inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the transition matrix \(P\) for a Markov chain:
\[
P = 
\begin{pmatrix}
0 & 1 & 0 & 0 \\
\frac{1}{3} & 0 & \frac{2}{3} & 0\\
0 & \frac{2}{3} & 0 & \frac{1}{3}\\
0 & 0 & 1 & 0\\ \end{pmatrix}
\]
and the limiting matrix \(\Pi = \lim_{n \to \infty} P^n\):
\[
\Pi = 
\begin{pmatrix}
0.25 & 0 & 0.75 & 0 \\
0 & 0.75 & 0 & 0.25\\
0.25 & 0 & 0.75 & 0\\
0 & 0.75 & 0 & 0.25\\ \end{pmatrix}
\]
Calculate the long-run portion of time that the first urn is empty, denoted as \(\pi_0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{8}}
Given that \(a, b, c, d > 0\) and \(a + b = 3\), and \(\frac{a}{c} + \frac{b}{d} = 1\), find the minimum value of \(c + d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the largest positive integer \( N \) such that the expression \( \sqrt{64 + 32^{403} + 4^{N+3}} \) is an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2008}
Find the number of functions \( f : \{ 1,2,\ldots ,n\}  \rightarrow  \{ {1995},{1996}\} \) such that
 \( f\left( 1\right)  + f\left( 2\right)  + \ldots  + f\left( {1996}\right) \) is odd.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2^{1995}$}
Two cars, A and B, start at different positions on a road. Car A drops its first marker 5 feet from its starting position, and car B drops its first marker 9 feet from its starting position. Both cars then drop subsequent markers at increasing distances, with each subsequent marker being 2 feet further than the previous one. Determine the distance from the starting position where both cars drop a marker at the same location for the first time.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Given that $X$ and $Y$ are independent random variables with a continuous uniform distribution on [0,1], and $Z = \max\{X, Y\}$, calculate $\mathbb{E}[X | Z]$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}Z}
Find the dimension of the subspace \(W\) of \(\operatorname{Hom}(\mathbb{R}^3, \mathbb{R}^3)\), the space of all linear transformations from \(\mathbb{R}^3\) to \(\mathbb{R}^3\), defined by:
\[
W = \{L \in \operatorname{Hom}(\mathbb{R}^3, \mathbb{R}^3) : \operatorname{Im}(L) \subset S\}\]
where \(S = \{(x^1, x^2, x^3) \in \mathbb{R}^3 : x^1 + x^2 + x^3 - 1 \leq 0, \ x^1 + x^2 + x^3 + 1 \geq 0\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the probability that a random variable $X$ from a Gamma distribution with parameters $(3,3)$ is greater than a random variable $Y$ from an Exponential distribution with rate $1$. That is, find $P(X > Y)$. Assume $X \sim \text{Gamma}(3,3)$ and $Y \sim \text{Exp}(1)$. Express your answer as a probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{37}{64}}
What is the limit, as \( n \to \infty \), of the expected distance between two points chosen uniformly at random within a unit-radius hypersphere in \( \mathbb{R}^n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Let $X, Y, Z \sim \mathcal{N}(0,1)$ be independent standard normal random variables. Calculate the probability $\mathbb{P}(X > Y \mid Y > Z)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
What is the least common multiple of the orders of all possible move sequences in a Rubik's cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1260}
Let \( X_1, X_2, \ldots, X_{40} \) be independent random variables, each uniformly distributed on \([-1/2, 1/2]\). What is the probability that the sum \( Y = X_1 + X_2 + \cdots + X_{40} \) is greater than 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.050}
Calculate the floor of the sum \(\left\lfloor{\sum_{A \subseteq S}^\ \frac {F(A)} {G(A)}}\right\rfloor\), where \(S = \{1, 2, 3, ..., 8\}\), \(F(A)\) is the sum of all elements in set \(A\), and \(G(A)\) is the product of all elements in set \(A\). Determine the floor value of this sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Determine the interval of convergence for the power series:
\[ \sum_{n=2}^\infty(-1)^n\frac{(x-3)^n}{(\sqrt[n]{n}-1)n} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 4]}
Determine the minimum number of terms required in the series \( \sum_{n=1}^\infty \frac{1}{3+n^2} \) so that the error in approximating the sum of the series by the sum of the first \( N \) terms is less than \( 10^{-5} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100000}
Calculate the surface integral of the vector field $\underline{F} = 3x\underline{i} -5z\underline{k}$ over the cylindrical region $x^2 + y^2\leq 1$, where $0\leq z \leq 1$, using parameterizations for all three surfaces.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi}
What is the maximum number of bottles that can be tested for poison using 50 rats within one hour, assuming that rats die exactly one hour after consuming poison and testing time is negligible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{50}}
Determine the greatest integer \( k \) such that the inequality \( \left\lfloor \frac{n}{\sqrt{3}} \right\rfloor + 1 > \frac{n^2}{\sqrt{3n^2 - k}} \) holds for all positive integers \( n \geq 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a quintic polynomial $g(x)$, it leaves a remainder of $-2$ when divided by $(x+2)^3$ and a remainder of $2$ when divided by $(x-2)^3$. Find the value of $g''(-1)$. \( \text{(The second derivative of } g \text{ at } x = -1) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{45}{32}}
ABCD is a unit square. One vertex of a rhombus lies on side AB, another on side BC, and a third on side AD. Determine the area of the set of all possible locations for the fourth vertex of the rhombus.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
Find the exact value of \( C \) such that the sequence defined by the recursive relation \( x_{i} = \frac{x_{i-1} + x_{i-2}}{Cx_{i-1}} \), with initial conditions \( x_1 = 1 \) and \( x_2 = 2 \), converges to \( \frac{2}{C} \) as \( i \) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to
$
\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.
$
Proposed by Daniel Zhu.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
167
}
Find the dimension of the vector space \( \mathbb{Q}(t) \) over the field \( \mathbb{Q}(t^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
There are 100 towns labeled 1 through 100. The cost to build a bridge between town "i" and town "j" is given by max{|i − j|, 4}. However, a bridge between town "i" and town "2i" can be built at a cost of 2. Determine the minimum cost required to connect all the towns in a single connected graph.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{296}
Given the cyclic group \( \mathbb{Z}_7 \) with generator 3, determine the smallest non-negative integer \( n \leq 5 \) such that \( 3^n \equiv 6 \pmod{7} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the values of \(\beta\) for which the function \(g(x) = |x|^\beta\) is weakly differentiable on the unit ball in \(\mathbb{R}^m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\beta > 1 - m}
Let \( \phi: L \to L \) be a linear operator with a minimal polynomial \( m_\phi(x) = (x-1)x^8 \). Find the minimal polynomial of \( \phi^3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x - 1)x^3}
Evaluate the integral \( \int_{-1/2}^{1/\sqrt{2}} \frac{f(x)}{1-x^2} \, dx \) given that \( f:[0,1]\to [0,1] \) is a continuous function satisfying \( x^2 +(f(x))^2\le 1 \) for all \( x\in [0,1] \) and \( \int_0^1 f(x) \, dx=\frac{\pi}{4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5\pi}{12}}
Determine the minimum number of proper subgroups required to cover a noncyclic group \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the equation of a circle that passes through the intersection of the circle \( S: x^2 + y^2 - 12x - 4y - 10 = 0 \) and the line \( L: 3x + y = 10 \), and has the same radius as the circle \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^2 + y^2 = 50}
Determine the order of the automorphism \( \sigma \) in the Galois group \( \text{Gal} \left( \frac{\mathbb{Q}(\sqrt[3]{3},\sqrt{-3})}{\mathbb{Q}} \right) \), where \( \sigma(\sqrt[3]{3}) = \omega\sqrt[3]{3} \) and \( \sigma(\sqrt{-3}) = -\sqrt{-3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the condition on the parameters $b$ and $k$ such that the lines $x-y=5$ and $bx^2+2kxy+by^2=0$ (where $b \neq 0$) form an isosceles triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k^2 > b^2}
Determine the number of homomorphisms from the symmetric group $S_7$ to the alternating group $A_8$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{316}
Determine the non-trivial real numbers $x$ for which the $n \times n$ matrix $A$ with entries $A_{ij} = x^{|i-j|}$ is invertible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x \neq \pm 1}
Evaluate the limit \( \lim_{r\rightarrow 0} \frac{1}{r} \int_{C_{r}} \frac{\sin(z)}{z}dz \), where \( C_r \) is the contour \( re^{i\theta} \) for \( 0 \leq \theta \leq \pi \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Compute the tensor product \( \mathbb{Z}[1/2] \otimes_{\mathbb{Z}} \mathbb{Z}/3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/3\mathbb{Z}}
Two circles with radii in the ratio 1:2 touch each other externally. Let the center of the smaller circle be $C$ and the center of the larger circle be $D$. The point of contact is $A$. A straight line $\overline{PAQ}$ is drawn such that $P$ is on the smaller circle and $Q$ is on the larger circle, and $\overline{PAQ}$ does not pass through $C$. Given that $\angle QAD = \alpha$, find the angle between the tangent at $Q$ to the larger circle and the diameter of the smaller circle that passes through $P$ (extended if necessary).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Determine the values of \(\alpha \in \mathbb{R}\) for which the integral \(\int_{0}^{\infty} \frac{\sin u}{(u+1)^\alpha} \, du\) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 0}
Evaluate the integral \( \int_0^1 \frac{2x^3+x}{\tan x+x^{\frac{1}{3}}}\, \mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the polynomial $p(z)$ in the Hadamard product factorization of the function $e^z - 1$, given that the zeros are $\{2 \pi i n ,n\in \mathbb{Z}\}$, they are all simple, and the function is of order 1. The factorization is given by:
\[ e^z - 1 = e^{p(z)} z \prod_{n=1}^\infty \left(1 + \frac{z^2}{4n^2\pi^2}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{z}{2}}
Evaluate the limit: \[ \lim_{n\to\infty}\left(\sqrt[n]{n+2}\cdot\sqrt[n^2+2n]{\binom{n}{1}\binom{n}{2}\cdots\binom{n}{n}}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
Let $G$ be a 5-regular graph with $m$ edges and a girth of at least 6. Find the number of distinct paths of length 5 in $G$. A regular graph is a graph in which each vertex has the same degree, and the girth of a graph is the length of the shortest cycle within the graph.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256m}
In a school, there are \( n \) students, each assigned a unique student number. Each student number is a positive factor of \( 60^{60} \), and the highest common factor (H.C.F.) of any two student numbers is not a student number in the school. Determine the greatest possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3721}
In a circle with center A and radius R, point C lies on the circle. Segment DE connects C to the center A, with CE having a fixed length H and AE = AD = R. Angle ECF intersects the circle at point F, and segment CF is formed. Given that EC × CD = GC × CF, where CF = x and GC = y, and the equation H(2R - H) = x(y) holds, find the length of CF.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2R - H}
How many distinct triangles $\triangle ABC$ exist with $AB = 2011$ units such that both $\cos(2\angle A + 3\angle C)$ and $\sin(2\angle B + \angle C)$ are integers? Assume that $\angle A$, $\angle B$, and $\angle C$ are the interior angles of the triangle and satisfy the triangle inequality theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the integral \( \int_0^1 x^{k - 1} e^{-x} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\gamma(k, 1)}
A sequence $ a_1, a_2, \ldots$ of non-negative integers is defined by the rule $ a_{n + 2} = |a_{n + 1} - a_n|$ for $ n\ge 1$. If $ a_1 = 999, a_2 < 999,$ and $ a_{2006} = 1$, how many different values of $ a_2$ are possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{324
}
Given the polynomial equation $x^4 - 2x^3 + 3x^2 - 4x + 2 = 0$ with distinct roots $a$, $b$, $c$, and $d$, find the value of $a^4 + b^4 + c^4 + d^4$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the smallest integer $n$ such that $e^{\sqrt{n}} > n^4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{681}
Given integers \( x, y, z \) such that \( x + y + z = 3 \) and \( x^3 + y^3 + z^3 = 3 \), find the value of \( x^2 + y^2 + z^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
On a \( 4 \times  4 \times  3 \) rectangular parallelepiped,vertices \( A,B \) ,and \( C \) are adjacent to vertex
 \( D \) . The perpendicular distance from \( D \) to the plane containing
 \( A,B \) ,and \( C \) is closest to
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
2.1
}
Calculate the line integral of the vector field \( \vec{F} = (e^x \sin y + 24 y) \vec{i} + (e^x \cos y) \vec{j}\) along the curve \(C\), which is the upper half of the circle \(x^2 + y^2 = \frac{\pi^2}{4}\) oriented counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3\pi^3}
Evaluate the limit: $$\lim_{x\to0}\left(\frac {\csc(x)}{x^3} - \frac{\sinh(x)}{x^5}\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{90}}
In a diagram, two lines intersect with two circles at 9 distinct points. Determine the number of ways to select 4 points from these 9 such that no 3 points among the selected 4 are collinear or lie on the same circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let \( \textbf{Z}_n \) be the set \( \{0, 1, \ldots, n - 1\} \) with addition mod \( n \). Consider subsets \( S_n \) of \( \textbf{Z}_n \) such that \((S_n + k) \cap S_n\) is nonempty for every \( k \) in \( \textbf{Z}_n \). Let \( f(n) \) denote the minimal number of elements in such a subset. Determine \( \lim_{n \to \infty} {\ln f(n) \over \ln n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
A $24 \times 60$ rectangle is divided by lines parallel to it sides, into unit squares. How many parts will this rectangle be divided into if you also draw a diagonal in it?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1512}
Determine the minimum number of lines required to touch every square of an \(n \times n\) grid, given that the lines must pass inside the squares and not on their edges or corners.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
In the plane, circle \(\omega\) is externally tangent to circle \(\Omega\) at point \(P\). The center \(O\) of \(\Omega\) is located inside \(\omega\). A diameter \(XY\) of \(\Omega\) is tangent to \(\omega\). Given that \(PY > PX\) and \(PY\) intersects \(\omega\) at \(Z\) such that \(PZ = 3YZ\), determine the measure of angle \(PYX\) in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Evaluate the integral \( \int_{S}(x+y+z)\,dS \), where \( S \) is the region defined by \( 0\leq x,y,z\leq 1 \) and \( x+y+z\leq 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{8}}
Determine the number of solutions to the equation \(\cos(97x) = x\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61}
Determine the number of ways to arrange red balls in 6 places such that in any 3 consecutive places, there are at least 2 red balls. Use the minimum number of red balls possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find an integer \( n \) , where \( {100} \leq  n \leq  {1997} \) , such that
 \( \frac{{2}^{n} + 2}{n} \)
 is also an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{

n = 946

}
Given sequences of measurable sets $A_1, A_2, \ldots$ and $B_1, B_2, \ldots$ in a sigma-algebra $Q$, suppose $P(A_k \text{ infinitely often }) = 1$ and $P(B_k^c \text{ infinitely often }) = 0$. What is the probability that infinitely many of the joint events $A_k \cap B_k$ will occur?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the recurrence relation with initial conditions $a_0 = a_1 = a_2 = a_3 = a_4 = 0$ and $a_5 = 1$, and the formula:
\[ a_{n+6} = \frac{a_{n+5} + a_{n+4} + a_{n+3} + a_{n+2} + a_{n+1} + a_{n}}{6} \]
Find the limit of $a_n$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{7}}
Evaluate the limit: $$\lim_{n\to\infty}\left(\sum_{0<i,j<n}{\binom{2i}{i}\binom{2j}{j}}\right)^\frac{1}{n}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine the values of \( \alpha > 0 \) for which the series \( \sum_{k=1}^{\infty} \frac{1}{(k+1)\ln(k+1)^\alpha} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 1}
Given the expansion \((3x^2 + 2x + c)^{12} = \sum_{r=0}^{24} A_r x^r\), if \(\frac{A_{19}}{A_5} = \frac{1}{2^7}\), find the value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the set of all numbers \( C \) such that the inequality \( |\alpha \sin x + \beta \cos 4x| \leq C \) is satisfied for all \( x \) and for any \( (\alpha, \beta) \) where \( |\alpha| \leq 1 \) and \( |\beta| \leq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2, \infty)}
Let \( {c}_{1},{c}_{2},\ldots ,{c}_{6030} \) be 6030 real numbers. Suppose that for any 6030 real numbers \( {a}_{1},{a}_{2},\ldots ,{a}_{6030} \) ,there exist 6030 real
 numbers \( \left\{  {{b}_{1},{b}_{2},\ldots ,{b}_{6030}}\right\} \) such that
 \( {a}_{n} = \mathop{\sum }\limits_{{k = 1}}^{n}{b}_{\gcd \left( {k,n}\right) } \)
 and
 \( {b}_{n} = \mathop{\sum }\limits_{{d \mid  n}}{c}_{d}{a}_{n/d} \)
 for \( n = 1,2,\ldots ,{6030} \) . Find \( {c}_{6030} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
528

}
Find the largest natural number such that all its digits are different, and the sum of any two adjacent digits is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6310972}
A mixture of 30 liters of paint is $25\%$ red tint, $30\%$ yellow tint, and $45\%$ water. Five liters of yellow tint are added to the original mixture. What is the percent of yellow tint that is the mixture?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
40
}
Determine the capacity of the channel defined by the convolution operator \( C: \ell^1(G) \to \ell^1(G) \), where \( f \mapsto \mu * f \) and \( \mu \) is a probability measure on a finite group \( G \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log |G| - H(\mu)}
Evaluate the integral \( \displaystyle \int\limits_{0}^{\infty} \dfrac{x^x}{x^{1+\frac{1}{x}}}\, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given a Gaussian random variable X with covariance matrix $C_{XX}$, determine a matrix A such that the random variable Y = AX has a covariance matrix $C_{YY} = I$. Provide the matrix A.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C_{XX}^{-1/2}}
Using the maximum likelihood method, estimate the total number of fish in a pond. Initially, 20 fish are caught, marked, and released back into the pond. Later, 50 fish are caught, and 7 of them are found to be marked. Determine the total number of fish in the pond, \( N \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{142}
Let \(\left(a_{n}\right)\) be a sequence defined by \(a_{1}=3\), \(a_{2}=2\), and for \(n \geqslant 1\), \(a_{n+2}\) is the remainder of the Euclidean division of \(a_{n}+a_{n+1}\) by 100. Compute the remainder of the Euclidean division of:

\[ a_{1}^{2}+a_{2}^{2}+\cdots+a_{2007}^{2} \]

by 8.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Determine the set of statistical cluster points of sub-sequences of the sequence \( \frac{n}{e}-\left[\frac{n}{e}\right] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0,1]}
Given an irreducible representation $V$ of a semisimple Lie algebra with highest weight $\lambda$, determine the highest weight of the dual representation $V^*$. Provide your answer in terms of $\lambda$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-w_0(\lambda)}
Determine all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the equation \( f(x^3) - f(y^3) = (x^2 + xy + y^2)(f(x) - f(y)) \) for all real numbers \( x \) and \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = ax + b}
In triangle ADC, G is the centroid, and AE is perpendicular to FC. Given that BD = DC, AC = 12 units, and the midpoint of AC is H, with D, G, and H being collinear, find the length of AB.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}