inclusionAI/Ring-lite-rl-data huggingface dataset:
Square \(ABCD\) has an area of \(36\), and \(\overline{AB}\) is parallel to the x-axis. Vertices \(A\), \(B\), and \(C\) are on the graphs of \(y = \log_{a}x\), \(y = 2\log_{a}x\), and \(y = 3\log_{a}x\), respectively. What is the value of \(a\)? The original answer is in the form \(k^{m/n}\), please give the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the following product of sequences: $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{2187} \cdot \frac{6561}{1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
If the function $f(x)=2x^{2}-\ln x$ is defined and not monotonic in the interval $(k-1,k+1)$, then the range of values for the real number $k$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1\leq k<\frac{3}{2}}
Factorize the expression $x^5 + x^4 + 1$. Express your answer as a product of factors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x^2 + x + 1)(x^3 - x + 1)}
A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( A=(0,0) \), \( B=(b, 2) \), and \( ABCDEF \) be a convex equilateral hexagon such that \( \angle FAB=120^\circ \). Suppose \( AB \parallel ED \), \( BC \parallel FE \), \( CD \parallel AF \), and the \( y \)-coordinates of \( C, D, E, F \) are some permutation of \( 4, 6, 8, 10 \). If the area of the hexagon can be written in the form \( m \sqrt{n} \), where \( m \) and \( n \) are positive integers and \( n \) is not divisible by the square of any prime, find \( m+n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Given that $f'''(x)$ is continuous everywhere and \(\lim_{x \to 0}(1+x+ \frac{f(x)}{x})^{1/x}=e^5\), find the value of $f^{(4)}(0)$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{500}
If we consider $p$ be a prime number such that $p \equiv 3 \pmod{4}$. How many elements of order $4$ are in $GF(p^n)^*$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
已知正四棱雉 $\Gamma$ 的高为 3, 侧面与底面所成角为 $\frac{\pi}{3}$ 。先在 $\Gamma$ 内放入一个内切球 $O_{1}$, 然后依次放入球 $O_{2}, O_{3}, O_{4}, \cdots$, 使得后放入的各球与前一个球及 $\Gamma$ 的四个侧面均相切，则放入的所有球的体积之和的原始答案为$\frac{m\pi}{n}$，请给出$m + n$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
The sequence starts with 12, 13, 15, 17, 111, followed by a number, then 117, 119, 123, 129, and 131. What is the sixth number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{113}
Consider a sequence of $n$ random variables $X_1, X_2, \ldots, X_n$, not necessarily independent, with the property that the expected value of $X_i$ is greater than that of $X_j$ whenever $i < j$. Determine a lower bound for the probability of the event that $X_1$ is the maximum among all variables, i.e., $P[ (X_1 > X_2) \wedge (X_1 > X_3) \wedge \ldots \wedge (X_1 > X_n) ]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$|x-y|=y-x$, solve for $x$ \_\_\_ $y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x \leq y}
Define: If a positive integer can be expressed as the difference of the squares of two positive integers $m$ and $n$, and $m-n \gt 1$, then this positive integer is called a "difference optimal number". For example, $12=4^{2}-2^{2}$, so $12$ is a "difference optimal number". By using $m^{2}-n^{2}=\left(m+n\right)\left(m-n\right)$ for research, if the "difference optimal numbers" are arranged in ascending order, then the $10$th "difference optimal number" is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Compute  $\left\lceil\displaystyle\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$ . (The notation  $\left\lceil x\right\rceil$  denotes the least integer  $n$  such that  $n\geq x$ .)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019}
Determine the values of \(\alpha\) for which the improper integral \(\int_{\Bbb R^2} \frac{\sin(x^2+y^2)}{(1+x^2+y^2)^\alpha}dxdy\) converges. Use polar coordinates to analyze the convergence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > 0}
Let  $ABCD$  be a square and  $O$  is your center. Let  $E,F,G,H$  points in the segments  $AB,BC,CD,AD$  respectively, such that  $AE = BF = CG = DH$ . The line  $OA$  intersects the segment  $EH$  in the point  $X$ ,  $OB$  intersects  $EF$  in the point  $Y$ ,  $OC$  intersects  $FG$  in the point  $Z$  and  $OD$  intersects  $HG$  in the point  $W$ . If the  $(EFGH) = 1$ . Find: $(ABCD) \times (XYZW)$ Note  $(P)$  denote the area of the polygon  $P$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given $A=\{y\in \mathbb{N}|y=-x^{2}+4x,x\in \mathbb{Z}\}$ and $B=\{x|\ln x > 1\}$, find the intersection of $A$ and the complement of $B$ in $\mathbb{R}$. Express your answer as a set of numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0\}}
Determine the number of Boolean functions F: {0, 1}n → {0, 1} that satisfy the property F(x) ⊕ F(y) = F(x ⊕ y) for all x, y ∈ {0, 1}n, where ⊕ denotes the component-wise exclusive-or operation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
设整数 N\geq2，1=d_1<d_2<\cdots<d_s=N 是 N 的所有正因数. 已知 (d_1,d_2)+(d_2,d_3)+\cdots+(d_{s-1},d_s)=N-2, 求 N 的所有可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
An isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has $AB=13$, $CD=17$, and height $3$. Let $E$ be the intersection of $AC$ and $BD$. Circles $\Omega$ and $\omega$ are circumscribed about triangles $ABE$ and $CDE$. Compute the sum of the radii of $\Omega$ and $\omega$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
How much more is 7/12 of 450 than 3/5 of 320 and 5/9 of 540 combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{229.5}
A runner, two cyclists, and a motorcyclist are moving with constant speeds on a circular highway. The runner and one of the cyclists are moving in one direction, while the motorcyclist and the other cyclist are moving in the opposite direction. The runner meets the second cyclist every 12 minutes, the first cyclist overtakes the runner every 20 minutes, and the motorcyclist overtakes the second cyclist every 5 minutes. How often does the motorcyclist meet the first cyclist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \text{ minutes}}
We have the ring $R=\mathbb{Z}_5[x]$ and the element $b=(x-4)(x-3)^2$ of $R$. Let $A$ be the set of $a$ in $R$ such that $at=b$ has a solution $t\in R$. How can we determine the number of elements of $A$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Let $A$ and $B$ be matrices of order $3\times 2$ and $2\times 3$ respectively. Suppose that their product $AB=\begin{bmatrix} 8 & 2 &-2\\ 2 & 5 &4\\ -2 & 4 &5\\ \end{bmatrix}$ and $BA$ is non-singular. Find $\det(BA)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
A square with an initial side length of $4m$ increases in length at a constant rate of $3m/s$. One corner of the square follows a line $L$ at a speed of $2m/s$ for $5$ seconds, causing the square to rotate one full revolution about $L$ while maintaining a perpendicular orientation to $L$. Assuming the line $L$ is perpendicular to the plane of the square, calculate the volume of the solid generated during this process.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1510}
Find the value of \(a \in \mathbb{R}\), with \(a \neq 1\), for which the following limit holds:
\[\lim_{{n \to \infty}} \frac{{1^a} + {2^a} + \cdots + {n^a}}{{\left( {n + 1} \right)^{a - 1}\left[ {\left( {na + 1} \right) + \left( {na + 2} \right) + \cdots + \left( {na + n} \right)} \right]}} = \frac{1}{60}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Evaluate the integral: $$\int_{0}^{2} \mathrm{~d} z \int_{0}^{\left(2 z-z^{2}\right)^{\frac{1}{2}}} \mathrm{~d} y \int_{0}^{\left(2 z-z^{2}-y^{2}\right)^{\frac{1}{2}}}\left(x^{2}+y^{2}+z^{2}\right)^{-\frac{1}{2}} \mathrm{~d} x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
Let $N$ be a random variable uniformly distributed over the set \( \{0, 1, 2, \ldots, 19\} \). Compute the expected value:
\[ \mathbb{E}\left(\sum_{k=0}^{N} {N-k \choose k} (-1)^k \right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{10}}
Determine the Galois group of the polynomial $x^6 + 1$ over the finite field $\mathbb{F}_2 = \{0, 1\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/2\mathbb{Z}}
A three-digit number $\overline{xyz}$ satisfies $1 \leq x \leq 9$, $0 \leq y, z \leq 9$, and $\overline{xyz} = x! + x! + z!$. Find the value of $x + y + z$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
A man can commute either by train or by bus. If he goes to work on the train in the morning, he comes home on the bus in the afternoon; and if he comes home in the afternoon on the train, he took the bus in the morning. During a total of $x$ working days, the man took the bus to work in the morning 8 times, came home by bus in the afternoon 15 times, and commuted by train (either morning or afternoon) 9 times. Find the total number of working days $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine the cardinality of a set of real numbers such that no number in the set can be obtained by combining the others using the operations of addition, subtraction, multiplication, or division.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
The intersection of two squares with perimeter  $8$  is a rectangle with diagonal length  $1$ . Given that the distance between the centers of the two squares is  $2$ , the  perimeter of the rectangle can be expressed as  $P$ . Find  $10P$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
In an isosceles trapezoid \(ABCD\) with \(AB = CD\), an inscribed circle is drawn. Let \(M\) be the point where the circle touches side \(CD\), \(K\) be the intersection point of the circle with segment \(AM\), and \(L\) be the intersection point of the circle with segment \(BM\). Find the value of \(\frac{AM}{AK} + \frac{BM}{BL}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
From the digits 0, 1, 2, 3, 4, 5, 6, select 2 even numbers and 1 odd number to form a three-digit number without repeating digits. The number of such three-digit numbers that are divisible by 5 is ____. (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Let a,b,c, and d be real numbers such that\[a^2 + 3b^2 + \frac{c^2+3d^2}{2} = a + b + c+d-1.\]Find $1000a + 100b + 10c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
527
}
The letter T is formed by placing two $3\:\text{inch}\!\times\!5\:\text{inch}$ rectangles to form a T shape. The vertical rectangle is placed in the middle of the horizontal one, overlapping it by $1.5$ inches on both sides. What is the perimeter of the new T, in inches?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Find the number of group homomorphisms from the multiplicative group of units modulo 12, denoted as $\mathbb{Z}_{12}^{\times}$, into the multiplicative group of units modulo 8, denoted as $\mathbb{Z}_8^{\times}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
For a given \( n \in \mathbb{N} \), determine the values of \( p > 0 \) for which the function \( \frac{1}{|x|^p} \) is integrable over the region \( \mathbb{R}^n \setminus B_r(0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > n}
In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles $A$, $B$, and $C$ respectively. Given the inequality $\frac{b}{{b+c}}+\frac{c}{{a+b}}≥1$, determine the range of angle $A$ in radians.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \frac{\pi}{3}]}
Given $\overrightarrow{a}$ and $\overrightarrow{b}$ are two non-collinear vectors in a plane, $\overrightarrow{AB}=m\overrightarrow{a}+2\overrightarrow{b}$, $\overrightarrow{BC}=3\overrightarrow{a}+m\overrightarrow{b}$. If points $A$, $B$, and $C$ are collinear, then $m=$____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm\sqrt{6}}
If $ a > 1$, then the sum of the real solutions of\[\sqrt{a - \sqrt{a + x}} = x\]is equal to
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{4a - 3} - 1}{2}}
18. Find the sum of all real roots of the equation $3 \tan ^{2} x+8 \tan x+3=0$ in the range $0<x<2 \pi$.求方程 $3 \tan ^{2} x+8 \tan x+3=0$ 在區間 $0<x<2 \pi$ 內所有實根之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5\pi}
Evaluate \( \lim_{n \rightarrow \infty} \int_0^n \left(1 + \frac{-x}{n}\right)^n \cos\left(\frac{x}{\sqrt{n}}\right) e^{x/2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Square $ABCD$ has side length $36$.  Point $E$ is on side $AB$ a distance $12$ from $B$, point $F$ is the midpoint of side $BC$, and point $G$ is on side $CD$ a distance $12$ from $C$.  Find the area of the region that lies inside triangle $EFG$ and outside triangle $AFD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $0.0001$ of the correct answer will be accepted.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
If real numbers $a>b>1$ satisfy the inequality$$(ab+1)^2+(a+b)^2\leq 2(a+b)(a^2-ab+b^2+1)$$what is the minimum possible value of $\dfrac{\sqrt{a-b}}{b-1}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1

}
The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately what fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{9}}
In the tetrahedron \( P-ABC \), \( PA=BC=\sqrt{6} \), \( PB=AC=\sqrt{8} \), and \( PC=AB=\sqrt{10} \). Find the radius of the circumscribed sphere of the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3}}
Given a function $f(x)$ defined on $R$ such that $f(x) + f(x+4) = 23$. When $x \in (0,4]$, $f(x) = x^2 - 2^x$. Find the number of zeros of the function $f(x)$ on the interval $(-4,2023]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{506}
Determine the values of $p$ for which the function \( \rho(x, y) = \int_a^b \lvert x(t) - y(t) \rvert^p \, \mathrm{d} t \) is a metric on the set of continuous functions on \([a, b]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 1]}
Two bullets are fired independently at the origin of an $(x,y)$ coordinate system, and the points hit, say $(X,Y)$ are random variables. The random variables $X,Y$ follow standard Normal. What is the distribution of the distance between the two points hit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Rayleigh with parameter 2}
$174 S-A B C D-S^{\prime}$ is a regular octahedron with edge length 1. Sphere $O$ is outside this octahedron, tangent to face $S A B$, and tangent to the extended planes of faces $S B C$, $S A D$, and $S^{\prime} A B$. Find the radius $r$ of sphere $O$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4} \sqrt{\frac{2}{3}}}
Evaluate the contour integral \( \oint_{|z|=3} \frac{1}{2-z} e^{\frac{1}{z}} \,dz \) by computing the residue at the singularities inside the contour, particularly focusing on the residue at the essential singularity at \( z=0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\pi i}
You are given a square  $n \times n$ . The centers of some of some  $m$  of its  $1\times 1$  cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer  $n \geq 3$ , find the largest value of  $m$  for which it is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n-1}
Determine the prime ideals of the ring \( \mathbb{F}_p[x]/(x^2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(x)}
The shorter leg $AC$ of the right triangle $ABC$ has length $b$. On hypotenuse $AB$, a point $D$ is chosen such that $BD = BC$. On the leg $BC$, a point $E$ is chosen such that $DE = BE = m$. Find the perimeter of the quadrilateral $ADEC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2m + b}
A frog jumps on lily pads numbered 0 to 10. If it reaches pad 0, it dies, and if it reaches pad 10, it survives. When the frog is on pad $N$, where $0 < N < 10$, it jumps to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. Find the probability that the frog reaches pad 10 starting from pad 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{63}{146}}
b takes some more days than a to finish a task. b and a start this task and a leaves the task 12 days before the task is finished. b completes 60% of the overall task. b would have taken 36 days to finish the task if he had worked independently. How many more days does b take than a to finish the task?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the radius of the sphere that touches three faces of a regular tetrahedron with edges of unit length, as well as the three sides of the fourth face.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{8}}
Let \( S = \{1, 2, \cdots, 2025\} \). A subset of \( S \) with 51 elements is called a "good subset" if the sum of its elements is divisible by 5. Find the number of good subsets of \( S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5}\binom{2025}{51}}
Consider the symmetric group $S_5$ of permutations of five symbols. Let $S$ be the set of subgroups of $S_5$ that are isomorphic to the non-cyclic group of order 4. Two subgroups $H$ and $K$ in $S$ are considered equivalent if they are conjugate in $S_5$. Determine the number of equivalence classes in $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a stationary ARMA(1,1) process \(x_t = 0.7 x_{t-1} + u_t + 0.2 u_{t-1}\), where \(u_t\) is white noise with standard deviation \(\sigma(u_t) = 4\), find \( \Bbb Var(x_{t+1}|x_t, u_t)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
How many solutions in real numbers does the equation

$$
\sin (\sin (\sin (\sin (\sin (x)))))=x / 3 ?
$$

have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a room lit by two bulbs, the bulbs are replaced only when both burn out. The lifetimes of the bulbs are independent and exponentially distributed with parameter \( \lambda = 1 \). What fraction of the time is the room only half-lit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
If the function \( f(x) = -\frac{1}{2} x^2 + \frac{13}{2} \) has a minimum value of \( 2a \) and a maximum value of \( 2b \) in the interval \( (a, b] \), find \( [a, b] \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[1, 3]}
From milk with a fat content of $5\%$, cottage cheese with a fat content of $15.5\%$ is produced, while there remains whey with a fat content of $0.5\%$. How much cottage cheese is obtained from 1 ton of milk?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.3}
Consider the Hilbert space $l_2[0,\infty)$ with the right shift operator $S_r$ and the left shift operator $S_l$ defined as follows:

$$S_r (x_0, x_1, x_2, \dots) \mapsto (0, x_0, x_1, \dots)$$
$$S_l (x_0, x_1, x_2, \dots) \mapsto (x_1, x_2, x_3, \dots)$$

The combined operator $T = S_r + S_l$ acts as:

$$T(x_0, x_1, x_2, \dots) \mapsto (x_1, x_0 + x_2, x_1 + x_3, \dots)$$

Describe the spectrum of the operator $T$. Provide your answer as a set of complex numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-2, 2]}
Determine the range of values for $p$ such that the surface area of the solid generated by revolving the function $f(x) = \frac{1}{x^p}$ around the x-axis is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 1}
A square is drawn in the Cartesian coordinate plane with vertices at $(2, 2)$, $(-2, 2)$, $(-2, -2)$, $(2, -2)$. A particle starts at $(0,0)$. Every second it moves with equal probability to one of the eight lattice points (points with integer coordinates) closest to its current position, independently of its previous moves. In other words, the probability is $\frac{1}{8}$ that the particle will move from $(x, y)$ to each of $(x, y + 1)$, $(x + 1, y + 1)$, $(x + 1, y)$, $(x + 1, y - 1)$, $(x, y - 1)$, $(x - 1, y - 1)$, $(x - 1, y)$, or $(x - 1, y + 1)$. The particle will eventually hit the square for the first time, either at one of the $4$ corners of the square or at one of the $12$ lattice points in the interior of one of the sides of the square. The probability that it will hit at a corner rather than at an interior point of a side is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
What is the maximum possible degree of an irreducible polynomial in the ring of polynomials with real coefficients in three variables, \( \mathbb{R}[x_1, x_2, x_3] \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
5. 5.1. Find the largest natural number in which all digits are different and any two adjacent digits differ by 6 or 7.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60718293}
Determine the Krull dimension of the ring $K[t^{-1},t]$, where $K$ is a field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Points $X$, $Y$, and $Z$ lie on a circle with center $O$ such that $XY=12$. Points $A$ and $B$ lie on segment $XY$ such that $OA=AZ=ZB=BO=5$. Compute $AB$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$2\sqrt{13}$}