inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the transition matrix of a Markov chain:
\[
\begin{bmatrix}
1&0&0&0\\
0.1&0.6&0.1&0.2\\
0.2&0.3&0.4&0.1\\
0&0&0&1
\end{bmatrix}
\]
with states \( \{0, 1, 2, 3\} \), calculate the probability that the Markov chain, starting from state 1, will end in state 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{21}}
Assume a remailer reorders messages and maintains a pool of \( n-1 \) messages at all times. When the \( n \)th message arrives, one of the \( n \) messages is selected at random and forwarded. An attacker sends enough messages to force the \( n-1 \) messages in the original pool to be sent. Assuming the message to be sent is chosen according to a uniform random distribution, what is the expected number of messages the attacker would have to send to achieve this goal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n H_{n-1}}
Last year’s high school graduates started high school in year $n- 4 = 2017$, a prime year. They graduated high school and started college in year $n = 2021$, a product of two consecutive primes. They will graduate college in year $n + 4 = 2025$, a square number. Find the sum of all $n < 2021$ for which these three properties hold. That is, find the sum of those $n < 2021$ such that $n -4$ is prime, n is a product of two consecutive primes, and $n + 4$ is a square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{514}
In triangle \( \triangle ABC \), let \( A_1 \) be the midpoint of \( BC \), and \( G \) be the centroid of the triangle. Squares \( GBKL \) and \( GCMN \) are constructed on the left with respect to rays \( GB \) and \( GC \), respectively. Let \( A_2 \) be the midpoint of the segment connecting the centers of these squares. The circumcircle of triangle \( \triangle A_1A_2G \) intersects \( BC \) at points \( A_1 \) and \( X \). Find the ratio \( \frac{A_1X}{XH} \), where \( H \) is the foot of the altitude from \( A \) to \( BC \) in triangle \( \triangle ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the limit: \[ \lim_{n \to \infty} \left\{ \prod_{k=1}^{n} \frac{k}{H_k + 1} \right\}^{\frac{1}{n}} \] where \( H_k = \sum_{i=1}^{k} \frac{1}{i} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the Fréchet derivative of the function \( F[u] = u^2 \) at a point \( u \) in a commutative Banach algebra.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2u \cdot h}
Consider a function \( f(z) \) with an expansion of the form \( f(z) = \sum_{n\geq 1} a_n {\rm e}^{2\pi i n z} \). If \( f(z) \) is bounded on the upper half-plane, determine the decay rate of the terms \( a_n {\rm e}^{2\pi i n z} \) in the series as the imaginary part of \( z \) tends to infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e^{-2\pi n y}}
Evaluate the integral \( \int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \ln 2}
Determine the best known bound on the index \(m\) of an abelian subgroup \(\Gamma'\) in a discrete subgroup \(\Gamma\) of the group of isometries in \(\mathbb{R}^n\), where \([\Gamma : \Gamma']\) denotes the index of \(\Gamma'\) in \(\Gamma\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n n!}
Consider the integral \( \int_{-1}^1 x^n f(x) \, dx \) for every \( n \)-th degree polynomial \( f \) with integer coefficients. Let \( \alpha_n \) denote the smallest positive real number that such an integral can give. Determine the limit value \( \lim_{n\to \infty} \frac{\log \alpha_n}{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
What is the cardinality of the set of all non-decreasing functions defined on the interval \([0,1]\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Calculate the work done by the force field \( \mathbf{F}(x, y, z) = (x^2 + y)\mathbf{i} + (y^2 + z)\mathbf{j} + (z^2 + x)\mathbf{k} \) along the curve parametrized by \( \mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j} + t^3\mathbf{k} \) from \( t = 0 \) to \( t = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{149}{60}}
Determine the values of \( \lambda > 0 \) for which the integral \( \int_0^\infty e^{\sin x} \frac{\sin 2x}{x^\lambda} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2)}
Determine the number of distinct Cayley tables that can be constructed for the set \( \{0, 1, \ldots, n-1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^{n^2}}
In the product \( 24^{a} \cdot 25^{b} \cdot 26^{c} \cdot 27^{d} \cdot 28^{e} \cdot 29^{f} \cdot 30^{g} \), the exponents \(a, b, c, d, e, f, g\) are assigned the numbers \(1, 2, 3, 5, 8, 10, 11\) in some order. Determine the maximum number of zeros that can appear at the end of the decimal representation of this product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Two independent random variables $A$ and $B$ are uniformly distributed over $[0,1]$, representing the moments two bulbs are ignited. Two other independent random variables $S$ and $T$ are uniformly distributed over $[0,2]$, representing the durations for which the bulbs remain lit. Calculate the probability that both bulbs are lit simultaneously.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{6}}
Given a $5 \times 7$ matrix $B$, a basis for $\operatorname{Nul} B$ is given by $$\mathcal{B_N} = \left\{\begin{pmatrix} 1 \\ 2 \\ -1 \\ 0 \\ 1 \\ 0 \\ 1 \end{pmatrix} , \begin{pmatrix} 2 \\ 1 \\ 0 \\ 1 \\ -1 \\ 1 \\ 0 \end{pmatrix}\right\},$$ and $\vec c\in\operatorname{Col}B$. If a particular solution to $\operatorname{proj}_{\operatorname{Col}B}\vec c = \begin{pmatrix} 2 \\ 1 \\ 0 \\ 1 \\ 1 \end{pmatrix}$ is given by $\vec y = \begin{pmatrix} 1 \\ 0 \\ 1 \\ 1 \\ 0 \\ 1 \\ 0 \end{pmatrix}$, find the sum of the elements in $\vec c$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In a first-price auction with two bidders, each bidder may have either a low valuation $V_L$ or a high valuation $V_H$. Bidders do not know the valuations of others. Determine the minimum bid (i.e., the lowest bid in the support $[\underline{b_i},\overline{b_i}]$) for a bidder with a low valuation $V_L$. Assume that strategies involve a probability distribution over the domain $[\underline{b_i},\overline{b_i}]$. What is $\underline{b_i}$ for a $V_L$ type bidder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the least real number $L$ such that$$\dfrac{1}{a}+\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}\leqslant L$$for all quadruples $(a,b,c,d)$ of integers satisfying $1<a<b<c<d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$L = \dfrac{163}{60}$}
Let real numbers \(x\) and \(y\) be such that \(x-y\), \(x^2-y^2\), and \(x^3-y^3\) are all prime numbers. Determine the value of \(x-y\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Find the possible minimal polynomials of a matrix $B$ such that $B^3 = I$ and $B$ has only $-1$ as an eigenvalue.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x + 1}
Evaluate the integral \( \int_{1}^{2} \frac{e^x}{x} \, dx \) to five decimal places using the Taylor series.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.05912}
Given 10 subsets of a set X, determine the maximum number of unique subsets that can be generated using the operations of union, intersection, complement, and symmetric difference, even if some subsets might be empty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{1024}}
Let \( X \) be a normally distributed random variable with mean \( \mu \) and standard deviation \( \sigma = \mu \). Given that \( P(-2 < X < 17) = 0.4375 \), find the value of \( \mu \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.6}
In a $2023 \times 2023$ square grid, each cell is colored with one of $n$ colors. It is known that for any six cells of the same color in a single row, there are no cells of the same color above the leftmost or below the rightmost of these six cells. Determine the smallest possible value of $n$ for which this configuration is achievable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{405}
Let $V$ be the vector space of all $2\times 2$ matrices over the field of real numbers and let \(B=\begin{pmatrix}2&-2\\-1&1\end{pmatrix}.\) Let $W$ be the subspace of $V$ consisting of all $A$ such that $AB=0.$ Let $f$ be a linear functional on $V$ which is in the annihilator of $W.$ Suppose that $f(I)=0$ and $f(C)=3,$ where $I$ is the $2\times 2$ identity matrix and \(C=\begin{pmatrix}0&0\\0&1\end{pmatrix}.\) Find $f(B).$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the KKT point for the following optimization problem:

Minimize \( f(x_1, x_2) = (x_1 - 3)^2 + (x_2 - 2)^2 \)

Subject to:

\( x_1^2 + x_2^2 \le 5 \)

\( x_1 + 2x_2 = 4 \)

\( x_1 \ge 0, x_2 \ge 0 \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(2, 1)}
Determine the value of $k$ for which 100 arcs can be placed on a circle such that each arc intersects exactly $k$ other arcs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
Find the maximum value of the function \( A(x,y) = \frac{1}{2}(x(12-x) + y(13-y)) \) subject to the constraint \( x^2 + (12-x)^2 - y^2 - (13-y)^2 = 0 \) using Lagrange multipliers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
Determine the possible values of the integral \( \frac{1}{2\pi i}\int_{\gamma}\frac{2i}{z^2+1}dz \), where \( \gamma \) is a closed continuously differentiable path in the upper half-plane that does not pass through \( i \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Determine the number of points \( c \) such that for the function \( f(x) = x^3 + 3x + 2 \), the condition \((b-a)f'(c) \neq f(b) - f(a)\) holds for any two real numbers \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Calculate the volume of the tetrahedron with vertices at $A_{1}(-1, 2, -3)$, $A_{2}(4, -1, 0)$, $A_{3}(2, 1, -2)$, and $A_{4}(3, 4, 5)$. The height is dropped from vertex $A_{4}$ onto the face $A_{1} A_{2} A_{3}$. Provide your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{20}{3}}
The sequence \( x_1, x_2, x_3, \ldots \) is defined by \( x_1 = 2022 \) and \( x_{n+1} = 7x_n + 5 \) for all positive integers \( n \). Determine the maximum positive integer \( m \) such that \[ \frac{x_n(x_n - 1)(x_n - 2) \cdots (x_n - m + 1)}{m!} \] is never a multiple of 7 for any positive integer \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{404}
Let $\Omega = D(0,2)/\{\frac{1}{2}\}$, where $D(0,2)$ is a disc, and let $f$ be holomorphic in $\Omega$. Given that $\frac{1}{2}$ is a simple pole for $f$ with residue 1, calculate the limit:
\[ \lim_{n \to \infty} \frac{f^{(n)}(0)}{2^{n}n!} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Evaluate the expression \((\sin^{-1}a)^2-(\cos^{-1}b)^2+(\sec^{-1}c)^2-(\csc^{-1}d)^2\) given that:
\[(\sin^{-1}a)^2+(\cos^{-1}b)^2+(\sec^{-1}c)^2+(\csc^{-1}d)^2=\frac{5\pi^2}{2}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Identify a Banach space that is isometric to the quotient space $C[0,1]/M$, where $M=\{f\in C[0,1]:f(0)=0\}$ and $C[0,1]$ is equipped with the supremum norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Find the number of solutions to the equation \( \{f(f(x))\} = 0.5 \), where the function \( f(x) \) is defined as follows: \( f(x) = 4x - 2 \) for \( 0 \leq x \leq 1 \) and \( f(x) = -x^2 - 2x \) for \( -2 \leq x < 0 \). Here, \( \{x\} \) denotes the fractional part of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In a finite discrete probability space $(\Omega, P)$ with $|\Omega| = n$, determine the maximum number of mutually independent random variables that can exist on this space.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lfloor \log_2 n \rfloor}
Let $C = H \cap S$, where $S$ is a surface of general type canonically embedded in $\mathbb{P}^N$ for some $N > 0$, and $H$ is the divisor of a hyperplane in $\mathbb{P}^N$. Determine the degree of the divisor $H_{|C}$ on the curve $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{K_S^2}
In a particular water sample, ten bacteria are found, of which three are of type A. What is the probability of obtaining exactly six type A bacteria in a second independent water sample containing 12 bacteria in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.0792}
Given a Hamiltonian action of a torus on a symplectic 4-manifold \( M \) with a moment map \( \mu: M \to \mathbb{R}^2 \), where \( \mu = (\mu_1, \mu_2) \), determine the expression for the moment map \( \phi \) for the circle action on \( M \) induced by the homomorphism \( S^1 \to T^2 \) given by \( \theta \mapsto (2\theta, 3\theta) \). Express \( \phi \) in terms of \( \mu_1 \) and \( \mu_2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\mu_1 + 3\mu_2}
Determine the number of sets \( A \) containing 9 positive integers such that for any positive integer \( n \leq 500 \), there exists a subset \( B \subset A \) with \( \sum_{b \in B} b = n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A right circular cone has base of radius 1 and height 3.  A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone.  What is the side-length of the cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{6}{2 + 3\sqrt{2}}}
In a triangle divided into four smaller triangles and three quadrilaterals, each smaller triangle has an area of 1. Calculate the area of the quadrilateral \( C A_{0} C_{0} A_{1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $f$ be a continuous function defined on $[0,1]$ such that \( \int_{0}^1f^2(x)dx=\left(\int_{0}^1f(x)dx\right)^2 \). Determine the range of $f$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{c\}}
Evaluate the integral \( \int_0^1 \{2x\}\{3x\} \, dx \), where \( \{x\} \) denotes the fractional part of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{19}{72}}
Determine the radius of convergence of the power series \(\sum_{n \geq 0}a_n z^{n!}\) given that the radius of convergence of \(\sum_{n \geq 0}a_n z^{n}\) is \(L\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of equivalence classes under logical equivalence for the set of all formulas that can be constructed using the atoms \( \{A_1, A_2, \ldots, A_n\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^n}}
Determine the cardinality of the set of all obtuse triangles on the Cartesian plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathfrak{c}}
Let $X_1, X_2, \ldots$ be a sequence of infinite independent and identically distributed random variables, where $X_i \sim U[0,1]$. Define $D_n = \sqrt{X_1^2 + X_2^2 + \ldots + X_n^2}$. Determine the value of the following cumulative distribution function at the point $\frac{1}{\sqrt{3}}$: $$\lim_{n\to \infty} F_{\frac{D_n}{\sqrt{n}}} \left(\frac{1}{\sqrt{3}}\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the value of \( k \) such that \( \int^{\infty}_{0}\frac{\ln^2(x)}{(1-x)^2}dx + k\int^{1}_{0}\frac{\ln(1-x)}{x}dx = 0. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Dave is shaped like two hemispeheres placed on either side of a vertical cylinder. If Dave is $4$ feet across in width (excluding arms) , and has total volume $20\pi$ feet$^3$ , then how tall is he? (excluding legs and hair)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{19}{3}$}
Calculate the determinant of the linear transformation \( T(X) = \frac{1}{2}(AX + XA) \), where \( V \) is the vector space of all \( 3 \times 3 \) matrices and \( A \) is the diagonal matrix given by:
\[ A = \begin{pmatrix} 1 & 0 & 0\\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{8}}
Find the points of continuity for the function \( f(x,y) = \frac{\sin(xy)}{\sqrt{4-x^2-y^2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x, y) \mid x^2 + y^2 < 4}
Determine the values of \(a > 0\) for which the expected value of the random variable \(X_1\), with distribution \(\mathbb{P}(X_k \geq x) = \frac{1}{x^a}\) for \(x \in \mathbb{N}\), is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 1}
Find the function \( f(x) \) that satisfies the following conditions: \( f(ax) = \frac{1}{a}f(x) \) and \( \int_{a}^{ab} f(x) \, dx = \int_{1}^{b} f(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{C}{x}}
In triangle $\Delta ABC$, let $O$ be a point inside the triangle. Extend $AO, BO, CO$ to meet $BC, CA, AB$ at $D, E, F$ respectively. Given that $AO = 30$, $FO = 20$, $BO = 60$, $DO = 10$, and $CO = 20$, find the length of $EO$. \(\boxed{\text{EO}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find a function $g(n)$ such that $f(n) \in \Theta(g(n))$ for the given recurrence relation: $$ f(n) = f(5n/13) + f(12n/13) + n \;\;\; \forall n \geq 0 $$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2}
Find the value of the series \( \sum_{p \in \mathbb{P}} \frac{1}{p^2} \), where \( \mathbb{P} \) is the set of all prime numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.452247}
What is the maximum number of teams that could have participated in a quiz with 50 questions, if the score for each correct answer can be adjusted such that the teams can be ranked in any order preferred by the experts?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Evaluate the definite integral \( I = \int_0^\pi \frac{\sin^2(\theta) \, d\theta}{10-6\cos(\theta)}. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{18}}
Determine the minimum value of the parameter \( c \) such that the following system of equations has a unique solution:

\[
\begin{cases}
8(x+7)^{4}+(y-4)^{4}=c \\
(x+4)^{4}+8(y-7)^{4}=c
\end{cases}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given that the numbers \( x, y, z \) form an arithmetic progression with a common difference \( \alpha = \arccos \left(-\frac{3}{7}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{7}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression, find the value of \( \cos^{2} y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{10}{13}}
Draw cards one at a time, with replacement, from a standard deck of playing cards. What is the expected number of draws until four successive cards of the same suit appear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
In a simple, connected planar graph where all regions have 8 edges, determine the minimum possible degree of a vertex.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Inside the tetrahedron \(ABCD\), there is a point \(O\) such that line segments \(AO, BO, CO, DO\) intersect the faces \(BCD, ACD, ABD, ABC\) at points \(A_1, B_1, C_1, D_1\) respectively, with \(\frac{AO}{A_1O} = \frac{BO}{B_1O} = \frac{CO}{C_1O} = \frac{DO}{D_1O} = k\). Determine all possible values of \(k\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The Shannon formula C=Wlog2(1+S/N) indicates that the maximum information transmission rate C depends on the bandwidth W, the average power of the signal S in the channel, and the power of Gaussian noise N in the channel. Due to technological advancements, the bandwidth W increases by 20% from its original value, and the signal-to-noise ratio S/N increases from 1000 to 4000. Calculate the approximate percentage increase in C. Express your answer as a percentage, rounded to the nearest whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
From the ten digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$, select nine digits to form a two-digit number, a three-digit number, and a four-digit number such that the sum of these three numbers equals 2010. What is the digit that is not selected?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
If the lateral surface of a cone is unfolded into a semicircle, then the apex angle of the cone (the angle between two generatrices in the axial section of the cone) is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
In a sports complex, stations $A, B, C, D, E$ formed the vertices of a regular pentagon. These stations were connected by straight paths. Additionally, on the path from $A$ to $B$ there was a fountain $F$, which was connected to station $C$ by a path perpendicular to the path from $B$ to $E$. Pat and Mat met at station $E$ and decided to sweep some of the paths. Pat swept the path from $E$ to $B$. Mat swept the path from $E$ to $A$ and also from $A$ to $F$.

Determine the difference in the lengths of the paths swept by Pat and Mat.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
已知 $a$ 为实数, 且对任意 $k \in[-1,1]$, 当 $x \in(0,6]$ 时, $6 \ln x+x^{2}-8 x+a \leqslant k x$ 恒成立, 则 $a$的最大值是$m-m \ln m$，求$m$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
99 identical balls lie on a table. 50 balls are made of copper, and 49 balls are made of zinc. The assistant numbered the balls. Once a spectrometer test is applied to 2 balls, it allows determining whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What is the minimum number of tests required to determine the material of each ball if all the tests should be performed today?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
An isosceles triangle \(ABC\) has an inscribed rhombus \(DECF\) such that vertex \(E\) lies on side \(BC\), vertex \(F\) on side \(AC\), and vertex \(D\) on side \(AB\). Find the side length of the rhombus if \(AB = BC = 12\) and \(AC = 6\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}