inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the number of non-unital ring homomorphisms from $\mathbb{Z}_{30}$ to $\mathbb{Z}_{42}$. Given that the number of group homomorphisms from $\mathbb{Z}_{30}$ to $\mathbb{Z}_{42}$ is $6$, find the specific count for non-unital ring homomorphisms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In triangle $ABC$ we have $|AB|=1$ and $\angle ABC=120^\circ.$ The perpendicular line to $AB$ at $B$ meets $AC$ at $D$ such that $|DC|=1$. Find the length of $AD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$ \sqrt[3]{2}$}
Calculate the flux of the vector field \( \vec{F} = 2y\vec{j} + z\vec{k} \) over the surface \( S \), which is the portion of the surface defined by \( y = \ln(x) \) in the first octant. The projection of \( S \) onto the \( xz \)-plane is the rectangle defined by \( 1 \le x \le e \) and \( 0 \le z \le 1 \). The unit normal vector \( \vec{n} \) to \( S \) points outwards from the \( xz \)-plane. Compute the flux of \( \vec{F} \) over \( S \) in the direction of \( \vec{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the maximum number of different types of trees that can be placed at the vertices of equilateral triangles, given that each type of tree must form a complete triangle and only one tree can grow in each hole.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find a value of \( R \) such that a circle of radius \( R \) centered at the origin has exactly 32 integer points on its circumference.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{1105}}
Calculate the volume of the solid enclosed by the surface \((x^2 + y^2 + z^2)^3 = 3xyz\) using a triple integral in spherical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the minimum number of subspaces required to cover an entire vector space over a finite field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q + 1}
Find the number of non-isomorphic semi-direct products of two cyclic groups, one of order 4 and the other of order 9.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $A$ be an infinite set of positive integers. Find all natural numbers $n$ such that for each $a \in A$,
\[a^n + a^{n-1} + \cdots + a^1 + 1 \mid a^{n!} + a^{(n-1)!} + \cdots   + a^{1!} + 1.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$n=1$ or $2$}
Given that the moment generating function of a random variable \(X\) is \(M_X(t) = e^{2e^t - 2}\) and the moment generating function of a random variable \(Y\) is \(M_Y(t) = (0.2e^t + 0.8)^7\), and assuming \(X\) and \(Y\) are independent, compute \(\Pr(X + Y = 3)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2272}
Given a deck of $n$ cards numbered $1$ to $n$, calculate the probability of forming the maximum possible number of piles when drawing cards according to a specified rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{n!}}
Determine the smallest integer $k$ such that the following conditions are satisfied for 14 students at the IMO training camp: (a) The students can be arranged in a circle such that every two adjacent students have different favorite numbers. (b) Among the students, 7 are boys and the rest are girls, and no boy and girl share the same favorite number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In triangle \( \bigtriangleup {ABC},D \) lies between \( A \) and \( C \) and \( {AC} = {3AD},E \) lies between \( B \) and
 \( C \) and \( {BC} = {4EC} \) .
 \( B,G,F,D \) in that order, are on a straight line and \( {BD} = {5GF} = {5FD} \) . Suppose the
 area of \( \bigtriangleup {ABC} \) is 900,find the area of the triangle \( \bigtriangleup {EFG} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
90
}
Calculate the volume of the solid enclosed by the cone \( z = \sqrt{x^2 + y^2} \), the cylinder \( (x + \frac{3}{2})^2 + y^2 = \frac{9}{4} \), and the plane \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$, such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
738
}
For $t > 0$, how many times does the equation $S(t) = C(t)$ hold true, where $S(t) = -1200\cos\left(\frac{\pi}{6}t\right) + 1500$ and $C(t) = 1200(1.025)^{t}$? Here, $t$ represents the number of months since January 1, 2004.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the integral \( \iint (x^2-2x+y^2+2)\,dA \) using the change of coordinates \((x-1)=r\cos{\theta}\) and \(y=r\sin{\theta}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{2}}
Determine the number of equivalence classes for statements formed with 5 variable letters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{32}}
Let $N$ be the product of all divisors of $2013^{2013}$. Determine the last two digits of $N$ in its decimal notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
Three circles are given by the equations: \( S_1: x^2+y^2+4y-1=0 \), \( S_2: x^2+y^2+6x+y+8=0 \), and \( S_3: x^2+y^2-4x-4y-37=0 \). These circles touch each other at points \( P_1, P_2, \) and \( P_3 \). Let \( C_1, C_2, \) and \( C_3 \) be the centers of these circles. Calculate the ratio \( \frac{\Delta P_1P_2P_3}{\Delta C_1C_2C_3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
Evaluate the limit: \[ \lim_{n \to \infty} \frac{n+n^{\frac{1}{2}}+n^{\frac{1}{3}}+\ldots+n^{\frac{1}{n}}}{n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In triangle $OAB$, where $\angle A = 90^\circ$, the altitude from $A$ meets the side of angle $O$ at point $M$. The distances from $M$ and $B$ to the second side of angle $O$ are 2 and 1, respectively. Determine the length of $OA$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $U$ be a normally distributed random variable with mean $3$ and variance $16$ (i.e., $U \sim N(3, 16)$), and let $V$ be a chi-squared distributed random variable with $9$ degrees of freedom ($V \sim \chi_{9}^{2}$). Given that $U$ and $V$ are independent, calculate the probability $P(U - 3 < 4.33\sqrt{V})$. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.995}
What is the probability of never losing an infinite game where you start with a single four-sided die numbered 1 to 4? In each round, rolling a 1 loses the die, a 2 keeps the die, and a 3 or 4 gains an additional die. You lose the game when you have no dice left.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let $X = \{(x,y) : x^2 + y^2 < 5\}$ and $K = \{(x,y) : 1 \leq x^2 + y^2 \leq 2 \text{ or } 3 \leq x^2 + y^2 \leq 4\}$. Determine the number of relatively compact connected components of $X \setminus K$ in $X$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit \( \lim_{n \to \infty} \int_0^1 \frac{n \sin x}{x (1+n^2 x^2)} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Let \( A \) be a 2x2 square matrix with \( |A| \neq 0 \) such that \( |A + |A| \text{adj}(A)| = 0 \). Find the value of \( |A - |A| \text{adj}(A)| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $T$ be the linear transformation represented by the matrix 
$$ \left( \begin{array}{cccc}
1 & 1 & 0 & 3 \\
1 & 1 & 1 & 5 \\
2 & 2 & 1 & 8 \end{array} \right) $$
from $\mathbb{R}^4$ to $\mathbb{R}^3$. Consider the space $V$ of linear maps $B$ from $\mathbb{R}^2$ to $\mathbb{R}^4$ satisfying $AB=0$. Determine the dimension of the space $V$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the limit: $$\lim_{x\to 2}\frac{\cos(\frac \pi x)}{x-2} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Let $ a_{n}$ is a positive number such that $ \int_{0}^{a_{n}}\frac{e^{x}-1}{1+e^{x}}\ dx =\ln n$.

Find $ \lim_{n\to\infty}(a_{n}-\ln n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\ln 2}
You have 6000 red marbles, 6000 blue marbles, and 6000 green marbles. Determine the number of ways to select 3003 marbles such that either the number of red marbles is a multiple of 3 or the number of blue marbles is a multiple of 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2258257}
In a directed graph with 2013 vertices, there is exactly one edge between any two vertices, and for every vertex, there exists an edge outwards from this vertex. Given that from every vertex, you can reach exactly k vertices using at most two edges, determine the maximum value of k.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2012}
Determine the curve of minimum length between the points (2, 5) and (6, 1) that passes through the point (4, 3) and has an integral from x = 2 to x = 6 equal to 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = -x + 7}
Two bugs start at a distance of one foot away from each other on a beanstalk that is continuously growing at a constant rate, doubling its length every hour. The bugs crawl towards each other at a constant speed of x ft/hr. Determine the values of x for which the bugs will meet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x > \dfrac{\ln 2}{2}}
Determine the highest order for which all groups have been classified, excluding groups of order 1024 and cube-free groups of order 50000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
The three sides of the quadrilateral are equal, the angles between them are equal, respectively $90^o$ and $150^o$. Find the smallest angle of this quadrilateral in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\(45^\circ\)
}
In a wrestling tournament, there are $100$ participants, all of different strengths. The stronger wrestler always wins over the weaker opponent. Each wrestler fights twice and those who win both of their fights are given awards. What is the least possible number of awardees?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1
}
Determine the number of homomorphisms from the alternating group $A_4$ to the cyclic group $\mathbb{Z}_{12}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let \( \gcd(p, q) = 1 \) and consider the set \( Y = \{(t^p, t^q) \in \mathbb{C}^2 \} \). Determine the ideal \( I(Y) \) in \( \mathbb{C}[x,y] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x^q - y^p)}
Determine the minimum number of actions required to make a given triangle equilateral. The action $P(X)$ involves moving vertex $X$ to the intersection of the perpendicular bisector of the opposite side and a line parallel to that side passing through $X$. How many such actions are needed to transform the triangle into an equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Calculate the greatest common divisor of the polynomial \( p(x) = x^6 + x^5 + 2x^4 + x^3 + 2x^2 + x + 1 \) and its derivative \( p'(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + x + 1}
Calculate the probability that the set of independent random variables $E_1, E_2, \cdots$, where $E_n \sim \text{Unif}\{1, 2, \cdots, 2n\}$, equals the set of all even numbers $\mathbb{E}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$\text{Positive integers }a,b,c,\mathrm{and~}d\mathrm{~satisfy~}a>b>c>d,a+b+c+d=2010,\mathrm{and~}a^2-b^2+c^2-d^2=2010.\mathrm{Find~the}$
$\text{number of possible values of }a.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
501
}
In a base-3 number system represented by the symbols A, B, and C, each symbol corresponds to one of the digits 0, 1, or 2, but their exact values are unknown. You can ask questions involving A, B, and C, such as "What is A + B + C?", and receive answers in the same ABC base-3 system. What is the minimum number of questions required to determine the values of A, B, and C?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of group homomorphisms from the symmetric group \( S_3 \) to the quotient group \( \mathbb{Z}/6\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A particle of mass \(m\) starts from rest at time \(t=0\) and is moved along the \(x\)-axis with constant acceleration \(a\) from \(x=0\) to \(x=h\) against a variable force of magnitude \(F(t)=t^2\). Find the work done in moving the particle from \(x=0\) to \(x=h\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{h^2}{a}}
Find the equation of the directrix of a parabola with focus at $F(-1, -1)$ and a point $P(7, 13)$ on the parabola, where the equation of the tangent line at $P$ is $3x - y - 8 = 0$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x + 8y + 19 = 0}
Find the coefficient of \(x^{46}\) in the expansion of \(\frac{1}{1 - x^3 - x^4 - x^{20}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3224}
Compute \( a(x)b(x) + c(x) \) in \( \mathrm{GF}(2^4) \) using the irreducible generator polynomial \( x^4 + x + 1 \). Given:

\[ a(x) = x^3 + x^2 + 1, \]
\[ b(x) = x^2 + x + 1, \]
\[ c(x) = x^3 + x^2 + x + 1 \]

Find the result of \( a(x)b(x) + c(x) \) modulo \( x^4 + x + 1 \) in \( \mathrm{GF}(2^4) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^3 + x}
Evaluate the triple integral \( \iiint_T y^2 \, dx \, dy \, dz \), where \( T \) is the region bounded by \( 9y^2 + z^2 = 9 \), \( x + y = 3 \), and \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9\pi}{4}}
A total area of \( 2500 \, \mathrm{m}^2 \) is available for building identical houses. The construction cost for a house with an area \( a \, \mathrm{m}^2 \) includes material cost \( 100 p_{1} a^{\frac{3}{2}} \) yuan, labor cost \( 100 p_{2} a \) yuan, and other costs \( 100 p_{3} a^{\frac{1}{2}} \) yuan, where \( p_{1} \), \( p_{2} \), and \( p_{3} \) are consecutive terms of a geometric sequence. The sum of these terms is 21, and their product is 64. If building 63 houses results in the material cost being less than the sum of the labor and other costs, determine the maximum number of houses that can be built to minimize the total construction cost.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{156}
Evaluate the following expression: \[ \sum_{m=0}^\infty \lim_{n\rightarrow\infty}\int_{2x^2+y^2<n}\left(1-\frac{2x^2+y^2}{n^2}\right)^{n^2}x^{2m}\,dx\,dy \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Determine the limit of the characteristic function \( \varphi_{\bar{\xi}} \) as \( n \to \infty \), where \( \bar{\xi} \) is the sample mean of a sequence of independent and identically distributed random variables \( \xi_1, \xi_2, \ldots, \xi_n \) with finite second moments.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{i t \mu}}
What is the maximum distance between any two positions of a Rubik's cube, denoted as \( \max_{A,B \in P} d(A,B) \), where \( d(A,B) \) is the minimal number of moves required to reach position \( B \) from position \( A \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Calculate the remainder when $666^{666}$ is divided by $1000$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine the range of values for \( s \in \mathbb{R} \) such that the series \( \left(\sum_{k \ge 1} |k^s|^p \right)^{1/p} \) converges for \( 1 \le p < \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, -1)}
Evaluate the limit: \[ \lim_{x \to 1} \frac{x^x - x^{x^x}}{(1-x)^2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the double integral \( \int\int (x^{4}-y^{4})xy \, dx \, dy \) over the region \( G = \{(x,y)\in\mathbb{R}^{2}: 0 \leq x, 1 \leq x^{2}-y^{2} \leq 2, 1\leq xy \leq 2\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{8}}
If a function $h$ is both $a$-strongly convex and $b$-strongly convex, determine the strong convexity constant of $h$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max\{a, b\}}
An investor buys a stock worth $100. Each month, the stock has a 50% chance of increasing by $1.10 and a 50% chance of decreasing by $0.90, with changes being mutually independent. What is the probability that, after 100 months, the stock's value exceeds $91?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.9713}
If $x+y^{-99}=3$ and $x+y=-2$, find the sum of all possible values of $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
-195
}
Determine the largest possible cardinality of a set of mutually disjoint perfect matchings in the complete graph $K_\omega$, where $K_\omega$ is the complete graph on an infinite set of vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\omega}
Consider 2016 points arranged on a circle. You are allowed to jump ahead by either 2 or 3 points in a clockwise direction. What is the minimum number of jumps required to visit all points and return to the starting point?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Determine the number of distinct subgroups of order 10 in a non-cyclic abelian group of order 20.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the range of values for \(\lambda\) such that \(\lambda < \sin(\sqrt{4n^2 + 1}\pi)\) for all natural numbers \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0]}
Given a stack of 10 cards, where 3 are red, 5 are blue, and 2 are green, what is the expected number of draws, without replacement, before you draw both a red and a blue card? Assume that it does not matter whether the red or blue card is drawn first.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{121}{36}}
Determine the range of values for the real number \( a \) such that the function \( f(x) = x^2 - 2x - |x - 1 - a| - |x - 2| + 4 \) is always non-negative for any real number \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 1]}
Evaluate the definite integral: $$ \int_{0}^{\infty}\frac{x^{2}\ln x}{(1+x^{3})^4}dx $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{18}}
Evaluate \( \frac{\partial z}{\partial x}(1, 2) \) where \( z \) is a function of \( (x, y) \) defined implicitly by the equation \( z^2f + zg + x - y = 0 \). Given that \( f(1, 2) = 2 \), \( g(1, 2) = 1 \), and at \( (1, 2) \), \( df = -2dx + 5dy \), \( dg = 4dx - 3dy \), and \( z(1, 2) = -1 \). What is \( \frac{\partial z}{\partial x}(1, 2)? \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{5}{3}}
Evaluate the integral: $$\frac{1}{2\pi} \int_{0}^{2\pi} e^{\cos(\theta)}\cos(n\theta) \,d\theta,$$ where $n \in \mathbb{N}$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_n(1)}
Determine the number of ways to express 100 as a sum of positive even integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{204226}
Calculate the principal value of the integral \( \int_{-\infty}^\infty \frac{\sin(2x)}{x^3} \, dx \) considering the third-order pole at \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi}
Evaluate the sum \(\sum_{d|n}\tau(d)\mu(d)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1)^{\omega(n)}}
Evaluate the integral: $$ \int_{-1}^1 xde^{|x|} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the minimum number of charts needed to parametrize the subset \( C \) of \( \mathbb{R}^{3} \) defined by \( C = \{(x,y,z) \in \mathbb{R}^{3} \mid 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z = x^{2} + y^{2} \} \). Consider the circular nature of the boundary where \( z = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the number of distinct subgroups of order 8 in the symmetric group $S_5$. Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the value of $f(d)$, the smallest possible size for an efficient substitute for $\Omega = \{0,1\}^d$, where $d$ is an integer $\geq 2$. An efficient substitute $\Omega'$ satisfies $p(\Omega',i,j,A)=p(\Omega,i,j,A)$ for all $i,j$ and $A$, with $p(\Omega,i,j,A)$ being the probability that $(x_i,x_j) \in A$ for a randomly selected $(x_1,x_2, \ldots, x_d) \in \Omega$. The trivial bound is $f(d) \leq 2^d$, and it is known that $f(3)=4$. What is $f(d)$ in general?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\lceil \log_2 (d+1) \rceil}}
Determine the number of 25th power residues modulo 343, given that \(\gcd(25, 342) \neq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{294}
Calculate the expected residual service time for an arbitrary job in a queueing system with two types of arrivals. Type A arrivals occur at a rate of \(\lambda_A = 1/20\) per second and have a service time of exactly 8 seconds. Type B arrivals occur at a rate of \(\lambda_B = 1/30\) per second and have a service time of exactly 12 seconds. Use the formula for expected residual time: \[E(R) = \frac{E(B^2)}{2E(B)}\] where \(E(B)\) is the expected service time of an arbitrary job. Provide the expected residual time \(E(R)\) as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let \( f(x) \) be a function that is defined and has a continuous derivative on \((2, \infty)\). Assume that \( f(a) = b \), \( \vert f(x) \vert < x^9 + 9 \), and \( \int_{4}^{\infty} f(x) e^{-x/5} \, dx = -2 \). Determine the value of \( \int_{4}^{\infty} f'(x) e^{-x/5} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{2}{5}}