inclusionAI/Ring-lite-rl-data huggingface dataset:
There are 7 dwarfs, D1 through D7, and 7 tasks, T1 through T7, in a mine. Each dwarf can perform exactly one task. However, D1 cannot perform T1, D4 cannot perform T2, D5 cannot perform T6, and D6 cannot perform T3 or T7. Determine the number of ways to assign the tasks to the dwarfs under these constraints.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2364}
Consider a Brownian motion $B_t$ starting at $0$. Compute the probability $P(T_1 < T_{-1} < T_2)$, where $T_a = \inf\{t \geq 0 \mid B_t = a\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Solve the equation \( x = e^{\frac{1}{x}} \) for \( x \) without using the Lambert \( W \)-Function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.763}
A stock trader has 100 shares of a company and uses a random strategy for buying or selling shares based on the outcome of a die roll at each trading session. The strategy is as follows:

- Outcome = 1: Do nothing
- Outcome = 2 or 4: Buy one share
- Outcome = 3 or 5: Sell one share
- Outcome = 6: Buy two shares

Calculate the approximate probability that after 18 independent trading sessions, the trader will have more than 110 shares of the company.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.169}
Evaluate the limit: \[ l = \lim_{t \to 1^{-}} (1-t) \sum_{r=1}^\infty \frac{t^r}{1+t^r} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\ln 2}
Consider a sequence $X_1, X_2, \dots$ of independent exponential random variables, each with a mean of 1. Define $N$ as the minimum value of $n$ such that $\sum_{i=1}^n X_i > k$, where $k$ is a positive real number. Compute the expected value of $N$, $E[N]$, using Wald's equation $E[\sum_{i=1}^N X_i]=E[N]E[X]$. What is $E[N]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k + 1}
Evaluate the line integral \( \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} \) where \( S \) is the region of the plane bounded by the graph of \( x^2-y^2=4 \) and the lines \( y=2 \) and \( y=-2 \), and \( \mathbf{F}(x, y) = \left(-\frac{y}{x^2 + y^2}, \frac{x}{x^2 + y^2}\right) \). Use Green's theorem to simplify the integral.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Let \( \psi \) be a 7th primitive root of unity. Determine the number of elements in the set \( \{a = a_1\psi + a_2\psi^2 + a_3\psi^3 + a_4\psi^4 + a_5\psi^5 + a_6\psi^6 : a_i \in (0,1)\} \) such that \( \mathbb{Q}(a) = \mathbb{Q}(\psi) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
Find the number of elements of order 6 in the automorphism group of the cyclic group \( \mathbb{Z}_{720} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the exponential generating function for the number of different ways to seat \( n \) people around a round table, denoted as \( r(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln(1 - x)}
Price of the book increased by \( {20}\% \) , and then decreased by \( {10}\% \) . How many percents should we decrease current price so we
 get a price which is \( {54}\% \) percent of an original one?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50%}
Given a floating-point number $x$ with $e$ bits for the exponent and $m$ bits for the mantissa, and an elementary function $f$ with domain and range in the interval $[1,2)$, consider an algorithm $\psi$ that approximates $f$ in the same floating-point system. Assuming that $f(x)$ is not a floating-point number, determine the minimum accuracy achievable with $\psi$. Express your answer in terms of $e$ and $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{-(m+1)}}
Given a sample $X$ of one hundred observations $X_i$ such that $\forall i: X_i \in \mathbb{N}[-101, 101]$, with unique minimum $A$ and unique maximum $B$, the mean of the sample is $M = \bar{X}$. After negating $A$ and $B$, the mean of the modified sample becomes $-M$. What is the maximum possible value of $M$ in the original sample?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
求 $a+b+c+d$ such 的所有可能值之和，这些 $(a, b, c, d)$ 值是满足  $a \cdot b \cdot c \cdot d = 4792$  的（不一定不同的）素数的四元组。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
605}
Given two non-zero square matrices \( A \) and \( B \) such that \( A^2 B = BA \), find the value of \( k \) if \( (AB)^{10} = A^k B^{10} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1023}
Find the coefficient of \(x^{46}\) in the power series 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}
Let \(S\) be the set of ordered triples \((x,y,z)\) of real numbers for which
\[ \log_{10} (x + y) = z \text{ and } \log_{10} (x^2 + y^2) = z + 1. \]
There are real numbers \(a\) and \(b\) such that for all ordered triples \((x,y,z)\) in \(S\) we have 
\[ x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z}. \]
The original answer is in \(\frac{m}{n}\) format. Please find the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Let $p=2017$ be a prime. Find the remainder when\[\left\lfloor\dfrac{1^p}p\right\rfloor + \left\lfloor\dfrac{2^p}p\right\rfloor+\left\lfloor\dfrac{3^p}p\right\rfloor+\cdots+\left\lfloor\dfrac{2015^p}p\right\rfloor \]is divided by $p$. Here $\lfloor\cdot\rfloor$ denotes the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1010}
$\text{Define an ordered quadruple(a,b,c,d)as interesting if }1\leq a<b<c<d\leq10,\text{and a+d-b+c. How many ordered quadruples are there?}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80
}
Evaluate the integral \( I = \int_{|z-a|=r} \frac{e^{2z}}{(z-a)^3} \, dz \) using Cauchy's Integral Formula, given \( r > 0 \) and \( a \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi i e^{2a}}
Find the directional derivative of the function \( f(x, y) = \begin{cases} \frac{xy^2}{x^2+y^2} & \text{if $(x, y) \neq (0, 0)$} \\ 0 & \text{if $(x, y) = (0, 0)$} \end{cases} \) at the point \((0, 0)\) in the direction of the vector \(<2, 3>\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{18\sqrt{13}}{169}}
A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, we lift $P$ off the grid and then we place it back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of moves, we can free up all cells initially occupied by $P$.
How many caterpillars of area $n=10^{6}+1$ are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2
}
Determine the degree of the canonical divisor of the curve $C = \mathbb{P}^2$ using the differential form $\omega = dx \wedge dy$ on $\mathbb{C}^2$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Alice, Bob, Chad, and Denise decide to meet for a virtual group project between 1 and 3 PM, but they don't decide what time. Each of the four group members sign on to Zoom at a uniformly random time between 1 and 2 PM, and they stay for 1 hour. The group gets work done whenever at least three members are present. What is the expected number of minutes that the group gets work done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
48 
}
Evaluate the value of the following double summation: \[ \sum_{r=0}^{2019} \sum_{k=0}^{r} (-1)^k (k+1)(k+2) \binom{2021}{r-k} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2019}}
Find all $t\\in \\mathbb R$, such that $\\int_{0}^{\\frac{\\pi}{2}}\\mid \\sin x+t\\cos x\\mid dx=1$ 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
t=0}
A trough is 3 feet long and 1 foot high, with a vertical cross-section parallel to an end shaped like the graph of $x^2$ from -1 to 1. The trough is full of water. Calculate the amount of work required to pump all the water over the top of the trough. Assume the weight of water is 62 pounds per cubic foot.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{496}{5}}
In a regular triangular pyramid, the diameter of the circumscribed sphere is 9 times the height of the pyramid. Determine the angle between two of its lateral faces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
Given the linear transformation $T:V\to V$ defined by $T(X) = AX + XA$, where $A = \begin{pmatrix}1&1&0\\0&2&0\\0&0&-1 \end{pmatrix}$, find the determinant $\det(T)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Sally has five red cards numbered1through5and four blue
 cards numbered 3 through 6 . She stacks the cards so that the
 colors alternate and so that the number on each red card
 divides evenly into the number on each neighboring blue carc
 What is the sum of the numbers on the middle three cards?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Evaluate the limit:
\[ \lim_{x\to 0}\frac{\int_0^{x^2}{\frac{1-\cos{t^2}+at^4}{t}}dt}{(1-\cos{(\frac{x}{3})})^4} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13122(1 + 2a)}
Find all non negative integer pairs $(a,b)$ such that $3^a5^b-2024$ is a square of a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
(a, b) = (4, 2)}
Evaluate the limit of the series: $$\lim_{n \to \infty} \sum_{k=0}^{\infty} \frac{(2\pi)^{2k}(-1)^k (n+1)!}{(n+2k+1)!}$$ where $n$ is an integer greater than or equal to 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For each permutation \( \left( {{a}_{1},{a}_{2},\ldots ,{a}_{11}}\right) \) of the numbers1,2,3,4,5,6,7,8,9,10,11,we can determine at least \( k \) of \( {a}_{i} \) s when
 we get \( \left( {{a}_{1} + {a}_{3},{a}_{2} + {a}_{4},{a}_{3} + {a}_{5},\ldots ,{a}_{8} + {a}_{10},{a}_{9} + {a}_{11}}\right) \) . \( k \) can be at most ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5
}
Evaluate the Stieltjes integral: $$\int_{-\pi}^{\pi} (x+2) \, d(x \cdot \text{sign}(\sin x))$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi^2}
Evaluate the integral \( I = \int_{0}^{\infty} \frac{\sin\left(x\right)\sin\left(\frac{x}{3}\right)}{x\left(\frac{x}{3}\right)}\,dx \) using Laplace Transforms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
Given a circle of radius 2, there are many line segments of length 2 that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{π
}
Evaluate the contour integral \( \oint_{|z|=1} \sqrt{z} \, dz \) around the unit circle, where \( z = e^{2\pi it} \) for \( t \) from 0 to 1. Assume a branch cut along the positive real axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{4}{3}}
Evaluate the improper integral \( \int_{0}^{\infty}{\frac{\cos 2x - \cos x}{x^2}dx} \) using the method of indented contour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\pi}{2}}
Find the volume bounded by the surface defined by the parametric equations: $$(a(t-\sin(t)),a(1-\cos(t))\cos(\theta),a(1-\cos(t))\sin(\theta))$$ where both $\theta$ and $t$ vary between $[0,2\pi]$. Express your answer in terms of $a$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\pi^2 a^3}
Evaluate the integral: \[ \int_0^\pi \cos(a \sin(x)) e^{a \cos(x)} \, dx \] where \( a \neq 0 \) and \( a \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Evaluate the contour integral \( \frac{1}{2\pi i}\oint_{|z|=1}\frac{e^{\frac{c(z-z^{-1})}{2}}}{z^{n+1}}\,dz \) using the unit circle \( z=e^{i\theta} \), and simplify the expression by exploiting the \( 2\pi \)-periodicity and even-odd symmetries of the integrand.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{J_n(c)}
Find the minimal value of the expression: \[ \log_{\frac{3}{2}x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac{3}{2}x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac{3}{2}x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right) \] where \(x_{1}, x_{2}, \ldots, x_{n}\) are real numbers in the interval \(\left(\frac{1}{4},\frac{2}{3}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
Consider a $9\times 9$ matrix $B$ that satisfies $3B^2-B^4=I$. If $B$ has three distinct eigenvalues, each with geometric multiplicity $2$, determine the number of non-zero entries in the Jordan Canonical Form of $B$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find all functions \( g: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation:

\[ g(x+g(y))+g(2y)=2x+6y \] for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{g(x) = 2x}
Evaluate the definite integral: $$\int_{1}^{3} \frac{2x^3}{1+e^{-x}}\mathrm dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Given in the polar coordinate system, point P moves on the curve $\rho^2\cos\theta-2\rho=0$, the minimum distance from point P to point $Q(1, \frac{\pi}{3})$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{3}{2}"]}
Find the value of the limit: \[ \lim_{n\to\infty} \frac{a_n}{b_n} \] where \( a_n = \sum_{m=1}^{\lfloor n/2 \rfloor} \frac {n!}{2^m(n-2m)!} \) is the number of elements of order 2 in the symmetric group \( S_n \), and \( b_n = \sum_{m=1}^{\lfloor n/4 \rfloor} \frac {n!}{2^{2m}(n-4m)!} \) is the number of elements of order 2 in the alternating group \( A_n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of invertible elements in the quotient ring \(\mathbb{F}_5 [x]/\langle x^4 + 2x^2 \rangle\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{480}
Determine all positive integers \( p \) so that \( \frac{{3p} + {25}}{{2p} - 5} \) is also a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$p \in \{3, 5, 9, 35\}$}
Find the remainder when $35!$ is divided by $1457$ using Wilson's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{248}
Determine the number of integer pairs \((x, y)\) that satisfy the equation \(x^{2} + 6xy + 5y^{2} = 10^{100}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19594}
Let $\mathbb{X} = C^1([0,1], \mathbb{R})$ with the norm $\|f\|_{\mathbb{X}} = \|f\|_{\infty} + \|f'\|_{\infty}$. Define the operator $A: \mathbb{X} \rightarrow \mathbb{R}$ by $Af = f'\left(\frac{1}{2}\right)$. Find the operator norm of $A$. \[ \|A\|_{op} = \inf\{c \geq 0: \|Af\| \leq c \|f\|_{\mathbb{X}} \} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X$, $Y$, and $Z$ the midpoints of sides $\overline{AB},\overline{CD},\overline{EF}$, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle{ACE}$ and $\triangle{XYZ}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \dfrac{15}{32}\sqrt{3} \]}
Evaluate the surface integral \( \iint (xz\cos\alpha + yz\cos\beta + \cos\gamma) \, dS \) over the spherical cap \( S_1 \) formed by the intersection of the sphere \( x^2 + y^2 + z^2 = 25 \) with the plane \( z = 3 \). The unit normal is denoted by \( n = (\cos\alpha, \cos\beta, \cos\gamma) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144\pi}
Calculate the surface integral of the curl of the vector field \( F = \langle 2y, xz, 1 \rangle \) over the surface defined by \( z = f(x,y) = 3 - 2x - \frac{3}{2}y \). Use a normal vector obtained from the cross product of two vectors on the surface.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Evaluate the limit: \[ \lim_{n\to +\infty}\frac{1}{n^2}\sum_{i=1}^{n}\log\binom{n}{i}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the kernel of the ring homomorphism \( g: k[x_1,x_2] \to k[u,v] \) defined by \( x_1 \mapsto u^2+v^2 \) and \( x_2 \mapsto uv \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0)}
A line passes through the point (1, 2). The abscissa of the intersection of the tangent to this line with the x-axis is half the abscissa of the intersection of the tangent with the line itself. Determine the abscissa of the intersection point of the tangent with the x-axis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the number of group homomorphisms from $\Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z$ to $S_{3}$, where $S_{3}$ is the symmetric group of permutations of three elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the integral \( \int_{0}^{\infty} \frac{\cos x - e^{-x^2}}{x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{\gamma}{2}}
Find a primitive root of 71.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In a doctor's clinic, patients arrive at an average rate of 10 per hour. The consultation time per patient is exponentially distributed with an average of 6 minutes per patient. The doctor does not admit any patient if at any time 10 patients are waiting. Calculate the expected number of patients waiting in the steady state of this M/M/1/R queue.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{55}{12}}
Determine the number of roots of the equation \(2\cos^{2} \theta = 1 + 3^{\sin \theta} + 3^{\csc \theta}\) within the interval \([0, 2\pi]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
How many labeled trees with 10 vertices have exactly one vertex of degree 4?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29841840}
Consider a Markov chain on the state space \( S = \{1, 2, \ldots, 21\} \) with the given transition matrix. The initial distribution is \( P(X(0) = 11) = 1 \). State 20 is in a closed recurrent class, while state 11 is in a transient class. Determine \( \lim_{n \to \infty} P(X(n) = 20) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $A = \left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ be a set of numbers, and let the arithmetic mean of all elements in $A$ be denoted by $P(A)\left(P(A)=\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right)$. If $B$ is a non-empty subset of $A$ such that $P(B) = P(A)$, then $B$ is called a "balance subset" of $A$. Find the number of "balance subsets" of the set $M = \{1,2,3,4,5,6,7,8,9\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
For a standard Brownian motion $W_t$, compute the probability $P(T_{1} < T_{-3} < T_3)$, where $T_a$ denotes the hitting time of level $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Determine the number of elements in the conjugacy class of the permutation \( \tau = (56789)(3456)(234)(12) \) in the symmetric group \( S_9 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15120}
Evaluate the limit: \( \displaystyle\lim_{z\to3i}\frac{-3z}{z^2-9-6iz} \). Determine whether the limit is \( \infty \) or does not exist.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Johnny buys a pack of gum each day, aiming to collect all 6 different prizes, with each pack having an equal probability of containing any of the prizes. Calculate the standard deviation of the number of days Johnny needs to buy gum packs to collect all the different prizes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{3899}}{10}}
Evaluate the limit: $$\lim_{x\to 0}\frac{\sin x^4-x^4\cos x^4+x^{20}}{x^4(e^{2x^4}-1-2x^4)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Given a set of linearly independent vectors $\{v_1, v_2, \ldots, v_{2014}\}$, find the scalar $\lambda \in \mathbb{R}$ such that the vectors $\{v_1+v_2, v_2+v_3, \ldots, v_{2014}+\lambda v_1\}$ are also linearly independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For two independent Brownian motions \( W_t \) and \( \tilde{W}_t \), and a constant \( \rho \) such that \( -1 \leq \rho \leq 1 \), define \( X_t = \rho W_t + \sqrt{1-\rho^2} \tilde{W}_t \). Find the conditions on \( \rho \) for which \( X_t \) is a martingale.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1 \leq \rho \leq 1}
Find the norm formula for the ring of Kleinian integers, \( \mathbb{Z}[\frac{1+\sqrt{-7}}{2}] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^2 + ab + 2b^2}
Find an equivalent expression for the remainder of the series \( \sum_{n=1}^{\infty} \frac{1}{n^3} \) given by \( R_n = \sum_{k=n+1}^{\infty} \frac{1}{k^3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2n^2}}
Evaluate the limit: \[ \lim_{x\to1^{-}} (1-x)\left(\sum_{i=0}^{\infty} x^{i^2}\right)^{2} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}}
Evaluate the limit: \[ \lim _{x\to \infty }\left(x\sqrt{x^2-x}-x^2\cos\left(\frac{1}{\sqrt{x}}\right)\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{6}}
Let $n$ be a positive integer. Consider a random ordered triple $(a, b, c)$ of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples. Let $M_n$ be the expected value of the largest of $a$, $b$, and $c$. Determine the value that $\frac{M_n}{n}$ approaches as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{11}{18}}
Evaluate the line integral \( \int_{C} F \cdot \, dr \) where \( C \) is the curve consisting of line segments from \((4,1)\) to \((4,3)\) to \((1,3)\) to \((1,1)\), and \( F(x,y) = (x+y)i + (y-1)^3 e^{\sin(y)}j \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{33}{2}}
Find the value of \( r \) for which a non-trivial fixed point appears in the equation \( x = r\sin\pi x \), given that \( 0 \leq x \leq 1 \) and \( 0 \leq r \leq 1 \). The trivial fixed point is \( x = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{\pi}}