inclusionAI/Ring-lite-rl-data huggingface dataset:
7. Let $R S T U V$ be a regular pentagon. Construct an equilateral triangle $P R S$ with point $P$ inside the pentagon. Find the measure (in degrees) of angle $P T V$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( M = \{1, 2, \cdots, 17\} \). If there exist four distinct numbers \( a, b, c, d \in M \) such that \( a + b \equiv c + d \pmod{17} \), we call \( \{a, b\} \) and \( \{c, d\} \) a "balanced pair" of the set \( M \). How many balanced pairs are there in the set \( M \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{476}
Determine if 11 is a quadratic residue modulo 839 using quadratic reciprocity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ ($a > b > 0$) with eccentricity $\frac{\sqrt{3}}{2}$, a line is drawn through a point $M$ on the ellipse to intersect the ellipse at points $A$ and $B$, with slopes $k_1$ and $k_2$ respectively. If points $A$ and $B$ are symmetric with respect to the origin, find the value of $k_1 \cdot k_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{4}}
Let $X_n$ be a random variable with an exponential distribution with mean $n$ for each $n \geq 1$. Define $Y_n = X_n - 10 \left[\frac{X_n}{10}\right]$, where $[u]$ denotes the greatest integer less than or equal to $u$. Determine the distribution function of $Y_n$ and find the limit of this distribution function as $n \to \infty$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{y}{10}}
On a chessboard, $n$ white rooks and $n$ black rooks are arranged such that rooks of different colors do not attack each other. Find the maximum possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Evaluate the change in $z$, denoted as $\Delta z$, when moving from the point $(0,0)$ to the point $(1,1)$ along the curve $y = x^2$. Given the differential $dz$ as:

$$dz = xy\,dx + xy\,dy$$

Express $dz$ in terms of $d(xy(x+y))$ and use the path to determine $\Delta z$. Assume $f(x, y) = \frac{xy(x+y)}{2}$. What is $\Delta z$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{20}}
Given positive numbers $a$ and $b$ satisfying $a+2b=6$, find the minimum value of $\frac{1}{a+2}+\frac{2}{b+1}$. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{10}}
In the Cartesian coordinate system, define $d(P, Q) = |x_1 - x_2| + |y_1 - y_2|$ as the "polyline distance" between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$. Then, the minimum "polyline distance" between a point on the circle $x^2 + y^2 = 1$ and a point on the line $2x + y - 2 \sqrt{5} = 0$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{5}}{2}}
Find an expression for the coefficients of the polynomial given by the falling factorial $x^{\underline n} = x (x-1) \dots (x-n+1)$, which is a polynomial of degree $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^{n - k} \left[{n \atop k}\right]}
Circle \(\omega\) has a radius of 5 and is centered at \(O\). Point \(A\) lies outside \(\omega\) such that \(OA = 13\). The two tangents to \(\omega\) passing through \(A\) are drawn, and points \(B\) and \(C\) are chosen on them (one on each tangent), such that line \(BC\) is tangent to \(\omega\) and \(\omega\) lies outside triangle \(ABC\). Compute \(AB + AC\) given that \(BC = 7\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the sum of all positive integers $n$ between $1$ and $100$ inclusive such that \(\gcd(n,2^n - 1) = 3.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
Determine $w^2+x^2+y^2+z^2$ if

<center>$ \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 $</center>
<center>$ \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 $</center>
<center>$ \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 $</center>
<center>$ \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 $</center>
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{036}
The numbers $1978^{n}$ and $1978^{m}$ have the same last three digits. Find the positive integers $n$ and $m$ such that $m+n$ is minimized, given that $n > m \geq 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
Compute the dimension of the Zariski Tangent space of the affine scheme \(\operatorname{Spec} A\), where \(A = \mathbb{Z}[2i] \cong \mathbb{Z}[x]/(x^2 + 4)\), at the point corresponding to the maximal ideal \((2, x)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Suppose the sequence \((x_n)_{n \ge 1}\) converges to \(x \in \mathbb{R}\). Does the sequence \(\frac{1}{n}\{x_{mn+1}+x_{mn+2}+ \cdots +x_{2mn-1}+x_{2mn}\}\), where \(m \in \mathbb{Z^+}\), converge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{mx}
Find the area of quadrilateral $EFGH$, given that $m\angle F = m \angle G = 135^{\circ}$, $EF=4$, $FG=6$, and $GH=8$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18\sqrt{2}}
Evaluate the expression \( \frac {(x + \frac{1}{x})^6 - (x^6 + \frac{1}{x^6}) - 2} {(x + \frac{1}{x})^3 - (x^3 + \frac{1}{x^3})} \) and determine its minimum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the integral $$\int_0^3 \frac{x^3}{1-x^4}\, dx.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{4} (\ln 5 + 4 \ln 2)}
Olya wrote fractions of the form \( \frac{1}{n} \) on cards, where \( n \) includes all possible divisors of \( 6^{100} \). She then arranged these cards in some order. After that, she wrote on the board the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, and so on, finally the sum of the numbers on all the cards. Each sum Olya wrote on the board as an irreducible fraction. What is the minimum number of different denominators that could appear on the board?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
Let $X$ be a connected oriented triangulation (polyhedron) space, homeomorphic to a geometric realization of an oriented simplicial complex $S$ with dimension $n$, where the boundary $\partial S$ of $S$ is 0. Is the homology group $H_n(X, \mathbb{Z})$ isomorphic to $\mathbb{Z}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Calculate the discriminant of the number field \(\mathbb{Q}(\sqrt{-3})\) using the basis \(\{1, \sqrt{-3}\}\) and the complex embeddings \(\{\sigma_1, \sigma_2\}\), where \(\sigma_1\) is the identity map and \(\sigma_2\) is the conjugate map. Verify your result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-12}
Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$32$}
Find all the solutions to
\[\sqrt[3]{15x - 1} + \sqrt[3]{13x + 1} = 4 \sqrt[3]{x}.\]Enter all the solutions, separated by commas.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, \frac{1}{14}, -\frac{1}{12}}
The circle inscribed in a right trapezoid divides its larger lateral side into segments of lengths 1 and 4. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$. Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$. Find $100p+q$.

[i]Proposed by Michael Ren[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{301}
A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
Suppose S_n is the sum of the first n terms of a geometric sequence {a_n} with common ratio q > 0. Compare the products S_{n+1}a_n and S_na_{n+1}. Express your answer as a comparison (e.g., >, <, =).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_{n+1}a_n > S_na_{n+1}}
Find the sum of all $x$ from $2$ to $1000$ inclusive such that$$\prod\limits_{n=2}^x \log_{n^n}(n+1)^{n+2}$$is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{739}
Find the number of vertical asymptotes in the graph of
\[y = \frac{(x + 8) (x + 5)^2 (x + 1)^3 x^5 (x - 3)^2}{(x + 7) (x + 5)^2 (x + 1) x (x - 3)^3 (x - 4)}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A circle is inscribed in trapezoid $ABCD$. The extensions of the lateral sides of the trapezoid $AD$ and $BC$ beyond points $D$ and $C$ intersect at point $E$. The perimeter of triangle $DCE$ and the base of the trapezoid $AB$ are 60 and 20 respectively. The angle $ADC$ is $\beta$. Find the radius of the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{30 \sin \beta}{4 + \cos \beta}}
Determine the set of all accumulation points of the sequence of sets in \(\mathbb{R}^2\) defined by \(K_n = \left\{(x,y) \in \mathbb{R}^2: \left\|(x,y) - \left(\frac{1}{n},\frac{1}{n}\right)\right\|_\infty = \frac{1}{n}, xy \neq 0\right\}\), where \(\| \cdot \|_\infty\) denotes the infinity norm, and \(K = \bigcup\limits_{n\in\mathbb{N}} K_n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{(0, 0)\}}
There are $2014$ students from high schools nationwide sitting around a round table in an arbitrary manner. The organizers want to rearrange the students so that those from the same school sit next to each other. This is done by performing swaps between two adjacent groups of students. 

The task is to find the smallest number $k$ such that the desired arrangement can be achieved with no more than $k$ swaps.

Illustration of a swap: 
$$...\underbrace{ABCD}_\text{1}\underbrace{EFG}_\text{2}... \to ...\underbrace{EFG}_\text{2}\underbrace{ABCD}_\text{1}...$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2013}
Task B-2.4. A square and an equilateral triangle are inscribed in a circle of radius 1 so that they share one vertex. Determine the area of the common part of the square and the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}-\frac{9}{4}}
Determine the dimension of the symplectic group \( \text{Sp}(2n, F) \) where \( F \) is the algebraic closure of a finite field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(2n + 1)}
During a commercial break in the Super Bowl, there were three commercials with varying lengths: one 5-minute commercial, one 6-minute commercial, and one 7-minute commercial. Additionally, there were eleven 2-minute commercials, but two of them overlapped with a 3-minute interruption causing them to restart after the interruption ends. How many minutes was the commercial break?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Let  $S$  be the set of points inside and on the boarder of a regular haxagon with side length 1. Find the least constant  $r$ , such that there exists one way to colour all the points in  $S$  with three colous so that the distance between any two points with same colour is less than  $r$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{2}}
Two robots start at position X=0 on the X-axis. Each turn, each robot moves one unit forward or backward with equal probability (0.5), independently of each other and of previous turns. After 450 turns, let D be the distance between the two robots. Calculate approximately the probability that D is less than or equal to 6, using the Central Limit Theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.159}
Calculate the volumes of the solids formed by rotating the figures bounded by the graphs of the functions about the $Oy$ axis.

$$
y=\arccos \frac{x}{5}, \quad y=\arccos \frac{x}{3}, \quad y=0
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi^2}
Given that the function f(x) = ax + sin(x) has extreme points, find the range of the real number a. Express your answer as an inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 < a < 1}
A man sold oranges at the rate of 12 a rupee and experienced a certain loss. To gain 44%, he must sell them at the rate of 7.5 a rupee. What was the percentage of his loss?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10\%}
Let \( \Omega \) be a bounded open set in \( \mathbb{R}^n \), and let \( f, g \in L^2(\Omega) \) with constants \( C_1, C_2 > 0 \). Given the inequality:
\[ \int_{\Omega} fg \, dx \leq C_1 + C_2 \|f\|_2^2 \]
find an estimate for \( \|g\|_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{C_1 C_2}}
Suppose $f(x)$ is such that \( \int_{-\infty}^\infty e^{tx} f(x) \, dx = \arcsin(t - \sqrt{1/2}) \) for all \( t \) where the right-side expression is defined. Compute \( \int_{-\infty}^\infty x f(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Find the expected value of a random variable X with the following probability density function:

$$f(x) = \frac{1}{\lambda} \left(1 + \frac{\xi}{\lambda}(x-\mu)\right)^{-(1 + \frac{1}{\xi})}$$

where $\lambda > 0$, $\xi \neq 0$, and $\xi < 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mu + \frac{\lambda}{1-\xi}}
Three married couples arrange a party. They arrive at the party one at a time, the couples not necessarily arriving together. They all, upon arriving, shake the hand of everyone already there, except their own spouse. When everyone has arrived, someone asks all the others how many hands they shook upon arriving, and gets five different answers. How many hands did he himself shake upon arriving?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
3. Given is $\triangle A B C$ where $\measuredangle C$ is obtuse. Points $D \in A C, E \in A B$, and $F \in B C$ are marked on its sides such that quadrilateral $C D E F$ is a rhombus. If $A E=140, B E=84$, and $r=15 \sqrt{3}$, where $r$ is the radius of the circle inscribed in the rhombus $C D E F$, calculate the area of $\triangle A B C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3840\sqrt{3}}
Given the Markov chain with transition matrix \( \begin{pmatrix} 0 & 0 & 0.2 & 0.8 & 0 \\ 0 & 0 & 0 & 0.9 & 0.1 \\ 0.6 & 0 & 0 & 0 & 0.4 \\ 0.2 & 0.8 & 0 & 0 & 0 \\ 0 & 0.9 & 0.1 & 0 & 0 \end{pmatrix} \), calculate the expected number of visits to state B before reaching state A, starting from state C.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{300}{149}}
Given the function $y=x^3$ and the graph of $y=2^x+1$ intersect at the point $(x_0, y_0)$, determine the interval that $x_0$ falls into. Express your answer in standard interval notation (e.g., (a, b)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,2)}
Evaluate the integral \( \int_{S^2} \sin(\theta) \, d\theta \wedge d\phi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
A table tennis team has 4 players, among whom player A is not suitable for doubles matches. The team's coach needs to arrange for 3 players to participate in 5 matches, with the third match being a doubles match and the rest being singles matches. Each player plays two matches, and no player plays consecutive matches in the first two or last two matches. Calculate the total number of ways the coach can arrange for the players to participate in the matches.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
A standard die is rolled eight times. What is the probability that the product of all eight rolls is odd and consists only of prime numbers? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{6561}}
Triangle  $ABC$  with  $\measuredangle A = 90^\circ$  has incenter  $I$ . A circle passing through  $A$  with center  $I$  is drawn, intersecting  $\overline{BC}$  at  $E$  and  $F$  such that  $BE < BF$ . If  $\tfrac{BE}{EF} = \tfrac{2}{3}$ , then  $\tfrac{CF}{FE} = \tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the largest positive integer  $N$  so that the number of integers in the set  $\{1,2,\dots,N\}$  which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
If $\mathcal{A}$ is an antichain in $2^{\mathbb{N}}$, measurable with respect to the measure $\mu$ generated from cylindrical sets, does it follow that $\mu(\mathcal{A}) = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $R$ be a commutative ring with identity, and let $P$ be a prime principal ideal of $R$. Suppose there exists a proper principal ideal $I$ of $R$ such that $R \supsetneq I \supsetneq P$. Determine if $P^2 = P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{P^2 = P}
Given that all vertices of a regular triangular prism $ABC-A_1B_1C_1$ are on the surface of a sphere with radius 1, when the volume of the regular triangular pyramid is at its maximum, the height of the regular triangular pyramid is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{3}}{3}}
Determine the values of \( k \) for which the second partials test is inconclusive for the function \( f(x, y) \) at the point \((0, 0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 2}
A particle moves under a central force inversely proportional to the $k$-th power of the distance. If the particle describes a circle ( the central force proceeding from a point on the circumference of the circle ), find $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$k = 5$}
Let $A \in M_n$ be a doubly stochastic matrix. Is the largest singular value of $A$, denoted as $\sigma_1(A)$, equal to 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
$ABC$ is a triangle with $AB = 33$, $AC = 21$, and $BC = m$, where $m$ is an integer. There are points $D$ and $E$ on sides $AB$ and $AC$ respectively such that $AD = DE = EC = n$, where $n$ is also an integer. Find the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
How many ways are there to put five beads on a necklace if there are eight distinct beads to choose from, and rotations and reflections of the necklace are considered the same?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{672}
Calculate the value of the sum \( \sum_{r=0}^5 \binom{32}{6r} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{708653429}
A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of $ 3$ kilometers per hour, bikes at a rate of $ 20$ kilometers per hour, and runs at a rate of $ 10$ kilometers per hour. Which of the following is closest to the triathlete's average speed, in kilometers per hour, for the entire race?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
6
}
Determine the number of field homomorphisms from a finite field with 9 elements to a finite field with 4 elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Two pipes A and B separately fill a cistern in some minutes and 5 minutes respectively, and a third pipe C can empty at a rate of 14 L/min. If all the pipes are open when the cistern is full, it is emptied in 1 hour. The cistern holds 39.99999999999999 litres. How many minutes does it take for pipe A to fill the cistern?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.5}
Compute the nearest integer to $$100 \sum_{n=1}^{\infty} 3^{n} \sin ^{3}\left(\frac{\pi}{3^{n}}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{236}
A long rope is folded in half, then folded in half again along the middle of the folded rope, and this process is repeated for a total of 6 folds. After the 6th fold, the rope is cut along the middle. How many segments will the rope be cut into? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
Assume that $X_1, X_2, X_3, X_4$ are independent and identically distributed random variables such that $P(X_1=0)=0.3$, $P(X_1=1)=0.1$, and $X_1$ has a density $f(x)=0.6$ for $x$ in $(0,1)$. Calculate $P(X_1 + X_2 + X_3 + X_4 \leq 1)$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2295}
Determine whether the set \( S = \mathbb{C} \setminus \{z = x + iy \in \mathbb{C} | \, \text{Re}(z) = 0 \text{ and } \text{Im}(z) > 0\} \) forms a star domain. If it does, identify the point at which it is centered.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In how many ways can you divide $n$ students into 4 distinct groups such that each student is in exactly one group and each group is assigned to a unique project?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4^n - 4 \cdot 3^n + 6 \cdot 2^n - 4}
Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \sum_{k=0}^{22} \frac{1}{1+z^{k}+z^{2 k}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46 / 3}
If $x+2 y-3 z=7$ and $2 x-y+2 z=6$, determine $8 x+y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}