inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$ be a set with $2n$ distinct elements, and $B_i\subseteq A$ for any $i=1,2,\cdots,m.$ If $\bigcup\limits_{i=1}^m B_i=A,$ we say that the ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A.$ If $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A,$ and for any $i=1,2,\cdots,m$ and any $j=1,2,\cdots,n,$ $\{a_j,b_j\}$ is not a subset of $B_i,$ then we say that ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A$ without pairs. Define $a(m,n)$ as the number of the ordered $m-$coverings of $A,$ and $b(m,n)$ as the number of the ordered $m-$coverings of $A$ without pairs.
Let $m\geq2,$ and there is at least one positive integer $n,$ such that $\dfrac{a(m,n)}{b(m,n)}\leq2021,$ Determine the greatest possible values of $m.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
Let \( A = \{1, 2, \ldots, 2002\} \) and \( M = \{1001, 2003, 3005\} \). A non-empty subset \( B \) of \( A \) is called an \( M \)-free set if the sum of any two elements in \( B \) does not belong to \( M \). If \( A = A_1 \cup A_2 \), \( A_1 \cap A_2 = \varnothing \), and both \( A_1 \) and \( A_2 \) are \( M \)-free sets, then the ordered pair \(\left(A_{1}, A_{2}\right)\) is called an \( M \)-partition of \( A \). Determine the number of all \( M \)-partitions of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{501}}
Randy has a deck of $29$ distinct cards. He chooses one of the $29!$ permutations of the deck and then repeatedly rearranges the deck using that permutation until the deck returns to its original order for the first time. What is the maximum number of times Randy may need to rearrange the deck?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2520
}
Determine the maximum order of the automorphism group of an elliptic curve over a field of characteristic 0, when restricted to torsion subgroups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Consider an infinitely differentiable function $y=f(x)$ defined on the real numbers, such that $f(0) \neq 0$ and the $n$th derivative of $y$ with respect to $x$ evaluated at $x=0$ is nonzero for $n=1,2,3,4$. Determine the value of $25a + 50b + 100c + 500d$, where $a$, $b$, $c$, and $d$ are constants, given that the following limit exists:

$$\lim_{x \to 0} \frac{f(4x) + af(3x) + bf(2x) + cf(x) + df(0)}{x^4} = L$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Let $A$ be a finite abelian group of order $360$ which does not contain any elements of order $12$ or $18$. How many elements of order $6$ does $A$ contain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Let $f$ be a function from the set $\{1,2,3,\dots,10\}$ to the real numbers $\Bbb{R}$, satisfying the condition:

$$\left(\sum\limits_{i=1}^{10}{\frac{|f(i)|}{2^i}}\right)^2 = \left(\sum\limits_{i=1}^{10}{|f(i)|^2}\right)\left(\sum\limits_{i=1}^{10}{\frac{1}{4^i}}\right)$$

Determine the number of distinct functions $f$ that satisfy this condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
Find the minimum value of the function:

$$f(x, y) = \sqrt{4 + y^2} + \sqrt{(x - 2)^2 + (2 - y)^2} + \sqrt{(4 - x)^2 + 1}$$

where \(x\) and \(y\) are real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the limit: $$\lim _{x\to \infty }\left(x\sqrt{x^2-x}-x^2\cos\left(\frac{1}{\sqrt{x}}\right)\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{6}}
Given a ring \( R \) and an ideal \( \mathbf{a} \), let \( \pi: R \rightarrow R/\mathbf{a} \) be the natural projection map. Define \( \pi^*:\mathrm{Spec}(R / \mathbf{a}) \rightarrow \mathrm{Spec}(R) \) as the pullback of \( \pi \). Determine the image of \( \pi^* \) in \( \mathrm{Spec}(R) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{V(\mathbf{a})}
Evaluate the integral \( \int_0^1 \frac{x^{10}-1}{\ln x} \, dx \) using Feynman's method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 11}
Find the value of \(a\) for which the integral \(\int_{0}^{\infty}\frac{dx}{a^2+\left(x-\frac{1}{x}\right)^2}\) equals \(\frac{\pi}{5050}\) for \(a \geq 2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2525}
Evaluate the integral \( \int_R (y+2x^2)(y-x^2) \, dA \), where the region \( R \) is bounded by the curves \( xy=1 \), \( xy=2 \), \( y=x^2 \), and \( y=x^2-1 \), with both \( x \) and \( y \) positive.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
In triangle $ABC$, a point $O$ is located such that $OA = 15$, $OB = 12$, and $OC = 20$. The feet of the perpendiculars from $O$ to the sides of triangle $ABC$ form an equilateral triangle. Determine the measure of angle $BAC$. \( \text{(in degrees)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
A unit circle is centered at(0,0)on the(x,y)plane. A regular hexagon passing through(1,0)is inscribed in the circle. Two
 points are randomly selected from the interior of the circle and horizontal lines are drawn through them, dividing the hexagon
 into at most three pieces. The probability that each piece contains exactly two of the hexagon's original vertices can be
 written as
 \( \frac{2{\left( \frac{m\pi }{n} + \frac{\sqrt{p}}{q}\right) }^{2}}{{\pi }^{2}} \)
 for positive integers \( m,n,p \) ,and \( q \) such that \( m \) and \( n \) are relatively prime and \( p \) is squarefree. Find \( m + n + p + q \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
11}
Determine the smallest integer \( n \) such that for any two sets \( \{ x_1, x_2, \ldots, x_n \} \) and \( \{ y_1, y_2, \ldots, y_n \} \) of \( n \) distinct points on the unit circle \( \mathbb{S}^1 \), there exists a unique automorphism \( f: \mathbb{D} \rightarrow \mathbb{D} \) satisfying \( f(x_i) = y_i \) for all \( i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the number of terms needed in the series \( \sum_{n=0}^\infty \frac{(-1)^n(7)^{2n+1}}{(2n+1)(2n+1)!} \) to approximate the integral \( \int_0^1 \frac{\sin(7x)}{x} \, dx \) to four decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
The inclination angle of the line $x-y+1=0$ is what value?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{\\pi}{4}"]}
Calculate the minimum value of the function \( f(z) = |z^2 + z + 1| + |z^2 - z + 1| \) for \(z \in \mathbb{C}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Solve the equation for real numbers $x_1, x_2, \ldots, x_n$:

$$
\sqrt{x_{1}-1^{2}}+2 \sqrt{x_{2}-2^{2}}+\cdots+n \sqrt{x_{n}-n^{2}}=\frac{x_{1}+x_{2}+\cdots+x_{n}}{2}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_k = 2k^2}
Determine all values of \( a \) for which the equation \( a\sin{x}+(a+1)\sin^2{\frac{x}{2}} + (a-1)\cos^2{\frac{x}{2}} =1 \) has at least one solution for \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, \infty)}
Evaluate the integral: $$\int x^{x^2+1}(2\ln x+1)dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^{x^2}}
A mouse is trapped in a maze and can initially choose to go right or left. If it goes left, it will walk for 3 minutes and return to the initial position. If it goes right, with probability \( \frac{1}{3} \) it will exit the maze after 2 minutes, and with probability \( \frac{2}{3} \) it will return to the initial position after 5 minutes. Assuming the mouse randomly chooses left or right, calculate the variance of the time it remains in the maze.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{438}
Determine the convergence domain of the series \( \sum_{n=1}^{\infty}\frac{(2n-1)^{n}(x+1)^n}{2^{n-1}n^n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 0)}
Consider a random sample of size \(n\) from a distribution with the probability density function given by:
\[
f(x;\theta)=\begin{cases}
 \sqrt{\frac{2}{\pi}}e^{-\frac{1}{2}(x-\theta)^2} & \text{if } x \geq \theta \\
 0 & \text{elsewhere}
\end{cases}
\]
Find the maximum likelihood estimator of \(\theta\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X_{(1)}}
For a given natural number $n$, determine the greatest possible value of $k$ such that there exist subsets $A, B, C, D$ of the set of the first $n$ natural numbers with the property that any two elements of the set are in at least one of the subsets, and at least one of $|A|,|B|,|C|,|D|$ is greater than $k|S|$. Find the maximum value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the value of \(a\) such that for \(a \in \mathbb{R}\) and \(a \neq 1\), the following limit holds:
\[ \lim_{n \to \infty} \frac{1^a + 2^a + \cdots + n^a}{(n+1)^{a-1} \left[ (na+1) + (na+2) + \cdots + (na+n) \right]} = \frac{1}{60}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the number of continuous involutions on the set of real numbers, where an involution is a function \( f \) such that \( f = f^{-1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Determine the number of inequivalent complete norms that exist on an infinite-dimensional vector space $X$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Calculate the probability that five randomly chosen points form a hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
A quadrilateral that has consecutive sides of lengths $70,90,130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length 130 divides that side into segments of length $x$ and $y$. Find $|x-y|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13

}
Evaluate the limit: $$L = \lim_{x\rightarrow\infty}xe^{x^2} \left(\int_0^xe^{-t^2}dt-\int_0^\infty e^{-t^2 } dt\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Solve the equation for real numbers \(x\), given that \(a\) and \(b\) are real:
\[
\left| a^2-2a-b^2-4b-x \right| + \left| a^2-2a-b^2-4b-3x+2 \right| + \left| a^2-2a+b^2+4b+2x \right| + a^2-2a+b^2+4b+18 \left| x-2 \right| + 11x = 20.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of matrices in $GL(2,\mathbb{F_{25}})$ that have a determinant equal to 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15600}
Find the codimension of the maximal ideal $J = (x,y,z)$ in the ring $R = k[x,y,z]/(x^2y^2z^2, xy^3z^2)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral: $$\int_0^{2\pi}\cot(e^{it})ie^{it}dt$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Calculate the limit of the sequence defined by:
\[ x_n = \frac{1}{n}\left(n + \frac{n-1}{2} + \frac{n-2}{3} + \ldots + \frac{2}{n-1} + \frac{1}{n} - \log(n!)\right) \]
Use asymptotic equivalences to find the limit, considering that \( \log(n!) \sim n\log(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gamma}
Let $A$ be a complex unital Banach algebra and consider the matrix $R_t=\begin{pmatrix} \cos\frac{\pi t}{2} & -\sin\frac{\pi t}{2} \\ \sin\frac{\pi t}{2} & \cos\frac{\pi t}{2} \end{pmatrix}$. Determine the norm of $R_t$ as an element of $M_2(A)$, where $M_2(A)$ is equipped with the operator norm derived from $A^2$ with the norm $||(a,b)||=\sqrt{||a||^2+||b||^2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the value of the real constant \(a\) for which the series \[ 1 + \frac{1}{\sqrt{3}} - \frac{a}{\sqrt{2}} + \frac{1}{\sqrt{5}} + \frac{1}{\sqrt{7}} - \frac{a}{\sqrt{4}} + \frac{1}{\sqrt{9}} + \frac{1}{\sqrt{11}} - \frac{a}{\sqrt{6}} + \cdots \] converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Let $G=(\Sigma,N,R,S)$ be a context-free grammar where $n = |N|$ is the number of non-terminal symbols and $k$ is the maximal length of a production rule. Assume $L(G)$, the language generated by $G$, is not empty. Find a tight upper bound expression $f(n,k)$ for the length of the shortest word in $L(G)$. The expression should satisfy that the shortest word in $L(G)$ is at most $f(n,k)$, and there exists another context-free grammar $F$ such that the shortest word in $L(F)$ is exactly $f(n,k)$. Provide the expression $f(n,k)$. \( \text{Express your answer in terms of } n \text{ and } k. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k^{n-1}}
Consider the sequence \( \{ a_{n} \} \) in the Cantor space with respect to the Cantor metric, where each term \( a_n \) consists of the first \( n \) digits being 0 followed by 1s. Does this sequence converge? If so, determine the limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{000\ldots}
John’s birthday cake is a scrumptious cylinder of radius $6$ inches and height $3$ inches. If his friends cut the cake into $8$ equal sectors, what is the total surface area of a piece of birthday cake?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{27\pi}{2}+36}
Find all increasing functions \( g: \mathbb{Z}^+ \to \mathbb{Z} \) such that \( g(1) = 1 \) and \( g(ab) = g(a) + g(b) + g(a)g(b) \) for all positive integers \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(n) = 2^n - 1}
Calculate the integral \( \oint_{L}{\frac{e^{\frac{1}{z-a}}}{z}}dz \) using the Residue Theorem, where \( L=\{z\in\mathbb{C}:|z|=r\} \) for some \( r>|a| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
$k$ stones are put into $2012$ boxes in such a way that each box has at most $20$ stones. We are chosing some of the boxes. We are throwing some of the stones of the chosen boxes. Whatever the first arrangement of the stones inside the boxes is, if we can guarantee that there are equal stones inside the chosen boxes and the sum of them is at least $100$, then $k$ can be at least?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
349}
Find a function $g(t) \in L^1(\mathbb{R})$ that satisfies the integral equation:

$$ g(t) = \int_{-\infty}^{t}e^{u-t}g(u)\,du + f'(t) $$

where $f \in L^1(\mathbb{R})$ and its derivative $f'$ is both continuous and in $L^1(\mathbb{R})$. Use the Fourier transform to determine $g(t)$. Provide your answer as a function $g(t)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(t) = f(t) + f'(t)}
A rhombus \( \mathcal{R} \) has a short diagonal of length 1 and a long diagonal of length 2023. Let \( \mathcal{R}' \) be the rotation of \( \mathcal{R} \) by \( 90^\circ \) about its center. Define \( \mathcal{U} \) as the set of all points contained in either \( \mathcal{R} \) or \( \mathcal{R}' \) (the union of \( \mathcal{R} \) and \( \mathcal{R}' \)), and \( \mathcal{I} \) as the set of all points contained in both \( \mathcal{R} \) and \( \mathcal{R}' \) (the intersection of \( \mathcal{R} \) and \( \mathcal{R}' \)). Compute the ratio of the area of \( \mathcal{I} \) to the area of \( \mathcal{U} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2023}}
Given two random vectors \( X = (X_1, \ldots, X_n) \) and \( Y = (Y_1, \ldots, Y_n) \), with a covariance matrix \( G \) such that the covariance between \( X_i \) and \( Y_j \) is \( G_{ij} \), and two \( n \times n \) matrices \( A \) and \( B \), find the covariance matrix between \( AX \) and \( BY \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A G B^\top}
Determine the number of terms, n, that need to be summed in the Maclaurin series for the arctangent function to ensure that \(|4P_n(1) − π| < 10^{-3}\), given that \(\tan \frac{π}{4} = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Find the maximum area bounded by the curves \(y^2=4ax\), \(y=ax\), and \(y=\frac{x}{a}\) for \(a \in [1,2]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
Determine the number of non-isomorphic semisimple $\mathbb{C}$-algebras of dimension 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Solve the equation: \[ 8x^2 + 2\ln x - 6 = 0 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8837}
Determine the number of ways to arrange the first 9 positive integers on a 3x3 chessboard such that the largest of the smallest numbers in each column is equal to the smallest of the largest numbers in each row, and both are equal to 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25920}
Determine the minimum number of bits Alice must transmit to Bob so that he can compute the approximate Hamming distances between a binary string $A$ of length $n$ and every substring of length $n$ of a binary string $B$ of length $2n$, within a factor of 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\log n)}
Determine the smallest integer $n$ for which the number of pairs $(x, y)$ satisfying $x \ge y$, $\operatorname{lcm}(x, y) = n!$, and $\gcd(x, y) = 2023$ is greater than 2023. Note that $2023 = 7 \times 17^2$. What is this value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Evaluate the limit: $$\lim_{\alpha\to\infty} \int_{0}^{\delta}\frac{\sin(\alpha x) \sin(\lambda\alpha x)}{x^2}dx$$ where $\lambda,\delta \in \mathbb{R}, \lambda,\delta > 0$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the complexity of the recurrence relation \( T(n) = T(n - \sqrt{n}) + n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^{3/2})}
Given $a,b,c,d \in \mathbb{R}$ such that $c^2+d^2 \neq 0$, consider the Cauchy problem $au_x+bu_y=e^{x+y}$ for $x,y\in \mathbb{R}$ with the condition $u(x,y)=0$ on $cx+dy=0$. Using Lagrange's equations, we have $bx-ay=C_1$ and $u-\frac{a}{a+b}e^{x+y}=C_2$. Determine the condition for a unique solution to this problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ac + bd \neq 0}
Consider a Markov chain with state space $S = \{0, 1, 2, \dots \}$ and given transition probabilities. Determine the values of $p$ for which the chain is recurrent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p \leq \dfrac{1}{2}}
Evaluate the limit: $$\lim_{x \to 0} \frac{(x-\sqrt{2})^4}{\sin^3(x)\cos^2(x)}\int_{0}^x \frac{t^2}{t^4+1}dt$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Evaluate the limit: $$\lim_{x \to 0^+} \lim_{n \to \infty} \frac{1}{1 + n \sin^2 x}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the integer \( \lambda \) such that the group homomorphism \( \mathbb{Z} \cong \pi_{n+1}(S^{n+1}) \rightarrow \pi_n(SO_{n+1}) \rightarrow \pi_n(S^n) \cong \mathbb{Z} \) is multiplication by \( \lambda \). The map is induced by the matrix multiplication map \( \alpha: SO_{n+1} \rightarrow S^n \) and the connecting homomorphism of the fibration \( SO_{n+1} \rightarrow SO_{n+2} \rightarrow S^{n+1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
## 11. "Wipers"

On the windshield of a car, there are two "wipers" of length $L$ each, rotating around two points that are also $L$ apart. Each "wiper" "sweeps" one semicircle. What area do both wipers sweep?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["(2\\pi/3+\\sqrt{3}/4)L^{2}"]}
Determine the constant $p$ such that the geometric mean of any positive number $N_0$ multiplied by successive random numbers between $0$ and $p$ will, on average, neither diverge to infinity nor converge to zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Calculate the conditional expectation \( E[W|X>Y] \) where \( W = X+Y \) and \( X, Y \sim \exp(2) \) are independent exponential random variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the line integral \( \int_C y^2 \, dx + x \, dy + z \, dz \), where \( C \) is the curve formed by the intersection of the surfaces \( x^2 + y^2 = x + y \) and \( 2(x^2 + y^2) = z \). The curve \( C \) is oriented in the positive direction as viewed from the point \( (0, 0, 2R) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_{|z|=2} \log\frac{z+1}{z-1} \, dz \), assuming that for \( w \in \mathbb{C} - \{z; \text{Re}(z) \leq 0\} \), we have \(-\pi < \text{Im}(\log w) < \pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi i}
For integers $n > 1$, let $a_n$ represent the number of trailing zeroes in the factorial $n!$ when expressed in base $n$. Determine the maximum value of the ratio $\frac{a_n}{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
The plane given by the equation \(x + 2y + 3z = 7\) is rotated about the line where it intersects the yz-plane by an angle \(m\). In its new position, the plane contains the point \((-1, 0, 2)\). Find \(\cos m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{6}{7}}
Given a stationary and asymptotic Markov chain $\{X_t; t = 0,1,...\}$ with state space $S_x = \{1,2,3\}$ and transition matrix $P = \begin{bmatrix}0.2&0.5&0.3\\0.4&0.3&0.3\\1&0&0\end{bmatrix}$, and initial distribution $\overline{p} = \begin{bmatrix}1/3\\1/3\\1/3\end{bmatrix}$, compute $P(X_3 = 3, X_2 = 1 | X_1 = 2, X_0 = 2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{25}}
Determine the values of $p$ and $q$ for which the inequality \((2^{q-1}x^q+2^{q-1}y^q)^{1/q} \leq (2^{p-1}x^p+2^{p-1}y^p)^{1/p}\) holds for all positive real numbers $x$ and $y$. Provide your answer in terms of $p$ and $q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p \geq q}
Given the recurrence relations:

\[ a_{2n} = a_{n} - 1 \]
\[ a_{2n+1} = 2a_{n} + 1 \]

and the condition \( a_{20} = 1 \), find the value of \( \sum_{n=1}^{63} a_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{728}
Determine the number of subgroups isomorphic to \( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z} \) in the group \( G \) isomorphic to \( \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/9\mathbb{Z} \times \mathbb{Z}/81\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}