inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $ABC$ be a triangle with $AB=20, AC=21,$ and $\angle BAC = 90^{\circ}.$ Suppose $\Gamma_1$ is the unique circle centered at $B$ and passing through $A,$ and $\Gamma_2$ is the unique circle centered at $C$ and passing through $A.$ Points $E$ and $F$ are selected on $\Gamma_1$ and $\Gamma_2,$ respectively, such that $E, A, F$ are collinear in that order. The tangent to $\Gamma_1$ at $E$ and the tangent to $\Gamma_2$ at $F$ intersect at $P$. Given that $PA \bot BC$, compute the area of $PBC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1261}{2}}
A positive integer is written on each vertex of a pentagon, with a different integer on each vertex. On each side of the pentagon, the least common multiple (LCM) of the numbers at the vertices forming that side is written. If the same integer $n$ is written on each side, determine the minimum possible value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the area of the intersection between the sphere \(x^2 + y^2 + z^2 = 1\) and the cylinder \(x^2 + y^2 - y = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi - 4}
Compute the Lefschetz number \( L(f) \) for the map \( f : S^n \to S^n \) defined by \( f(x_0, x_1, \dots, x_n) = (x_0, -x_1, \dots, -x_n) \), where \( S^n \) is the unit sphere in \( \mathbb{R}^{n+1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the limit: \[ \lim_{n\to\infty} \int_{0}^{\infty} \frac{x^n}{1+x^n} \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In the space \(\mathbb{R}^3\), let \(S\) be the set of all spheres and \(C\) be the set of all circles. Consider the topologies \(\tau_3\) and \(\tau_4\) on \(\mathbb{R}^3\) generated by the subbases \(S\) and \(C\), respectively. Determine the relationship between the topologies \(\tau_3\) and \(\tau_4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\tau_3 = \tau_4}
A chain consists of $2k$ white beads and $2m$ black beads. Determine the minimal number of cuts required to divide the chain into two parts, each containing exactly $k$ white beads and $m$ black beads.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For a positive integer \( k \), define the set \( R_{n} \) as follows:
\[ 
R_{n} = 
\begin{cases} 
\{-k, -(k-1), \cdots, -1, 1, \cdots, k-1, k\}, & \text{if } n = 2k; \\
\{-k, -(k-1), \cdots, -1, 0, 1, \cdots, k-1, k\}, & \text{if } n = 2k+1.
\end{cases} 
\]

A device contains balls connected by red or white strings. A "marking" is a coloring of each ball with elements from \( R_n \). A marking is "good" if any two balls connected by a string have different colors. A marking is "insightful" if any two balls connected by a white string have different colors and any two balls connected by a red string have colors whose sum is not zero.

For a fixed positive integer \( n \geq 3 \), assume any device has a "good" marking regarding \( R_n \) and an "insightful" marking regarding \( R_m \). Find the smallest positive integer \( m = m(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n - 1}
For the differential equation \( y' = a - 2b \sin{y} \), where \( a \) and \( b \) are constants, determine the approximate value of \( \frac{y(2T) - y(T)}{T} \) for sufficiently large \( T \), given that \( 2b < a \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{a^2 - 4b^2}}
In a group $G$, given that $a^5 = e$ and $aba^{-1} = b^2$ for elements $a$ and $b$, find the order of the element $b$, denoted by $o(b)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Given the linear regression model with 100 observations:
\[ y_i = b_0 + b_1 x_i + \epsilon_i \]
The estimated coefficients are:
\[ \hat{\beta} = \begin{pmatrix} 9 \\ 1 \end{pmatrix} \]
And the variance-covariance matrix of the estimated coefficients is:
\[ V(\hat{\beta}) = \hat{\sigma}^2 (X'X)^{-1} = \begin{pmatrix} 3 & 0.12 \\ 0.12 & 0.064 \end{pmatrix} \]
Calculate the sum of squared residuals, \( e'e \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27195}
Find the Lebesgue measure of the interval \([n, n + \frac{1}{n^2}]\) for \(n \in \mathbb{N}\) using the definition of the n-dimensional Lebesgue measure: \( \lambda^{n}([[a,b))) = \prod_{j=1}^{n}(b_{j}-a_{j}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{n^2}}
Find the particular solution to the non-homogeneous differential equation \(y'' + \cos(t) y' = \cos(t)\) given that the homogeneous solution is \(y_h (t) = A + Be^{- t \cos (t)}\) with \(A,B \in \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t}
The sum \( 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{45} \) is expressed as a fraction with the denominator \( 45! = 1 \cdot 2 \cdot \ldots \cdot 45 \). Determine how many zeros (in decimal notation) the numerator of this fraction ends with.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a real periodic signal with period $T_0=2$ and Fourier coefficients $$X_k=\left [\frac{2}{3}, \ \frac{1}{3}e^{j\pi/4}, \ \frac{1}{3}e^{-i\pi/3}, \ \frac{1}{4}e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$, calculate the integral $\int_0^{T_0}x^2(t)\, dt$ using the Fourier coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{217}{36}}
Find the measure of the angle \( \delta = \arccos \left( \left( \sin 2195^\circ + \sin 2196^\circ + \cdots + \sin 5795^\circ \right)^{\cos 2160^\circ} + \cos 2161^\circ + \cdots + \cos 5760^\circ \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55^\circ}
Determine the smallest cardinality of a basis for the topology \( \tau \) on \( \mathbb{R}^2 \), where \( \tau \) consists of the empty set, \( \mathbb{R}^2 \), and all open disks \( \{ (x, y) \in \mathbb{R}^2 \mid x^2 + y^2 < r \} \) for all \( r > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_0}
In triangle \( ABC \), \( \angle BAC = 75^\circ \), \( AB = 1 \), and \( AC = \sqrt{6} \). A point \( M \) is chosen on side \( BC \) such that \( \angle BAM = 30^\circ \). The line \( AM \) intersects the circumcircle of triangle \( ABC \) at a point \( N \) different from \( A \). Find the length of \( AN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. A plane is tangent to these two spheres and intersects the surface of the cylinder, forming an ellipse. Find the sum of the lengths of the major axis and the minor axis of this ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Let \( R \) be the set of real numbers and \( C \) be the set of all functions \( f : [0, 1] \to R \) that have a continuous derivative and satisfy \( f(0) = 0 \) and \( f(1) = 1 \). Determine \( \inf_{f \in C} \int_{0}^{1} | f'(x) - f(x) | \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{e}}
Determine the values of $s$ for which the integral \[ \int\limits_{e_k}^{\infty} \frac{\mathrm{d} x}{x \cdot \ln x \cdot \ln^{(2)} x \cdot \ln^{(3)} x \cdots (\ln^{(k)} x)^s } \] converges, where $\ln^{(k)}$ denotes the $k$-fold iterated logarithm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s > 1}
Evaluate the limit: $$\lim_{x\to 0}\left(\frac{1}{\sin(x)\arctan(x)}-\frac{1}{\tan(x)\arcsin(x)}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral \( \int_{-\infty}^{\infty} dk\left[\Im\left( \frac{1}{(\omega+i\eta+k)^2+(\omega+i\eta+k+q)^2}\right) \right] \) for an infinitesimal \( \eta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an entire function satisfying $|f(z)| \leq C|\cos(z)|$ for all $z \in \mathbb{C}$, where $C \in \mathbb{R}$ is a constant. Determine the possible form of $f(z)$. Provide your answer in the form of a function or expression.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A \cos(z)}
Calculate the flux of the vector field \( \mathbf{F} = (x^2+y^2+2+z^2)\mathbf{i} + (e^{x^2} + y^2)\mathbf{j} + (3+x)\mathbf{k} \) outward across the part of the spherical surface \( x^2 + y^2 + z^2 = 2az + 3a^2 \) that is above the \( xy \)-plane, where \( a > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\pi a^2}
Evaluate the line integral \( \int_\mathcal{E} F \cdot dr \) where \( F(x,y,z) = x\,i+xy\,j+x^2yz\,k \) and \( \mathcal{E} \) is the ellipse \( x^2 + 4y^2 - 8y + 3 = 0 \) in the xy-plane, without using parameterization.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Evaluate the limit: $$\lim_{x\to\infty}\frac{\sum_{r=1} ^x re^\frac{r}{x}}{x^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many points at infinity are there in the complex plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that \( \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix} = -4 \), find the determinant of the matrix \( \begin{vmatrix} 2a_3 & 2a_2 & 2a_1 \\ b_3-a_3 & b_2-a_2 & b_1-a_1 \\ c_3+3b_3 & c_2+3b_2 & c_1+3b_1 \end{vmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the largest integer \( n \) such that \( 7^n \) divides \( \binom{2016}{1003} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Alan, Barb, Cory, and Doug are on the golf team, Doug, Emma, Fran, and Greg are on the swim team, and Greg, Hope, Inga, and Alan are on the tennis team. These nine people sit in a circle in random order. The probability that no two people from the same team sit next to each other is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2521}
Solve the equation $x^x = 10$ for $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.506}
Calculate the total work required to pump all the water out of a cylindrical tank with a radius of 2 meters and a length of 10 meters. Assume the tank is filled with water, the acceleration due to gravity is 9.8 m/s², and the density of water is 1000 kg/m³. Express your answer in terms of Joules.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{784000\pi}
Under ideal conditions of uniform gravitational acceleration and no friction, what shape would the top ends of a set of straight ramps form if they were arranged such that they all ended at the same base point and took an equal amount of time for a point-like body to slide down from the top to the base?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a circle}
Determine the values of $z$ for which the generating function $\sum_{n=1}^{\infty} g_{n} z^{n}$, where $g_{n}$ is the number of isomorphism classes of groups of order $n$, converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|z| < 1}
Find the minimum value of the function \(f(x)=12x^2\int_0^1yf(y)dy+ 20x\int_0^1y^2f(y)dy+4x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the time complexity of the recursive function given by:

$$T(n) =\left\{\begin{aligned} &2T((n-3)/3) + k_3n^3&&\text{ when }n > 1\\ &k_1 &&\text{ when } n = 1 \end{aligned}\right.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^3)}
Determine the value of $n$ for which a cyclic $n$-gon can be divided by non-intersecting diagonals into $n-2$ triangles, such that each triangle is similar to at least one other triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the number of integer solutions for \( n \) such that \( 0 \leq n < 23^2 \) and \( n^5 + 2n^4 + n^3 - 3n + 2 \equiv 0 \pmod{23^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Ana, Bob, Cho, Dan, and Eve want to use a microwave. In order to be fair, they choose a random order to heat their food in (all orders have equal probability). Ana's food needs $5$ minutes to cook, Bob's food needs $7$ minutes, Cho's needs $1$ minute, Dan's needs $12$ minutes, and Eve's needs $5$ minutes. What is the expected number of minutes Bob has to wait for his food to be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{37}{2}$}
Determine the values of \( \lambda \) for which the function \( N_\lambda(x,y) = \sqrt{x^2 + y^2 + 2\lambda xy} \) defines a norm on \( \mathbb{R}^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Given a composite system in a pure state with its Schmidt decomposition, determine the rank of the reduced density matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r}
Determine the maximum number of students who can participate in a math competition where each student solves 3 problems, each scored from 0 to 7, such that no student scores equal to or higher than another student on all three problems.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Find the limit of \( \frac{{{k-1}\choose n}}{{{n+k-1}\choose n}} \) as \( k/n^2 \to a > 0 \) and both \( k, n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-1/a}}
Let $a$, $b$, and $c$ be nonzero real numbers such that $a + \frac{1}{b} = 5$, $b + \frac{1}{c} = 12$, and $c + \frac{1}{a} = 13$. Find $abc + \frac{1}{abc}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
750
}
The lateral edge length of a regular n-sided prism is equal to the side length of its base. Find the maximum possible value of n. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the range of values for $r$ such that the function $f$ is continuous on the interval $[0, \infty)$. Provide your answer as an interval or union of intervals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \infty)}
Calculate the integral \( \int_{0}^{3}\sqrt{4-x^2}\,dx \) using a Riemann sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Calculate the sum \[ \sum_{n=1}^{\infty}\left((2n-1)\left(\sum_{k=n}^{\infty}\frac{1}{k^2}\right)-2\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
Determine the set \( S \) given that it consists of all positive real numbers, is closed under addition, and for any interval \([a, b] (a>0)\), there exists an interval \([c, d] \subseteq [a, b]\) such that \([c, d] \subseteq S\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}^+}
Determine the number of roots of the polynomial \(z^4 + z^3 - 4z + 1 = 0\) within the annular region \(1 < |z| < 2\) using Rouché's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the function $f(u)$ in the system of equations:
\[ (u_x)^2 - (u_t)^2 = 1 \\
   u_{xx} - u_{tt} = f(u) \]
given that $u = \sqrt{x-t}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Compute the intersection number of the divisors \(\iota^*\mathcal{O}(3,2)\) and \(\iota^*\mathcal{O}(1,4)\) on the smooth subvariety \(\iota:X=V_+(f)\subset\mathbb{P}^2\times\mathbb{P}^1\), where \(f\in H^0(\mathcal{O}(1,2))\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find a second degree polynomial $t(x)$ such that for every second degree polynomial $r(x)$, there exist constants $a$, $b$, and $c$ satisfying the equation $r(x) = a \cdot p(x) + b \cdot q(x) + c \cdot t(x)$, where $p(x) = x^2 + 5x - 3$ and $q(x) = 4x^2 + 18x + 4$. Determine $t(x)$. \(t(x)\) should be a second degree polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2}
Determine the limiting distribution of the sequence of random variables $Y_n = \frac{X_n - n}{n^2}$, where $X_n$ is uniformly distributed over the interval $[n, n^2]$ for $n = 1, 2, 3, \dots$. What does $Y_n$ converge to in distribution as $n \to \infty$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{U(0,1)}
Evaluate the limit: $$\lim _{x \to 0} \left[\frac{3}{x^4}(\cos^{-1}x - \sin^{-1}x )\right]^{\frac{3}{x^3}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Calculate the integral \( \int_E z(x^2+y^2)\ dx\ dy\ dz \), where \( E \) is the region in \( \mathbb{R}^3 \) defined by the inequalities \( x^2+y^2+z^2 \le 4 \), \( \frac{1}{3}(x^2+y^2) \le z^2 \le 3(x^2+y^2) \), and \( z\ge 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8\pi}{3}}
Let $X_1 \sim N(3,\sqrt{5})$ and $X_2 \sim \chi^2(15)$. Assume $X_1$ and $X_2$ are independent. Determine the distribution of \[ Z := \sqrt{\frac{3}{X_2}}(X_1-3). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t(15)}
Determine whether there exists a natural number $n$ such that $n^n + (n+1)^n$ is divisible by 1987.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{993}
Determine the number of zeros of the polynomial \( g(z) = z^4 + iz^3 + 1 \) that lie in the region \( \{ z \in \mathbb{C} : \text{Re}(z) > 0, \text{Im}(z) > 0 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the smallest number of subsets, $n$, into which the set of all integers can be partitioned such that none of the subsets contain an infinite arithmetic progression?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral \( \int_{|z|=2} \frac{dz}{z\sqrt{1-\frac{1}{z^2}}} \) over the circle \(|z|=2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Determine the values of $p$ for which the function \[ f(x,y) = \begin{cases} \frac{(x+y)^{p}}{x^2-y^2} & \text{if $(x,y) \neq (0,0)$} \\ 0 & \text{if $(x,y)= (0,0)$} \end{cases} \] is differentiable at the point $(0,0)$. Provide your answer as a set of values for $p$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 3}
Compute the size of the symmetry group of a configuration consisting of two congruent regular \(n\)-gons in the plane. Use the orbit-stabilizer theorem to find your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4n}
Given a function $g(t)$ such that its Fourier transform satisfies \( \int_{-\infty}^{\infty} g(t)e^{-j\omega t}dt=\omega e^{-2\omega ^2} \) for any real \( \omega \), find the value of \( \int_{-\infty}^{\infty} y(t)dt \), where \( y(t)=\int_{-\infty}^{t}g(\tau)d\tau \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-j}
Find the general solution to the Clairaut differential equation: \( y = xy' - \ln y' \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = Cx - \ln C}
Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$, what is $\beta$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi}{2} - 2\alpha$
}
What is the largest number of starting clues in a Sudoku puzzle that does not guarantee a unique solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{77}
Determine the time complexity of the conjugate gradient method when solving the system of linear equations $Ax = b$, where $A$ is a sparse, symmetric, and positive definite matrix.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n^2)}
Five points, no three of which are collinear, are given. What is the least possible value of the numbers of convex polygons whose some corners are from these five points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Let \(f(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n\), where \(a_i\) are non-negative integers. Given that \(f(1) = 21\) and \(f(25) = 78357\), find the value of \(\frac{f(10) + 3}{100}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
A positive integer $x$ is randomly generated with equal probability from the set $\{1, 2, \cdots, 2016\}$. What is the probability that the sum of the digits of $x$ in binary representation does not exceed 8?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{655}{672}}
For a projective curve $X$ over a field $k$, determine if there exists a positive integer $N$ such that the equation $h^0(\mathcal{O}_X(D)) = \deg D + 1 - p_a(X)$ holds for any Cartier divisor $D$ with $\deg D > N$. Provide the value of $N$ if it exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2p_a(X) - 2}
In a single server queue, the arrival rate is 2 when the queue is empty and 1 otherwise, while the departure rate is a constant 3. Using Little's Law and the invariant distribution, calculate the average waiting time in the queue.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Determine the number of arrangements of the letters in the word INSTRUCTOR such that: (a) the vowels appear in alphabetical order (IOU), (b) there are at least 2 consonants between each vowel, and (c) the arrangement begins or ends with the two consecutive Ts.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
Find the limit of the expression as \( n \) approaches infinity: $$\lim_{n\to \infty}\frac{1}{2n}\log\left(\binom{2n}{n}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log 2}
Evaluate the double integral \( \iint \frac{x^2}{y^3} \, dA \) using a change of variables over the region defined by the curves \( y=2x, \; y=x, \; y=x^2, \; y=2x^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \[ f(x + y) + f(z) = f(x) + f(y + z) \] for all real numbers \( x, y, z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = ax + b}
Determine the number of complex roots of the equation $x^n = a$, where $a \ge 0$ and $n$ is not an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given a smooth closed curve $\mathcal{C}$ in the plane $\mathbb{R}^2$, with a curve $\mathcal{C'}$ located at a constant distance $d > 0$ from $\mathcal{C}$ and orthogonal to it at all points, if $\mathcal{C}$ has length $L$, find the length $L'$ of $\mathcal{C'}$. Express your answer in terms of $L$ and $d$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{L + 2\pi d}