \) exists, find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}}
Students arrive at a help center according to a Poisson process with rate \( r \). When there are \( n \geq 1 \) students in the center, the first one to leave does so at a random \( \text{Exp}(2r) \) time. Suppose that there are currently no students in the center. What is the expected time until there are two students in the center?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{r}}
Given that the derivative of the function $F(x)$ is $\frac{d}{dx}F(x) = \frac{e^{\sin x}}{x}$ for $x > 0$, and if \( \int_1^4 \frac{2e^{\sin x^2}}{x} \, dx = F(k) - 1 \), find a possible value for $k$ by employing an appropriate substitution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}