inclusionAI/Ring-lite-rl-data huggingface dataset:
Let the functions \( f(\alpha, x) \) and \( g(\alpha) \) be defined as
\[ f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} \]
\[ g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2} \]

Then \( g(\alpha) \) is a polynomial in \( \alpha \). Find the leading coefficient of \( g(\alpha) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/16}
Find the value of the following limit:

$$\lim_{q\to 1^-}\frac{\pi K'} {K} \cdot\frac{K} {2\pi^2}\cdot(E-k'^2K)$$

where $K, K', E$ are elliptic integrals and $k, k'$ are moduli corresponding to the nome $q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.25}
Find the greatest common divisor (GCD) of all numbers of the form $p^{12} - 1$, where $p$ is a prime number with a units digit of 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65520}
Find an upper bound for \( \mathbb{E}\left[\frac{1}{1+M}\right] \), where \( M \) follows a Binomial distribution \( B(n,p) \). Express the upper bound in terms of \( n^{-1} \), and consider the case where \( n \) is sufficiently large.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{pn}}
The 600 students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. What is the probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group? Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{9}}
A permutation  $(a_1, a_2, a_3, \dots, a_{2012})$  of  $(1, 2, 3, \dots, 2012)$  is selected at random. If  $S$  is the expected value of 
		\[
		\sum_{i = 1}^{2012} | a_i - i |,
		\]
		then compute the sum of the prime factors of  $S$ .

*Proposed by Aaron Lin*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2083}
In a right-angled triangle $ABC$ with $\angle A = 90^{\circ}$, it is given that $AB = AC = 2AD$. The line $AE$ is perpendicular to $BD$, and the side lengths are denoted by $a$. Let point $O$ be the intersection of $BD$ and $AE$. Given that $AO = \frac{a}{\sqrt 5}$, determine the ratio $BE : CE$. Express your answer as a simplified ratio.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:1}
Determine the necessary condition for the function $f(x)$ to have an inverse, given the functional relationship \(2(f(x) - 2x) = -\left(f\left(\frac{1}{x}\right) - \frac{2}{x}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = 2x}
Determine the change of variable that can be used to show the equality \( \int_0^{\infty}\frac{dx}{1+x^2}=2\int_0^1\frac{dx}{1+x^2} \) without performing direct integration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x = \frac{1}{t}}
In Rabin's cryptosystem, given the public and private keys with \( p = 127 \), \( q = 131 \), \( N = 16637 \), and \( B = 12345 \), the ciphertext \( c \) is computed as follows for the plaintext \( m = 4410 \):

\[ c = m(m + B) \mod N = 4633 \]

To decrypt, compute:

\[ t = \left(\frac{B^2}{4} + c\right) \mod N \]

Find the value of \( t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1500}
Determine the largest natural number \( m \) such that \[ \int_0^\infty \frac{\ln^m x}{e^x} \, \mathrm{d}x < 2. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that the two sides of a triangle are factors of 72. Then, the number of different such triangles is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
In a class, there are \( a_{1} \) students who received at least one failing grade during the year, \( a_{2} \) students who received at least two failing grades, ..., and \( a_{k} \) students who received at least \( k \) failing grades. How many failing grades are there in total in this class?

(It is assumed that no one has more than \( k \) failing grades.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_1 + a_2 + \ldots + a_k}
In an arithmetic sequence with a common difference of 3, find the first term if the sum of the squares of the first 1001 terms is equal to the sum of the squares of the next 1000 terms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3000}
Evaluate the integral \( \int_{0}^{\infty}\frac{dv}{v\cosh v} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
设$M = \{ 1 , 2 , \cdots , 2 0 \},A _ { 1 }A _ { 2 } , \cdots A _ { n }$是M的互不相同的非空子集，当$i \neq j$时，$A _ { i } \cap A _ { j }$至多有两个元，求n的最大值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1350}
Suppose that $X_1, X_2, \ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S=\sum\limits_{i=1}^kX_i/2^i,$ where $k$ is the least positive integer such that $X_k<X_{k+1},$ or $k=\infty$ if there is no such integer. Find the expected value of $S.$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$ 2e^{\frac 12} - 3$}
A girl, who is an acquaintance of a young man sitting by the window of a moving tram, was walking towards the tram. Eight seconds after she passed by the window, the young man got off the tram and started following her. How much time passed from this moment until he caught up with the girl? The speed of the young man is twice the speed of the girl and one-fifth of the speed of the tram.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88 \text{ s}}
Find the value of $k$ such that the line $y=k$ intersects the curve $y=2x-3x^3$ in the first quadrant, and the areas of the shaded regions formed are equal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{9}}
Call a lattice point *visible* if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$, denote by $S_k$ the set of all visible lattice points $(x, y)$ such that $x^2 + y^2 = k^2$. Let $D$ denote the set of all positive divisors of $2021 \cdot 2025$. Compute the sum \[ \sum_{d \in D} |S_d| \]

Here, a lattice point is a point $(x, y)$ on the plane where both $x$ and $y$ are integers, and $|A|$ denotes the number of elements of the set $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the derivative of an $m \times m$ real matrix $A$ with respect to its vectorized transpose $(\mathrm{vec}(A))^T$, without using tensor notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_{m^2}}
Evaluate the limit: $$\lim_{n\to \infty} \int_{0}^{n} \frac{1}{1+n^2\cos^2x}\, dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \(ABCD\) be a circumscribed quadrilateral such that \(m(\widehat{A})=m(\widehat{B})=120^\circ\), \(m(\widehat{C})=30^\circ\), and \(|BC|=2\). Find the length of \(|AD|\). The original answer is in the form \(\sqrt{k} - m\). Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \(H\) be a subgroup of a finite group \(G\) with \(|H| = 3\). Determine the value of \(|N_{G}(H) : C_{G}(H)|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a soccer league with \( n \) teams (\( n > 3 \)), each pair of teams played at most one match, and there were no draws. Exactly one team remained unbeaten. Additionally, for any two teams \( A \) and \( B \), there exists another team \( C \) such that either \( A \) defeated \( C \) and \( B \) defeated \( C \), or \( C \) defeated \( A \) and \( C \) defeated \( B \). What is the minimum possible number of matches \( k \) that were held?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(n-1)}
Determine the values of $p$ for which the triangle inequality fails for the function $\lVert (x_1,x_2) \rVert_p = (|x_1|^p+|x_2|^p)^{1/p}$ on $\mathbb{R}^2$. Provide your answer as a range or set of values for $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1)}
Find the value of \( k \) for which the solution to the differential equation \( y'' + 4y' + ky = 0 \) vanishes at \( t = 2 \) and \( t = 3 \), but not in between.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 + \pi^2}
For which real numbers \(\alpha\) does the sequence \(a_n = \sin(\alpha n)\) converge to a limit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha = k\pi}
Determine the number of partial orders on a set \( S \) where \( |S| = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Evaluate the limit: $$ \lim _{x\to \infty \:}\left(\frac{\left(\left(2x\right)!\right)}{x^xx!}\right)^{\frac{1}{x}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{e}}
Consider a Markov chain where $X_0 = x$ and each subsequent state $X_i$ is drawn from a discrete uniform distribution on the set $\{0, \dots, X_i-1\}$. Determine the expected absorption time for the process to reach the state 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sum_{k=1}^{x} \frac{1}{k}}
Let $L$ be a Lie algebra of dimension $n$ and let $M(n \times n)$ denote the space of square matrices of size $n$. Determine the dimension of $\text{Hom} \big( L, M(n \times n) \big)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^3}
Evaluate the integral \( \int_0^{+\infty} \frac{\sin^4{x}}{x^4} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Determine the largest integer $k$ for which the equation \[ 25\lfloor x\rfloor - \lfloor x\lfloor x\rfloor \rfloor = 50k - 13\lceil x\rceil \] has a real solution for $x$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let \( \alpha \) be an element in an algebraic closure of \( GF(64) \) such that \( \alpha^4 = \alpha + 1 \). Determine the natural number \( r \) for which \( GF(64)(\alpha) \) is isomorphic to \( GF(2^r) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the stabilizer of the coset $aH$ for the action of the group $G$ on the set $X = G/H$ by left multiplication.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{aHa^{-1}}
Evaluate the integral: $$\int_{0}^{\infty}\frac{\ln x}{(x+1)^{3}}\mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Determine the number of steps required to construct the centroid of 10 points using a compass with a fixed opening.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Given integers $a$ and $b$, and odd prime numbers $p$ and $q$, suppose there exists a solution to the equation $a^2 + 2b^2 = p^{11}q^{13}$. Determine the number of solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
Given that $4x^2 + 2xy + 9y^2 = 100$, determine all possible values of the expression $x^2 + 2xy + 3y^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ \dfrac{100}{7}, 40 \right]}
Suppose there are 3 bank tellers, each currently occupied with a customer, and you are next in line. The service times are independent and identically distributed exponential random variables with parameter \( \lambda \). What is the probability that you will be the last customer to complete service?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Find the constant \( c \) such that \( \text{P}(\limsup\limits_{n\to \infty} X_n/\sqrt{\log n}=c)= 1 \), where \( X_n \) are a sequence of independent random variables with \( X_k\sim\mathcal{N}(0,1) \). Use the fact that \( \sqrt{2\pi}\xi\exp\left({\xi^2\over 2}\right)\text{P}(X\geq \xi)\to 1 \) as \( \xi \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Given a positive integer N, calculate the number of integers from 1 to N (inclusive) whose binary representation contains exactly K zeroes. For example, if N = 18 and K = 3, determine how many such numbers exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
If three lines \( a, b, c \) in space are pairwise skew lines, determine the number of lines that intersect with all three lines \( a, b, \) and \( c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral: $$\int_{-2}^{1} \int_{-2}^{1} \int_{-2}^{1} \frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the contour integral \( \oint_{|z|=2} z^n(1 - z)^m\ dz \) where \(m\) and \(n\) are negative integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find $$\sum^{k=672}_{k=0} \binom{2018}{3k+2} \pmod{3}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the sequence $\{a_n\}$, $(a_1=1, a_2=2)$, and $(a_{n+2} - a_n = 1 + (-1)^n (n \in \mathbb{N}^*))$, find the value of $(a_1 + a_2 + \dots + a_{51} = \_\_\_\_\_\_\_)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{676}
There are  $2022$  equally spaced points on a circular track  $\gamma$  of circumference  $2022$ . The points are labeled  $A_1, A_2, \ldots, A_{2022}$  in some order, each label used once. Initially, Bunbun the Bunny begins at  $A_1$ . She hops along  $\gamma$  from  $A_1$  to  $A_2$ , then from  $A_2$  to  $A_3$ , until she reaches  $A_{2022}$ , after which she hops back to  $A_1$ . When hopping from  $P$  to  $Q$ , she always hops along the shorter of the two arcs  $\widehat{PQ}$  of  $\gamma$ ; if  $\overline{PQ}$  is a diameter of  $\gamma$ , she moves along either semicircle.

Determine the maximal possible sum of the lengths of the  $2022$  arcs which Bunbun traveled, over all possible labellings of the  $2022$  points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2042222}
Given \(\triangle ABC\). On ray \(BA\), take a point \(D\) such that \(|BD| = |BA| + |AC|\). Let \(K\) and \(M\) be two points on rays \(BA\) and \(BC\) respectively, such that the area of \(\triangle BDM\) is equal to the area of \(\triangle BCK\). Find \(\angle BKM\) if \(\angle BAC = \alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\alpha}{2}}
Find the total number of non-negative integer solutions to the equation $x_1+2x_2+x_3+x_4+\ldots+x_{10}=3$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{174}
Determine the value of $v$ that minimizes the integral \[ \int_{-\infty}^{\infty} p(x) \log \frac{p(x)}{g(x,v)} \, dx \] where $p(x)$ and $g(x,v)$ are probability density functions given by \[ p(x) = \frac{1}{2} \left( \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x-2)^2} + \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x+1)^2} \right) \] and \[ g(x,v) = \frac{1}{3} \left( \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x-1)^2} + \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x+3)^2} + \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x-v)^2} \right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Evaluate the limit: $$ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin(x)}{4+n^2x^2} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Two circles with radii \( R \) and \( r \) intersect at point \( A \). Let \( BC \) be a common tangent to both circles at points \( B \) and \( C \). Find the radius of the circumcircle of triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{R r}}
In multivariable calculus, find a constant $c$ such that the tangent planes of the spheres $(x-c)^2 + y^2 + z^2 = 3$ and $x^2 + (y-1)^2 + z^2 = 1$ are perpendicular at any intersection point.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\pm \sqrt{3}\)}
Determine the number of homomorphisms from the dihedral group $D_{21}$ to the dihedral group $D_{15}$. Use the fact that any element in $D_{21}$ can be expressed as $s^ir^j$ and that for homomorphisms, $\phi(s^{i + i'}r^{j' - j}) = \phi(s^ir^j)\phi(s^{i'}r^{i'})$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers \[ N = a + (a+1) +(a+2) + \cdots + (a+k-1) \] for $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Given the group $G = S_7$ and a subgroup $H = \langle (1\ 2\ 3\ 4\ 5\ 6\ 7) \rangle$, determine the number of elements in the intersection of $A_7$ and the normalizer $N_G(H)$. It is given that $|N_G(H)| = 42$ and $|A_7| = 2520$. Use the internal product formula to find $|A_7 \cap N_G(H)|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
On Monday, 5 students in the class received A's in math, on Tuesday 8 students received A's, on Wednesday 6 students, on Thursday 4 students, and on Friday 9 students. None of the students received A's on two consecutive days. What is the minimum number of students that could have been in the class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $C$ be the linear code over $\mathbb{F}_5$ with parity check matrix $H$ whose columns are vectors of the form $\begin{bmatrix}1\\ \alpha\\ \alpha^{2} \end{bmatrix}$, where $\alpha\in\mathbb{F}_5$. Determine the number of codewords in $C$ that have a Hamming weight of exactly 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a regular $2 n+1$-sided polygon. How many triangles are there whose vertices are among the vertices of the polygon and the triangle contains the center of the polygon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n+1)(2n+1)}{6}}
Determine the number of non-isomorphic random graphs that can be formed with a set of cardinality \( \kappa \), where \( \kappa \) is a regular cardinal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^\kappa}
Three customers, A, B, and C, arrive simultaneously at a bank with two tellers on duty. The tellers were idle when the customers arrived. Customer A goes to one teller, B goes to the other, and C waits until either A or B finishes before receiving service. If the service times are exponentially distributed with a mean of 4 minutes, what is the probability that customer A is still in the bank after customers B and C have left?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
A regular quadrilateral pyramid is intersected by a plane passing through a vertex of the base perpendicular to the opposite lateral edge. The area of the resulting cross-section is half the area of the pyramid's base. Find the ratio of the height of the pyramid to the lateral edge.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{33}}{8}}
Evaluate the limit: $$\lim_{x \to 0}\lim_{n \to \infty} \frac{n^{\operatorname{cosec}^{2}x}}{(1^{\tan^{2}x}+2^{\tan^{2}x}+\cdots+n^{\tan^{2}x})^{\cot^{2}x}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Find the remainder when the infinite power tower \(7^{7^{7^{7^{.^{.^{.^{\infty}}}}}}}\) is divided by 13.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the sum of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}$ with an accuracy of $10^{-3}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.973}
Determine the number of surjective functions from the set \( \{1, 2, \ldots, 10\} \) to the set \( \{1, 2, \ldots, 9\} \) such that \( f(2i+1) > f(2i-1) \) and \( f(2i) > f(2i+2) \) for \( i = 1, 2, 3, 4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{630}
Given $F_1$ and $F_2$ are the two foci of the ellipse $E$ on the left and right, respectively, and the parabola $C$ has $F_1$ as its vertex and $F_2$ as its focus. Let $P$ be a point of intersection between the ellipse and the parabola. If the eccentricity $e$ of the ellipse satisfies $|PF_1| = e|PF_2|$, then the value of $e$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac { \sqrt {3}}{3}}
Determine the values of \(a > 0\) and \(b > 0\) for which the integral \(\int_{1}^{\infty} \frac{1}{x^a(1+x^b)} \,dx\) is convergent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a + b > 1}
The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
For circles $C_1: x^2+y^2+2ax+a^2-9=0$ and $C_2: x^2+y^2-4by-1+4b^2=0$ having only one common tangent line, given $a\in \mathbb{R}, b\in \mathbb{R}$, and $ab\neq 0$, the minimum value of $\frac{4}{a^2} + \frac{1}{b^2}$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Flight number 608 of "Aeroflot" departs from Moscow at 12:00 and arrives in Bishkek at 18:00 (local time).

The return flight number 607 departs at 8:00 and arrives at 10:00. How long does the flight last?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4 \text{ hours}}
Determine the limit of the series \( S = 1 + r + \frac{1}{2}r^2 + \frac{1}{3}r^3 + \cdots \) if it exists, given that the series converges for \( r < 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \ln(1 - r)}
Reduce the lambda calculus expression \((\lambda x. (\lambda y. x \, y) \, x) \, (\lambda z.p)\) to its simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p}
In the complex plane, the points \( 0, z, \frac{1}{z}, z+\frac{1}{z} \) form a parallelogram with an area of \( \frac{35}{37} \). If the real part of \( z \) is greater than 0, find the minimum value of \( \left| z + \frac{1}{z} \right| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5 \sqrt{74}}{37}}
Determine the number of 8-tuples \((a_1, a_2, \ldots, a_8)\) from the set \([4]^8 = \{1, 2, 3, 4\}^8\) such that each number in \([4]\) appears exactly twice, and for each \(i \in [4]\), the number \(i\) does not appear in the \(i\)-th position.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{864}
Given that the roots of the polynomial \(x^4 + 2x^3 - 5x^2 + 7x + 10 = 0\) are \(\alpha, \beta, \gamma, \delta\), and the roots of the polynomial \(x^4 + ax^3 + bx^2 + cx + d = 0\) are \(\alpha+\beta+\gamma, \alpha+\beta+\delta, \alpha+\gamma+\delta, \beta+\gamma+\delta\), find the value of \(a + b - c - d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}