inclusionAI/Ring-lite-rl-data huggingface dataset:
A court astrologer calls a moment in time "good" if the hour, minute, and second hands of a clock are all on the same side of any diameter of the clock face (the hands rotate on a common axis and do not make jumps). Is there more "good" or "bad" time in a day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Good}
Consider an integer $n \ge 4$ and a sequence of real numbers $x_1, x_2, x_3, \ldots, x_n$. An operation consists of eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3, x_7, x_{11}, \ldots$. For example, the sequence $4, 5, 9, 3, 6, 6, 1, 8$ produces the sequence $9, 1$. 

Upon the sequence $1, 2, 3, \ldots, 1024$, the operation is performed successively $5$ times. Show that at the end only one number remains and find this number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{683}
Compute the expected value $E(N_1N_2)$ for a Poisson process $N_t$ with rate $\lambda = 2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $S$ be a subset of the set $\{1,2,3, \ldots, 2015\}$ such that for any two elements $a, b \in S$, the difference $a-b$ does not divide the sum $a+b$. Find the maximum possible size of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}
In triangle  $ABC$ ,  $AB>AC$ . The bisectors of  $\angle{B},\angle{C}$  intersect the sides  $AC,AB$  at  $P,Q$ , respectively. Let  $I$  be the incenter of  $\Delta ABC$ . Suppose that  $IP=IQ$ . How much isthe value of  $\angle A$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Equilateral $\triangle A B C$ has side length 6. Let $\omega$ be the circle through $A$ and $B$ such that $C A$ and $C B$ are both tangent to $\omega$. A point $D$ on $\omega$ satisfies $C D=4$. Let $E$ be the intersection of line $C D$ with segment $A B$. What is the length of segment $D E$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{20}{13}}
Let a, b, c, and d represent the ages of four individuals. The total age of a and b is 10 years more than the total age of b and c, and the total age of c and d is 15 years less than the total age of a and d. If the ratio of a's age to d's age is 7:4, how many years younger is c compared to a?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the upper bound for numbers such that the sum of the fifth powers of their digits equals the number itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{354294}
Find the largest possible value in the real numbers of the term   $$ \frac{3x^2 + 16xy + 15y^2}{x^2 + y^2} $$   with  $x^2 + y^2 \ne 0$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
Find the point \( w_0 \) that is excluded from the circle when the line \( x+y=1 \) is mapped to the circle minus a point under the mapping \( w=\frac{1}{z+2+3i} \), where \( z=x+iy \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $P$ be a point selected uniformly at random in the cube $[0,1]^{3}$. The plane parallel to $x+y+z=0$ passing through $P$ intersects the cube in a two-dimensional region $\mathcal{R}$. Let $t$ be the expected value of the perimeter of $\mathcal{R}$. If $t^{2}$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12108}
Given the joint probability density function of the bivariate random variable $(X,Y)$:

$$f_{XY}(x,y) = \frac{1}{\sqrt{3 \pi } } e^{ -\frac{2}{3} ( x^2 - xy + y^2 ) }$$

Calculate the variance of the random variable $Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of half an hour at the same speed, calculate the total distance she would have ridden. Express your answer in miles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In 1978, a person introduced their family by saying: "I have one son and one daughter, and they are not twins. The cube of my son's age plus the square of my daughter's age equals the year of my birth. I was born after 1900. Both of my children are under 21 years old. I am 8 years older than my wife." Find the ages of each family member in 1978.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12, 14, 46, 54}
Consider the sum
\[
  S_n = \sum_{k = 1}^n \frac{1}{\sqrt{2k-1}} \, .
\]
Determine  $\lfloor S_{4901} \rfloor$ .  Recall that if  $x$  is a real number, then  $\lfloor x \rfloor$  (the *floor* of  $x$ ) is the greatest integer that is less than or equal to  $x$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Find the smallest normal subgroup of a group $G$ of order $p^{\alpha}m$, where $p$ is a prime and $p$ does not divide $m$, such that the quotient group is a $p$-group and the subgroup does not contain any Sylow $p$-subgroups of $G$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O^p(G)}
In an abelian group $G$, given that $(r + 1)|A| > |G|$ for some subset $A$, determine the smallest possible value of $k$ such that $kA = \langle A \rangle$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r + 1}
Solve for:

$$
\frac{(1+\operatorname{tg} \alpha)(1+\operatorname{tg} \beta)(1+\operatorname{tg} \gamma)}{1+\operatorname{tg} \alpha \operatorname{tg} \beta \operatorname{tg} \gamma}=2
$$

given that $\alpha+\beta+\gamma=45^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
## Task B-3.4.

While preparing for the competition, Matko discovered a bookstore with good mathematical literature. The bookstore offers 7 different books with problems only in geometry, 4 only in number theory, and 5 only in combinatorics. Furthermore, the store also offers books with problems from exactly two areas. Thus, there are 6 different books with problems in number theory and combinatorics, and 7 with problems in geometry and combinatorics. In how many ways can Matko choose literature from two of all the mentioned areas if he can buy at most two books?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["270"]}
Let \( \sum _{n=-\infty}^\infty a_nz^n \) be the Laurent series expansion of \( f(z)=\dfrac{1}{2z^2-13z+15} \) in the annulus \( \frac{3}{2}<|z|<5 \). Find the value of \( \frac{a_1}{a_2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at $\text{8:00 AM}$, and all three always take the same amount of time to eat lunch. On Monday the three of them painted $50\%$ of a house, quitting at $\text{4:00 PM}$. On Tuesday, when Paula wasn't there, the two helpers painted only $24\%$ of the house and quit at $\text{2:12 PM}$. On Wednesday Paula worked by herself and finished the house by working until $\text{7:12 PM}$. How long, in minutes, was each day's lunch break?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
48
}
The equation \(z(z+i)(z+3i)=2002i\) has a zero of the form \(a+bi\), where \(a\) and \(b\) are positive real numbers. The original answer is in the form \(\sqrt{m}\). Please find the value of \(m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{118}
设 $A$ 是以 $B C$ 为直径的圆上的一点, $D, E$ 是线段 $B C$ 上的点, $F$ 是 $B C$ 延长线上的点, 已知 $B F=4, B D=2, B E=5, \angle B A D=\angle A C D, \angle B A F=\angle C A E$, 求 $B C$ 的长。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given that the function f(x) is differentiable on R with its derivative f'(x), and f(x) satisfies: (x-1)[f'(x)-f(x)] > 0, f(2-x) = f(x)e^(2-2x), determine the solution set of the inequality e^2f(ln x) < xf(2). Express your answer as an interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, e^2)}
The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7.$ How many of these four numbers are prime? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
$22$  football players took part in the football training. They were divided into teams of equal size for each game ( $11:11$ ). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the set of points where the function \( f(x,y) = \begin{cases} x^{4/3} \sin(y/x) & \text{if } x \neq 0 \\ 0 & \text{if } x = 0 \end{cases} \) is differentiable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}^2}
A frog makes $ 3$ jumps, each exactly $ 1$ meter long. The directions of the jumps are chosen independently and at random. What is the probability the frog's final position is no more than $ 1$ meter from its starting position?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
A 2022-digit natural number starts with the digit 4, and any two consecutive digits form a two-digit number divisible by either 19 or 23. Determine the last digit of this number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the periodic points of period 5 for the function \( f(x) = x^2 + 1 \) where \( f: \mathbb{R} \rightarrow \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Determine all the possible non-negative integer values that are able to satisfy the expression: $\frac{(m^2+mn+n^2)}{(mn-1)}$  
if  $m$  and  $n$  are non-negative integers such that  $mn \neq 1$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, 4, 7}
Find all natural numbers which are divisible by  $30$  and which have exactly  $30$  different divisors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11250, 4050, 7500, 1620, 1200, 720}
Find the volume of the region bounded by the hemisphere \( y=\sqrt{4-x^2-z^2} \) and the planes \( y=x \) and \( y=\sqrt{3}x \) using spherical polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4\pi}{9}}
A square is called proper if its sides are parallel to the coordinate axes. Point $P$ is randomly selected inside a proper square $S$ with side length 2012. Denote by $T$ the largest proper square that lies within $S$ and has $P$ on its perimeter, and denote by $a$ the expected value of the side length of $T$. Compute $\lfloor a \rfloor$, the greatest integer less than or equal to $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1676

}
Given the general term formula of the sequence $\{a_{n}\}$ as $a_{n}=(-1)^{n-1}(n-1)$, let $S_{n}$ denote the sum of its first $n$ terms. Calculate the value of $S_{15}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A, B, C)$ of distinct chosen points such that $A$ and $B$ have the same $y$-coordinate, and $B$ and $C$ have the same $x$-coordinate. Find the greatest value that $N$ can attain considering all possible ways to choose the points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8100}
Determine the cardinality of the set of all partially ordered sets of natural numbers that have one least element and an infinite number of maximal elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
What is $o-w$, if $gun^2 = wowgun$ where $g,n,o,u,w \in \{0,1,2,\dots, 9\}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
Let $f(x)$ be an odd function defined on $\mathbb{R}$ with $f(2)=0$. When $x > 0$, we have $\frac{x f''(x)-f(x)}{x^{2}} < 0$ holds true. Determine the solution set for $\frac{f(x)}{x} > 0$. Express your answer as an interval or union of intervals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2,0) \cup (0,2)}
Let $S$ be the region in the first quadrant bounded by the curves $xy=1$, $xy=3$, $x^2-y^2=1$, and $x^2-y^2=4$. Compute \[ \iint _S (y^2+x^2)\, dA. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Big Head Son and Small Head Dad opened a McDonald's together. When they were calculating the day's revenue in the evening, they found that there was an extra $32.13 on the books. Later, they realized that the decimal point of a sum of money was misplaced by one digit. What was the original amount of this sum of money? Express your answer in dollars, rounded to two decimal places.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.57}
The straight line $y=kx+1$ and the parabola $y^{2}=2x$ have at most one common point. Determine the range of values for $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0\}\cup[ \dfrac {1}{2},+\infty)}
The stadium seats 60,000 fans, but only a certain percentage of the seats were sold for the music show. Because of the threat of rain, 5,000 fans stayed home. 40,000 fans attended the show. What percentage of the seats were sold for the music show?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75\%}
There are two distinct pairs of positive integers $a_1 < b_1$ and $a_2 < b_2$ such that both $(a_1 + ib_1)(b_1 - ia_1) $ and $(a_2 + ib_2)(b_2 - ia_2)$ equal $2020$, where $i =\sqrt{-1}$. Find $a_1 + b_1 + a_2 + b_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{714}
Evaluate the limit: $$\lim_{x \to 0}\lim_{n \to \infty} \frac{n}{\left(1^{\left(\cot^2x \right)}+2^{\left(\cot^2x \right)}+\ldots+n^{\left(\cot^2x \right)}\right)^{\left(\tan^2x \right)}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the determinant:
\[\begin{vmatrix} \cos 2019^{\circ} & \cos 1131^{\circ}  & \cos 1869^{\circ}  & \cos  1101^{\circ}\\ -\cos 1131^{\circ} & \cos 2019^{\circ}  & -\cos 1101^{\circ}  & \cos  1869^{\circ}\\  -\cos 1869^{\circ} & \cos 1101^{\circ}  & \cos 2019^{\circ}  & -\cos  1131^{\circ} \\  -\cos 1101^{\circ} & -\cos 1869^{\circ}  & \cos 1131^{\circ}  & \cos  2019^{\circ}\end{vmatrix}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let  $p = 9001$  be a prime number and let  $\mathbb{Z}/p\mathbb{Z}$  denote the additive group of integers modulo  $p$ .  Furthermore, if  $A, B \subset \mathbb{Z}/p\mathbb{Z}$ , then denote  $A+B = \{a+b \pmod{p} | a \in A, b \in B \}.$  Let  $s_1, s_2, \dots, s_8$  are positive integers that are at least  $2$ . Yang the Sheep notices that no matter how he chooses sets  $T_1, T_2, \dots, T_8\subset \mathbb{Z}/p\mathbb{Z}$  such that  $|T_i| = s_i$  for  $1 \le i \le 8,$   $T_1+T_2+\dots + T_7$  is never equal to  $\mathbb{Z}/p\mathbb{Z}$ , but  $T_1+T_2+\dots+T_8$  must always be exactly  $\mathbb{Z}/p\mathbb{Z}$ . What is the minimum possible value of  $s_8$ ?

[i]Proposed by Yang Liu
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A large playlist consists of songs with lengths that have a mean of 2 minutes and 10 seconds and a standard deviation of 15 seconds. What is the probability that more than 36 randomly chosen songs are required to fill a program that is 76 minutes long?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0912}
Evaluate the integral by changing the order of integration:
\[ \int_{0}^{1} \int_{1-\sqrt{1-y^2}}^{1+\sqrt{1-y^2}} \frac{(x^2+y^2)x}{\sqrt{4x^2-(x^2+y^2)^2}} \, dx \, dy \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
已知四面体 \(A B C D\) 内接于球 \(O\)，且 \(A D\) 是球 \(O\) 的直径。若 \(\triangle A B C\) 和 \(\triangle B C D\) 都是边长为 1 的等边三角形，则四面体 \(A B C D\) 的体积是多少？原始答案的形式为 \(\frac{\sqrt{c}}{b}\)，请给出a+b+c的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
For a set  $ P$  of five points in the plane, no three of them being collinear, let  $ s(P)$  be the numbers of acute triangles formed by vertices in  $ P$ .
Find the maximum value of  $ s(P)$  over all such sets  $ P$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Let \( n=2017 \) and \( x_1,\dots,x_n \) be boolean variables. A \( 7 \)-CNF clause is an expression of the form \( \phi_1(x_{i_1})+\dots+\phi_7(x_{i_7}) \), where \( \phi_1,\dots,\phi_7 \) are each either the function \( f(x)=x \) or \( f(x)=1-x \), and \( i_1,i_2,\dots,i_7\in\{1,2,\dots,n\} \). For example, \( x_1+(1-x_1)+(1-x_3)+x_2+x_4+(1-x_3)+x_{12} \) is a \( 7 \)-CNF clause. Determine the smallest number \( k \) for which there exist \( 7 \)-CNF clauses \( f_1,\dots,f_k \) such that \[f(x_1,\dots,x_n):=f_1(x_1,\dots,x_n)\cdots f_k(x_1,\dots,x_n)\] is zero for all values of \((x_1,\dots,x_n)\in\{0,1\}^n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
John makes $30 a week from his job. He earns a 10% raise, followed by another 15% raise, and finally a 5% raise on his salary. What is the total percentage increase in John's weekly income after all three raises?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32.83\%}
Find the expectation of the random variable \( \frac{X_i}{\bar{X}} \), where \( X_i \) are IID random variables and \( \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \). Express \( E\left[\frac{X_i}{\bar{X}}\right] \) in terms of \( n \) and the distribution of \( X_i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A convex pentagon $ABCDE$ is inscribed in a circle. The distances from point $A$ to the lines $BC$, $CD$, and $DE$ are $a$, $b$, and $c$, respectively. Determine the distance from point $A$ to the line $BE$. \( \text{Express your answer in terms of } a, b, \text{ and } c. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{ac}{b}}
Let $A_1, A_2, A_3, A_4$ be events such that for every $i, j, k = 1, 2, 3, 4$, $P(A_i) = \frac{1}{2}$, $P(A_i \cap A_j) = \frac{1}{3}$ for $i \ne j$, $P(A_i \cap A_j \cap A_k) = \frac{1}{4}$ for $i \ne j, j \ne k, k \ne i$, and $P(A_1 \cap A_2 \cap A_3 \cap A_4) = \frac{1}{5}$. Find $P(A^c_1 \cap A^c_2 \cap A^c_3)$. \( \text{Express your answer as a fraction.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
已知x、y∈(0,+∞)，且$\frac { 1 9 } { x } + \frac { 9 8 } { y } = 1$，则x+y的最小值是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$x+y$ 的最小值是 $117+14 \sqrt{38}$.}
Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x+3} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{16}}
Anakin's age and the ages of his three children are all prime numbers, and their combined ages sum to 50. After n years, the sum of the children's ages will be twice Anakin's age. Given that Anakin's age is a prime number, and the ages of his three children form a unique set of primes, determine the value of n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{43}
8、In tetrahedron $A B C D$, it is known that $D A \perp$ base $A B C$, side $A B D \perp$ side $B C D, B D=B C=2$, the sum of the squares of the areas of the three sides $D A B, D B C, D C A$ is 8, then $\angle A D B=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4}}
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$ , $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$ . Compute the area of the pentagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\[ 1 \]}
Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$, and $E$ appear to be equally spaced along the horizon. The perimeter in miles of pentagon $ABCDE$ can be written as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{95}
A common chord of two circles subtends arcs of $60^{\circ}$ and $120^{\circ}$. Find the ratio of the areas of these circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:1}
Evaluate the integral \( \int_\Gamma \frac{1}{z} \, dz \), where \( \Gamma \) is the semicircular contour in the right half-plane, traversed from \( -i \) to \( i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{i\pi}
Given 25 vessels with capacities $1, 2, \ldots, 25$ liters, determine the number of ways to select ten of them such that any two selected vessels can be used to measure exactly 1 liter of water. Assume an unlimited amount of water is available, and water can be transferred between vessels.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
一只小虫在正八面体的表面上爬行，每秒从某一个顶点等可能地爬往四个相邻顶点之一。则小虫在第8秒爬回初始位置的概率为_______。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{299}
Given there are ten steps from the first floor to the second floor, calculate the total number of ways Xiao Ming can go from the first floor to the second floor.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Evaluate the integral \( I = \int_{C(0,2)^+} \frac{z^3}{z^3+z^2-z-1} \, dz \), where \( C(0,2)^+ \) denotes the positively oriented circle centered at the origin with radius 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
How many elements are in the conjugacy class of $(123)(456)$ in $A_7$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
Find the interval(s) where the function $y = \sqrt{-x^2 + 2x + 3}$ is decreasing.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, 3]}
Given a Markov chain with state space $E = \{1, 2, 3, 4\}$ and transition matrix $$P = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0.4 & 0.6 & 0\\ 0.8 & 0 & 0.2 & 0\\ 0.2 & 0.3 & 0 & 0.5\\ \end{bmatrix},$$ compute the expected value of the product $E[f(X_5)f(X_6)|X_4 = 4]$, where $f$ assigns values $2, 4, 7,$ and $3$ to states $1, 2, 3,$ and $4$, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.41}
Let \( f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} \) be a function such that for all \(n > 1\) there is a prime divisor \(p\) of \(n\) such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \] Furthermore, it is given that \( f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 \). Determine \( f(2014^2)+f(2015^3)+f(2016^5) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{49}{3}}