inclusionAI/Ring-lite-rl-data huggingface dataset:
Cagney can frost a cupcake every 18 seconds and Lacey can frost a cupcake every 40 seconds. Lacey starts working 1 minute after Cagney starts. Calculate the number of cupcakes that they can frost together in 6 minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
A cylinder of height h is 6/7 full of water. When all of the water is poured into an empty cylinder with a certain percentage larger radius than that of the original cylinder, the new cylinder is 3/5 full. The height of the new cylinder is 70 percent of h. How much larger is the radius of the new cylinder compared to the original cylinder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3/7,}
Determine the generating function for the summation: \[ \sum_{k=0}^m\binom{m}{m-k}\binom{-n-1}{m-k} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1 - x)^n}
Given the hyperbola \( C: \frac{x^2}{2} - y^2 = 1 \), let \( P \) be a point on its right branch. Line \( l \) is an asymptote of the hyperbola, and \( Q \) is the projection of \( P \) onto \( l \). Let \( F_1 \) be the left focal point of the hyperbola. Determine the minimum value of \( |PF_1| + |PQ| \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 2\sqrt{2}}
Given that $X_t$ is a standard Brownian motion, calculate the probability $P(X_2 > 0 \mid X_1 > 0)$. Provide your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Compute the number of intersection points between $\mathbb{R}P^2$ and $\mathbb{C}P^1$ in $\mathbb{C}P^2$. Consider $\mathbb{R}P^2 = \{ [x_0:y_0:z_0] \mid x_i \in \mathbb{R} \}$ and $\mathbb{C}P^1 = \{ [z_0:z_1:z_2] \mid a_0 z_0 + a_1 z_1 + a_2 z_2 = 0 \}$ where $a_i \in \mathbb{C}$. Determine the number of points satisfying $a_0 x_0 + a_1 x_1 + a_2 x_2 = 0$ with $x_i \in \mathbb{R}$. Provide the number of such intersection points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the value of the exterior 2-form \( \theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1} \) on \( \mathbb{R}^n \) at the vectors \( A = (1, 1, 1, \dots, 1) \) and \( B = (-1, 1, -1, \dots, (-1)^n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + (-1)^n}
Determine the number of distinct monic quadratic polynomials with integer coefficients such that they have at least one root, all roots are powers of 3 with non-negative integer exponents, and the absolute value of their coefficients does not exceed $27^{47}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5111}
Given a tetrahedron with five edges of length 2, the maximum volume of the tetrahedron is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player".

(Condition) For each player $B$ who won $A$, there is a player $C$ who won $B$ and lost to $A$.

It turned out there are exactly $N(\geq 0)$ skilled player. Find the minimum value of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3
}
In a round-robin tournament with 2010 players, each player plays against every other player exactly once. A win awards the player 1 point, while a loss gives 0 points. Determine the last three digits of the maximum possible score difference between the first and second highest scorers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{005}
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$
What is the least number of colours which suffices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Solve the differential equation using Laplace transform: $$xy'' + y' + 4xy = 0$$ with initial conditions $y(0) = 3$ and $y'(0) = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 J_0(2x)}
Let \( X_1, X_2, \) and \( X_3 \) be three independent exponential random variables with rate \( \lambda \). Find the probability that \( \min(X_1, X_2, X_3) \leq X_1 + X_2 + X_3 - \min(X_1, X_2, X_3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of bases in the vector space \( \mathbf{Z}_5^4 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{116064000000}
Evaluate the integral \( \int_0^{2\pi} \frac{\cos(3\theta)}{5-4\cos(\theta)}\mathrm{d}\theta \) using complex analysis techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{12}}
Given right triangle $\triangle ABC$ has a hypotenuse $AC$ of length 10, where $A$ is at the right angle. Point $M$ is the midpoint of hypotenuse $\overline{AC}$, and $D$ is a point on leg $\overline{AB}$ such that $AD:DB = 1:2$. Determine the area of $\triangle ADM$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Given that the maximum value of \((x+2y+3z)^2\) is 70, find the maximum value of \(x^2 + (2y)^2 + (3z)^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Determine the exact value of the norm \( \Vert g \Vert \) of the functional \( g \) defined on the normed space \( C[-1,1] \) by \[ g(x) = \int_{-1}^1 x(t)t \, \mathrm{d}t \] for all \( x \in C[-1,1] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral \( I = \int_{0}^{2\pi} \frac{\cos^2(\theta) - \sin^2(\theta)}{\sin^4(\theta) + \cos^4(\theta)} \, d\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
An ant is on the circumference of the base of a cone with radius $2$ and slant height $6$. It crawls to the vertex of the cone $X$ in an infinite series of steps. In each step, if the ant is at a point $P$, it crawls along the shortest path on the exterior of the cone to a point $Q$ on the opposite side of the cone such that $2QX = PX$. What is the total distance that the ant travels along the exterior of the cone?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$$
6\sqrt3
$$}
It is known that \(4 \operatorname{tg}^{2} Y + 4 \operatorname{ctg}^{2} Y - \frac{1}{\sin ^{2} \gamma} - \frac{1}{\cos ^{2} \gamma} = 17\). Find the value of the expression \(\cos ^{2} Y - \cos ^{4} \gamma\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{25}}
If $a_1, a_2, \ldots, a_{2013} \in [-2, 2]$ and $a_1 + a_2 + \cdots + a_{2013} = 0$, find the maximum of $a_1^3 + a_2^3 + \cdots + a_{2013}^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4026}
In figure 1, \(ABCD\) is a rectangle, and \(E\) and \(F\) are points on \(AD\) and \(DC\), respectively. Also, \(G\) is the intersection of \(AF\) and \(BE\), \(H\) is the intersection of \(AF\) and \(CE\), and \(I\) is the intersection of \(BF\) and \(CE\). If the areas of \(AGE\), \(DEHF\), and \(CIF\) are 2, 3, and 1, respectively, find the area of the grey region \(BGHI\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the interval of convergence for: $$\sum_{i=1}^\infty \frac{(x+2)^n}{(n^{1/2})(3^n)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-5,1)}
A gambler starts with an initial fortune of $2. Calculate the probability that the gambler goes broke on the 10th bet, given that their wealth reaches $6 at some point before eventually decreasing to $0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{1024}}
Find the value of Ack(3,1) using the Ackermann function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Evaluate the double integral \( \iint 3x^2 \, dA \) over the region in the first quadrant bounded by the hyperbola \( xy = 9 \), the line \( y = x \), the y-axis (\( y = 0 \)), and the x-axis (\( x = 6 \)). Round your answer to the nearest tenth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{425.3}
Determine the maximum positive integer $r$ such that for any five 500-element subsets of the set $\{1, 2, \ldots, 1000\}$, there exist two subsets with at least $r$ common elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Given the transition matrix \( M = \begin{pmatrix} 0.25 & 0.5 & 0.25 \\ 0.5 & 0.25 & 0.25 \\ 0.5 & 0.25 & 0.25 \end{pmatrix} \) of a Markov chain with states \( S = \{0, 1, 2\} \), calculate the expected number of steps to reach state 1 starting from state 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{20}{7}}
In triangle $ABC$, let $I$ be the incenter, and $D, E, F$ be the points of tangency of the incircle with sides $BC, CA, AB$, respectively. Point $K$ is the intersection of line $DI$ and line $EF$. Given that $BC = 10$, $AC = 8$, and $AB = 7$, find the ratio of the areas $\frac{S_{\triangle BKC}}{S_{\triangle ABC}}$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Find the minimum variance unbiased estimator for the total number of people in a group, given that 5 people had a birthday today, assuming birthdays are uniformly distributed over 365 days.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1825}
In a convex 1950-gon, all diagonals are drawn, dividing it into smaller polygons. Determine the maximum number of sides that one of these polygons can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1950}
The function $f(x)=(m^{2}-m-1)x^{m^{2}+m-1}$ is a power function, and it is decreasing on $(0,+\infty)$. Find the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-1"]}
Solve the equation for the unknown \( x \) where \( n \geq 2 \) is a given natural number:

$$
\sum_{i=0}^{n-2} \frac{1}{(x+i)(x+i+1)}=x(x+1)(x+2) \cdot \ldots \cdot (x+n)+\frac{n-1}{x(x+n-1)}
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-n}
Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e., a uniformly random bearing $\theta \in [0, 2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{190}
In an $8 \times 8$ table, 23 cells are black, and the rest are white. In each white cell, the sum of the black cells located in the same row and the black cells located in the same column is written. Nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can take?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{234}
What is the characteristic path length of the Petersen graph?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{3}}
Calculate the sum \( S = \lim_{N\rightarrow \infty} \sum_{n=-N}^N \frac{1}{n+e} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \cot(\pi e)}
Given $f(x) = \sin \left( \frac{\pi}{3}x \right)$, and the set $A = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Now, choose any two distinct elements $s$ and $t$ from set $A$. Find out the number of possible pairs $(s, t)$ such that $f(s)\cdot f(t) = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Colleen has three shirts: red, green, and blue; three skirts: red, green, and grey; three scarves: red, blue, and grey; and three hats: green, blue, and grey.

How many ways are there for her to pick a shirt, a skirt, a scarf, and a hat, so that two of the four clothes are one color and the other two are one other color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Find an explicit formula for the operator $TT^*$, where $T: H \rightarrow \ell^{2}$ is the analysis operator $T x=\left\{\left\langle x, e_{n}\right\rangle\right\}_{n \in \mathbb{N}}$ and $T^{*}: \ell^{2} \rightarrow H$ is the synthesis operator $T^{*} c=\sum_{n=1}^{\infty} c_{n} e_{n}$, given an orthonormal sequence $\{e_{n}\}_{n \in \mathbb{N}}$ in a Hilbert space $H$.}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{TT^* = I}
The sides of rectangle $ABCD$ have lengths 10 and 11.  An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$  The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r,$ where $p, q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any prime number.  Find $p+q+r.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{554}
Each time you click a toggle switch, the switch either turns from [i]off[/i] to [i]on[/i] or from [i]on[/i] to [i]off[/i]. Suppose that you start with three toggle switches with one of them [i]on[/i] and two of them [i]off[/i]. On each move you randomly select one of the three switches and click it. Let $m$ and $n$ be relatively prime positive integers so that $\frac{m}{n}$ is the probability that after four such clicks, one switch will be [i]on[/i] and two of them will be [i]off[/i]. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142}
In the ancient Chinese mathematical text "Jiuzhang Suanshu" (The Nine Chapters on the Mathematical Art), there is a problem: "Given a metal rod, 5 chi in length, if one cuts 1 chi from the thicker end, it weighs 4 jin. If one cuts 1 chi from the thinner end, it weighs 2 jin. Assuming the rod varies uniformly in thickness from one end to the other, how much does each chi section of the rod weigh?" Suppose the rod varies evenly from the thick end to the thin end, we denote the weight of the rod by $M$. Now the rod is cut into 10 equal segments, and we denote the weight of the $i^{th}$ segment as $a_i$ (for $i=1,2,\ldots,10$), such that $a_1 < a_2 < \ldots < a_{10}$, and if $48a_i = 5M$, then $i = \underline{\hspace{1em}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( ABC \) be an acute triangle and \( O \) be its circumcenter. Let \( D \) be the midpoint of \( [AB] \). The circumcircle of \( \triangle ADO \) meets \( [AC] \) at \( A \) and \( E \). If \( |AE|=7 \), \( |DE|=8 \), and \( m(\widehat{AOD}) = 45^\circ \), what is the area of \( \triangle ABC \) in the form \( x \sqrt{2} \)? Find the value of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
In the trapezoid \(ABCD\), the bases are given as \(AD = 4\) and \(BC = 1\), and the angles at \(A\) and \(D\) are \(\arctan 2\) and \(\arctan 3\) respectively.

Find the radius of the circle inscribed in triangle \(CBE\), where \(E\) is the intersection point of the diagonals of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{18}{25 + 2 \sqrt{130} + \sqrt{445}}}
Determine the number of solutions to the equation \(X^\top X = B^\top B\), where \(B \in \mathbb{R}^{d \times n}\) with \(d \geq n\) and \(B^\top B\) is non-singular.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Dr. Jones earns $6,000 a month. His house rental is $640 each month; his monthly food expense fluctuates between $350 and $400 depending on his needs; his electric and water bill costs 1/4 of what he makes, and his insurances cost 1/5 of what he makes. He spends 10% of his salary on taxes and 3% on transportation expenses. Additionally, Dr. Jones needs to allocate 2% of his salary for unexpected emergency costs. Dr. Jones also has a student loan payment of $300 per month and he invests 5% of his income into a retirement plan. If his food expense for a particular month is $385, how much money does he have left after paying all these expenses and making investments for that month?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{775}
Evaluate the double integral: $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_0^{a\sin\theta} r^2 \, dr \, d\theta$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2a^3}{9}}
A fair six-sided die is labeled $1-6$ such that opposite faces sum to $7$. The die is rolled, but before you can look at the outcome, the die gets tipped over to an adjacent face. If the new face shows a $4$, what is the probability the original roll was a $1$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{4}$}
In a school with 200 teachers, each teacher sends a greeting message to exactly one other teacher. Determine the minimum number of representatives, \( k \), that can be selected for a recognition ceremony such that no representative has sent a message to another representative in the group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Evaluate the integral: $$ \int \cos x \cdot \cos^2(2x) \cdot \cos^3(3x) \cdot \cos^4(4x) \cdot \ldots \cdot \cos^{2002}(2002x) \, dx $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \(\int_{-1}^1 \frac{d}{dx}\ \frac{1}{1 + 2^{1/x}}\ dx\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Determine the number of distinct bracelets that can be formed using 12 beads, consisting of 3 red beads, 4 green beads, and 5 blue beads.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1170}
Calculate the volume of the solid formed by rotating the region bounded by the curve \( y = e^{-x} \sin x \) and the line \( x \ge 0 \) about the x-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
Calculate the probability that a random undirected graph G(5, 1/2) is connected, where G(n, p) denotes a graph with n vertices and probability p of an edge between any two vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{91}{128}}
Determine the maximum achievable Hamming distance between $k$ bitvectors of length $n$, where $n$ is a power of 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n}{2}}
Evaluate the line integral \( \int_{K}\left(x^{2}+y z\right) d x+\left(y^{2}+x z\right) d y+\left(z^{2}-x y\right) d z \) where \( K \) is a closed curve, oriented positively, consisting of an arc defined by the parametric equation \( x=a\cos t, y=a\sin t, z=\frac{1}{2\pi}t \) and the segment \( BA, A=(a,0,0), B=(a,0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: $$\lim_{x \to 0} \frac{x-\sin(\sin(\cdots\sin(x)))}{x^3},$$ where $\sin(\sin(\cdots\sin(x)))$ denotes the n-fold composite sine function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{n}{6}}
The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as
\[P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.\]
Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\dbinom{2014}{1007}$}
Find the fundamental group of the space obtained by removing a finite number of points from \( \mathbb{R}^n \) for \( n \geq 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Define a sequence of polynomials $P_0\left(x\right)=x$ and $P_k\left(x\right)=P_{k-1}\left(x\right)^2-\left(-1\right)^kk$ for each $k\geq1$. Also define $Q_0\left(x\right)=x$ and $Q_k\left(x\right)=Q_{k-1}\left(x\right)^2+\left(-1\right)^kk$ for each $k\geq1$. Compute the product of the distinct real roots of\[P_1\left(x\right)Q_1\left(x\right)P_2\left(x\right)Q_2\left(x\right)\cdots P_{2018}\left(x\right)Q_{2018}\left(x\right).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}-1}
Find the smallest positive integer \( n \) for which the following limit holds true for two linearly independent solutions \( \phi_1(x) \) and \( \phi_2(x) \) of the differential equation \( 3x^2y''-(2x+x^2)y'+(x^2-3)y=0 \), given that \( \phi_1(0)=0 \): \[ \lim_{x \to 0}x^n\frac{\phi_2(x)}{\phi_1(x)}=0 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a chess tournament, a team of schoolchildren and a team of students, each consisting of 15 participants, compete against each other. During the tournament, each schoolchild must play with each student exactly once, with the condition that everyone can play at most once per day. Different numbers of games could be played on different days.

At some point in the tournament, the organizer noticed that there is exactly one way to schedule the next day with 15 games and $N$ ways to schedule the next day with just 1 game (the order of games in the schedule does not matter, only who plays with whom matters). Find the maximum possible value of $N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Solve the equation $2^x = -2x + 11$ for $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.56}
Given the system of equations:
\[\begin{cases} x^2 + xy - y^2 = u \\ 2xy + y^2 = v \end{cases}\]
find the value of \( \left(\frac{\partial x}{\partial u}\right)_{v} \) at the point where \( x = 2 \) and \( y = -1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{7}}
Evaluate the contour integral \( \int_C \frac{dz}{\sin^3 z} \), where \( C \) is the unit circle in the complex plane with positive orientation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi i}
Calculate the net flux of the vector field \( F(x, y, z)=\frac{r}{|r|^{3}}+zi-xk \) through the upper half of the ellipsoid defined by \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1, \quad z\geq0,\) in the direction where the component of the outward normal along the \(y\)-axis is positive (\(N \cdot j>0\)).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
In a continuous-time Markov chain, Sleeping Beauty (SB) can be in one of three states while she sleeps: State 0 (flat on her back), State 1 (foetal position), and State 2 (foetal position and sucking her thumb). The transition rate matrix is given by:

\[ G = \begin{bmatrix} -\frac{1}{2} & \frac{3}{8} & \frac{1}{8} \\
\frac{1}{12} & -\frac{1}{3} & \frac{1}{4} \\
\frac{1}{4} & \frac{3}{4} & -1 \end{bmatrix} \]

What is the probability that Prince Charming finds SB in State 0 (looking truly radiant) after a long time? Assume SB has been sleeping for a very long time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{36}}