inclusionAI/Ring-lite-rl-data huggingface dataset:
Find the integrating factor that makes the differential equation \( y^{2}dy = x\left ( x dy - y dx \right ) e^{\frac{x}{y}} \) exact.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{y^3}}
Find the positive constant \( c \) such that the sum \( \frac{S_n}{n^c} \) of independent and identically distributed random variables \( X_1, X_2, \ldots, X_n \) converges in distribution to a random variable \( A \), where the characteristic function of each \( X_i \) is given by \( \phi(t) = 1 - \sqrt{|t|(2-|t|)} \) for \( t \in [-1, 1] \) and zero elsewhere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given the quadratic equation x^2 - 3x = a^2 and the root x = 4, find the value of the constant a. Express your answer as ±value, where value is a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±2}
Given the force of interest \( F(t) = \frac{0.02 + 0.01t}{1 + 0.02t + 0.005t^2} \), where \( t \) is the number of years starting from March 31, 2001, calculate the accumulated value of two investments: $100 invested on March 31, 2002, and $200 invested on March 31, 2006. Find the total accumulated value of these investments on March 31, 2010.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{413.41}
Consider the operator \( T: L^2([0,1]) \to L^2([0,1]) \) defined by \( Tf(t) = \int_0^t f(s) \, ds \). Find \( \alpha < 1 \) such that \( ||T|| \leq \alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{\pi}}
Determine the number of integer solutions \((x, y, z)\) to the congruence equation \(x^5 + y^5 \equiv z^5 \pmod{25}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1625}
Let \(a, b\) be nonzero complex numbers such that \(\frac{a}{b}\) is not a real number. Define the set:
\[ 
L_{a, b}=\{r a+s b \mid r, s \in \mathbf{Z}\}, 
\]
and the set:
\[ 
R_{a, b}=\left\{z \mid z \text{ is a nonzero complex number, and } L_{a, b} = L_{z a, z}\right\}.
\]
Determine the maximum number of elements in the set \(R_{a, b}\) as \(a, b\) vary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the area enclosed by the curve defined by the equation \( \sin^4(x) + (\cos(y) - 3)^2 - 16 = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let \( \mathbf{G}: \mathbb{R}^2 \to \mathbb{R}^2 \) be defined by \( G(x,y) = (\cos(x-y), \sin(x+y)) \). Assume \( f: \mathbb{R}^2 \to \mathbb{R} \) is differentiable with gradient \( \nabla f(0,1) = (2,3) \). Define \( h: \mathbb{R}^2 \to \mathbb{R} \) by \( h(x) = f(\mathbf{G}(\mathbf{x})) \). Find the gradient \( \nabla h\left( \frac{\pi}{2}, 0 \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-2, 2)}
Determine the order of the element $x + x^2$ in the multiplicative group of the field $\mathbb{F}_2[x]/(x^4 + x^3 + 1)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given a random sample $X_1$ and $X_2$ of size $n=2$ from an exponential population, find the efficiency of the estimator $2Y_1$ relative to the estimator $\bar{X}$, where $Y_1$ is the first order statistic and both $2Y_1$ and $\bar{X}$ are unbiased estimators of the parameter $\theta$. Provide your answer as a numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the smallest positive integer \( n \) such that:
\[ 1,000,000^{{1,000,000}^{1,000,000}} < n^{n^{n^n}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the total charge on the spherical surface defined by \(x^2+y^2+z^2=R^2\) with a charge density given by \(\rho(x,y,z)=|z|\) C/m\(^2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi R^3}
Evaluate the integral \( I = \int_0^{\pi} e^{\cos\theta}\cos(\sin\theta)\ d\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Find the maximum value of the function \( f(x) = 4\sin x + 48\sin x\cos x + 3\cos x + 14\sin^2x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
What maximal number of the men in checkers game can be put on the chess-board \( 8 \times  8 \) so,that every man can be taken by at
 least one other man?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64
}
Find the minimal polynomial of $\sqrt{3}$ over the field $\mathbb{Q}(\sqrt[6]{3})$. Provide your answer as a polynomial with rational coefficients.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 - 3}
Let $P = (0, q)$ be the hyperbolic midpoint of $A = (0, a)$ and $B = (0, a + b)$ in the Hyperbolic/Half-Plane Model. Compute the limit of $q$ as $a \rightarrow \infty$, as a function of the fixed number $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{b}{2}}
Find an integer $n$ such that the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^*$ contains elements of order $4$ and $5$, but no elements of order $3$. What is the value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Evaluate the line integral \( \int_C G(x,y,z) \, ds \) for the curve parameterized by \( x= \frac{1}{3} t^3, \quad y=t^2, \quad z=2t \), where \( t \) ranges from 0 to 1, and \( G(x, y, z) = 4xyz \). Round your answer to the nearest hundredth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.06}
Given a sequence \( a_1, a_2, a_3, \ldots, a_n \) of non-zero integers such that the sum of any 7 consecutive terms is positive and the sum of any 11 consecutive terms is negative, what is the largest possible value for \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Approximate \( \sqrt{8} \) using the Taylor series expansion of \( \sqrt{1+x} \) around zero up to the fourth term. Use the Lagrange Remainder to ensure the approximation is accurate to within \( 10^{-4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.8284}
Determine the largest natural number \( n \) such that the group \( \mathbb{Z}_{180} \times \mathbb{Z}_{30} \) is isomorphic to the direct product of \( n \) cyclic groups.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the ratio of the bases of the trapezoid, given that its midline is divided into three equal parts by the diagonals.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:2}
Count the number of 10-digit numbers using the digits from \( \{1, 2, 3, 4\} \) such that no two adjacent digits differ by 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21892}
## Task Condition

Find the derivative of the specified order.

$$
y=(5 x-1) \ln ^{2} x, y^{\prime \prime \prime}=?
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6-2(5x+2)\lnx}{x^{3}}}
Determine all integer values of \( n \) such that the equation \( 2\sin{nx} = \tan{x} + \cot{x} \) has real solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \equiv 2 \pmod{8}}
An equilateral triangle with a side length of 8 is divided into smaller equilateral triangles with side lengths of 1 (see the figure). What is the minimum number of these smaller triangles that need to be shaded so that every intersection point of the lines (including those on the edges) is a vertex of at least one shaded triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the minimum layer width required for universal approximation in deep feedforward networks with ReLU activation function, given that the input dimension is $\nu$. Provide your answer in terms of $\nu$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\nu + 1}
Determine the number of automorphisms of the symmetric group $S_6$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1440}
Determine the largest value of $C$ such that any shape in the plane with an area less than $C$ can be translated and rotated to avoid touching any point in the integer lattice $\mathbb{Z}^2$. What is this largest value of $C$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the surface integral \( \int_M (\nabla \times F) \cdot N \, dS \) where \( F = (e^x + z^2, \sin(yz), x^3) \) and \( M = \{(x,y,z) : y = \frac{x^2}{2} + z^2 - 4, y \leq 0\} \). The vector \( N \) is the outward-pointing unit normal to the surface \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24\sqrt{2}\pi}
Two circles with radius 1 intersect at points $X$ and $Y$, with a distance of 1 between them. From point $C$ on one circle, tangents $CA$ and $CB$ are drawn to the other circle. Line $CB$ intersects the first circle again at point $A'$. Find the distance $AA'$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Let $V$ be a finite dimensional vector space over the finite field $\mathbb{F}_q$ with $q$ elements, and let $U$ be a subspace of $V$. Determine the number of subspaces $W \subset V$ such that $W \cap U = \{0\}$ and $V = W + U$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{q^{k(n - k)}}
Consider a function \( f : \mathbb{N} \rightarrow  \mathbb{N} \) such that for any two positive integers \( x,y \) , the equation \( f\left( {{xf}\left( y\right) }\right)  = {yf}\left( x\right) \) holds. Find the smallest possible value of \( f\left( {2007}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
18
}
In trapezoid \(ABCD\), \(\overrightarrow{AB} = 2 \overrightarrow{DC}\), \(|\overrightarrow{BC}| = 6\). Point \(P\) is a point in the plane of trapezoid \(ABCD\) and satisfies \(\overrightarrow{AP} + \overrightarrow{BP} + 4 \overrightarrow{DP} = 0\). Additionally, \(\overrightarrow{DA} \cdot \overrightarrow{CB} = |\overrightarrow{DA}| \cdot |\overrightarrow{DP}|\). Point \(Q\) is a variable point on side \(AD\). Find the minimum value of \(|\overrightarrow{PQ}|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4 \sqrt{2}}{3}}
Compute the intersection multiplicity of the curves $V(f)$ and $V(g)$ at the point $p = [-2:1:4]$ in $\mathbb{P}^2$, where $f = x^2 - yz$ and $g = (x+z)^2 - yz$. Use the fact that the intersection multiplicity is the Hilbert polynomial of the $I(p)$-primary component of the ideal $(f) + (g)$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How many distinct necklaces can be made using exactly three red beads and three black beads?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the limit: \[ \lim_{n\rightarrow\infty} n^2 \int_{\frac{1}{n+1}}^{\frac{1}{n}}\frac{\tan^{-1}nx}{\sin^{-1}nx}dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Let \( \alpha \) denote \( \cos^{-1}(\tfrac 23) \). The recursive sequence \( a_0,a_1,a_2,\ldots \) satisfies \( a_0 = 1 \) and, for all positive integers \( n \), \[ a_n = \dfrac{\cos(n\alpha) - (a_1a_{n-1} + \cdots + a_{n-1}a_1)}{2a_0}. \] Suppose that the series \[ \sum_{k=0}^\infty\dfrac{a_k}{2^k} \] can be expressed uniquely as \( \tfrac{p\sqrt q}r \), where \( p \) and \( r \) are coprime positive integers and \( q \) is not divisible by the square of any prime. Find the value of \( p+q+r \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Determine the oscillation, $O(f, (0,0))$, of the function $f$ defined by:

$$ f(x,y)= \begin{cases} \frac{xy}{x^2+y^2} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \end{cases} $$

at the point $(0,0)$. Use the definition of oscillation at a point $c$:

$$ O(f,c) = \inf_{c\in U}\sup_{x_1,x_2\in U}|f(x_1)-f(x_2)| $$

where $U$ is an open subset containing $c$. Note that $f$ is not continuous at $(0,0)$ since:

$$ \lim_{x\rightarrow0}f(x,0) = 0 \neq \frac{1}{2} = \lim_{x \rightarrow 0}f(x,x) $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate \( \Gamma\left(-\frac{1}{2}\right) \) directly from the definition \( \Gamma(s) = \int_0^\infty e^{-t}t^{s-1}dt \), without using the known values \( \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} \) and \( \Gamma(s+1) = s\Gamma(s) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\sqrt{\pi}}
Construct a sequence \((y_n)\) that is not bounded, but satisfies \(\lim_{n\to+\infty}|y_n-y_{n+1}|=0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y_n = \sqrt{n}}
Find the conditions on a matrix \( A \) such that the matrix \( (I+A)(I-A)^{-1} \) is orthogonal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{A^\top = -A}
Given the equation \( x^a \cdot y^b = (x+y)^{(a+b)} \), where \( a \) and \( b \) are positive constants, find an expression for \( \frac{dx}{dy} \) by differentiating implicitly, assuming that \( ay \neq bx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{x}{y}}
Evaluate the integral \( \int_{|z| = 106} \cos\left(\frac{1}{6z+1}\left(\frac{1}{(z-1)(z-2)\cdots(z-105)}\right)\right)\,dz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the values of $x$ for which the series \( \sum_{n=1}^{\infty}\frac{x^n}{1-x^n} \) converges absolutely.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Find the angle \( \theta \) in the range \( 0 \leq \theta < 360 \) degrees that minimizes the x-coordinate of the Archimedean spiral given by \( r = 1 - \frac{0.5}{2\pi}\theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{174}
Consider a rectangle with dimensions $m \ge 3$ and $n \ge 3$, consisting of $mn$ unit squares. In each unit square, exactly one number is written such that any 3x3 sub-square forms a magic square. Determine the maximum number of different numbers that can be written in the entire rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the maximum step size $h$ for the forward Euler method such that the solution's norm approaches zero for the system of ODEs given by:
\[
a' = -a - 2b \\
b' = 2a - b
\]
with initial conditions $a(0)=1$ and $b(0)=-1$. The eigenvalues of the coefficient matrix for the forward Euler method are $(-1 + 2i)(1 + h)$ and $(-1 - 2i)(1 + h)$. Find the largest $h$ that ensures stability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{5}}
Suppose $x$ and $y$ are elements in a group $G$ where the order of $y$ is not equal to $1$, $yx = xy^2$, and the order of $x$ is $3$. Determine the order of $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Given the differential equation \( x' + x^3 = 2x \), find the differential equation for \( v = \frac{1}{x^2} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{v' + 4v = 2}
Evaluate the limit: $$\lim_{n \to \infty} \left[\frac{(n+2)^{n + 2}}{(n+1)^{n+1}} - \frac{(n+1)^{n+1}}{n^n} \right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Determine the oscillation of the function \( f(x,y) = \begin{cases} \frac{xy}{x^2+y^2} & (x,y)\neq(0,0) \\ 0 & (x,y)=(0,0) \end{cases} \) at the point \((0,0)\). The oscillation of \(f\) at a point \(c\) is defined as: \[ O(f,c) = \inf_{c\in U}\sup_{x_1,x_2\in U}|f(x_1)-f(x_2)| \], where \(U\) is an open subset containing \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
How to integrate: \(\int_0^T\cos(T-s)\sin(s)ds\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac12 T\sin{(T)}\)}
A sequence \( \{b_n\} \) is defined by the recurrence relation \( b_n = 2b_{n-1} - b_{n-2} + b_{n-3} \) with initial conditions \( b_1 = 1 \), \( b_2 = 1 \), and \( b_3 = 2 \). Calculate the value of the infinite series:
\[
b_1 + \frac{b_2}{3} + \frac{b_3}{3^2} + \cdots
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{11}}
For a given fixed integer $N$, find the largest integer $n$ such that the symmetric group $\Sigma_n$ can be embedded as a subgroup of the orthogonal group $O(N)$. What is the maximum value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{N + 1}
A set $S$ consisting of $2019$ different positive integers has the following property: the product of every $100$ elements of $S$ is a divisor of the product of the remaining $1919$ elements. What is the maximum number of prime numbers that $S$ can contain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1820}
我们称置换群 $S_{9}$ 中的两个元素 $\sigma_{1}, \sigma_{2}$ 是共轭的，如果存在 $S_{9}$ 中的元素 $\tau$使得 $\tau^{-1} \sigma_{1} \tau=\sigma_{2}$ 。设 $\sigma=(12345)$ (4567), 则 $S_{9}$ 中有 $\qquad$个元素与 $\sigma$ 共轭。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15120}
In a school, the decision to balance the menu with both meat and vegetables was agreed upon by the following percentage of students from different grade levels: 

- 60% of the third-grade students
- 45% of the fourth-grade students
- 35% of the fifth-grade students
- 55% of the sixth-grade students 

There are 256 third-graders, 525 fourth-graders, 410 fifth-graders, and 600 sixth-graders in the school. Calculate the total number of students from all four grade levels who agreed with the decision.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{864}
Let \( f:\mathbb{N}\mapsto\mathbb{R} \) be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that \( f(1)=\tfrac{\pi^2}{6} \). Determine the smallest positive integer \( m \) such that \( m\cdot f(10) \) is the square of a rational multiple of \( \pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Given a random variable xi that follows a normal distribution with mean 1 and variance 36, and P(-2 <= xi <= 1) = 0.4, calculate the probability P(xi > 4). Express your answer as a decimal value between 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.1}
Bonnie constructs a frame for a cube using 12 pieces of wire that are each eight inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to create a collection of unit cube frames that are not connected. The total volume of Roark's cubes is the same as the volume of Bonnie’s cube. What is the ratio of the total length of Bonnie's wire to the total length of Roark's wire?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{128}}
7. If the solution set of the inequality $k x^{2}-2|x-1|+6 k<0$ with respect to $x$ is an empty set, then the range of values for $k$ is $\qquad$

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k\geq\frac{1+\sqrt{7}}{6}}
Given the system of differential equations \( x' = xy + t^2 \) and \( y' = \frac{-y^2}{2} \), with initial conditions \( x(1) = 3 \) and \( y(1) = 2 \), find \( \frac{d x}{d y_0} \) at \( x_0 = 3 \) and \( y_0 = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
在 $1,2, \cdots, 10$ 中随机选出三个不同的数, 它们两两互素的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Let \( X \sim \mathrm{Geometric}(1/11) \). Use Jensen's inequality to determine a lower bound on \( \mathrm{E}(X^4) \) by applying Jensen's inequality to \( X^2 \) with \( f(x) = x^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53361}
Determine the number of six-digit numbers of the form $abcabc$ that have exactly four prime factors, with one of these factors having a power of 3. Express your answer as an integer $n$. Note that $abcabc$ can be expressed as $7 \times 11 \times 13 \times (100a + 10b + c)$. The prime factor with a power of 3 must be $100a + 10b + c$. Find the number of such numbers $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the degree of the algebraic number $\xi_{7} = e^{2\pi i/7}$ over the field $\mathbb{Q}(\xi_{3})$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the degree of the polynomial \( \deg(\rho_7+\rho_7^2+\rho_7^4) \) over the field of rational numbers \( \mathbb{Q} \), where \( \rho_7 \) is a primitive seventh root of unity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For the equation $6 x^{2}=(2 m-1) x+m+1$ with respect to $x$, there is a root $\alpha$ satisfying the inequality $-1988 \leqslant \alpha \leqslant 1988$, and making $\frac{3}{5} \alpha$ an integer. How many possible values are there for $m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2385}
For a given integer $n \geq 2$, determine how many integers $x$ in the set $\{1, 2, \ldots, n\}$ satisfy the congruence $x^2 \equiv x \pmod{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\omega(n)}}
Let $V = M_{3 \times 3}(\mathbb{C})$, the set of $3 \times 3$ matrices with complex entries. Consider the matrix $A = \begin{pmatrix} 0 & -2 & 0 \\ 1 & 3 & 0 \\ 0 & 0 & 2 \end{pmatrix}$. Define $W = \{ p(A) \mid p(t) \in \mathbb{C}[t] \}$, where $\mathbb{C}[t]$ is the set of polynomials with complex coefficients. Determine the dimension of the subspace $W$ in $V$. \(\dim W\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $M$ be a Riemannian manifold with submanifolds $S$ and $T$. Suppose $	heta$ is a differential $k$-form on $M$ such that \( \oint_{[S]} \theta = r \neq 0 \). If \( \eta \) is the Hodge star of \( \theta \) and the \( \mathbb{Z}\pi \)-intersection number of \( S \) and \( T \) is \( \pm 1 \), determine the value of \( \oint_{[T]} \eta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm r}
Let  $ A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$  ,  $ 1\leq a_i\leq i \plus{} 1$  for each  $ i \equal{} 1,2\dots,8\}$ .A subset  $ X\subset A$  is called sparse if for each two distinct elements  $ (a_1,\dots,a_8)$ , $ (b_1,\dots,b_8)\in X$ ,there exist at least three indices  $ i$ ,such that  $ a_i\neq b_i$ .
   Find the maximal possible number of elements in a sparse subset of set  $ A$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7!}
Let $a,b,c$ be positive real numbers such that $a+b+c=2005$. Find the minimum value of the expression:
$$E=a^{2006}+b^{2006}+c^{2006}+\frac{(ab)^{2004}+(bc)^{2004}+(ca)^{2004}}{(abc)^{2004}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
4010
}