inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$, and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$. The perimeter of hexagon $ABCDEF$ can be expressed as $m +n\sqrt{p}$, where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Given that the function $f(x)= \frac{1}{2}(m-2)x^{2}+(n-8)x+1$ is monotonically decreasing in the interval $\left[ \frac{1}{2},2\right]$ where $m\geqslant 0$ and $n\geqslant 0$, determine the maximum value of $mn$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Let $n$ be a positive integer and let $f_{1}(x), \ldots, f_{n}(x)$ be affine functions from $\mathbb{R}$ to $\mathbb{R}$ such that, amongst the $n$ graphs of these functions, no two are parallel and no three are concurrent. Let $S$ be the set of all convex functions $g(x)$ from $\mathbb{R}$ to $\mathbb{R}$ such that for each $x \in \mathbb{R}$, there exists $i$ such that $g(x)=f_{i}(x)$.

Determine the largest and smallest possible values of $|S|$ in terms of $n$.
(A function $f(x)$ is affine if it is of the form $f(x)=a x+b$ for some $a, b \in \mathbb{R}$. A function $g(x)$ is convex if $g(\lambda x+(1-\lambda) y) \leq \lambda g(x)+(1-\lambda) g(y)$ for all $x, y \in \mathbb{R}$ and $0 \leq \lambda \leq 1$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n}-1}
A zoo has some pandas, paired into mates (in the same zoo). Only 25% of the panda couples get pregnant after mating. They each have one baby, and 2 panda babies are born. How many pandas are there in the zoo?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Kid and Karlsson play a game. Initially they have a square piece of chocolate $2019\times 2019$ grid with $1\times 1$ cells . On every turn Kid divides an arbitrary piece of chololate  into three rectanglular pieces by cells, and then Karlsson chooses one of them and eats it. The game finishes when it‘s impossible to make a legal move. Kid wins if there was made an even number of moves, Karlsson wins if there was made an odd number of moves.Who has the winning strategy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Karlsson

}
Find a six-digit number that is both a perfect square and a perfect cube. If 6 is subtracted from this number, the result is a prime number. What is the original number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{117649}
Determine whether the relative homology group $H_n(M, \partial M)$ is always isomorphic to $\mathbb{Z}$ for an oriented, compact, connected $n$-dimensional smooth manifold $M$ with boundary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Baker Ram is storing cake boxes in cartons that measure 25 inches by some inches by 60 inches. If the measurement of each cake box is 8 inches by 7 inches by 5 inches, and the maximum number of cake boxes that can be placed in each carton is 210, what is the width of the carton?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39.2}
Let  ${(a_n)_{n\ge1}} $  be a sequence with   ${a_1 = 1} $   and   ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$  for all  ${n \ge 1}$ , where  ${\lfloor x \rfloor}$  denotes the greatest integer less than or equal to  ${x}$ . Find all  ${n \le 2013}$  such that  ${a_n}$  is a perfect square
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ n = 1 }
Let $G$ be a Lie group acting smoothly and freely on a smooth manifold $M$. If the action is not necessarily proper, but $M/G$ is a topological manifold, is the dimension of $M/G$ equal to $\dim M - \dim G$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dim M - \dim G}
Given four points $x_1, x_2, x_3, x_4$ on the unit circle $\mathbb{S}^1$ in the plane $\mathbb{R}^2$ such that $\sum_{i=1}^4 x_i = 0$, does this imply that the points form a rectangle or two pairs of antipodal points? Answer "yes" or "no."
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Let $R$ be a commutative ring and $J_1, J_2$ be two non-zero proper ideals of $R$. Suppose there exist $r_1, r_2 \in R$ such that for any $a \in R$, the conditions $r - r_1 \in J_1$ and $r - r_2 \in J_2$ are equivalent to $r \in J_2$ and $r - a \in J_1$. Does this imply that $J_1 + J_2 = R$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{J_1 + J_2 = R}
Let  $N$  be the smallest positive integer  $N$  such that  $2008N$  is a perfect square and  $2007N$  is a perfect cube.  Find the remainder when  $N$  is divided by  $25$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Using the implicit function theorem, determine if the equation $x\cos(x-y) = 0$ implicitly defines $y$ as a function of $x$ near the point $(\pi/2, \pi/2)$. If it does, calculate the derivative $\frac{dy}{dx}$ at $x=\pi/2$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine all positive integers $n$ such that $n^2+3$ divides evenly (without remainder) into $n^4-3n^2+10$ ?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$n \in \{1, 2, 5\}$}
Compute the 100$^{th}$ smallest positive integer $ n $ that satisfies the three congruences

$$
\left\lfloor \frac{n}{8} \right\rfloor \equiv 3 \pmod{4},
$$

$$
\left\lfloor \frac{n}{32} \right\rfloor \equiv 2 \pmod{4},
$$

$$
\left\lfloor \frac{n}{256} \right\rfloor \equiv 1 \pmod{4}.
$$

Here $\left\lfloor \cdot \right\rfloor$ denotes the greatest integer function.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6491}
Determine whether the sequence of functions \( g_n(\alpha) = \int_a^b \alpha(x) \sin(nx) \cos(nx) \, dx \) converges uniformly for all functions \( \alpha \) in the set \( X = \{ \alpha : [a,b] \to \mathbb{R} \} \), where \( \alpha'' \) exists and is continuous, and there exists a constant \( K > 0 \) such that \( |\alpha(x)|, |\alpha''(x)| \le K \) for all \( x \in [a,b] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
在一条公路上每隔100千米有一个仓库，共有五个仓库，一号仓库存有10吨货物，二号仓库有20吨货物，五号仓库存有40吨货物，其余两个仓库是空的。现在想把所有的货物集中存放在一个仓库里，如果每吨货物运输一千米需要一元的运费，那么最少要花多少运费？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
Let $T_1$ be a triangle with sides $2011$, $2012$, and $2013$. For $n \geq 1$, if $T_n = \Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\Delta ABC$ to the sides $AB$, $BC$, and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE$, and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $\left(T_n\right)$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{1509}{128}$}
Determine whether the quantity \( \frac{\log(\|A^{-1}\|)}{\log(\|A\|)} \) is bounded by a constant that depends only on the size \( n \) of the matrix, as \( A \) ranges over \( SL(n,\mathbb{R}) \) (the determinant one matrices).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 1}
Let $A = \{1, 2, \cdots, 2002\}$ and $M = \{1001, 2003, 3005\}$. For any non-empty subset $B$ of $A$, if the sum of any two elements in $B$ is not in $M$, then $B$ is called an $M$-free set. If $A = A_{1} \cup A_{2}$, $A_{1} \cap A_{2} = \emptyset$, and both $A_{1}$ and $A_{2}$ are $M$-free sets, then the ordered pair $\left(A_{1}, A_{2}\right)$ is called an $M$-partition of $A$. Find the number of all $M$-partitions of $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{501}}
Consider the system of partial differential equations in cylindrical coordinates:
\[ \partial_r b_2 - \partial_{\theta}b_1 = 0, \\
\partial_{\theta} b_3 - \partial_{z}b_2 = 0, \\
\partial_rb_3 - \partial_{z}b_1 = \xi(r, z), \]
where $\xi(r, z)$ is a given function of $r$ and $z$, and $b_1, b_2, b_3$ are unknown functions of $(r, \theta, z)$. Assume that this system has a unique solution. Determine whether $b_2 = 0$ for this solution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b_2 = 0}
Determine the convergence of the integral \( J_n=\int^\infty_0 e^{-t}\big|\cos(nt+c_n)\big|^n\, dt \) as \( n\rightarrow\infty \), where \( c_n \) is an arbitrary numerical sequence. For which \( p>0 \) does \( n^pJ_n \) converge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Let $\Lambda_A$ be a lattice with a basis $A= \{(1,1,1)^T, (1,0,2)^T, (0,1,1)^T\}$. Determine if $\Lambda_B$, with a basis $B= \{(1,2,6)^T, (3,1,3)^T, (3,3,7)^T\}$, is a sublattice of $\Lambda_A$. If it is, calculate the index $[\Lambda_A : \Lambda_B]$. Provide your answer as a single number or state that $\Lambda_B$ is not a sublattice.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the smallest integer \( n \) such that the remainder estimate \( |R_n| \leq \frac{M}{(n+1)!} (x-a)^{n+1} \) is satisfied, where \( M \) is the maximum value of \( |f^{(n+1)}(z)| \) on the interval \([-1,1]\) for the function \( f(x) = e^{-2x} \) expanded around \( a = 0 \). Determine \( M \) and ensure that \( |R_n| \leq \frac{1}{1000} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
If $xf(x)$ approaches a limit as $x$ approaches infinity, does it necessarily follow that $f(x)$ approaches 0? Specifically, if $f: \mathbb{R} \to \mathbb{R}$ satisfies $\lim_{x \to \infty} xf(x) = \lambda$ for some $\lambda \in \mathbb{R}$, does it imply that $\lim_{x \to \infty} f(x) = 0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$ C_{1}$ and $ C_{2}$ be two externally tangent circles with diameter $ \left[AB\right]$ and $ \left[BC\right]$, with center $ D$ and $ E$, respectively. Let $ F$ be the intersection point of tangent line from A to $ C_{2}$ and tangent line from $ C$ to $ C_{1}$ (both tangents line on the same side of $ AC$). If $ \left|DB\right| ={}\left|BE\right|={}\sqrt{2}$, find the area of triangle $ AFC$. 


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$ 2\sqrt{2}$}
Determine if the limit \( \lim_{x\rightarrow\infty}\prod_{p\leqslant x}p^{1/(2x)} \) is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
In 2009, 108 U.S. skydivers fell headfirst from a height of 18,000 ft. How far had they fallen 11 seconds after having jumped from the plane?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1936}
Circles $\omega_1$, $\omega_2$, and $\omega_3$ are centered at $M$, $N$, and $O$, respectively. The points of tangency between $\omega_2$ and $\omega_3$, $\omega_3$ and $\omega_1$, and $\omega_1$ and $\omega_2$ are tangent at $A$, $B$, and $C$, respectively. Line $MO$ intersects $\omega_3$ and $\omega_1$ again at $P$ and $Q$ respectively, and line $AP$ intersects $\omega_2$ again at $R$. Given that $ABC$ is an equilateral triangle of side length $1$, compute the area of $PQR$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2\sqrt{3}$}
Find the Nyquist frequency for the signal \( x(t) = \sum_{n=1}^{3} \sin\left(2\pi \frac{n}{8}\frac{t}{T}\right) \), where \( T > 0 \) is a positive number. Determine the smallest frequency \( u_0 \) such that for all \( u > u_0 \), the Fourier transform \( \hat{x}(u) = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8T}}
Given four positive integers $a < b < c < d$ that satisfy $a + d = b + c = n$, where $n$ is a constant, determine if there exist integers $p, q, r, s$ such that $a = p + q$, $b = p + r$, $c = q + s$, and $d = r + s$. Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
Determine the limit of the characteristic function $\chi_{B(x,r)}$ of the ball $B(x,r)$ as the radius $r$ approaches 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi_{\{x\}}}
Evaluate the integral \( \int_{-\pi}^{\pi} \frac{\cos(x)}{\sqrt{4+3\sin(x)}} \, dx \). Determine if the integral equals zero due to symmetry.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine whether the first continuous cohomology group \(H^1_{cont}(\mathrm{Gal}(L/K), L^{\times})\) vanishes for algebraic extensions \(L/K\) of \(\mathbb{Q}_p\) with \(L^{\times}\) having the \(p\)-adic topology.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
[ Sphere inscribed in a pyramid ]
[Volume helps solve the problem.]

In a triangular pyramid, two opposite edges are equal to 12 and 4, and the other edges are equal to 7. A sphere is inscribed in the pyramid. Find the distance from the center of the sphere to the edge equal to 12.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{13}}{\sqrt{5}+\sqrt{13}}}
Inside the right triangle \(ABC\) (with angle \(B\) being the right angle), a point \(D\) is placed such that the areas of triangles \(ABD\) and \(BCD\) are, respectively, three times and four times smaller than the area of triangle \(ABC\). The segments \(AD\) and \(DC\) have lengths \(a\) and \(c\), respectively. Find \(BD\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\frac{8c^2 + 3a^2}{35}}}
19 In a carriage, any $m(m \geqslant 3)$ passengers have a unique common friend (if A is a friend of B, then B is also a friend of A, and no one is a friend of themselves), how many friends does the person with the most friends have in this carriage?


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
Determine whether the integral \( \int_0^\infty \frac{xe^{-bx}}{\sqrt{x^2 +a^2}}dx \) can be solved analytically, where \(a\) and \(b\) are constants.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a K_1(a b)}
A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of each square is 24, and the perimeter of each small rectangle is 16. What is the perimeter of the large rectangle?

The perimeter of a figure is the sum of the lengths of all its sides.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52}
Given a set of linearly independent $d \times d$ complex unitary matrices $\{U_i\}_{i=1}^n \subseteq M_d$ with $n \geq d$, determine if there exists a vector $v \in \mathbb{C}^d$ such that the span of $\{U_1v, U_2v, \ldots , U_nv\}$ is $\mathbb{C}^d$. Answer with 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
If each of the three operation signs, $+$, $-$, $\times $, is used exactly ONCE in one of the blanks in the expression

\[5\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}4\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}6\hspace{1 mm}\underline{\hspace{4 mm}}\hspace{1 mm}3\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
19
}
Alice is thinking of a number. She gives the following 3 clues. "My number has 243 as a factor. My number is a multiple of 36. My number is between 1000 and 3000." What is Alice's number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2916}
What is the maximal number of solutions can the equation have   $$ \max  \{a_1x+b_1, a_2x+b_2, \ldots, a_{10}x+b_{10}\}=0 $$  where  $a_1,b_1, a_2, b_2, \ldots , a_{10},b_{10}$  are real numbers, all  $a_i$  not equal to  $0$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For a linear operator $T$ on $\mathbb{R}^5$ where $T^4 \neq 0$ and $T^5 = 0$, determine the dimensions of the range of $T^i$ for $i = 1, 2, 3, 4$. Provide your answers as a list of dimensions for each power.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[4, 3, 2, 1]}
Calculate the probability of being in state $3$ after $500,000$ steps in a Markov chain, given that the chain starts in state $1$. Assume the transition probabilities are defined as in the original problem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Determine whether the zeros of a holomorphic function can have an accumulation point on the boundary of its domain. Answer 'yes' or 'no'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{yes}
In the Cartesian coordinate system \( xOy \), the area of the region corresponding to the set of points \( K = \{(x, y) \mid (|x| + |3y| - 6)(|3x| + |y| - 6) \leq 0 \} \) is ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Given a positive integer \( k \), find the largest positive integer \( m \) such that if no more than \( m \) of any set of \( 3k \) points in the plane are collinear, then the points can be divided into \( k \) sets of three points each, where the points in each set form the vertices of a triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2k}
How many numbers not more than 100 whose number at tens place and ones place is different have 0, 1, 7, and 9?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine whether the function \( f(x, y) = \frac{xy}{\sqrt{x} + y^2} \) for \( x, y \neq 0 \) and \( f(0, 0) = 0 \) has a continuous limit at the origin \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. The difference in their money in dimes is:

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$10(q-1)$}
The parabola $y = x^2$ is tangent to the graph of $y = x^4 + ax^3 + x^2 + bx + 1$ at two points.  Find the positive difference between the $x$-coordinates of the points of tangency.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6}}
Determine the exact value of $a > 1$ for which the periodic function $g$ defined by $g(x) = f(x)$ for $0 \le x < a$ and $g(x) = g(x+a)$ for all $x \in \mathbb{R}$ is continuous.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
For a given positive integer \( k \) denote the square of the sum of its digits by
 \( {f}_{1}\left( k\right) \) and let \( {f}_{n + 1}\left( k\right)  = {f}_{1}\left( {{f}_{n}\left( k\right) }\right) \) . Determine the value of \( {f}_{1991}\left( {2}^{1990}\right) \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{

The value of \( f_{1991}(2^{1990}) \) is 256.
}
Find the splitting field $K$ of the polynomial $x^{12} - 9$ over the field of rational numbers $\mathbb{Q}$, and determine the degree of the extension $[K: \mathbb{Q}]$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
5. Let $A B C D E F$ be a regular hexagon. A frog starts at vertex $A$ and can randomly jump to one of the two adjacent vertices each time. If it reaches point $D$ within 5 jumps, it stops jumping; if it does not reach point $D$ within 5 jumps, it stops after 5 jumps. How many different possible jumping sequences are there from the start to the stop? $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
If $b \gt a \gt 1$, and $3\log _{a}b+2\log _{b}a=7$, then the minimum value of ${a^2}+\frac{3}{{b-1}}$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}+1}
Decrypt the ElGamal encrypted message given the parameters: \( p = 29 \), \( g = 2 \), and a public key of 28. The intercepted encrypted message is: (28, 28), (23, 1), (18, 1), (16, 6), (22, 13). Provide the decrypted message.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28, 1, 28, 6, 13}
The Pell numbers $P_n$ satisfy $P_0 = 0$, $P_1 = 1$, and $P_n=2P_{n-1}+P_{n-2}$ for $n\geq 2$. Find $$\sum \limits_{n=1}^{\infty} \left (\tan^{-1}\frac{1}{P_{2n}}+\tan^{-1}\frac{1}{P_{2n+2}}\right )\tan^{-1}\frac{2}{P_{2n+1}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \left( \tan^{-1} \frac{1}{2} \right)^2 }
Leon has cards with digits from 1 to 7. How many ways are there to glue two three-digit numbers (one card will not be used) so that their product is divisible by 81, and their sum is divisible by 9?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
If $\cos(α + \frac{π}{3}) = -\frac{\sqrt{3}}{3}$, find the value of $\sin α$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6} + 3}{6}}
Compute the Galois group of the polynomial $x^4+12x^3+44x^2+48x+40$ over $\mathbb{Q}$, given that its discriminant is $2^{14}3^27^2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$V_4$}
Find the smallest positive integer $k$ such that $z^{10}+z^{9}+z^{6}+z^{5}+z^{4}+z+1$ divides $z^{k}-1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
(6?3) + 4 - (2 - 1) = 5. To make this statement true, the question mark between the 6 and the 3 should be replaced by
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\div}
Given a stochastic variable $x(t)$ with a steady-state Fokker-Planck distribution $P(x)$, where the expectation value for any function $f(x)$ is given by
$$\overline{(f(x))} = \int dx f(x) P(x),$$
and the stochastic differential equation is $\dot{x}(t) = -ax(t) + b\eta(t)$ with $\overline{\eta(t)} = 0$ and $\overline{\eta(t)\eta(t')} = \delta(t-t')$, calculate $\overline{\dot{x}(t)}$ and determine its relationship to $\overline{x(t)}$. What is the value of $\overline{\dot{x}(t)}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given that vehicles are not allowed to turn back at a crossroads, calculate the total number of driving routes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine whether the coefficient \( m_{a,b,c} \) of the monomial \( x_1^{a+c+1}x_2^{a+b+1}x_3^{b+c+1} \) in the expansion of the polynomial \( f_{a,b,c}(x_1,x_2,x_3) = (x_1-x_2)^{2a+1}(x_2-x_3)^{2b+1}(x_3-x_1)^{2c+1} \) is zero when \( a, b, c \) are pairwise unequal nonnegative integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given the set $X=\{1,2,5,7,11,13,16,17\}$, let $x_{i}$, $x_{j}\in X$. If the equation $x_{i}-x_{j}=k\left(k \gt 0\right)$ has at least three different solutions, write down all possible values of $k$ as ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4, 6}
In the figure, points $A$, $B$, $C$, and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$, and segment $CD$ measures $4$. The circumferences with centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length of segment $AD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}