inclusionAI/Ring-lite-rl-data huggingface dataset:
Determine the function \( h(t) \) for the partial differential equation \( u_{xt}(x,t) + u(x,t)u_{xx}(x,t) = h(t) \) with the boundary conditions \( u(\pi,t) = u_x(0,t) = u_x(\pi,t) = 0 \), where \( x \in [0,\pi] \) and \( t > 0 \). Assume all computations and regularity conditions are valid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine all positive integers \( n \) such that if \( 1 = d_1 < d_2 < \ldots < d_k = n \) are the divisors of \( n \), then \( n = d_6^2 + d_7^2 - 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
在 $4 n \times 4 n\left(n \in \mathbf{N}_{+}\right)$实数表中, 每个数的绝对值均不超过 1 , 所有数的和为 0 . 已知数表中的每行各数之和的绝对值、每列各数之和的绝对值均不小于 $c$. 求 $c$ 的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2 n$}
Given proposition $p$: The equation $\frac{x^2}{m} + \frac{y^2}{6-m} = 1$ represents an ellipse with foci on the $x$-axis; proposition $q$: The eccentricity $e$ of the hyperbola $\frac{y^2}{5} - \frac{x^2}{m} = 1$ belongs to $(\frac{\sqrt{6}}{2}, \sqrt{2})$. If the proposition “$p \lor q$” is true, and “$p \land q$” is false, find the range of values for $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{5}{2},3] \cup [5,6)}
Let $U$ and $V$ be the null spaces of matrices $A = \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & 3 & 2 \\ 0 & 1 & 2 & 1 \end{bmatrix}$, respectively. Determine the dimension of the vector space $U + V$. \( \text{Provide your answer as a single integer.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
现有 $6$ 人会英语， $4$ 人会日语， $2$ 人都会（共 $12$ 人），从其中选出 $3$ 人做翻译，若要求两种语言都有人翻译，则符合条件的选法共 $\qquad$种。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{196}
In triangle \(ABC\), if the triangle formed by its medians and the triangle formed by its altitudes are both similar to triangle \(ABC\), what is the relationship between the side lengths of triangle \(ABC\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{AB = BC = CA}
Given the function f(x) = x^2 * e^|x|, a = f(log_2(3)), b = f(-log_5(8)), and c = f(-2^1.001), determine the relationship between a, b, and c. Express your answer using > or < to describe the order of a, b, and c.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c > a > b}
Given a triangle \( \triangle ABC \), which fulfills the equations \( 6\cos{A}+2\cos{B}=7 \) and \( 6\sin{A}-2\sin{B}=\sqrt{3} \), find the value of \( \angle C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Determine the amortized time complexity of performing a series of union and find operations on disjoint sets, assuming that all union operations occur after all find operations.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\alpha(n))}
Calculate the limit \( \lim_{x \to \infty, n \to \infty} f^n(x) \), where \( f(x) = \ln(1+x) \) and \( f^n(x) = f(f^{n-1}(x)) \) for all integers \( n \geq 2 \) with \( f^1(x) = f(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $B$ be a $3\times3$ invertible matrix with real entries such that $\det(B)=m$. Given that $\det(B+madj(B))=0$, find the value of $\det(B-madj(B))$. \(\text{Express your answer as a single number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Suppose $x$ is a real number such that $\sin(1 + \cos^2 x + \sin^4 x) = \tfrac{13}{14}$. Compute $\cos(1 + \sin^2 x + \cos^4 x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$-\dfrac{3\sqrt{3}}{14}$}
Find the sum of the values of $x$ such that $\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x$, where $x$ is measured in degrees and $100< x< 200.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{906}
Find the volume of the solid generated by rotating the region bounded by $x^2-y^2 = 7$, $x = 4$, and $y = 5$ about the $y$-axis using the shell method.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{44}{3}\pi}
Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{92}
Find the value of \(x\) such that \(e^{e^{10^{10^{2.8}}}} = 10^{10^{10^{10^{x}}}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.8}
Evaluate the following limit.$\begin{aligned} \lim_{n\to\infty}\sqrt{n}\int_0^\pi\sin^nxdx \end{aligned}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{${\sqrt{2\pi}}$}
What conditions must the elements of a $3 \times 3$ matrix $\mathbf{R}$ satisfy for the matrix to be orthogonal, thereby preserving the length of any vector $\mathbf{A}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbf{R}^\top \mathbf{R} = \mathbf{I}}
Evaluate the integral \( \int^{\pi}_0 \log{(\sin{x})} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi \ln 2}
Find the third term in the Taylor series expansion of \( f(x) = \frac{(1+x)^4}{(1+2x)^3(1-2x)^2} \) around \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
From $31$ students sitting around a circular table, how many different selections of $3$ students can be made such that there are at least $4$ students between any two selected students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1581
}
Let $p(x)$ and $q(x)$ be two polynomials such that $p(2)=5$, $p(3)=12$, and $p(q(x))=p(x)q(x)-p(x)$. Find the value of $q(10)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{114}
Evaluate \( \lim_{x\to 0}\frac{\frac{1}{1+x^2}-\int_{0}^{x}\arctan u\,du}{3\tan^{-1}x-3x-\frac{x^3}{3}} \) using L'Hopital's Rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3}{4}}
Calculate the limit \(\lim_{n \to \infty} \frac{f^{n}(0)}{2^{n}n!}\) given that \(\Omega = D(0,2) \setminus \left\{\frac{1}{2}\right\}\), \(f\) is a holomorphic function in \(\Omega\), and \(\frac{1}{2}\) is a simple pole with residue 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Through the median $BE$ of the base $ABC$ of the pyramid $ABCD$ and the midpoint $F$ of the edge $DC$, a plane is drawn. Find the volume of the figure $ADBF E$ if the volume of the pyramid $ABCD$ is $40 \text{ cm}^3$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30 \ \text{cm}^3}
Find the angle of intersection between the logarithmic spiral given by the equation \( r = Ce^{k\theta} \) and any line passing through the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arctan\left(\frac{1}{k}\right)}
Determine the number of different ways to completely cover a regular hexagonal chessboard, consisting of 24 small equilateral triangles, using 12 rhombus-shaped paper pieces, each composed of 2 small equilateral triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given that $f(x)$ is a twice differentiable function with $f(a)=0$, $f(b)=2$, $f(c)=-1$, $f(d)=2$, and $f(e)=0$, where $a<b<c<d<e$, determine the minimum number of zeros of the function $g(x) = (f'(x))^2 + f(x)f''(x)$ in the interval $[a, e]$. Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Evaluate the limit \( \lim_{{n \to \infty}} n\int_1^2 \frac{dx}{x^2(1+x^n)} \) without using Taylor expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Evaluate the limit: \[ \lim_{n\to\infty}\sum_{i=1}^{n}{\frac{2n}{(n+2i)^2}}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
A piece is placed in the lower-left corner cell of a $15 \times 15$ board. It can move to cells that are adjacent either by sides or corners of its current cell. The piece must alternate between diagonal and horizontal moves, starting with a diagonal move. What is the maximum number of moves the piece can make without stepping on the same cell twice?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{224}
There are 4 people standing in a line numbered 1 through 4 in a left to right order.
Each person has the following attributes:  Job, Hobby, Food, Music-Genre, Pet, Transport.
The attributes have the following possible values:
- Job: dancer, dressmaker, engineer, nurse
- Hobby: baking, filmmaking, hiking, puzzles
- Food: carrot, cauliflower, lettuce, mango
- Music-Genre: ambient, classical, jazz, rock
- Pet: fish, frog, lizard, turtle
- Transport: boat, car, helicopter, trike

Given the following premises about the line of people:
- Music-Genre:ambient == Hobby:puzzles
- Job:nurse is somewhere between Job:dancer and Job:engineer
- Hobby:filmmaking is on the left of Music-Genre:rock
- Pet:lizard is on the right of Music-Genre:jazz
- Job:dancer == Music-Genre:classical
- Pet:fish is on the right of Food:cauliflower
- Pet:frog is somewhere to the left of Transport:trike
- Transport:helicopter is somewhere to the right of Pet:turtle
- Job:dressmaker != Food:lettuce
- Transport:boat is on the left or right of Pet:turtle
- Transport:helicopter is on the far left or far right
- Music-Genre:rock is somewhere to the left of Hobby:hiking
- Food:carrot is on the left of Food:cauliflower
- Hobby:hiking == Food:cauliflower

Answer the following question:

Question: Question: What is the Music-Genre of the person who has/is filmmaking??
 Please provide your answer in the format: \boxed{X}, where X is the answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{jazz}
Given that each time you play a game, you win a prize, and the probability of winning a new type of prize is \((N-a)/N\), where \(N\) is the total number of unique prizes and \(a\) is the number of unique prizes already won, determine the formula to find the most likely number of unique prizes \(N\) based on the sequence of wins: (1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Napoleon has 56 jelly beans and Sedrich has 9 more jelly beans than 3 times what Napoleon has. Daphne has twice as many jelly beans as the difference between Sedrich's and Napoleon's jelly beans. Alondra has 8 fewer jelly beans than the average of Napoleon, Sedrich, and Daphne's jelly beans. If the sum of all four people's jelly beans is equal to 5 times the number of jelly beans that Mikey has, how many jelly beans does Mikey have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
Determine the values of $b$ for which the series \( \sum_{k=1}^\infty \frac{k!}{(b+1)(b+2)...(b+k)} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b > 1}
Given the function $f(x)=\ln x- \frac {1}{2}ax^{2}-2x$ $(a\neq 0)$ has a decreasing interval, the range of the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1,0)\cup(0,+\infty)}
Evaluate the polynomial congruence $\sum_{j=(p-1)/2}^{p-1}\left(\frac{j}{p}\right)j(j-1)...(j-(p-1)/2+1) \ (\textrm{mod} \ p)$, where $p$ is an odd prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
What is the product of all odd positive integers less than 10000? 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( \frac{{10000}!}{{2}^{5000} \cdot  {5000}!} \)}
Determine the radius of convergence of the series \( \sum a_n x^{n^2} \), given that the series \( \sum a_n x^n \) has a radius of convergence \( R \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of ways to choose $k$ numbers from the first $n$ natural numbers such that the longest string of consecutive numbers is exactly $m$ numbers long. For instance, if $k = 7$, $n = 14$, and $m = 3$, find the number of combinations of numbers that satisfy these conditions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1456}
Given that $21^{-1} \equiv 15 \pmod{61}$, find $40^{-1} \pmod{61}$, as a residue modulo 61.  (Provide a number between 0 and 60, inclusive.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
In a certain class of Fengzhong Junior High School, some students participated in a study tour and were assigned to several dormitories. If each dormitory accommodates 6 people, there are 10 students left without a room. If each dormitory accommodates 8 people, one dormitory has more than 4 people but less than 8 people. The total number of students in the class participating in the study tour is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{46}
Two points are chosen independently at random on the sides of a square with side length 1. The probability that the straight-line distance between the points is at least \( \frac{1}{2} \) is \( \frac{a-b\pi}{c} \), where \( a \), \( b \), and \( c \) are positive integers with \( \gcd(a,b,c)=1 \). Find \( a+b+c \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{59}
Find the limit of the sequence $(x_0 x_1 \cdots x_n)\sqrt{n}$ as $n$ approaches infinity, where $x_0 = \frac{1}{2}$ and $x_{n+1} = x_n^3 - x_n^2 + 1$ for all $n \in \mathbb{N} \cup \{0\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the limit: \[ \lim_{n\to \infty } \int_0^n \left(1-\frac{x}{n}\right)^n \cos\left(\frac{x}{n}\right)dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A random variable $Y$ has a moment generating function given by $e^{10t+2t^{2}}$. Calculate the probability that $25 < (Y - 10)^2 < 49$. Express your answer as a single probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.012}
A square is drawn inside a rectangle. The ratio of the width of the rectangle to a side of the square is $3:1$ and the ratio of the rectangle's length to its width is $3:2$. Calculate the percentage of the rectangle’s area that is inside the square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7.41\%}
Determine the leading order term of the asymptotic expansion for \( \int_{1/x}^1 e^{-t} \, dt \) as \( x \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \frac{1}{e}}
Determine for which real values of \(\alpha\) the improper integral \(\int_0^\infty x^\alpha \cos x\, dx\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 0)}
Evaluate the integral \( I = \int_0^1 \frac{\sin(\ln(|z|))}{\ln(|z|)} \, d|x| \), where \( z = x + iy \) and \( |z| \) denotes the modulus of \( z \). Express the result in terms of \( \pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Four points are chosen independently and at random on the surface of a sphere (using the uniform distribution). What is the probability that the center of the sphere lies inside the resulting tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{8}}
Given a set of identical regular pentagons, each with natural numbers from 1 to 5 written at its vertices. The pentagons can be rotated and flipped. They are stacked vertex to vertex such that the sums of numbers at each of the five vertices are equal. Determine the number of pentagons in this stack.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the joint density function \( f_{X,Y}(x,y)=6e^{-x}e^{-2y} \) for \( 0<x<y< \infty \), where \( X \) and \( Y \) are the fail times for two components of a machine, find the expected time of failure of the machine. The component \( Y \) starts working if and only if component \( X \) fails, and the machine fails when component \( Y \) fails.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{6}}
Given a hyperbola $x^{2}- \frac {y^{2}}{3}=1$ and two points $M$, $N$ on it are symmetric about the line $y=x+m$, and the midpoint of $MN$ lies on the parabola $y^{2}=18x$. Find the value of the real number $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8}
[ Law of Sines ]

In a convex quadrilateral $A B C D$, the segment $C M$, connecting vertex $C$ with point $M$ located on side $A D$, intersects diagonal $B D$ at point $K$. It is known that $C K: K M = 2: 1, C D: D K = 5: 3$ and $\angle A B D + \angle A C D = 180^{\circ}$. Find the ratio of side $A B$ to diagonal $A C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:9}
Let $V$ be an $n$-dimensional vector space over a finite field $F$ of order $p^m$. Let $V_1$ be a $k$-dimensional subspace of $V$. Determine the number of $(n-k)$-dimensional subspaces $V_2$ of $V$ such that $V_1 + V_2 = V$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^{m k (n - k)}}
Let $A$ be the area of the region in the first quadrant bounded by the line $y = \\frac{x}{2}$, the x-axis, and the ellipse $\\frac{x^2}{9} + y^2 = 1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y = mx,$ the y-axis, and the ellipse $\\frac{x^2}{9} + y^2 = 1.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$m = \frac{2}{9}$}
Determine the maximum possible dimension of the annihilator $W^\circ$ of the subspace $W = \{A^m; m \geq 0\}$ within the vector space $V = M_n(R)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2 - 1}
Find an equivalent expression for the remainder of the converging series \( R_n = \sum_{k=n+1}^{\infty} \frac{1}{k^3} \) given the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2n^2}}
Evaluate the surface integral \(\iint_{S}(z-x^2-y^2)\,dS\) over the surface \(S\) of the cylinder \(x^2+y^2=4\) for \(-2\le z\le2\) using symmetry and geometry.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-64\pi}
Determine the total length of intervals on the number line where the minimum of the numbers \( 8 - x^{2} \) and \( \operatorname{ctg} x \) is not less than -1. Round your answer to the nearest hundredth if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.57}
You roll a fair six-sided die repeatedly until you roll a 5. What is the expected value of the maximum number rolled before the first 5 appears? Exclude the roll of 5 from consideration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{20}}
Calculate the Frobenius norm \(||AA^+ - I||_F\), where \(A^+\) is the Moore–Penrose inverse of matrix \(A\). The matrix \(A\) has dimensions \(n \times m\) and rank \(r\). The Frobenius norm is defined as \(||A||_F = \sqrt{\text{Tr}(AA^T)}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{n - r}}
Given the function sequence defined by \( f_{1}(x)=|1-2x| \) for \( x \in [0,1] \) and \( f_{n}(x)=f_{1}(f_{n-1}(x)) \), find the number of solutions to the equation \( f_{2005}(x)=\frac{1}{2}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2005}}
Calculate the upper bound for the relative error in computing \((a+b)^2\) using floating point arithmetic, where \(a\) and \(b\) are arbitrary real numbers and \(\epsilon\) is the machine error.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\epsilon}
Determine the cardinality of the set of all power series with real coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
What is the probability that a randomly selected integer is greater than a given integer $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
A doctor gives a scientist a pack of 10 tablets for his absent-mindedness. Each time the scientist experiences absent-mindedness, he takes a tablet. If he sees only one tablet left, he orders a new pack. If the pack is empty, he discards it. What is the probability that at 10 a.m. on December 31st, the scientist will have exactly two packs of tablets in his cabinet?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{10}}
Solve the inequality for $x > 0$: $$ \Gamma(x) \leq \frac{1}{x} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1]}
How many ways are there to partition the set \([8]\) into 3 disjoint pairs? For example, one such partition is \((12)(45)(67)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
In a triangle \(\Delta ABC\) inscribed in a circle of radius one unit, the internal angle bisectors of angles \(\angle A, \angle B, \angle C\) meet the circle at the points \(A_1, B_1, C_1\) respectively. Find the value of the expression:
\[ S = \frac{\sum AA_1 \cos\left(\frac{A}{2}\right)}{\sum \sin A} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the standard wave equation for small amplitudes, determine the position of a string $y(x,t)$ with the initial conditions $y(x,0) = \sin(x)$ and $\frac{\partial y}{\partial t}(x,0) = \cos(x)$. Express your solution in terms of $x$ and $t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sin(x + t)}
Evaluate the integral of $z^{-1}$ over the contour defined by the square with vertices at $(1+i), (1-i), (-1-i), (-1+i)$ using Cauchy's Integral Formula and the calculus of residues.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Find the radius of convergence for the series \( \sum_{n\geq 1}\left(\frac{x}{\sin n}\right)^{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the matrix $A = I - uv^T$, determine the eigenvalues of $A^2$ in terms of the eigenvalues of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda^2}
Two particles start at the $x$-intercepts of a circle with a circumference of $12$ units, centered at the origin. Each second, both particles move independently one unit either clockwise or counterclockwise with equal probability. What is the expected number of seconds until the two particles simultaneously occupy the same point on the circle for the first time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}