inclusionAI/Ring-lite-rl-data huggingface dataset:
Assume that the target population is divided into three demographic groups with different response rates: Group A with a 70% response rate, Group B with a 55% response rate, and Group C with a 40% response rate. If the percentage of each group in the target population is 40% for Group A, 35% for Group B, and 25% for Group C, and a total of 300 responses are needed, what is the minimum number of questionnaires that should be mailed accounting for the weighted average of response rates?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{525}
Find a generator of the maximal ideal of the local ring of rational functions on the algebraic curve $C: Y^2 = X + X^3$ at the point $P = (0,0)$. This local ring is a discrete valuation ring and thus a principal ideal domain.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Y}
Through the right focus \( F \) of the ellipse \( C: \frac{x^2}{2} + y^2 = 1 \), draw a line \( l \) that intersects the ellipse \( C \) at two different points \( A \) and \( B \). Suppose \( \overrightarrow{FA} = \lambda \overrightarrow{FB} \) and \( T(2,0) \). If \( \lambda \in [-2, -1] \), find the range of values of \( |\overrightarrow{TA} + \overrightarrow{TB}| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[ 2, \frac{13 \sqrt{2}}{8} \right]}
Determine the number of real solutions to the equation \(\sin(4x) = \frac{x}{100}\) for \(x \in (0, 2\pi)\) without using graphical methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Determine the range of values of \(\alpha\) for which the Lyapunov condition is not satisfied for a collection of independent random variables \(\{X_{i}\}_{i\geq1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > \dfrac{1}{2}}
Evaluate the integral \( \int_{a}^{d} \delta(x-b)\delta(x-c)f(x)\,dx \), where \( a \le b \le d \) and \( a \le c \le d \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(b) \delta(b - c)}
Identify the original function \( f(x) \) for which the Newton's method iteration is given by:
\[ x_{n+1} = 2x_n - x_n^2y \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y - \dfrac{1}{x}}
Compute the contour integral \( \oint_{|z|=4} \frac{e^{1 /(z-1)}}{z-2} \, dz \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Two circles with centers \( P \) and \( Q \) intersect at points \( M \) and \( N \). The lines \( PM \) and \( QN \) intersect the first circle at points \( A \) and \( B \), and the second circle at points \( D \) and \( C \) respectively. The point \( M \) lies between \( A \) and \( D \), and the point \( N \) lies between \( B \) and \( C \). It is known that \( AB = CD \), and the area of the quadrilateral \( ABCD \) is \( 24 \sqrt{3} \). If the distance between the intersection points \( M \) and \( N \) is 4, find the distance between the centers \( P \) and \( Q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{3}}
Determine the order of convergence of the modified Newton's method given by \(x_{n+1}=x_n-m\frac{f(x_n)}{f'(x_n)}\), where \(m\) is the multiplicity of the root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the recursively defined sequence $T_n$ where $T_n = n + \frac{T_{n+2}}{n+1}$, starting with $T_1$. Express the result as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{e}}
Evaluate the integral \( \frac{1}{2\pi i}\int_C \frac{\sum_{n=0}^{15}z^n}{(z-i)^3}dz \), where \( C \) is the contour \(|z-i|=2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52 + 60i}
Alice and Bob are each secretly given a real number between 0 and 1 uniformly at random. Alice states, \u201cMy number is probably greater than yours.\u201d Bob repudiates, saying, \u201cNo, my number is probably greater than yours!\u201d Alice concedes, muttering, \u201cFine, your number is probably greater than mine.\u201d If Bob and Alice are perfectly reasonable and logical, what is the probability that Bob\u2019s number is actually greater than Alice\u2019s?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
11/12
}
Evaluate the integral \( \int_{0}^{2\pi} e^{-\sin^{2}{x}} \cos\left(6x - \frac{\sin{2x}}{2}\right) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{24 \sqrt{e}}}
A triangle is formed by selecting three lattice points (points with integer coordinates) at random with replacement within the square \([-99, 100] \times [-99, 100]\). Determine the probability that the area of this triangle (which may be degenerate) is an integer. Express this probability as a fraction \(\frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers, and calculate \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Calculate the expected time until there are two students in a help center, given that student arrivals follow a Poisson process with rate $r$, and the first student leaves after an exponential time with rate $2r$. Provide your answer in terms of $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{r}}
Find \(\alpha \in \mathbb{R}\) such that there exists a Lie algebra representation of \(\mathfrak{sl}_2 (\mathbb{R})\) on \(C^\infty(\mathbb{R})\) with image equal to \(\text{span}_\mathbb{C} \{ X , Y , Z_\alpha \}\), where the differential operators are defined as follows:

\[X f(t) = f''(t)\]
\[Y f(t) = t^2 f(t)\]
\[Z_\alpha f(t) = t f'(t) + \alpha f(t)\]

for \(f \in C^\infty(\mathbb{R})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the precise decay rate of the sequence \(a_n = \min_{1 \le N \le n} |\langle N \alpha \rangle|\) for an irrational number \(\alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{n}}
For discrete probability distributions \( P \) and \( Q \) defined on the same probability space, \( \mathcal{X} \), the relative entropy from \( Q \) to \( P \) is defined as \[ D_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \log \left(\frac{P(x)}{Q(x)}\right) \] Given the distributions \( P=[p, 0, \ldots,0, 1-p] \) and \( Q=\text{binomial}(n,q)=\left(\begin{array}{l} n \\ k \end{array}\right) q^{k}(1-q)^{n-k}, \forall k \in \{0,1,2,...n\} \), find the parameter \( q \) that minimizes \( D_{\mathrm{KL}}(P \| Q) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - p}
Evaluate the expression:
\[ \left(a_{0}-4 a_{2}+16 a_{4}+\cdots-2^{14} a_{14}\right)^{2}+\left(2 a_{1}-8 a_{3}+32 a_{5} \cdots-2^{15} a_{15}\right)^{2} \]
where for a complex number \(z\), we have the binomial expansion \((1+z)^{15}=a_{0}+a_{1} z+\cdots+a_{15} z^{15}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5^{15}}
Given that $i$ is an annual effective interest rate, $1$ accumulates to $3$ in $r$ years, $2$ accumulates to $7$ in $s$ years, and $3$ accumulates to $16$ in $t$ years. Determine the value of $10(1+i)^{(2r+s-t)}$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{945}{16}}
Professor Liyung wants to form a math club with his 40 students. Each student is enemies with two other students, and no student will join the club if any of their enemies is already a member. Let \( M \) be the maximum number of members the club can have. Find the sum of all possible values of \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{119}
What is the cardinality of a Banach space $X$ that has a countable dense subset?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Evaluate the limit: $$\lim_{n\to\infty} \int_0^\infty \frac{\sin(x/n)}{x+x^2}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a countable collection of pairwise disjoint sets $\{ X_n \mid n \in \omega \}$, each with cardinality $c$, where $c = 2^{\aleph_0}$ (the cardinality of the continuum), determine the cardinality of their union $\bigcup_n X_n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c}
Evaluate the integral \[ I = \frac{1}{\pi^2} \int_{0}^{\infty} \frac{(\ln{x})^2}{\sqrt{x}(1-x)^2} \, \mathrm{d}x \] using suitable substitutions and techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Suppose the power series \( \sum^\infty_{n=0}{ a_n x^n } \) converges with a radius of convergence \( R_a \) and the power series \( \sum^\infty_{n=0}{ b_n x^n } \) converges with a radius of convergence \( R_b \). Determine the radius of convergence of the power series \( \sum^\infty_{n=0}{ (a_n+b_n) x^n } \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\min(R_a, R_b)}
Let \( M \) be a \( 3 \times 3 \) matrix such that \( M \begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 6 \\ -3 \\ 0 \end{bmatrix} \). Suppose that \( M^3 \begin{bmatrix} 1 \\ -\frac{1}{2} \\ 0 \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} \) for some \( \alpha, \beta, \gamma \in \mathbb{R} \). Find the value of \( |\alpha| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Determine the number of zeros of the polynomial \( f(z) = z^{4}+z^{3}+4z^{2}+2z+3 \) that lie in the first quadrant of the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Bryan Ai has the following 8 numbers written from left to right on a sheet of paper: \( \textbf{1 4 1 2 0 7 0 8} \). In each of the 7 gaps between adjacent numbers, Bryan Ai places one of \( + \), \( - \), or \( \times \). If he picks a random placement out of the \( 3^7 \) possible placements, what is the expected value of the expression, considering the order of operations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{166}{81}}
Determine the smallest integer \( k \) such that if the Fourier coefficients \( a_n \) and \( b_n \) of a function \( f \) satisfy \( |a_n| \leq Cn^{-k} \) and \( |b_n| \leq Cn^{-k} \) for some constant \( C \), then \( f \) is in \( C^1(T) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the triple integral \( \int\int\int_D z(x^2+y^2+z^2)^{-\frac{3}{2}} \, dV \) over the solid \( D \) that lies above the plane \( z=3 \) and below the sphere \( x^2+y^2+z^2=36 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
Evaluate the limit: \( \lim_{x \to \infty} x^2\ln\left(x\cot^{-1}(x)\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Two painters are painting a fence that surrounds garden plots. They come every other day and paint one plot (there are 100 plots) in either red or green. The first painter is colorblind and mixes up the colors; he remembers which plots he painted, but cannot distinguish the color painted by the second painter. The first painter aims to maximize the number of places where a green plot borders a red plot. What is the maximum number of such transitions he can achieve (regardless of how the second painter acts)?

Note: The garden plots are arranged in a single line.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Calculate the norm \( ||T|| \) of the operator \( T \) defined on \( H = L^2(0,1) \) by:
\[ Tf(s) = \int_0^1 (5s^2t^2 + 2)f(t) \, dt \]
where \( f \in L^2(0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{3}}
There are 7 volunteers to be arranged to participate in the promotional activities for the Shanghai World Expo on Saturday and Sunday, with 6 people participating each day and 3 people arranged for each day. There are __________ different arrangements possible (answer with a number).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
Evaluate the limit: \[ \lim_{n\to\infty}\left[\dfrac{1}{2^{n/2}\Gamma(n/2)} \int_{n-\sqrt{2n}}^{\infty} t^{\frac{n}{2}-1}e^{-t/2}\,dt\right] \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Phi(1)}
Calculate the number of ways to obtain exactly one set of three of a kind when rolling 9 dice.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4082400}
Evaluate the contour integral \( \int_\gamma \frac{1+z}{1-\cos z} \, dz \), where \( \gamma \) is a circle centered at the origin with a radius of 7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\pi i}
An insurance company pays out claims following an exponential distribution. When a deductible \( d \) is applied, the expected claim payment decreases by 10\%. Calculate the percentage reduction in the variance of the claim payment.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1\%}
Given a uniform random variable $U$ over all $n$-bit binary strings, and another random variable $X$ that depends on $U$ and also takes values in $n$-bit binary strings, if the mutual information between $X$ and $U$ is bounded as $I(X;U) \le \epsilon$, determine the tightest lower bound on the entropy of the XOR of $X$ and $U$, denoted by $H(X \oplus U)$. Specifically, can we establish a bound like $H(X \oplus U) \ge n - \epsilon$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H(X \oplus U) \ge n - \epsilon}
Let $  a_1 = \sqrt{7}  $ and $  b_i = \lfloor a_i \rfloor  $. $  a_{i+1} = \frac{1}{b_i - [b_i]}  $ for each $  i \geq i  $. 

What is the smallest integer $  n  $ greater than 2004 such that $  b_n  $ is divisible by 4? 

($  [x]  $ denotes the largest integer less than or equal to $  x  $)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2005
}
Determine the number of Hamiltonian paths in a tournament that contains a cycle and has at least 4 vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the function $f(x)=\sin (2x+\varphi)$, if the graph is shifted to the left by $\dfrac {\pi}{6}$ units and the resulting graph is symmetric about the $y$-axis, determine the possible value of $\varphi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {\pi}{6}}
Initially, the men and women in a room were in the ratio of 7 : 9. Then, 5 men entered the room and 6 women left the room. Afterwards, 3 more men joined the room and 4 women returned. The number of women then doubled and now there are 28 men in the room. How many women are currently in the room?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Determine the number of real solutions in the interval \((0, \pi)\) for the equation:
\[ \sin(14u) - \sin(12u) + 8\sin(u) - \cos(13u) = 4 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: $$\lim \limits_{m\to \infty }\cos^2 \left(\pi \sqrt{m^2-m}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let y = f(x) be a function defined on R with a period of 1. If g(x) = f(x) + 2x and the range of g(x) on the interval [1,2] is [-1,5], find the range of the function g(x) on the interval [-2020,2020]. Express your answer as an interval [min, max].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-4043,4041]}
Given \( 0 < x < \frac{\pi}{4} \), let \( A = \cos (x^{\sin (x^{\sin x)})} \), \( B = \sin (x^{\cos (x^{\sin x)})} \), and \( C = \cos (x^{\sin (x x^{\cos x})}) \). Determine the relationship of magnitudes between \( A \), \( B \), and \( C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{B<A<C}
Quadrilateral \(ABCD\) is inscribed in a circle. Point \(X\) lies on its side \(AD\), with \(BX \parallel CD\) and \(CX \parallel BA\). Find \(BC\) if \(AX = \frac{3}{2}\) and \(DX = 6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
There are 6 spots for top students, planned to be distributed among three classes. The probability that exactly one class does not receive a spot for a top student is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{15}{28}}
5.1. At the vertices of a cube, numbers $\pm 1$ are placed, and on its faces - numbers equal to the product of the numbers at the vertices of that face. Find all possible values that the sum of these 14 numbers can take. In your answer, specify their product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-20160}
Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations \[a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2\] has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$. Then $\rho^2$ can be expressed as a fraction $\frac {m}{n}$, 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{$7$}
If $x > y > 0$, then $\frac{x^{y}y^{x}}{y^{y}x^{x}} = $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left(\frac{x}{y}\right)^{y-x}$}
Jerry went to a shop and bought several items worth a total of Rs. 120. Out of these, 40% of the items were taxed at a 6% tax rate, 30% of the items were taxed at an 8% tax rate, and the remaining items were tax-free. Additionally, Jerry received a discount of 5% on items in the second tax bracket. What was the cost of the tax-free items?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Determine the number of elements in the quotient group $\mathbb{Z}^2/\ker\varphi$ given the homomorphism $\varphi: \mathbb{Z}^2 \to S_{15}$ where \[ \varphi(1,0) = (2,7,3)(3,11,5)(12,13) \quad \text{and} \quad \varphi(0,1) = (14,15)(1,4,6,8,9,10). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Evaluate the limit \( \lim_{n\to\infty}{ \int_{1}^{\infty}{\dfrac{x^n}{2+x^n}}dx } \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Evaluate the line integral: $$ \int_{L} \frac{-y \, dx}{x^{2}+y^{2}}+\frac{x \, dy}{x^{2}+y^{2}} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Determine the cardinality of a basis for the vector space of the real numbers over the field of rational numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
The positive sequence $\{a_{n}\}$ satisfies the equation \(\frac{1}{a_{n} a_{n+1}}+\frac{1}{a_{n} a_{n+2}}+\frac{1}{a_{n+1} a_{n+2}}=1\) for all natural numbers \(n\). Given that \(a_{1} + a_{3} = 6\), and \(a_{1}\), \(a_{2}\), and \(a_{3}\) form a strictly increasing geometric progression, let \(S_{n}\) be the sum of the first \(n\) terms of the sequence \(\{a_{n}\}\). Find the value of \(\left\lfloor S_{2008} \right\rfloor\), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5352}
Given that \( \int_0^\infty (g(x) - f(x)) \, dx = 0 \), where both \( g(x) \) and \( f(x) \) are continuous, differentiable, positive, and monotone decreasing functions with \( \lim_{x \to \infty} g(x) = 0 \) and \( \lim_{x \to \infty} f(x) = 0 \), and assuming \( f(x) \) and \( g(x) \) are distinct such that the set \( S = \{x: f(x) \neq g(x) \} \) has a non-zero measure, determine the cardinality of the set \( X = \{x \in \mathbb{R}: f(x) = g(x), \text{ and for some } \epsilon > 0, \ [f(x-\epsilon) < g(x) < f(x+\epsilon)] \lor [f(x-\epsilon) > g(x) > f(x+\epsilon)] \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the volume of the $d$-dimensional $\ell_1$-surface $\mathbb{S}_d$ using the parameterization and integral approach.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2^d \sqrt{d}}{(d - 1)!}}
Consider the sequence defined by \(a_1 = 7\) and \(a_{n+1} = \frac{2^n + n^{10^{17}}}{(1 + \frac{1}{n})^{(n^2)} - \frac{1}{n}} \cdot a_n\). Given that \(\lim_{n \to \infty}{(1+\frac{1}{n})^{(n^2)} - \frac{1}{n}} = e^2\), determine the limit of this sequence as \(n\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Two chess players, X and Y, play a series of games with the following rules: The probability of X winning a particular game against Y is \( \frac{1}{3} \), and the probability of Y winning the game is \( \frac{2}{3} \). X wins the series if X wins two consecutive games, while Y wins the series if Y wins four consecutive games. They continue playing until one of them wins the series. What is the probability that Y wins the series?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{64}{129}}
How many positive integers $n\leq100$ satisfy $\left\lfloor n\pi\right\rfloor=\left\lfloor\left(n-1\right)\pi\right\rfloor+3$? Here $\left\lfloor x\right\rfloor$ is the greatest integer less than or equal to $x$; for example, $\left\lfloor\pi\right\rfloor=3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{86}
Let the incircle of  $\triangle ABC$  be tangent to  $AB,BC,AC$  at points  $M,N,P$ , respectively. If  $\measuredangle BAC=30^\circ$ , find  $\tfrac{[BPC]}{[ABC]}\cdot\tfrac{[BMC]}{[ABC]}$ , where  $[ABC]$  denotes the area of  $\triangle ABC$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 - \sqrt{3}}{4}}
Given the sequence $\{a_n\}_{n\in\mathbb{N}}$ defined by $a_0 = -1$, $a_1 = 1$, and $a_{n+2} = 6a_{n+1} - a_n$ for all $n\in\mathbb{N}$, find all indices $n$ such that $a_n = 8k^2 - 1$ for some $k\in\mathbb{N}$. Provide your answer as a set of all such $n$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{2\}}
Determine the precise value of the uniform density function $f(y_1, y_2)$ for the continuous bivariate random variable $(Y_1, Y_2)$ with support $S = \{(y_1, y_2) : y_2 \leq 1-y_1^2, y_1 \leq 0, y_2 \leq 0\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Determine the number of semi-direct products \(H \rtimes Q\) that can be constructed for \(H \cong C_{42}\) and \(Q \cong C_{3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the surface integral \( \iint_Y \mathbf{F} \cdot \mathbf{N} \, \mathrm{d}S \) where \( \mathbf{F}= (x^2 yz + x \sin z , x^2 + y(1 - \sin z ), x + y - xy z^2) \) and \( Y \) is the part of the conical surface \( 4x^2 + (y-1)^2 = z^2 \) which lies between \( z = 1 \) and \( z = 2 \). The normal vector points away from the \( z \)-axis. Use Gauss's theorem to assist in the calculation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{3}}