inclusionAI/Ring-lite-rl-data huggingface dataset:
In the triangle \(ABC\), \(AB = AC\), \(\angle A = 90^\circ\). \(D\) is the midpoint of \(BC\), \(E\) is the midpoint of \(AC\), and \(F\) is a point on \(AB\) such that \(BE\) intersects \(CF\) at \(P\). Points \(B\), \(D\), \(P\), and \(F\) lie on a circle. Let \(AD\) intersect \(CP\) at \(H\). Given that \(AP = \sqrt{5} + 2\), find the length of \(PH\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the cardinality of the set of all subsets \( A \subset \mathbb{R} \) such that \( \mathbb{R} \setminus A \) is countably infinite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Calculate the integral \( \frac{1}{2\pi i}\oint_{|z|=1}\frac{2(1-\cos(z))e^z}{z^4}dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If every $k-$element subset of $S=\{1,2,\dots , 32\}$ contains three different elements $a,b,c$ such that $a$ divides $b$, and $b$ divides $c$, $k$ must be at least ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
25}
Given the hyperbola $C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ (where $a > 0$ and $b > 0$), with its left and right foci being $F_1$ and $F_2$, respectively. Suppose that there exists a point $P$ on the hyperbola $C$ such that the triangle $PF_1F_2$ is isosceles with $\cos\angle PF_1F_2 = \frac{1}{8}$. Find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{3}}
Let $S$ be the set of all polynomials $Q(x,y,z)$ with coefficients in \{0,1\} such that there exists a homogeneous polynomial $P(x,y,z)$ of degree 2016 with integer coefficients and a polynomial $R(x,y,z)$ with integer coefficients so that 
\[P(x,y,z) Q(x,y,z) = P(yz,zx,xy) + 2R(x,y,z)\]
and $P(1,1,1)$ is odd. Determine the size of $S$.

Note: A homogeneous polynomial of degree $d$ consists solely of terms of degree $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{509545}
Evaluate the integral \( \frac{1}{2\pi}\int\limits_{-\infty}^{0}\int\limits_{x}^{\infty}e^{-\frac{(x^2+y^2)}{2}}dydx \) without using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. The volume of the set of points swept out by the larger log as it rolls over the smaller one can be expressed as $n \pi$, where $n$ is an integer. Find $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{250}
Example 1 A city has $n$ high schools, the $i$-th high school sends $c_{i}$ students $\left(1 \leqslant c_{i} \leqslant 39\right)$ to watch a ball game at the gymnasium, where $\sum_{i=1}^{n} c_{i}=1990$. Each row in the stand has 199 seats, and it is required that students from the same school sit in the same row. Question: What is the minimum number of rows that need to be arranged so that all students can definitely be seated?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given sets A = {x ∈ ℕ | -1 < x < 3} and B = {x | -2 ≤ x < 2}, find the intersection of sets A and B. Express your answer as a set, including all elements of the intersection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{0, 1}}
Given $x^3y = k$ for a positive constant $k$, find the percentage decrease in $y$ when $x$ increases by $20\%$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42.13\%}
Given points $P_{1}(2,-1)$, $P_{2}(0,5)$, and point $P$ is on the extension line of $P_{1}P_{2}$, with $| \overrightarrow{P_{1}P}|=2| \overrightarrow{PP_{2}}|$, find the coordinates of point $P$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2,11)}
Determine the number of ways to select a positive number of squares on an \(8 \times 8\) chessboard such that no two lie in the same row or the same column and no chosen square lies to the left of and below another chosen square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12869}
Find the mass percentage of H in Ammonia
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17.75\%}
A cylinder with a volume of 9 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 63. Find the volume of the original cone.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Determine the number of solutions to the equation \[{(\sqrt{3}\sin x+\cos x)}^{\sqrt{\sqrt{3}\sin{2x}-\cos{2x}+2}}=4.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the family of holomorphic functions $\mathcal F = \{f$ holomorphic in $\Bbb D$ with $f(\Bbb D)\subset\Bbb D,$ $f\left(\frac12\right)=f'\left(\frac12\right)=0\}$, where $\Bbb D$ is the unit disc. Compute $\sup_{f\in\mathcal F}|f(0)|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/4}
Johnny TV makes 25 percent more movies than L&J Productions each year, with a 5 percent annual growth rate in movie production. If L&J Productions produces 220 movies in the first year with a 3 percent annual growth rate, how many movies do the two production companies produce combined in five years?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2688}
Determine the number of natural numbers \( n \) such that \( 0 \leq n \leq 10,000,000 \) and the sum of the digits of \( n \) is at most 34.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6505246}
Find the counting formula for the number of n-covers of an n-element set, denoted as \( C_{n, n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2^n - 1)^n}
In $\triangle ABC$, $3\sin A + 4\cos B = 6$, $4\sin B + 3\cos A = 1$, find the size of angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
Find the smallest three-digit divisor of the number \(1\underbrace{00\ldots 0}_{100\text{ zeros}}1\underbrace{00\ldots 0}_{100\text{ zeros}}1.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{111}
Given $4^{a}=9^{b}=k$ and $\frac{1}{a} + \frac{1}{b} = 2$, find the value of $k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
In the "Joy Tonight" program, a TV station took out two mailboxes, which contained letters from audiences who performed well in two consecutive quizzes. There are 30 letters in mailbox A and 20 letters in mailbox B. Now, the host draws lots to determine the lucky audience. If one lucky star is determined first, and then one lucky partner is determined from each mailbox, how many different outcomes are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28800}
The polynomial $$P(x)=(1+x+x^2+\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\left[\cos(2\pi\alpha_k)
+i\sin(2\pi\alpha_k)\right]$, $k=1,2,3,\ldots,34$, with $0<\alpha_1\le\alpha_2\le\alpha_3\le\dots\le\alpha_{34}<1$ and $r_k>0$. Find $\alpha_1+\alpha_2+\alpha_3+\alpha_4+\alpha_5.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{159}{323}}
Given a regular pentagon with side length 1, draw the two diagonals from one vertex. This creates a net for a tetrahedron with a regular triangular base. What is the volume of this tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1 + \sqrt{5}}{24}}
How many triples of integers $(a, b, c)$ with $-10 \leq a, b, c \leq 10$ satisfy $a^2 + b^2 + c^2 = (a + b + c)^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{103}
Consider the Hopf bundle \( \mathbb{S}^3 \to \mathbb{S}^2 \) and the usual irreducible representations \( \rho_n: U(1) \to \operatorname{Gl}(\mathbb{C}) \) for \( n \in \mathbb{Z} \). Given that the first Chern class of the Hopf bundle is \( c_1(\operatorname{Hopf}) = -1 \), determine the first Chern class \( c_1\left(\operatorname{Hopf} \times_{\rho_n} \mathbb{C}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-n}
Given the polynomial

\[ P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_{1} x + a_{0} \]

where each coefficient \( a_i \) belongs to the interval \([100, 101]\). For what minimum \( n \) can such a polynomial have a real root?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
There is convex $2009$-gon on the plane.

b) Find the greatest number of vertices of $2009$-gon such that among any three of them there is one that is not connected with other two by side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1339}
A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows: 


- It has  $3$  digits.
- It is a palindrome.
- The middle digit is a multiple of  $4$ .
- It is divisible by  $11$ .

What is the sum of all possible numbers of cows that the herder has?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{726}
Given the function $f(x)=2x-\cos x$, the sequence $\{a_n\}$ is an arithmetic sequence with a common difference of $\frac{\pi}{8}$. If $f(a_1)+f(a_2)+f(a_3)+f(a_4)+f(a_5)=5\pi$, find the value of $[f(a_3)]^2-a_1a_5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13\pi^2}{16}}
Determine the leftmost three digits of the number \[1^1 + 2^2 + 3^3 + \ldots + 999^{999} + 1000^{1000}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
In $\triangle ABC$, $AB = BC = 2$, $\angle ABC = 120^\circ$. A point $P$ is outside the plane of $\triangle ABC$, and a point $D$ is on the line segment $AC$, such that $PD = DA$ and $PB = BA$. Find the maximum volume of the tetrahedron $PBCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/2}
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and the area is $S$. Given that $a^2 + b^2 - c^2 = 4\sqrt{3}S$, find the value of $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{6}}
Evaluate the limit: $$\lim_{x \to 0} \frac{(1 + \sin x + \sin^2 x)^{1/x} - (1 + \sin x)^{1/x}}{x}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Right triangle \( XYZ \) has a right angle at \( Y \) with \( XY = 228 \) and \( YZ = 2004 \). Angle \( Y \) is trisected, and the angle trisectors intersect \( XZ \) at \( P \) and \( Q \) such that \( X, P, Q, Z \) lie on \( XZ \) in that order. Find the value of \((PY + YZ)(QY + XY) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1370736}
Two transformations are said to commute if applying the first followed by the second
 gives the same result as applying the second followed by the first. Consider these four
 transformations of the coordinate plane:
 - A translation 2 units to the right
 - A \( {90}^{ \circ  } \) - rotation counterclockwise about the origin.
 - A reflection across the \( x \) -axis, and
 - A dilation centered at the origin with scale factor 2.
 Of the 6 pairs of distinct transformations from this list, how many commute?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3
}
Let \( p, q, r \) be the roots of the polynomial \( x^3 - 8x^2 + 14x - 2 = 0 \). Define \( t = \sqrt{p} + \sqrt{q} + \sqrt{r} \). Find \( t^4 - 16t^2 - 12t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8}
A line passes through the intersection of the medians of triangle \( ABC \) and intersects the sides \( AB \) and \( AC \).

The distances from vertices \( B \) and \( C \) to this line are \( b \) and \( c \) respectively. Find the distance from vertex \( A \) to this line.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b + c}
A robot starts at the origin of the number line and moves along the direction of the number line, following a pattern where it advances 4 steps then retreats 3 steps. The robot moves one step per second, each step being one unit length. Let $x_n$ denote the position of the robot on the number line at second $n$ (for example, $x_4=4$, $x_5=3$, $x_7=1$). What is the value of $x_{2007} - x_{2011}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
The edge of a cube is 7 cm. If the edge length is increased by 15%, find the new surface area of the cube.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{388.815}
For a positive integer $p$, define the positive integer $n$ to be $p$<i>-safe</i> if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$958$}
Determine the set of real numbers \( p \) for which the integral \( \int_{\mathbb{R}^n}|\log|x||^p \, dx \) converges in \( \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
设$a$、$b$、$c$、$a＋b-c$、$a＋c-b$、$b＋c-a$、$a＋b＋c$是7个两两不同的质数，且$a$、$b$、$c$中有两数之和是800．设$d$是这7质数中最大数与最小数之差．求$d$的最大可能值．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1594}
Given that the solution set of the inequality $ax^{2}+bx+c < 0$ is $((-∞,1)∪(3,+∞))$, determine the solution set of the inequality $cx^{2}+bx+a > 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{1}{3}, 1)}
All positive integers whose digits add up to 14 are listed in increasing order: $59, 68, 77, ...$. What is the fifteenth number in that list?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{266}
Let $K$ be a number field where the rational prime $2$ splits as $(2)\mathcal{O}_K = \mathfrak{P}_1\mathfrak{P}_2$. Determine the intersection $K_{\mathfrak{P}_1} \cap K_{\mathfrak{P}_2}$, where $K_{\mathfrak{P}_i}$ denotes the completion of $K$ at the prime $\mathfrak{P}_i$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Q}_2}
$n$ trains circulate in one direction at equal distances along a circular track. Stations $A$, $B$, and $C$ are located at the vertices of an equilateral triangle along the track, labeled in the direction of movement. Ira enters at station $A$, and Lexa enters at station $B$ at the same time to board the nearest trains. It is known that if they enter the stations when the driver Roma is passing through the forest, Ira will board the train before Lexa. In all other cases, Lexa will board before Ira or simultaneously with her. Determine the portion of the track that passes through the forest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
On the standard $12$ hour clock, each number from $1$ to $12$ is replaced by the sum of its divisors. On this new clock, what is the number of degrees in the measure of the non-reflex angle between the hands of the clock at the time when the hour hand is between $7$ and $6$ while the minute hand is pointing at $15$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Find the minimum value of the function \( f = \sin\theta_1 + \sin\theta_2 \), given that \( \theta_1 + \theta_2 + \phi = \pi \), where \( \phi \) is a variable in the range \((0, \pi)\). Here, \( \theta_1, \theta_2, \phi \) represent the interior angles of a triangle formed by connecting a point on an ellipse to its focus, with \( \phi \) being the angle that sees the major axis of the ellipse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( S \) be the surface defined by \( \{(x,y,z) : x^{2} + y^{2} + 2z = 2, z \geq 0 \} \). Given the vector field \( \mathbf{F} = (y, xz, x^{2} + y^{2}) \), find the net flux of \( \mathbf{F} \) through the surface \( S \). Assume \( \mathbf{n} \) is the outward normal vector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Calculate the Laplace Transform of the function \(-e^{-at}u(-t)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{s + a}}
How many mutually non-isomorphic graphs are there with 30 vertices and 3 edges?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the number of distinct integers that can be obtained from the function \( f(x) = [3x] + [4x] + [5x] + [6x] \) for \( 0 \leq x \leq 100 \), where \([x]\) denotes the greatest integer less than or equal to \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1201}
Determine the Euler characteristic of a sphere with three non-intersecting holes cut out.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Determine the dimension of the field of rational functions \( \mathbb{Q}(t) \) as a vector space over the field \( \mathbb{Q}(t^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the smallest integer $n$ such that for all $x \in \left(-\frac{1}{2}, \frac{1}{2}\right)$, the inequality \[ \left|e^x - \sum_{k=0}^n \frac{x^k}{k!}\right| \leq \frac{|e^x|}{10^{16}} \] holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Two players take turns drawing one number randomly from the set \( \{1, 2, 3, 4\} \), with each number having a probability of \( \frac{1}{4} \) of being drawn. The winner is the player who draws a number that makes the sum of all numbers drawn by both players divisible by 3. What is the probability that the first player will be the winner?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{23}}
Let $A$ be a square matrix of order $n$ with real entries such that $A^3 + I = 0$. Determine the value of $\operatorname{trace}(A)$. \(\text{Note: } I \text{ is the identity matrix of the same order as } A.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the asymptotic probability that the random matrix $A(p, n)$ is non-singular as $n$ approaches infinity, given that $p = 1/2$. Provide your answer as a probability expression or value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the condition number of the matrix \(A = \begin{bmatrix} 0 & 0 & -10^4 & 0 \\ 0 & 0 & 0 & -10 \\ 0 & 10^{-3} & 0 & 0 \\ 10^{-2} & 0 & 0 & 0 \\ \end{bmatrix}\) by determining \(\|A\|\), \(\|A^{-1}\|\), and their product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^7}
Pentagon \(ABCDE\) is inscribed in a circle. The distances from point \(A\) to the lines \(BC\), \(CD\), and \(DE\) are \(a\), \(b\), and \(c\) respectively. Calculate the distance from vertex \(A\) to the line \(BE\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{ac}{b}}
Determine the values of \( x > 0 \) for which the series \( \sum_{n=1}^\infty \frac{\ln(1+nx)}{nx^n} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, \infty)}
Calculate the expected length of runs in a sequence of 52 binary digits (0s and 1s) that contains an equal number of 0s and 1s.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{52}{27}}
Calculate the volume of the solid enclosed by the surface \((x^2 + y^2 + z^2)^2 = x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Find the values of \(\alpha\) for which the series \(\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k}}{(2k)!^{\alpha}}\) converges for all real numbers \(x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 0}
Evaluate the double integral \( \iint \frac{x^2}{y^3} \, dA \) over the region enclosed by the curves \( y = 2x \), \( y = x \), \( y = x^2 \), and \( y = 2x^2 \) using a change of variables.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
In a circular arrangement, particles are initially placed in the sequence 0, 1, -0, and -1. Each second, a new particle is randomly added to the sequence, chosen equally among the four types: 0, 1, -0, and -1. If a 0 particle neighbors a -0 or a 1 particle neighbors a -1, both particles disappear. What is the probability that eventually the entire sequence will vanish?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the intervals where the Fourier series of the function \(f(x) = \sin(x)\) for \(-2 < x \leq 2\) converges uniformly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 2)}
Determine the points in the complex plane where the function \( f(z) = e^z(z-\bar{z})^2 \) is differentiable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{z \in \mathbb{R}}
Evaluate the integral \( \int_a^\infty \frac{1}{x} e^{-x} \,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{E_1(a)}
Find the maximum value of \( u_x(0) \) for a harmonic function \( u: D \rightarrow [0,1] \), where \( D \subset \mathbb{R}^2 \) is the open unit disk centered at 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number\[ 1990^{1991^{1992}} + 1992^{1991^{1990}}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1991}
Determine the integer \( D \) such that the quadratic subfield of \( \mathbb{Q}(\zeta_7) \), where \( \zeta_7 = e^{i2\pi/7} \) is a 7th root of unity, can be expressed as \( \mathbb{Q}(\sqrt{D}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-7}
Consider all labeled trees with nodes \(\{1, 2, 3, \ldots, n\}\). How many of these trees contain the edge \(\{1, 2\}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n^{n-3}}
Given a random sample \(X_1, X_2, \dots, X_n\) from a distribution with probability density function (p.d.f.) \[f(x, \theta) = e^{-(x - \theta)}, \quad \theta < x < \infty, \quad -\infty < \theta < \infty,\] find the sufficient statistic for the parameter \(\theta\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{X_{(1)}}
Determine the minimum order \( n \) of an \( n \)-th order homogeneous differential equation with constant coefficients that has the particular solutions \( t^3 \) and \( t(e^t)(\sin(t)) \). What is the minimum value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the vectors $x_1=(1,-1,1,0)^T$, $x_2=(1,1,0,1)^T$, $x_3=(1,3,1,0)^T$, and $x_4=(1,1,1,0)^T$, determine the rank of the matrix $A = \sum_{i=1}^4 x_ix_i^T$ without explicitly computing $A$. Note that $x_1 = 2x_4 - x_3$. What is the rank of $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}