inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the limit: $$\lim_{u \rightarrow -\infty} \sum_{n=0}^{\infty} \frac{\alpha^{n}}{(n !)^u}$$ for a positive constant $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the volume integral \( \iiint_ E (x+y) \, dV \) where \( E \) is the pyramid with vertices \((1,1,1), (0,0,0), (0,1,1), (0,1,0), (0,0,1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{24}}
The minimum of $ \sin \frac{A}{2} - \sqrt3 \cos \frac{A}{2}$ is attained when $ A$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\pi}{3}}
Compute the probability density function of a standard normal random variable $Z$ conditional on $Z > 0$. Express the conditional density function $f(z|Z>0)$ in terms of the standard normal density function $\phi(z)$ and the cumulative distribution function $\Phi(z)$. Provide the simplified expression for $f(z|Z>0)$. \( Z \sim N(0,1) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\phi(z)}
Evaluate the integral \( \int_{-\infty }^{\infty }|x| \delta \left(x^2\right) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Compute the integral \( \int_{\gamma} \frac{\mathrm{e}^{-z}}{(z+2)^{3}}\mathrm{d}z \) using the Cauchy integral theorem and the Cauchy integral formula, where \( \gamma(t)= 3 \mathrm{e}^{2\pi i t } \) for \( t\in [0,1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi i e^{2}}
Calculate the limit: $$ \lim_{n \to \infty} E[\cos (\sqrt{S_n} - 2\sqrt{n})],$$ where $S_n = \sum_{i=1}^n X_i$ and $X_i$ are i.i.d non-negative random variables with $\mu = 4$ and $\sigma^2 = 16$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-1/2}}
In a 3D space with a Poisson distribution of points, calculate the probability that a randomly chosen ray will not intersect any points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of ways to partition an $n \times n$ matrix into cells.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4^{n-1}}
Consider the martingale defined by \(X_t = \exp(B_t - \frac{t}{2})\), where \((B_t)\) is a Brownian motion. Determine whether \(\lim_{t \rightarrow \infty} X_t\) exists, and if so, find its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose \( \mu \) and \( \nu \) are measures such that \( \nu(\Omega) = 2 \). Let \( f \) be the Radon-Nikodym derivative of \( \mu \) with respect to \( \mu + \nu \). Find \( \nu(\{x: f(x) < 1\}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate $b+c+d$ given the integral:
$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\left(\frac{\pi}{2}-x\right)\cos x}{3+\sin^2 x}\mathrm{d}x= \frac{\pi^a}{b\sqrt c+d}$$
where $a$, $c$, and $d$ are positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given a collection of N lines, where M subsets of these lines are parallel, with each subset containing m1, m2, m3, etc. parallel lines, such that 2 ≤ m1, m2, m3, ..., ≤ N and m1 + m2 + m3 + ... ≤ M. For instance, if N = 7 and M = 2, and we have two subsets with 2 parallel lines each, determine the number of distinct triangles that can be formed by these lines.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Let $m>1$ and $n>1$ be integers. Suppose that the product of the solutions for $x$ of the equation\[8(\log_n x)(\log_m x)-7\log_n x-6 \log_m x-2013 = 0\]is the smallest possible integer. What is $m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A circle has 2000 dots, each of which can be colored either black or gray. If two black dots are adjacent, we shout "bingo." If two gray dots are two apart (with one dot in between), we shout "bingo bingo." Determine the minimum number of times "bingo" can be shouted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
For positive integers $n$, let $\nu_{3}(n)$ denote the largest integer $k$ such that $3^{k}$ divides $n$. Find the number of subsets $S$ (possibly containing 0 or 1 elements) of $\{1, 2, \ldots, 81\}$ such that for any distinct $a, b \in S$, $\nu_{3}(a - b)$ is even.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6859000
}
Find the value of \( m \) such that the line \( y = mx \) intersects the curve \( y = |x(x-1)| \) at three points, and the areas of the two regions bounded by the line and the curve are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt[3]{2} - 1}
Determine the minimum number of 100W bulbs needed to ensure that an advertising board is lit for at least 20,000 hours with a probability of at least 0.9. Assume that the life of each bulb follows an exponential distribution with an average life of 600 hours. Use the central limit theorem in your calculations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Consider a 3x3 matrix \( A \) with elements chosen from the set \( \{0, 1, 2, 3\} \). Determine the maximum possible value of \( \det(A) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Calculate the limit: $$\lim_{x\to0} \left(\int_x^1 t^{-t} dt\right)^{\frac{1}{x}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the limit: \[ \lim_{n \to \infty} \frac{1}{2n}\log\binom{2n}{n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log 2}
David wanted to calculate the volume of a prism with an equilateral triangular base. He was given the height of the prism $H=15$ and the height of the base $h=6$. He accidentally swapped the values of $H$ and $h$ in his calculations. By what number should he multiply his result to get the correct volume?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
Find the minimum value of \( B \) such that the inequality \( e^{|y|} \leq B |\sin (x+iy)| \) holds for all complex numbers \( z = x + iy \) satisfying \( |z - n\pi| \geq \frac{\pi}{4} \) for all integers \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{1 - e^{-\pi/2}}}
Let \( S_n = \frac{1}{n^6} \sum_{k=1}^{n} k^5 \) and \( T_n = \frac{1}{n^6} \sum_{k=1}^{n-1} k^5 \). Determine whether \( S_n + T_n > \frac{1}{3} \) or \( S_n + T_n < \frac{1}{3} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_n + T_n > \frac{1}{3}}
Determine the minimum value of \( r \) in the Markov chain defined by \( X_n = \begin{cases} 1.5X_{n-1} & \text{with probability } 0.5 \\ rX_{n-1} & \text{with probability } 0.5 \end{cases} \) for \( X_0 > 0 \) such that the limiting value of \( X_n \) equals \( X_0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Find the non-zero singular value(s) of the matrix \( M = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{bmatrix} \) without using singular value decomposition (SVD).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{5}}
Given that $P$ is an idempotent operator satisfying $P^2 = P$ and $P \neq \mathbb{1}$, and that $P.L |\psi\rangle = L|\psi\rangle$ holds for all $|\psi\rangle$, determine the relationship between the operators $P$ and $L$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{PL = L}
3 [ Central symmetry helps solve the problem]

What is the maximum number of pawns that can be placed on a chessboard (no more than one pawn per square), if:

1) a pawn cannot be placed on the e4 square;
2) no two pawns can stand on squares that are symmetric relative to the e4 square?

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
Find the largest positive integer \(n\) for which there exist \(n\) finite sets \(X_{1}, X_{2}, \ldots, X_{n}\) with the property that for every \(1 \leq a<b<c \leq n\), the equation \(\left|X_{a} \cup X_{b} \cup X_{c}\right|=\lceil\sqrt{a b c}\rceil\) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the number of the connected graphs with 6 vertices. (Vertices are considered to be different)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
26704
}
Given a quadratic function f(x), we define a new function F(x) as follows:

$$F(x) = \int_0^x f(t)dt$$

Assume that a is a positive number and F(x) has extreme values at x = -2a and x = 2a. What is the local maximum value of the function \(\frac{F(x)}{F'(0)}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{4a}{3}\)}
Given the function f(x) = 1/3 ax^3 + cx, which is monotonically increasing on R, and the constraint a <= 4, calculate the maximum value of a/(c^2+4) + c/(a^2+4). Express your answer as a single fraction or number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $C$ and $D$ be points on the semicircle with center $O$ and diameter $AB$ such that $ABCD$ is a convex quadrilateral. Let $Q$ be the intersection of the diagonals $[AC]$ and $[BD]$, and $P$ be the intersection of the lines tangent to the semicircle at $C$ and $D$. If $m(\widehat{AQB})=2m(\widehat{COD})$ and $|AB|=2$, then what is $|PO|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\(\frac{2\sqrt{3}}{3}\)

}
A machine consisting of two components will fail if one of the components fails. Let $X$ be the time-to-failure random variable of the first component and $Y$ be the time-to-failure random variable of the second component, both measured in hours. The joint moment generating function of the time-to-failure of the two components is given by:
$$
M_{X,Y}(s,t)=\frac{1}{(1-3s-2t+6ts)}
$$
Calculate the probability that the machine will fail within the first hour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - e^{-5/6}}
Determine the minimum number of pieces needed to cut a larger Greek cross into three smaller Greek crosses of the same size.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the integral closure of the ring $k[t^2, t^3]$ in the ring $k[t]$, where $k$ is a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k[t]}
Below is a portion of the graph of a quadratic function, $y=p(x)=dx^2 + ex + f$:

The value of $p(12)$ is an integer. The graph's axis of symmetry is $x = 10.5$, and the graph passes through the point $(3, -5)$. Based on this, what is the value of $p(12)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
Find the 50th percentile \( \xi_Z^{0.5} \) for the random variable \( Z = \exp(-0.05T) \), where \( T \) follows a half-normal distribution with the probability density function \( \frac{2}{10\sqrt{2\pi}} e^{-t^2/200} \) for \( t > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.714}
Evaluate the limit: \[ \lim_{x \to -\infty} \frac{\sqrt[n]{x+1} + \sqrt[n]{x+2} - 2\sqrt[n]{x-3}}{\sqrt[n]{x-1} + \sqrt[n]{x-2} - 2\sqrt[n]{x+3}}, \quad \text{where } n \in \mathbb{N}, n>2, \text{ and } n \text{ is odd}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given the set \( S = \{1, 2, \dots, 7\} \), determine the number of distinct values that can be obtained by multiplying the elements of each subset of \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Determine the cardinality of the set $S$, where $S = \{T: \mathbb{R}^3 \to \mathbb{R}^3 \mid T \text{ is a linear transformation over } \mathbb{R}, T(1,0,1)=(1,2,3), \text{ and } T(1,2,3)=(1,0,1)\}$. Find $|S|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
What is the necessary and sufficient condition for the existence of a homomorphism between the finite fields $\mathbb{F}_{p^a}$ and $\mathbb{F}_{p^b}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \mid b}
Find the probability that the length of a randomly chosen chord of a circle lies between \(\frac{2}{3}\) and \(\frac{5}{6}\) of its diameter.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
How many distinct necklaces can be formed using 6 identical diamonds and 3 identical pearls?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The equation $x^3 - 3x^2 - 7x - 1 = 0$ has three distinct real roots $a$, $b$, and $c$. If\[\left( \dfrac{1}{\sqrt[3]{a}-\sqrt[3]{b}} + \dfrac{1}{\sqrt[3]{b}-\sqrt[3]{c}} + \dfrac{1}{\sqrt[3]{c}-\sqrt[3]{a}} \right)^2 = \dfrac{p\sqrt[3]{q}}{r}\]where $p$, $q$, $r$ are positive integers such that $\gcd(p, r) = 1$ and $q$ is not divisible by the cube of a prime, find $100p + 10q + r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1913
}
Given a matrix $A \in M_n(F)$ where each entry is either $1$ or $0$, determine the maximum number of $1$'s that can be present in $A$ while ensuring that $A$ is invertible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - n + 1}
Four ants are placed on the vertices of a square with side length 1. Each ant starts moving towards its left neighbor. What is the length of the path each ant travels until they meet at the center of the square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given \( x_{i} \in \{-1,1\}, i=1,2,\cdots,2021 \), and \( x_{1}+x_{2}+\cdots+x_{k} \geq 0 \) for \( k=1,2,\cdots,2020 \), with \( x_{1}+x_{2}+\cdots+x_{2021}=-1 \). How many ordered arrays \( (x_{1}, x_{2}, \cdots, x_{2021}) \) are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{1011} \binom{2020}{1010}}
Let \( A \) be a \( 7 \times 7 \) matrix satisfying \( 2A^2 - A^4 = I \). If \( A \) has two distinct eigenvalues, each with geometric multiplicity 3, determine the number of non-zero entries in the Jordan Canonical Form of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider a random walk starting at 1, where each step is determined by independent and identically distributed random variables $X_i$ with $P(X_i = 1) = \frac{1}{2}$, $P(X_i = -1) = \frac{1}{4}$, and $P(X_i = 0) = \frac{1}{4}$. The position of the random walk after $n$ steps is given by $S_n = 1 + X_1 + X_2 + \cdots + X_n$. What is the probability that this random walk ever reaches the position $0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the line integral \( \oint_C \vec{F} \cdot d\vec{R} \) for the vector field \( \vec{F} = (ye^x, x^2+e^x, z^2e^z) \) along the curve \( C \) given by \( \vec{r}(t) = (1+\cos t, 1+\sin t, 1-\cos t-\sin t) \) for \( 0 \leq t \leq 2\pi \). Use Stokes' theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Jo practices her tennis serve by serving until she achieves 50 successful serves. Each serve is successful with a probability of 0.4, independently of previous serves. What is the approximate probability that she will need more than 100 serves to accomplish her goal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.974}
Find a function $K^* : \mathbb{N} \to \mathbb{N}$ such that the expression \[ f(K^*(T), T) = \mathbb{E} \left[ \max_{k \in \{1, \dots, K^*(T)\}} \sum_{t=1}^T \mathbb{I}\{X_t = k\} \right] - \frac{T}{K^*(T)} \] grows asymptotically as $\Omega(T^\alpha)$ for the largest possible $\alpha > 0$. Determine the largest possible value of $\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the integral \( \int_C \text{Re }z\,dz \) where \( C \) is the lower half of the circle of radius 4 centered at the origin, from \(-4\) to \(4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\pi i}
Find the remainder when $51!$ is divided by $61$. Use Wilson's theorem or any other method to solve the problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Evaluate the integral: $$\int_{0}^{1} \int_{0}^{1} \frac{y}{1 - (xy)^2}\, dx \, dy$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
There exist primes $ p $ and $ q $ such that

$$
pq = 1208925819614629174706176 \times 2^{4404} - 4503599560261633 \times 134217730 \times 2^{2202} + 1.
$$

Find the remainder when $ p + q $ is divided by 1000.

Proposed by Evan Chen
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
358}
Let $X$ and $Y$ be two independent random variables, where $X \sim N(0,1)$ and $Y \sim \text{Uniform}(0,1)$. Compute the probability $P(Y > X)$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6844}
Determine the minimum degree of a four-regular graph with a diameter of 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the asymptotic relative efficiency (ARE) for the Poisson process given by \(\frac{\lambda^t \exp(-\lambda)}{t!} =  P(X=t)\). For \(X = t = 0\), the best unbiased estimator of \(e^{-\lambda}\) is \(((n-1)/n)^y\), where \(y = \Sigma{X_i}\), the complete sufficient statistic for \(\lambda\). For \(X = t = 1\), the best unbiased estimator of \(\lambda e^{-\lambda}\) is \(\frac{y}{n}\left(\frac{n-1}{n}\right)^{y-1}\). The ARE for \(t=0\) is given by: \[\left[\frac{e^{-\lambda}}{\left(\frac{n-1}{n}\right)^{n\lambda}\log\left( \frac{n-1}{n}\right)^n}\right]^2\]. Calculate this expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( {S}_{1} \) and \( {S}_{2} \) be sets of points on the coordinate plane \( {\mathbb{R}}^{2} \) defined as
 follows
 \( {S}_{1} = \left( {x,y}\right)  \in  {\mathbb{R}}^{2} : \left| {x + }\right| x\left| \right|  + \left| {y + }\right| y\left| \right|  \leq  2 \)
 \( {S}_{2} = \left( {x,y}\right)  \in  {\mathbb{R}}^{2} : \left| {x - }\right| x\left| \right|  + \left| {y - }\right| y\left| \right|  \leq  2 \)
 Find the area of the intersection of \( {S}_{1} \) and \( {S}_{2} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
Calculate the limit of the expression as \( n \to \infty \):

\[ \lim_{n\to\infty}\frac{(2(n+1))!n!n^n}{(2n)!(n+1)!(n+1)^{n+1}}\left(1+\frac{1}{n}\right )^{-n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{e^2}}
How can I express $1/\alpha$ in terms of $\alpha$, given that $\alpha = \sqrt{3+\sqrt{20}}$ and the minimal polynomial of $\alpha$ is $x^4-6x^2-11$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\alpha^3-6\alpha}{11}\)}
Given an urn with $b$ blue balls and $r$ red balls, where sampling is done sequentially without replacement until a blue ball is drawn. The expected number of red balls drawn before drawing a blue ball is $\mathbb{E}(\text{number of red balls drawn}) = 1 + \sum_{i=1}^r \mathbb{E}(Z_i) - 1 = r\left(\frac{1}{b+1}\right)$. What is the variance, $\text{Var}[\text{number of red balls drawn}]$, for the number of red balls drawn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{br(b + r + 1)}{(b + 1)^2 (b + 2)}}
Compute the group of automorphisms $\text{Aut}(S)$ where $S = \mathbb{Q}[x]/(x^2)$. Express your answer in terms of a known group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Q}^\times}
Determine the subset $A \subset [0,8]^2$ with the largest Lebesgue measure such that $(A + (3,4)) \cap A = \emptyset$. What is the measure of this subset?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Jason is counting the number of cars that pass by his window. He notices that there are twice as many blue cars as there are red cars. Furthermore, there are three times as many red cars as there are orange cars. He also notices that there are half as many yellow cars as there are orange cars, but three times as many yellow cars as there are purple cars. Lastly, there are five times as many green cars as there are purple cars. Of the total cars, 35% were electric. Jason also observed that there were at least 200 blue cars and at least 50 red cars passing by. If Jason counted a total of 1423 cars, how many of them were purple?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Let $m$ and $n$ be positive integers such that $\gcd(m,n)=1$ and $$\sum_{k=0}^{2020} (-1)^k {2020 \choose k}  \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}.$$ Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that $n=12=2\times 2\times 3$, then the answer would be $2+2+3=7$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{209601}
Determine the number of connected components of the quotient space \( \mathbb{R}^2 / \sim \setminus \{(0,0)\} \), where \( \sim \) is the equivalence relation defined by \([0,0] = [0,1] = \{(0,0),(0,1)\}\) and \([x,y] = \{(x,y)\}\) if \(x \neq 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For each nonnegative integer \( n \), there exists a polynomial \( p_n(x) \) such that
\[ \int x^n e^x \, dx = p_n(x) e^x + C \]
Let
\[ L = \lim_{n \to \infty} \frac{|p_n(2)|}{n!} \]
Find \( \lfloor 1000L \rfloor \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{135}
Calculate the probability of collecting all $n$ coupons in the Coupon Collector's Problem if you purchase $m = n \cdot \ln(n) + t$ coupons, where $t$ is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-1}}
Use the Divergence Theorem to compute the flux of the vector field \( F = \langle 0, 0, x+z \rangle \) across the surface \( S \), which is the part of the plane \( x+y+2z=4 \) inside the first octant. Assume the unit normal vector \( n \) to \( S \) has a positive third component. Find the flux of \( F \) across \( S \) by evaluating an appropriate triple integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Identify a subgroup $H$ of the set $\Bbb{R}\setminus\{0\}$ such that for any $x, y \in \Bbb{R}\setminus\{0\}$, the quotient $xy^{-1}$ belongs to $H$ if and only if $x$ and $y$ share the same sign.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}^+}
Consider the topology on \(\mathbb{R}\) generated by the basis \(\{[a, b) : a, b \in \mathbb{Q}\}\). What is the closure of the set \((0, \sqrt{2})\) in this topology?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \sqrt{2}]}