inclusionAI/Ring-lite-rl-data huggingface dataset:
Calculate the expected value of the maximum distance among three jumps. Each jump has a \( \frac{1}{3} \) probability of being disqualified, resulting in a distance of 0. If a jump is qualified, its distance is uniformly distributed on the interval \([2, 3]\].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{23}{9}}
Find the minimal polynomial of \( \beta \) over \( \mathbb{F}_5 \), where \( \beta \) is a root of the polynomial \( q(Y) = Y^2 - \alpha \) in the extension \( \mathbb{L} \) of \( \mathbb{K} = \mathbb{F}_5(\alpha) \), with \( \alpha \) being a root of the irreducible polynomial \( p(X) = X^2 - 2 \) over \( \mathbb{F}_5 \). The minimal polynomial \( s(Z) \) should be of degree 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Z^4 - 2}
Let \( n \) be a positive integer. Determine the first digit after the decimal point of the number

$$
\sum_{k=1}^{n} \frac{k(k+1)}{n}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $V$ be a finite-dimensional $
Q$-linear space, and $T: V \rightarrow V$ a linear operator such that $T^2 = -Id$. If $V$ has a $T$-invariant proper subspace $W$ with $\dim(W) \ge 1$, find the smallest possible dimension of $V$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A positive integer is called *Tiranian* if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the variables $a$ and $b$ satisfying $b=-\frac{1}{2}a^2+3\ln(a)$ (with $a>0$), if point $Q(m,n)$ is on the line $y=2x+\frac{1}{2}$, calculate the minimum value of $(a-m)^2+(b-n)^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{5}}
Let $p(n) = n^4 - 6n^2 - 160$. If $a_n$ is the least odd prime dividing $q(n) = |p(n-30) \cdot p(n+30)|$, find $\sum_{n=1}^{2017} a_n$. ($a_n = 3$ if $q(n) = 0$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7933}
Using the 24-hour clock, the time \(\ 16:11:22 \) on the date \(16 / 11 / 22\) (i.e. the 16th of November 2022) has hours, minutes, and seconds equal to the date, month, and (two-digit) year respectively. Let \(S\) be the number of seconds which will elapse after that date and time until this phenomenon next occurs. What is the value of \(\sqrt{S}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{300}
Given independent random variables $X, Y \sim \exp(1)$ and $U \sim U[0,1]$, find the distribution of the random variable $U(X+Y)$. Provide your answer in terms of a probability density function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-z}}
How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy 
$1-3\sin\theta+5\cos3\theta = 0?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the projective closure of the complement of the locus defined by $x_0x_1x_2$ in the projective space $\mathbb{P}^2$. Provide your answer as a geometric description or set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{P}^2}
Find the asymptotes for the curve given by the equation \(x_1=\frac{C}{x_{1}^{2}-3x_1x_2+3x_{2}^{2}}\), where \(C\in \mathbb{R}\), in the \((x_1,x_2)\)-plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_1 = 0}
Compute the limit: \[ \lim_{n \to \infty} \int_{0}^{\pi} \frac{\sin x}{1 + 3\cos^{2}(nx)} \, \mathrm{d}x. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the expected value of the random variable Y with the cumulative distribution function given by:
\[ F_Y(y) = 1 - 0.3e^{-0.5y} + 0.6e^{-0.25y} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A polygon is said to be friendly if it is regular and it also has angles that, when measured in degrees, are either integers or half-integers (i.e., have a decimal part of exactly 0.5). How many different friendly polygons are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
Determine the conditions on the initial value $x_0$ such that the differential equation $x' = \sqrt{|x|} + x^2$ has a global solution defined for all $t > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_0 \leq 0}
Given that $F(1,0)$ is the focus of the ellipse $\frac{x^2}{9} + \frac{y^2}{m} = 1$, $P$ is a moving point on the ellipse, and $A(1,1)$, find the minimum value of $|PA| + |PF|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6 - \sqrt{5}}
Given a matrix $A$ with a null trace, determine the minimum number of linear measurements, defined as $A\vec{x}_i \cdot \vec{h}_j$ for specified $\vec{x}_i$ and $\vec{h}_j$, necessary to uniquely determine the matrix $A$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the expected value of $\exp(Y)$, where $Y$ is a binomial random variable with parameters $x$ (number of trials) and $p$ (probability of success).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 - p + pe)^x}
Given that \( P \) and \( Q \) are points on the parabola \( y^{2} = x \) and the circle \( (x-3)^{2} + y^{2} = 1 \) respectively, find \(|PQ|_{\operatorname{mnn}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{11}}{2} - 1}
In the projective space $\mathbb{P}^2$, consider the curves defined by the polynomials $f = x^2 - yz$ and $g = (x+z)^2 - yz$. Compute the intersection multiplicity of $V(f)$ and $V(g)$ at the point $p = [-2:1:4]$. Use the fact that the intersection multiplicity at $p$ is equal to the Hilbert polynomial of the $I(p)$-primary component of $(f)+(g)$, where $I(p) = (X+2, Y-1, Z-4)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the radius of convergence for the power series $$\sum \left(\frac{n^3}{2^n}\right)^{n^2} x^{2n^2+1}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$ such as $\{3, 31\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the dimension of the tangent space of the submanifold $G = \{M \in M_4(\mathbb{R}) \mid M^{tr}PM = P\}$, where $P = \operatorname{diag}(1,1,1,-1)$, at the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the square matrix $A$ of minimal order such that the vector function \( \vec{X}_0(t) = \vec{W}(t) + \vec{V}e^{2t} \) satisfies the differential equation \( \vec{X}_0'(t) = A \cdot \vec{X}_0(t) \), where:

\[ \vec{W}(t) = \begin{pmatrix} \sin(t) + \cos(t) \\ \cos(t) - \sin(t) \\ 0 \\ \vdots \\ 0 \end{pmatrix} \quad \text{and} \quad \vec{V} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \\ \vdots \\ v_n \end{pmatrix} \]

Determine the smallest possible value of \( n \) for which such a matrix \( A \) exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a nice functional $F: \mathcal B \to \mathbb R$ on a Banach space $\mathcal B$, define the map $\phi_x: \mathbb R \to \mathbb R$ for a fixed $x \in \mathcal B$ and $c \in \mathbb R$ by $\phi_x(c) = F(cx)$. Find an expression for $\frac{d}{dc}\phi_x(c)$. Assume that $F$ is differentiable in a suitable sense.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{DF(cx)(x)}
A barn with a flat roof is rectangular in shape, $ 10$ yd. wide, $ 13$ yd. long and 5 yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
590
}
In the Cartesian coordinate system $xOy$, the figure formed by the line $y=x$, the line $x=1$, and the x-axis is rotated around the x-axis to form a cone. The volume $V_{\text{cone}}$ of the cone is $\int_{0}^{1} \pi x^{2} dx = \frac{\pi}{3}x^{3}\bigg|_{0}^{1} = \frac{\pi}{3}$. By analogy: rotating the figure formed by the curve $y=2\ln x$, the line $y=1$, and the x and y axes around the y-axis once, we obtain a solid of revolution. The volume $V$ of this solid is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi(e-1)}
Find the length of the primary component of the ideal \((xy, y^2)\) at the origin in the polynomial ring \(\mathbb{R}[x, y]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the smallest integer \( N \) such that all elements of the symmetric group \( S_{11} \) have orders dividing \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27720}
Find the number of distinct $2 \times 2$ matrices with complex entries that satisfy the equation $A^3 = A$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the flux of the vector field \( F = y\hat{i} + z\hat{k} \) across the boundary of the solid cone defined by \( 0 \le z \le 1 - \sqrt{x^2 + y^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Evaluate the double integral: \[ \int_{1}^{\infty} \int_{1}^{\infty} \frac{1}{x^4y^2\ln (xy)} \, dx \, dy \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\ln 3}{2}}
Find the smallest integer \( n \) such that the remainder estimate \( |R_n| \le \frac{M}{(n+1)!} (x-a)^{n+1} \), where \( M \) is the maximum value of \( |f^{(n+1)}(z)| \) on the interval between \( a \) and the indicated point, satisfies \( |R_n| \le \frac{1}{1000} \) for the function \( f(x) = e^{-2x} \) on the interval \([-1,1]\), with \( a = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In James' aquarium, there's a Bobbit worm, a predatory fish, and 60 other fish. The Bobbit worm and the predatory fish are nocturnal, feeding only at night. Every night, the Bobbit worm eats 2 fish, and the predatory fish eats 3 fish. The other fish in the aquarium reproduce at a rate of 2 new fish every 3 days, with reproduction occurring at noon. After two weeks, James adds 8 more fish to the aquarium. A week later, he discovers the Bobbit worm and the predatory fish. How many fish does the aquarium have when James discovers the Bobbit worm and the predatory fish?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given a sample $X_1, X_2, \ldots, X_{15}$ from a distribution with probability density $\exp(-(x-\theta))$ for $x \ge \theta$, test the hypothesis $H_0: \theta=0$ against $H_1: \theta>0$. The test statistic is $T = \min\{x_1, x_2, \ldots, x_{15}\}$. If the observed value of $T$ is $t=0.1$, what is the p-value of this test? (Hint: If $X_1, X_2,\ldots,X_n$ is a sample from an $\operatorname{Exp}(\lambda)$ distribution, then $\min\{X_1, X_2,\ldots,X_n\}$ has an $\operatorname{Exp}(n\lambda)$ distribution.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-1.5}}
The Kovács couple was visited by four other couples. After the introductions, Mr. Kovács observed that, besides himself, each of the remaining nine attendees had met a different number of people.

How many people did Mrs. Kovács meet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in the 3-by-3 array of boxes, then Carl places an O
 in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all 9 boxes are filled or
 one of the players has 3 of their symbols in a row - horizontal, vertical, or diagonal - whichever comes first, in which case that
 player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and tha
 Carl wins the game when he places his third O. How many ways can the board look after the game is over?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
148
}
Find the value of the limit: \[ \lim_{z\rightarrow \infty}\dfrac{\Gamma(k,iz)}{\Gamma(k)} \] where $z$ is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the residue of the function \( \frac{(z^6+1)^2}{(z^5)(z^2-2)(z^2-\frac{1}{2})} \) at \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{4}}
Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, 
$S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}$
 .Find the least positive integer $k$ making the number $k!\cdot  S_{2016}$ an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
Evaluate the limit \(\lim_{x \to 0} \frac{e^{\tan x} - e^x + \ln(\sec x + \tan x) - x}{\tan x - x}\) using L'Hopital's rule or alternative methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Given a diagonalizable square matrix $A$ with eigendecomposition $A = S_A D_A S_A^{-1}$, express the matrix $S_{A \oplus A}$ that diagonalizes the Kronecker sum $A \oplus A = A \otimes I + I \otimes A$ in terms of $S_A$. Here, $I$ is the identity matrix of the same size as $A$. Assume that $A \otimes I$ and $I \otimes A$ are simultaneously diagonalizable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_A \otimes S_A}
Find the maximum value of \( x_1^2 + x_2^2 + \cdots + x_{2014}^2 \) given that \( x_i \in [-11, 5] \) for \( i = 1, 2, \ldots, 2014 \) and \( x_1 + x_2 + \cdots + x_{2014} = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110710}
Determine the largest power of 7 that divides the binomial coefficient \( \binom{2016}{1003} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{343}
Let $M$ be a $7\times6$ real matrix, and let $X$ be the matrix defined as:
$$
X = \pmatrix{
0&0&x_{13}&x_{14}&0&0\\
0&0&0&0&0&0\\
0&0&x_{33}&x_{34}&0&0\\
0&0&0&0&0&0\\
0&0&0&x_{54}&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0}.
$$
If $N = M + X$ has rank $4$, what is the maximum possible rank of $X$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: $$\lim_{x \to 0} \frac{(x-\sqrt{2})^4}{\sin^3(x)\cos^2(x)}\int_{0}^x \frac{t^2}{t^4+1}dt$$ using L'Hopital's rule or other appropriate methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Let $H$ be a Hilbert space representing single-particle states, and let $\mathfrak{F}$ be the fermionic Fock space. Determine whether the creation operator $c^*(f)$ is a bounded linear operator on $\mathfrak{F}$ for $f \in H$. If it is bounded, find its norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\|f\|}
Tej is typing a string of $L$s and $O$s that consists of exactly $7$ $L$s and $4$ $O$s. How many different strings can he type that do not contain the substring ‘$LOL$’ anywhere? A substring is a sequence of consecutive letters contained within the original string.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Determine the number of conjugacy classes in the Dihedral group \( D_{11} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the matrix representation of $\exp(tE_{a,b})$ given that $E_{a,b}^2 = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I + tE_{a,b}}
Determine the values of $a$ for which the improper integral \( \int_{0}^{\infty} \frac{\sin x}{x^a} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
Define the sequence \(\{a_n\}\) as \(a_n = n^3 + 4\), where \(n \in \mathbb{N}_+\). Let \(d_n = \gcd(a_n, a_{n+1})\), the greatest common divisor of \(a_n\) and \(a_{n+1}\). Determine the maximum value of \(d_n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{433}
Let $a_1, \dots, a_n$ be a set of vectors in a vector space $V$. It is given that $a_1, \dots, a_n$ are linearly independent. Find all values of $\lambda$ such that the following set is linearly independent:

$$a_1 + a_2, \dots, a_{n-1} + a_n, a_n + \lambda a_1$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda \neq (-1)^n}
In a client-server scenario, a client masks a value $b \in \mathbb{F}_q$ as $M=(r_1\cdot b+r_2)\bmod q$, where $r_1$ and $r_2$ are chosen uniformly at random from the field such that $r_1, r_2 > \frac{q}{2}$. The client sends $M$ to the server and deletes $b$. The client can later check the integrity of $M$ by computing $s_1=(M- r_2)\bmod q$ and $s_2=(s_1\cdot (r_1)^{-1})\bmod q$. If $s_2$ is a uniformly random value, the client can detect a modification. The server can modify $b$ by finding $k=c\cdot r_1$ with $k<q$ and computing $M+k$, resulting in $s_2=b+c$. If $k \neq c\cdot r_1$, $s_2$ is uniformly random. What is the probability that the server modifies $M$ without being detected?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the number of non-isomorphic well-orderings of the set of natural numbers \( \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
Determine the minimal degree $d$ such that there exists a map of degree $d$ from $S^2 \times S^2$ to $\mathbb{CP}^2$. Consider the cohomology ring of $\mathbb{CP}^2$ and the classes in $H^2(S^2 \times S^2)$ to find the minimal $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given two independent and identically distributed random variables $X$ and $Y$ with moment generating function \( M(t) = \left( \frac{1}{1-1.5t}\right)^2 \) for \( t < \frac{2}{3} \), calculate the standard deviation of the sum $X + Y$. Assume $X$ and $Y$ are Erlang distributed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $\overline{X_n}$ denote the mean of a random sample of size $n$ from a Poisson distribution with mean $\lambda = 2$. Use the delta method to find the limiting distribution of $\sqrt{n}\left( (\overline{X_n})^2 -4 \right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathcal{N}(0, 32)}
Evaluate the expression \( 2^{2010}\frac{\int^{1}_{0}x^{1004}(1-x)^{1004}dx}{\int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4020}
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136\pi}
Determine the class to which the language \(L_3=\{\langle M\rangle\mid|\langle M\rangle|\le 2016\text{ and M is a TM that accepts }\varepsilon \}\) belongs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{R}
Given an initial boundary value problem (IBVP) for the heat equation with non-homogeneous Neumann boundary conditions, where the boundary conditions satisfy $A = B$, determine the equation that governs the function $v(x)$. Provide the resulting equation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{v''(x) = 0}
Given that \(a+b+c=10\), \(abc=2015\), and \(ab+bc+ca=1\), find the value of the expression \((bc-a^2)(ac-b^2)+(ac-b^2)(ba-c^2)+(ba-c^2)(bc-a^2)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-97}
Evaluate the integral \( \displaystyle\int_0^{2\pi} \frac{\cos2x}{5+ 4\cos x}\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6}}
Find the square of the perpendicular distance from the origin to the line L, which intersects perpendicularly both the lines \( \frac{x+2}{2}=\frac{y+6}{3}=\frac{z-34}{-10} \) and \( \frac{x+6}{4}=\frac{y-7}{-3}=\frac{z-7}{-2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the smallest integer $n$ such that the $n$th Taylor polynomial $P_n(x)$ of $f(x) = \tan^{-1}(x)$ about $x_0 = 0$ approximates $f(x)$ within $10^{-6}$ on the interval $[-0.5, 0.5]$. Provide the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Evaluate the line integral \( \int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz \) where \( \gamma \) is the curve formed by the intersection of the cone \( x^2+y^2=z^2 \) and the plane \( x+y+10z=10 \). The curve is oriented to make one clockwise revolution around the \( z \)-axis when viewed from above.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi}
A computer science laboratory has 15 workstations and 10 servers. A cable can be used to directly connect a workstation to a server, with only one direct connection to each server active at any time. Determine the minimum number of direct connections required to ensure that any set of 10 or fewer workstations can simultaneously access different servers via direct connections.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Suppose 20 real numbers are chosen independently from the interval [0, 20] with uniform probability. Find a lower bound for the probability that their average lies between 8 and 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{22}{27}}
Determine the number of ordered pairs \((F_1, F_2)\) such that \(F_1 \not\subseteq F_2'\), where \(F_1, F_2 \subseteq \{a, b, c, d\}\) and \(F_2' = F_2 \cup \{c \mid d \in F_2\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{157}
Evaluate the integral \( \int_0^\frac{\pi}{2} \frac{\cos x}{(1+\sqrt{\sin (2x)})^2}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
A vehicle is equipped with four regular tires and one spare tire. Each regular tire has an independent exponential lifetime with a mean of 4000 miles, and the spare tire has an exponential lifetime with a mean of 2000 miles. Calculate the expected mileage the vehicle can travel before a tire replacement service is required, assuming the spare tire is used when one of the regular tires fails.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1800}
Find the value of $r$ such that a Möbius transformation maps the annulus $\{z:r<|z|<1\}$ to the region between the circles $\{z:|z-1/4|=1/4\}$ and $\{z:|z|=1\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 - \sqrt{3}}