inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the joint probability mass function $p(x,y)$ for $x \in \{0, 1\}$ and $y \in \{0,1,2\}$, where $3p(1,1) = p(1,2)$ and $p(1,1)$ maximizes the variance of $XY$, calculate the probability that $X$ or $Y$ is zero.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{23}{49}}
In a Gaussian random walk, where $x_n$ is the sum of $n$ normal random variables, calculate the probability $P(x_1 > 0, x_2 > 0)$. Assume that the events $x_1 > 0$ and $x_2 > 0$ are not independent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Calculate the variance of $12(X_1 + X_2 + X_3)$, where $X_1$ is uniformly distributed on $(0,1)$, $X_2$ is uniformly distributed on $(0, X_1)$, and $X_3$ is uniformly distributed on $(0, X_2)$. You do not need to determine the distribution of $X_1 + X_2 + X_3$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{565}{12}}
There are 4 different types of coupons, divided into two groups: the first group consists of types 1 and 2, and the second group consists of types 3 and 4. Each new coupon obtained is type $i$ with probability $p_i$, where $p_1 = p_2 = 1/8$ and $p_3 = p_4 = 3/8$. Find the expected number of coupons that one must obtain to have at least all the types of either group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{123}{35}}
Calculate the surface area of the portion of the surface defined by \( z^2 = 2xy \) within the region \( 0 \le x \le 2 \) and \( 0 \le y \le 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given a figure composed of $3n^{2}$ rhombuses, determine the minimum number of rearrangement operations required to transform it into a specified target figure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
In a game involving a positive integer \( n \geqslant 2 \), Player A selects \( 2n \) non-negative numbers \( x_1, x_2, \ldots, x_{2n} \) such that their sum is 1. Player B arranges these numbers in a circle and calculates the product of each pair of adjacent numbers, writing the largest product on the board. Determine the number that will be written on the board if both players play optimally.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4n^2}}
Find all functions $f: \Bbb{R} \to \Bbb{R}$ that satisfy the functional equation $f(a^2 + b^2) = f(a^2 - b^2) + f(2ab)$ for all real numbers $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = kx^2}
How many even 11-digit numbers, with no leading zeros, contain at least three occurrences of the digit 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3244680630}
Determine the number of subgroups of $\mathbb{Z}_{p^2} \oplus \mathbb{Z}_{p^2}$ that are isomorphic to $\mathbb{Z}_{p^2}$, where $p$ is a prime.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p(p + 1)}
Determine the number of unique ways to distribute 5 blue, 4 green, 2 yellow, 1 red, and 1 black ball into 3 buckets such that each bucket has a distinct combination of balls and all balls must be placed in a bucket. An empty bucket is considered a distinct combination.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16956}
Given a sequence of independent identically distributed (i.i.d.) random variables $X_1, X_2, \cdots$, each following a normal distribution with mean 1 and variance 1, and a Poisson random variable $N$ with mean 2, independent of the sequence $X_i$'s, find the variance of the sum $X_1 + X_2 + \cdots + X_{N+1}$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the minimum distance of the ternary linear code with the following parity check matrix:
\[
H=\begin{pmatrix}
2 & 0 & 1 & 1 & 1 \\
0 & 2 & 2 & 0 & 0 \\
0 & 0 & 2 & 2 & 1 \\
1 & 1 & 2 & 2 & 0
\end{pmatrix}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the constant $c$ such that the solution to the differential equation $$(1-2x)y' = y$$ with boundary conditions $y(0) = -1$ and $y(1) = 1$ is real and finite for all $0 \leq x \leq 1$. The solution is given by $\ln y = \ln\left(\frac{2}{1-2x}\right) + c$. Find the value of $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\ln 2}
In a $11\times11$ grid, calculate the number of non-decreasing grid walks from $(0,0)$ to $(10,10)$ that do not pass through the points $(3,3)$, $(7,2)$, $(3,7)$, or $(7,7)$. The total number of unrestricted paths is $\binom{20}{10}$. Determine the number of valid paths by subtracting the number of paths that pass through any of the forbidden points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60256}
Calculate the probability of obtaining a sum of at least 100 when rolling 30 dice, each with 10 sides numbered from 1 to 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.99998}
An urn initially contains 6 black balls and 0 white balls. In each turn, one ball is drawn at random, removed, and replaced with a ball of the opposite color. Calculate the expected number of turns until all 6 balls in the urn are white.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{416}{5}}
Determine the sign of the second-order term in the Taylor expansion of the functional \( I[1+v] \), where \( I(u)=\int_{0}^{1}u^2(2-u)^2dx \) and \( u \in Y \), with \( Y=\{u\in C([0,1]):u(0)=1,u(1)=1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{negative}
How many times changes the sign of the function \[ f(x)=\cos x\cos\frac{x}{2}\cos\frac{x}{3}\cdots\cos\frac{x}{2009}\]at the interval $ \left[0, \frac{2009\pi}{2}\right]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
75}
Let \(n \in \mathbb{Q}\) be a number such that \(\sqrt{n} + \sqrt[3]{n} \in \mathbb{N}\). Determine the form of \(n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m^6}
Find a closed-form expression for the integral \( \int_0^\pi \log(\sin x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi \ln 2}
Consider a system where two repairmen serve three machines, meaning at most two machines can be under repair at the same time. The time until a machine needs repair is exponentially distributed with a mean of \(1/3\), and each repair time is exponentially distributed with a mean of \(1/2\). These times are independent of each other. Let \(X_t\) represent the number of machines under repair or needing repair at time \(t\). Given the transition rate matrix:
\[
Q = \begin{bmatrix}
 -9 &  9  &  0  &  0  \\
  2 & -8  &  6  &  0  \\
  0 &  4  & -7  &  3  \\
  0 &  0  &  2  & -2
\end{bmatrix}
\]
Determine \(\lim_{t \to \infty} \mathbb{P}_i(X_t = 0)\) for \(i = 0, 1, 2, 3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{179}}
Given a right circular cone with an isosceles right triangle as its axial cross-section, where \( P \) is the vertex, \( PA \) is a generatrix, and \( B \) is a point on the base, find the maximum value of \( \frac{PA + AB}{PB} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{4 + 2\sqrt{2}}}
Let $w$ be a positive integer. Determine the range of values for $w$ such that there exist $a, b, c$ with $\pi \le a < b < c \le 2\pi$ satisfying the equation:
\[ \sin(aw) + \sin(bw) + \sin(cw) = 3. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{w \geq 6}
Determine the radius of the sphere that touches the faces of the unit cube passing through vertex $A$ and the edges passing through vertex $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["2 - \\sqrt{2}"]}
Let $ABC$ be a triangle with $AB=17$ and $AC=23$. Let $G$ be the centroid of $ABC$, and let $B_1$ and $C_1$ be on the circumcircle of $ABC$ with $BB_1\parallel AC$ and $CC_1\parallel AB$. Given that $G$ lies on $B_1C_1$, the value of $BC^2$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
81803
}
Compute the expected value of the random variable \(\tau\), which represents the number of draws needed for the sum of values drawn from a uniform distribution on \((0, 1)\) to exceed 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^2 - e}
Determine the values of $p$ for which the $p$-norm on $\mathbb{R}^n$ is differentiable everywhere except at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 1}
Find the solution for $u_1(x, t)$ given the system of partial differential equations with $a = 0$ and the initial conditions $u_1(x, 0) = f(x)$ and $u_2(x, 0) = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{u_1(x, t) = f(x)}
An employer is hiring one new employee from a group of N candidates, each rated on a scale from 1 to N. Candidates are interviewed in random order, and a decision to hire is made immediately after each interview. The employer can hire the mth candidate only if they are better than all previous m-1 candidates. If a candidate is hired on the ith trial, what is the probability that this candidate is the best among all N candidates?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{i}{N}}
Suppose 55% of a large population of voters favor candidate A. Determine the minimum size of a random sample needed to ensure a 99% probability that the majority of voters in the sample will favor candidate A.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{556}
Evaluate the line integral \( \oint_C \vec{F} \cdot d\vec{R} \) using Stokes' theorem, where the vector field is \( \vec{F} = (ye^x, x^2+e^x, z^2e^z) \) and the curve \( C \) is given by \( \vec{r}(t) = (1+\cos t, 1+\sin t, 1-\cos t-\sin t) \) for \( 0 \leq t \leq 2\pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Given the surface parameterized by:
\[ x = u(3v^2 - u^2 - 1/3), \quad y = v(3u^2 - v^2 - 1/3), \quad z = 2uv \ . \]
Find the angle between the coordinate lines at a point on the surface.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{2}}
A cube is sectioned by 33 planes parallel to its faces, forming multiple parallelepipeds. If the number of parallelepipeds with faces coinciding with the faces of the cube is half of the total number of parallelepipeds, find the total number of parallelepipeds formed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1344}
Determine the number of solutions to the equation $e^z - z = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the eigenvalues of the following $4 \times 4$ matrix:
$$J(\epsilon)=\begin{bmatrix} -\mu & 0 & 0 & -\frac{\beta b}{\mu}\\ 0 & -\sigma & 0 & \frac{\beta b}{\mu}\\ 0 & \sigma & -(\phi + d) & 0\\ 0 & 0 & \alpha & -(\phi + \epsilon) \end{bmatrix}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\mu}
Find the equation of the directrix of a parabola with a focus at (-1, -1) and a tangent at the point (7, 13) given by the equation 3x - y - 8 = 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x + 8y + 19 = 0}
Find all values of \( p > 1 \) such that the series \( \sum_{n=1}^{\infty} f_n \) converges in \( L^p(\mathbb{R}) \), where:
\[
f_n(x) = \frac{1}{1+n^2\sqrt{x}}\chi_{[\exp(2n),2\exp(2n+2)]}(x)
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p \geq 2}
Determine the number of inequivalent two-dimensional complex representations of the group $Z_4$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the coefficient of \(x^{12}y^{24}\) in the expansion of \((x^3 + 2xy^2 + y + 3)^{18}\) using the multinomial theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111890128896}
A polynomial \( p(x) \) is called self-centered if it has integer coefficients and satisfies \( p(100) = 100 \). If \( p(x) \) is a self-centered polynomial, determine the maximum number of integer solutions \( k \) to the equation \( p(k) = k^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
In a $5$-regular graph $G$ with $m$ edges and a girth of at least $6$, determine the number of paths of length $5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256m}
Determine the operator norm of the linear operator $B: \ell^2 \to \mathbb{R}$ defined by \[ Bx = \sum\limits_{k=0}^\infty ((1+\sin(\pi(1/2)^{k+1}))\eta_{k+1}). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Let $x_1$ and $x_2$ be positive real numbers and define, for $n>2$: $$x_{n+1} = \sum_{k=1}^{n} \sqrt[n]{x_k}.$$ Evaluate $\lim_{n \to \infty} \frac{x_n - n}{\ln n}$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the triple integral: $$\int_{1}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty} \exp\left(-\frac{x^{2}}{y^{2}}-\frac{y^{2}}{z^{2}}-\left[z-1-\log(z)\right]\right) \,{\rm d}x\,{\rm d}y\,{\rm d}z$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{\pi}}
Find the value of the Lambert W function at the point where the function \(f(x) = e^x\) intersects the line \(y = x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{W(-1)}
A set \( T \) contains natural numbers \( a, b, c, d, e, \) and \( f \) such that the sum of their squares equals \( f^2 \). Another set \( S \) contains all possible values of the product \( a \times b \times c \times d \times e \times f \). Determine the greatest common divisor (GCD) of all elements of set \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Evaluate the infinite series \( S = \sum_{k=1}^\infty \frac{k^{k-1} e^{-k}}{k!} \) and determine its value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of symmetric, positive definite $10 \times 10$ matrices with a trace of 10 and a determinant of 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Suppose $g:\mathbb{D} \to \mathbb{D}$ is an analytic function with $g(0)=0$ and $|g'(0)|=\frac{1}{2}$. Determine the maximum possible value of $|g(\frac{1}{4})|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
In a convex quadrilateral \(ABCD\), points \(E, F, G, H\) are located on sides \(AB, BC, CD, DA\) respectively, such that:
\[ \frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA} = 1 \]

Another convex quadrilateral \(E_1F_1G_1H_1\) has points \(A, B, C, D\) on its sides \(H_1E_1, E_1F_1, F_1G_1, G_1H_1\) respectively, with \(E_1F_1 \parallel EF, F_1G_1 \parallel FG, G_1H_1 \parallel GH, H_1E_1 \parallel HE\). Given that \(\frac{E_1A}{AH_1} = \lambda\), determine the value of \(\frac{F_1C}{CG_1}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lambda}
For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a+\cos \theta)^2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{2 \pi |a|}{(a^2-1)^{3/2}}$.}
Determine the radius of convergence for the power series: $$ \sum_{n = 1}^{\infty} \left(1 + \frac{1}{n^{1/n}} \right)^{n} \sin^{2n}\left(\frac{n \pi}{3}\right) z^{n} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
In a Markov chain on the integers, the stochastic process \(X_n\) starts at \(X_0 = 3\). If \(|X_n|\) is even, the transition probabilities are \(P(X_{n+1} = i + 1|X_n = i) = \frac{3}{4}\) and \(P(X_{n+1} = i - 1|X_n = i) = \frac{1}{4}\). If \(|X_n|\) is odd, the transition probabilities are \(P(X_{n+1} = i + 1|X_n = i) = \frac{1}{4}\) and \(P(X_{n+1} = i - 1|X_n = i) = \frac{3}{4}\). Calculate the probability of reaching state 5 before reaching state 0, starting from state 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{9}}
For lines $l_1: x + ay + 3 = 0$ and $l_2: (a-2)x + 3y + a = 0$ to be parallel, determine the values of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-1"]}
Given the polynomial \((1-z)^{b_{1}}(1-z^{2})^{b_{2}}(1-z^{3})^{b_{3}} \cdots(1-z^{32})^{b_{32}}\), where \( b_{i} \) are positive integers for \( i = 1, 2, \ldots, 32 \), and this polynomial has the property that after expanding it and removing the terms with \( z \) raised to powers higher than 32, what remains is \( 1 - 2z \). Determine \( b_{32} \), and express your answer as the difference of two powers of 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{27} - 2^{11}}
Evaluate the surface integral \( \iint_S (x^2 + y^2 + z^2) \, dS \), where \( S \) is the part of the cone \( z = \sqrt{x^2+y^2} \) above the circle \( x^2 + y^2 \leq 2x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt{2}\pi}
Evaluate the surface integral \(\int_{S}\frac{dxdy}{z}+\frac{dydz}{x}+\frac{dxdz}{y}\) over the surface of the sphere \(S\) with radius \(R\) and outward orientation, without using the Gauss-Ostrogradsky formula.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi R}
Determine the minimum number of terms needed in the series \( \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \) to approximate \( \ln 2 \) within an error of 0.001, using the inequalities:

\[ 1 - \frac{1}{2} + \frac{1}{3} - \ldots + \frac{(-1)^{2k+1}}{2k} \approx \ln 2 \]

and

\[ 1 - \frac{1}{2} + \frac{1}{3} - \ldots + \frac{(-1)^{2k+2}}{2k+1} \leq \ln 2. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Given a function \( f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*} \) such that for any positive integers \( s \) and \( t \), the equation \( f\left(t^{2} f(s)\right) = s(f(t))^{2} \) holds, determine the minimum possible value of \( f(1998) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
In an obtuse triangle $\triangle ABC$, let $I$ be the incenter. Among the triangles $\triangle IAB$, $\triangle IBC$, and $\triangle IAC$, one of them is similar to $\triangle ABC$. Determine the ratio of the angles of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2:4}
Determine all real values of \(\lambda\) such that the inequality \(x + y + z \leq \frac{3}{2}\) holds for all non-negative real numbers \(x, y, z\) satisfying \(x^2 + y^2 + z^2 + 2\lambda xyz = 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, \infty)}
Find the last two digits of the sum \(a_{1}^{8} + a_{2}^{8} + \cdots + a_{100}^{8}\), where \(a_1, a_2, \ldots, a_{100}\) are 100 consecutive natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Evaluate the limit \( \lim_{n \to \infty} J_n \), where \( J_n = \int_{-\infty}^\infty \frac{|\cos \left( \frac{x}{n} \right) \cos(x)|}{\frac{x}{n} x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
An inspector selects every \( n \)-th item in a production line for a detailed inspection. The time between item arrivals follows an exponential distribution with a mean of 1 minute. It takes 2 minutes to inspect an item. Determine the smallest value of \( n \) such that the probability of completing the inspection before the arrival of the next item that requires inspection is at least 90%.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Compute the limit: $$ \lim_{N\rightarrow\infty}N^2\sum^{N-1}_{k=1}\left(\frac{k}{N}\right)^{N\ln N} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the dimension of the vector space of conics in the real projective plane with homogeneous coordinates \(\{x, y, z\}\) that are singular at the points \([1, i, 1]\) and \([i, -1, i]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let \( N = \min\{n \in \mathbb{N}_0: X_1 + \ldots + X_n > T\} \). Calculate \( E[N] \) for \( T > 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{T + 1}
Find the asymptotic upper and lower bounds for the recurrence relation \( T(n) = T(\sqrt{n}) + n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n)}
Calculate \( \lim_{n \to \infty} \int f_n \, d\mu \) where \( S = N, A = P(N), \mu = h \, dv \) with \( v \) being the counting measure, \( h(k) = \frac{1}{4^k} \), and \( f_n(k) = \left(\frac{k}{n} - 2\right)^k 1_{(1,2,3,...,n)}(k) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{3}}
Determine the order of the two-step Adams-Bashforth method given by the formula:
\[
u_{n+1} = u_n + \frac{h}{2}(3f_n - f_{n-1})\] where \(t_n = t_0 + nh\) is the \(n\)-th node, \(y_n = y(t_n)\), \(f_n\) stands for \(f(t_n,y_n)\), and \(u_n\) is an approximation of \(y_n\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the minimal polynomial of \( \alpha = \frac{1-\sqrt{5}}{2} \) over the finite field \( \mathbb{F}_{7} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 - x - 1}
Find the residue of the function \( f(z) = \frac{e^{-\alpha/z}e^{-\alpha z}}{z} \) at infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-I_0(2\alpha)}
Let $(Z_i)_{i \ge 1}$ be a sequence of i.i.d. random variables uniformly distributed on $\{1, -1\}$, with $P(Z_i = 1) = P(Z_i = -1) = \frac{1}{2}$. Define the Markov chain $X_n$ for $n \ge 1$ with $X_0 = 0$ as follows:
\[ X_n = X_{n-1} + Z_n \]
Let $a \in \mathbb{Z}$ be a non-zero integer and define the stopping time $T = \inf\{n \ge 1: X_n = a\}$, which is the first hitting time of $a$. Find $E[X_T]$. \(\text{Express your answer as a number.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a}
In a tetrahedron \(ABCD\) with a volume of 12, points \(E\), \(F\), and \(G\) are located on edges \(AB\), \(BC\), and \(AD\) respectively. It is given that \(AE = 2EB\), \(BF = FC\), and \(AG = 2GD\). A cross-section \(EFHG\) is formed through points \(E\), \(F\), and \(G\). If the perpendicular distance from point \(C\) to this cross-section is 1, find the area of the cross-section \(EFHG\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given a rectangle PQRS in the first quadrant with adjacent sides PQ and QR having slopes 1 and -1, respectively, and the function \( u(x,t) \) satisfying the wave equation \( \frac{\partial^2u}{\partial t^2}-\frac{\partial^2u}{\partial x^2}=0 \), where \( u(P)=1 \), \( u(Q)=1/2 \), and \( u(R)=1/2 \), find \( u(S) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}