inclusionAI/Ring-lite-rl-data huggingface dataset:
The area of the region bounded by the graph of \[x^2+y^2 = 3|x-y| + 3|x+y|\] is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Given the even numbers arranged in the pattern shown below, determine in which column the number 2008 will appear.

```
2  4  6  8
16 14 12 10
18 20 22 24
32 30 28 26
...
```
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A company is lining a 900-foot path with a fence. Every 6 feet, a fence pole is placed. The entire path is lined with a fence except for a certain length of a bridge. There are 286 fence poles placed along the whole path on both sides. What is the length of the bridge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
在世界杯足球赛前，F 国教练为了考察 A_1, A_2, \cdots, A_7 这七名队员，准备让他们在三场训练比赛（每场 90 分钟）中都上场. 假设在比赛的任何时刻，这些队员中有且仅有一人在场上，且 A_1,A_2,A_3,A_4 每人上场的总时间（以分钟为单位）均能被 7 整除，A_5,A_6,A_7 每人上场的总时间（以分钟为单位）均能被 13 整除. 如果每场换人次数不限，那么，按每名队员上场的总时间计算，共有多少种不同的情况？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42244}
The three sides of a triangle are $3$, $5$, and $1-2a$. Find the range of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{7}{2} < a < -\frac{1}{2}}
A kite-shaped field is planted uniformly with wheat. The sides of the kite are 120 m and 80 m, with angles between the unequal sides being \(120^\circ\) and the other two angles being \(60^\circ\) each. At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. Determine the fraction of the crop that is brought to the longest side of 120 m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Determine the value of the constant $λ > 0$ such that the equation $2\lfloor{\lambda n}\rfloor = 1 - n + \lfloor\lambda \lfloor\lambda n \rfloor\rfloor$ holds true for all positive integers $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + \sqrt{2}}
A printing company manufactures a board game that requires two types of figures: dogs and cats, cut from cardboard. In each set, the ratio of cats to dogs is \(5:3\). To minimize waste, they print two types of cardboard: one type contains 28 dogs and 28 cats, and the other type has 42 cats. In what proportion should they print the two types of cardboard so that after cutting, the ratio of the two figures is exactly as needed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9:4}
Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties:
[list]
[*]The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and
[*]It is possible to colour the edges of $\mathcal{P}$ with $k{}$ colours such that for every colour $c{}$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the colour $c{}$.
[/list]
[i]Viktor Simjanoski, Macedonia[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $n$ be a positive natural number. A sequence of $n$ positive integers (not necessarily distinct) is called a "four-group" sequence if it satisfies the following requirements: for any natural $k \geq 2$, if $k$ occurs in this sequence, then $k − 1$ occurs, and the first position at which the number $k − 1$ occurs is earlier than the last position where the number $k$ occurs. How many "four-group" sequences of length $n$ are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n!}
Petya conceived a natural number and wrote the sums of each pair of its digits on the board. Then he erased some sums, and the numbers $2,0,2,2$ remained on the board. What is the smallest number Petya could have conceived?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
How many 1000-digit positive integers have all digits odd, and are such that any two adjacent digits differ by 2?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8 \times 3^{499}}
Example 2 (17th All-Soviet Union Mathematical Olympiad, 9th Grade) One side of a rectangle is $1 \mathrm{~cm}$. It is known that it is divided into four smaller rectangles by two perpendicular lines, with the area of three of them being no less than $1 \mathrm{~cm}^{2}$, and the area of the fourth one being no less than $2 \mathrm{~cm}^{2}$. How long must the other side of the original rectangle be at least?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["3+2\\sqrt{2}"]}
Given the function \(g:\mathbb{R}^3 \to \mathbb{R}^3\) defined by \(g(x, y, z) = (3x + 4z, 2x - 3z, x + 3y)\) and the region \(S = \{(x, y, z) \in \mathbb{R}^3 : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1\}\), find the value of \(\alpha\) such that the following integral equality holds:

\[\iiint_{g(S)} (2x + y - 2z) \,dxdydz = \alpha \iiint_S z \,dxdydz\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{255}
Consider a Markov chain \(\{X_n\}\) with the following transition probability matrix:

\[
P=\left( \begin{array}{ccccccc} 
1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 \\
0 & 1/2 & 1/2 & 0 & 0 & 0 & 0 \\
1/4 & 0 & 0 & 1/2 & 1/4 & 0 & 0 \\
0 & 0 & 0& 1/2 & 1/2 & 0 & 0 \\
0 & 0 & 0 & 0 & 1/2 & 1/4 & 1/4 \\
0 & 0 & 0 & 0 & 1/4 & 1/2 & 1/4 \\
0 & 0 & 0 & 0 & 1/2 & 1/2 & 0 \\
\end{array} \right).
\]

Given that \(X_0=1\), calculate the expected number of times the chain visits state 3 before reaching any of the states 5, 6, or 7. Formally, find:

\[
\mathbb{E}\bigg[\sum_{k=1}^{T}\mathbb{I}_{\{X_k=3\}}\bigg],
\]

where \(T=\inf\{k\geq 0: ~X_k\in \{5,6,7\}\}\). Here, \(\mathbb{I}_A\) denotes the indicator function on set \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{3}}
Evaluate the surface integral \(\int_S (\nabla \times \vec{F}) \cdot d\vec{S}\) for the vector field \(\vec{F}(x, y, z) = (z, x, y)\) and the surface \(S\), which is the portion of the paraboloid \(z = x^2 + y^2\) lying below the plane \(z = 4\), oriented downward.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4\pi}
Given a group $G$ with $|G| = 40$ and a set $X$ with $|X| = 67$, where $G$ acts on $X$ resulting in 5 orbits, determine the number of possible lists of orbit lengths. Each orbit length must divide the order of the group $G$, and the sum of the orbit lengths must equal the size of the set $X$. How many such lists are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the values of \( \lambda > 0 \) for which the set \( I_{\lambda} \), consisting of all irrational numbers \( \rho = [a_{1}, a_{2}, \ldots, a_{n}, \ldots] \in (0,1) \) with continued fraction partial quotients satisfying \( a_{n} \leq \text{const} \cdot n^{\lambda} \), has a positive Lebesgue measure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda > 1}
Compute the differential of the multiplication map \( m: G \times G \rightarrow G \) at the identity element \((e,e)\) for a Lie group \( G \). Specifically, find \( dm_{(e,e)}: T_{e}G \oplus T_{e}G \rightarrow T_{e}G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X + Y}
Find the simplified form of the expression $E[E(ZX \mid Y) \mid Z,Y]$. Provide your answer in terms of $Z$ and $Y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E(ZX \mid Y)}
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
\[
D_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \log \left(\frac{P(x)}{Q(x)}\right)
\]
Assume the two distributions are \( P=[p, 0, \ldots,0, 1-p] \) and \( Q=\text{binomial}(n,q)=\binom{n}{k} q^{k}(1-q)^{n-k}, \forall k \in \{0,1,2,...n\} \).
Both distributions have \( n+1 \) mass points. 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}
Determine the positive integer values of \( k \) for which the expression \( N = 3^{6n-1} - k \cdot 2^{3n-2} + 1 \) is divisible by 7 for every positive integer \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Neville's Algorithm is used to approximate $f(0)$ using the values $f(-2)$, $f(-1)$, $f(1)$, and $f(2)$. If $f(-1)$ is understated by 2 and $f(1)$ is overstated by 3, determine the error in the original calculation of the interpolating polynomial used to approximate $f(0)$. What is the error in the approximation?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Determine the set of values for which the series \( \sum_{n=1}^\infty \left(\frac{n^2+1}{3n^2+4}\right)^n(x-2)^n \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 5)}
For a list of $9$ positive integers that are not necessarily all different, the mean, median, and (unique) mode are all $9$. What is largest possible positive difference between the largest element and the smallest element of the set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
32
}
A polygon has interior angles such that the difference between any two consecutive angles is 5 degrees. If the smallest angle is 120 degrees, determine the number of sides of the polygon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
For $u=2+2i$ and $A =\{z \in \mathbb{C}:|z-1|\leq|z-i|\text{ and }|z-u|\leq 1 \}$, determine the value of $|z|$ for which $z \in A$ and the argument of $z$, $\arg(z)$, is minimized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{7}}
Determine the number of possible rays that satisfy Snell's law in the lifeguard problem, given only the angle between the line connecting the two people and the normal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that $b + \frac{1}{b} + b^2 + \frac{1}{b^2} + 1 = 0$, where $b$ is a real number, find the value of $b^{4m} + b^{2m} + \frac{1}{b^{2m}} + \frac{1}{b^{4m}}$ for any positive integer $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the dimension of the quotient ring \( \mathbb{C}[x,y]/I \) over \( \mathbb{C} \), where \( I=\langle(x+2)^2,(x+2)(y+1),(y+1)^3\rangle \) and \( \mathbb{C}[x,y] \) is the polynomial ring over \( \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $F_X(x)$ be the cumulative distribution function (CDF) of a continuous random variable $X$, and let $F_X^{-1}(y)$ be its inverse function. If $V$ is a continuous random variable with a uniform distribution on the interval $[0, 1]$, determine the CDF of the random variable $Z = F_X^{-1}(V)$. Provide your answer in terms of $F_X(x)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_X(x)}
Evaluate the expression: \[ 3.444 A = \cos^6\left(\frac{3\pi}{2} - \frac{1}{2} \arcsin \frac{3}{5}\right) - \cos^6\left(\frac{5\pi}{2} + \frac{1}{2} \arcsin \frac{4}{5}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{492}}
For which values of \( p > 0 \) does the integral \( \int_0^{\infty} x \sin(x^p) \, dx \) converge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 2}
Determine the condition under which the function \( \max\{|p(a_j)| : 0 \leq j \leq k\} \), defined on the vector space \( V \) of real polynomials with degree less than or equal to \( n \), constitutes a norm. Assume \( a_j \) are distinct real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \geq n}
Let $ABCD$ be a convex quadrilateral with $\angle ABC = 120^{\circ}$ and $\angle BCD = 90^{\circ}$, and let $M$ and $N$ denote the midpoints of $\overline{BC}$ and $\overline{CD}$.  Suppose there exists a point $P$ on the circumcircle of $\triangle CMN$ such that ray $MP$ bisects $\overline{AD}$ and ray $NP$ bisects $\overline{AB}$.  If $AB + BC = 444$, $CD = 256$ and $BC = \frac mn$ for some relatively prime positive integers $m$ and $n$, compute $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3290111
}
At his craftwork store, Howard has a collection of wooden bowls where he rewards two to his customers for every 10 they buy. If he had 20 customers that day, half of whom bought 20 bowls each, there were 30 bowls remaining in the rewards collection. How many wooden bowls were in the collection initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Determine all quadruples $(x,y,z,t)$ of positive integers such that
\[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 1, 3, 1)}
Fifty gallons of tea were poured into 200 containers. Geraldo drank 44 containers. If Geraldo had consumed only the tea's 75% liquid content, how many pints of liquid did Geraldo actually drink, given that 1 gallon is approximately 8 pints and 1 container is 5% foam?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62.7}
52. On the side $AB$ of the rectangular trapezoid $ABCD (AB \perp BC)$, a semicircle (with the side as the diameter) is constructed, which touches the side $CD$ at point $K$. The diagonals of the trapezoid intersect at point $O$. Find the length of the segment $OK$, if the lengths of the bases of the trapezoid $ABCD$ are 2 and 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{6}{5}"]}
What is the minimum number of points that need to be marked inside a convex $n$-gon so that any triangle with vertices at the vertices of the $n$-gon contains at least one marked point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n-2}
Consider the polynomial \(P(x) = 5x^6 - ax^4 - bx^3 - cx^2 - dx - 9\), where \(a, b, c, d\) are real numbers. If the roots of \(P(x)\) are in arithmetic progression, determine the value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
3. A $4 \times 4 \times h$ rectangular prism can hold 8 small spheres with a radius of 1 and 1 large sphere with a radius of 2. Then the minimum value of $h$ is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2+2\sqrt{7}}
The last 5 digits of $99 \times 10101 \times 111 \times 1001001$ are _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{88889}
Evaluate the following limit using convergence theorems, such as Dominated Convergence: 

$$\lim_{n\rightarrow\infty}{\int\int}_{(0,\infty)\times (0,\infty)} \frac{\sin^2(nxy)xy}{e^{(x^2+y^2)}}\,dx\,dy.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
Determine the number of different connected graphs that can be formed with 6 vertices and 7 edges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5700}
Solve the exponential equation for \(x\): $$1000.00116=\frac{1000}{(1+x)^{16}}+\frac{1-(1+x)^{-16}}{x}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.001}
Coffee is poured into a truncated cone-shaped cup at a rate of $2 \text{ cc}/\text{sec}$. The cup has an upper radius of $4 \text{ cm}$, a lower radius of $2 \text{ cm}$, and a height of $6 \text{ cm}$. Determine the rate at which the coffee level is rising when it reaches the halfway point. The volume of the coffee in the cup is given by $V = \frac{h\pi}{3}(R^2 + Rr + r^2)$, where $r < R$ and $r = 2 \text{ cm}$. Note that $R$ varies linearly with $h$ as the coffee level rises.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{9\pi}}
Find the point of intersection of the functions \( y = e^{-x} \) and \( y = x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0.567, 0.567)}
Compute the triple integral \( \int_D f(x, y, z) \, dV \) where \( D \) is the region in the first octant \((0, \infty)^3\) between the cone \( z^2 = (x - 1)^2 + y^2 \) and the sphere \( x^2 + y^2 + z^2 = 1 \). The function \( f : D \to \mathbb{R} \) is given by \( f(x, y, z) = z \sqrt{x^2 + y^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{75}}
Evaluate the contour integral \( \oint_{|z|=1} |z-1|^2 \, dz \), where \( z \) is a complex number and the integration is performed along the circle of radius 1 centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
Evaluate the integral \( \int_0^{2\pi} \frac{\cos(t)}{4+\sin(t)} \, dt \) using complex analysis techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Example 10 Let $a, b, c$ be positive real numbers, find the minimum value of $\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-\frac{8 c}{a+b+3 c}$. (3rd China Girls Mathematical Olympiad)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-17 + 12\\sqrt{2}"]}
The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\text{a circle}$}
Let \((x_1, x_2, \ldots, x_n)\) be \(n\) independent random variables, each uniformly distributed as \(x_i \sim U(\theta, 3\theta)\) with an unknown parameter \(0.5 \le \theta \le 1\). Determine the domain of the minimal number of samples \(\min_n\) such that \(n \ge \min_n\) ensures \[ P(2\theta - 0.05 \le \hat{X} \le 2\theta + 0.05) \ge 0.97 \] where \(\hat{X} = \frac{1}{n} \sum_{i=1}^n x_i\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{628}
For a twice differentiable function \(f(x)\), it is given that \(|f(x)| \leq 3\) for all \(x \in \mathbb{R}\). Additionally, for some \(\alpha\), we have \(f(\alpha) + (f'(\alpha))^2 = 80\). Determine the number of integral values that \((f'(x))^2\) can take in the interval \((0, 77)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
Determine all natural numbers \( k \) such that there exists a set of \( k \) cards from a deck of 1024 cards, where each card has a unique set of digits from 0 to 9, including an empty card, with the property that no complete set of digits from 0 to 9 can be formed, but adding any additional card from the deck allows for a complete set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{512}
In the Cartesian coordinate system, consider the ellipse given by \(\frac{x^2}{2} + y^2 = 1\) with foci \(F_1\) and \(F_2\). A line \(l\) intersects the ellipse at two distinct points \(A\) and \(B\), and does not pass through \(F_1\). The distance from \(F_2\) to the line \(l\) is \(d\). If the slopes of the lines \(AF_1\), \(l\), and \(BF_1\) form an arithmetic sequence, determine the range of possible values for \(d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\sqrt{3}, 2)}
For a twice differentiable function \(f(x)\), if \(|f(x)| \leq 3\) for all \(x \in \mathbb{R}\) and there exists a point \(\alpha\) such that \(f(\alpha) + \left(f'(\alpha)\right)^2 = 80\), determine the number of integer values that \(\left(f'(x)\right)^2\) can take within the interval \((0, 77)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
Determine all values of $m$ such that the inequality \( \log(60x^2+120x+10m-10) > 1+3\log(x+1) \) has exactly 4 integer solutions for $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-9, 2]}
Determine the number of distinct ellipses that define the domain of the function: \[ f(x,y)=\frac{\sqrt{x}\sqrt{9-x^{2}-y^{2}}}{\ln(\frac{x^{2}}{4}+\frac{y^{2}}{16}-1)} \] without using a graphical representation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the area of the region in the first quadrant of the Cartesian plane defined by the inequalities $0 \leq x \leq \sqrt{y}$, $0 \leq y \leq \sqrt{x}$, and $x + y \leq \frac{3}{4}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{96}}
Determine the number of solutions in positive integers to the equation: \[ 37 + \frac{3(x-1)x}{2} + 3^{x+2} = y^2 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A lottery has $N$ tickets numbered $1, 2, \dots, N$, where $N$ is unknown. A random sample of 5 winning tickets is drawn, and their numbers are $\{a, b, c, d, e\}$. Determine the maximum likelihood estimator for $N$ based on this sample.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max\{a, b, c, d, e\}}
Evaluate \( x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y} \) for the function \( z = x\tan^{-1} \frac{y}{x} + y\sin^{-1} \frac{x}{y} + 2 \) at the point (1,1).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{4}}
In quadrilateral \(ABCD\), given \(\angle ABD = 70^\circ\), \(\angle CAD = 20^\circ\), \(\angle BAC = 48^\circ\), and \(\angle CBD = 40^\circ\), find the measure of \(\angle ACD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Let $\ell_1$ and $\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\ell_1$ while points $C$  and $D$ lie on $\ell_2$ such that $\angle BAC = 30^\circ$ and $\angle ABD = 60^\circ$. The minimum value of $AD + BC$ is $a\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Calculate the surface integral of the vector field \( \mathbf{G}(x,y,z)= ( y^2, xz, 1 ) \) over the portion of the paraboloid \( z=x^2+y^2 \) that lies below the plane \( z=4 \), where the normal vector points upwards. Use Gauss's Divergence Theorem to find the result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
A gas station receives cars at a rate of 20 vehicles per hour and has only one gas pump. If the pump is empty, it receives one client. If a new client arrives and finds the pump occupied, they wait. Each client takes approximately 6 minutes (1/10th of an hour) to fill their vehicle. If two clients are at the pump, one filling and the other waiting, new clients leave the station directly. Given the stationary distribution of the states as \((u_1,u_2,u_3) = (1/7,2/7,4/7)\), find the average number of clients served by the gas station every hour. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{60}{7}}
Calculate the flux of the vector field \( F(x,y,z) = (y, x, z) \) downwards through the surface \( z = 16 - x^2 - y^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-128\pi}
The volume of a rectangular solid each of whose side, front, and bottom faces are $12\text{ in}^{2}$, $8\text{ in}^{2}$, and $6\text{ in}^{2}$ respectively is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$24\text{ in}^{3}$}