inclusionAI/Ring-lite-rl-data huggingface dataset:
On a rectangular piece of paper, a picture in the form of a "cross" is drawn using two rectangles \(ABCD\) and \(EFGH\), the sides of which are parallel to the edges of the sheet. It is known that \(AB=9\), \(BC=5\), \(EF=3\), and \(FG=10\). Find the area of the quadrilateral \(AFCH\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{52.5}
Determine the number of terms required in the Maclaurin series for $\ln(1+x)$ to estimate $\ln(1.4)$ with an error less than $0.001$. Provide your answer as the number of terms needed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
How many words are there in a language that are 10 letters long and begin with a vowel, given that the language uses only the letters A, B, C, D, and E, where A and E are vowels, and B, C, and D are consonants, and a word does not include the same letter twice in a row or two vowels in a row?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{199776}
This year, the Second Son of the Gods’ age is 4 times the age of Chen Xiang. Eight years later, the Second Son’s age is 8 years less than three times Chen Xiang's age. How old was Chen Xiang when he saved his mother?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Determine the number of distinct ways to color a 3x4 rectangle using exactly 6 of each of two colors, considering that two colorings are equivalent if they can be obtained from each other by permuting rows or by cyclically permuting columns.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
7. $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns
$$
\begin{array}{l}
a_{11} a_{12} a_{13} a_{14} \cdots a_{1 n} \\
a_{21} a_{22} a_{23} a_{24} \cdots a_{2 n} \\
a_{31} a_{32} a_{33} a_{34} \cdots a_{3 n} \\
\cdots \cdots \cdots \cdots \cdots \\
a_{n 1} a_{n 2} a_{n 3} a_{n 4} \cdots a_{n n}
\end{array}
$$

where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find $a_{11}+a_{22}+a_{23}+\cdots+a_{n n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2-\frac{1}{2^{n-1}}-\frac{n}{2^{n}}}
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior angle of the trapezoid. What is $x$? 
[asy] unitsize(4mm); defaultpen(linewidth(.8pt)); int i; real r=5, R=6;  path t=r*dir(0)--r*dir(20)--R*dir(20)--R*dir(0); for(i=0; i<9; ++i) { draw(rotate(20*i)*t); } draw((-r,0)--(R+1,0)); draw((-R,0)--(-R-1,0)); [/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Given that $a, b \in \mathbb{R}$ and $a \neq 0$, when the polynomial $x^2 - 2ax + a^4$ is divided by $x + b$, the remainder is 1. Additionally, the polynomial $bx^2 + x + 1$ has $ax - 1$ as a factor. Determine the value of $b$. \( \text{(Express your answer as a real number.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Evaluate the surface integral \( \iint_S k \cdot \mathrm{d}S \), where \( S \) is the part of the cylinder \( z = \sqrt{1-x^2} \) that lies above the square with vertices \((-1,-1), (1,-1), (-1,1), (1,1)\) and has an upward orientation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $T:\mathbb{R}^{n} \rightarrow \mathbb{R}$ be a linear function defined as $T(x)=b_{1}x_{1}+\dots +b_{n}x_{n}$. Let $S=\left \{ \overline{x}\in \mathbb{R}^{n}:||x||_{1}\leq1 \right \}$. Determine the range of $T(S)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-M,M]}
Let $a, b, c, d$ be real numbers such that $a + b + c + d = 10$ and $ab + ac + ad + bc + bd + cd = 20$. Find the largest possible value of $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5 + \sqrt{105}}{2}}
Determine the values of $a \in \mathbb{R}$ for which the function \( \frac{1}{n^2+x^{a}+2y^{a}+3z^{a}} \) is integrable over \([0,+\infty)^3\) for all \(n \in \mathbb{N} \cup \{0\}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 3}
Suppose that a sequence $a_0, a_1, \ldots$ of real numbers is defined by $a_0=1$ and \[a_n=\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\cdots+a_0a_{n-1} & \text{if }n\text{ odd}\\a_{n-1}a_1+a_{n-3}a_3+\cdots+a_1a_{n-1} & \text{if }n\text{ even}\end{cases}\] for $n\geq1$. There is a positive real number $r$ such that \[a_0+a_1r+a_2r^2+a_3r^3+\cdots=\frac{5}{4}.\] If $r$ can be written in the form $\frac{a\sqrt{b}-c}{d}$ for positive integers $a,b,c,d$ such that $b$ is not divisible by the square of any prime and $\gcd (a,c,d)=1,$ then compute $a+b+c+d$.

[i]Proposed by Tristan Shin[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1923}
Calculate the line integral \(\int_\gamma \mathbf{f} \cdot d\mathbf{x}\) for the vector field \(\mathbf{f}(x_1, x_2, x_3) = (x_1^2 + 5x_2 + 3x_2x_3, 5x_1 + 3x_1x_3 - 2, 3x_1x_2 - 4x_3)\), where \(\gamma\) is the helix with radius 1 and pitch 1 from \((0, 1, 0)\) to \((0, 1, 2\pi)\). Use the parametrization \(\gamma(t) = (\cos{t}, \sin{t}, t)\) for \(t\) in the interval \([0, 2\pi]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8\pi^2}
There are 10,001 students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of \( k \) societies. Suppose that the following conditions hold:
1. Each pair of students are in exactly one club.
2. For each student and each society, the student is in exactly one club of the society.
3. Each club has an odd number of students. In addition, a club with \( 2m + 1 \) students (where \( m \) is a positive integer) is in exactly \( m \) societies.

Find all possible values of \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5000}
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$, respectively. If ${a}^{2}+{b}^{2}-{c}^{2}=\sqrt{3}ab$, find $\angle C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Define $n!!$ to be $n(n-2)(n-4)\cdots 3\cdot 1$ for $n$ odd and $n(n-2)(n-4)\cdots 4\cdot 2$ for $n$ even. When $\sum\limits_{i=1}^{2009} \frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\dfrac{ab}{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{401}
Consider a deck of cards consisting only of the jack, queen, king, and ace of each suit (clubs, diamonds, hearts, and spades). You start with the four jacks in your hand. In each round, you randomly select one of the four cards in your hand and replace it with one of the three other cards of the same suit, each with equal probability. What is the expected number of swaps needed to return to having all four jacks in your hand?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Determine the codimension of the set \( X = \{ A \in M_{m \times n}(\mathbb{R}) : \mathrm{rk}(A) = r \} \) as a submanifold of \( \mathbb{R}^{m \times n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(m - r)(n - r)}
Let $\gamma_1, \gamma_2, \gamma_3$ be three circles with radii $3, 4, 9,$ respectively, such that $\gamma_1$ and $\gamma_2$ are externally tangent at $C,$ and $\gamma_3$ is internally tangent to $\gamma_1$ and $\gamma_2$ at $A$ and $B,$ respectively. Suppose the tangents to $\gamma_3$ at $A$ and $B$ intersect at $X.$ The line through $X$ and $C$ intersect $\gamma_3$ at two points, $P$ and $Q.$ Compute the length of $PQ.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$PQ = \dfrac{72\sqrt{3}}{7}$}
Determine the condition on \( k \) such that the polynomial \( f(x) = x^4 + 4x^3 - 2x^2 - 12x + k \) has four real roots.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-7, 9]}
Let $z_1, z_2,$ and $z_3$ be complex numbers such that $z_1 + z_2 + z_3 = 2$, $z_1^2 + z_2^2 + z_3^2 = 3$, and $z_1z_2z_3 = 4$. Calculate the value of \( \sum_{i=1}^{3} \frac{1}{z_i^2 - z_i + 4} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{23}{36}}
Calculate the average distance between two randomly chosen points on the circumference of a circle with a large number of vertices (N).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4R}{\pi}}
Find the value of $$ \binom{2003}{1}+\binom{2003}{4}+\binom{2003}{7}+\cdots+\binom{2003}{2002} $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(2^{2003}-2\right) / 3}
Given a Markov chain with transition matrix \( M \), calculate the probability of transitioning from state 1 to state 3 in exactly \( n \) steps.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(M^n)_{1,3}}
Given points A(0,30), B(20,10), and O(0,0), use Pick's theorem to calculate the number of lattice points inside triangle ABO. Pick's theorem states that the area of a polygon with lattice points as vertices is S = N + 1/2L - 1, where N and L represent the number of lattice points inside and on the boundary of the polygon, respectively. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{271}
Determine the smallest non-commutative ring with unity, where "smallest" refers to having the least cardinality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
What is the sum of all possible \( \left( \begin{matrix} i \\  j \end{matrix}\right) \) subject to the restrictions that \( i \geq  {10},j \geq  0 \) ,and \( i + j \leq  {20} \) ? Count different \( i \) ,
 that yield the same value separately - for example,count both \( \left( \begin{matrix} {10} \\  1 \end{matrix}\right) \) and \( \left( \begin{matrix} {10} \\  9 \end{matrix}\right) \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27633

}
Determine the minimum number of open subsets required to form a base for the topological space $(Y, u)$, where $Y=\{p,q,r,s,t\}$ and $u=\{\emptyset,Y,\{p\},\{q\},\{p,q\},\{p,q,r\}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A man can row his boat with the stream at 16 km/h and against the stream at some speed. The man's rate is 4 km/h. What is his speed against the stream?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given two concentric circles with radii \(r\) and \(R\) (where \(R \geq r\)), forming an annulus, and \(n\) equidistant points \(p_0 = p_n, p_1, p_2, \dots, p_n\) on the outer circle, find \(n\) equidistant points \(q_1, \dots, q_n\) on the inner circle such that the triangles \(\Delta(p_i, p_{i+1}, q_i)\) for \(1 \leq i \leq n\) do not overlap and are adjacent to each other. Determine the angle \(\alpha\) that allows for this configuration.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{n}}
Given the vertices of triangle $\triangle ABC$ as $A(2,0)$, $B(0,4)$, and $C(4,1)$, where $AE$ is the angle bisector of $\angle BAC$, calculate the coordinates of a directional vector of line $AE$. Express your answer as a vector in the form $(x, y)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,3)}
已知函数$f(x)=(2-x){{e}^{x}}-ax-a$，若不等式$f(x)>0$恰好存在两个正整数解，则实数$a$的取值范围是__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$[-\frac{{{e}^{3}}}{4}\ ，\ \ 0)$.}
Evaluate the triple integral \( \iiint\limits_R{4y^2ze^{xy}dxdydz} \) where \( R \) is the region bounded by \( x=0,\; y=1,\; y=x,\; z=0,\; z=1. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e - 2}
Given the series $$ S=\sum_{k=0}^{\infty}\frac{1}{2^{2k+1}(2k+1)}\ \ \ (*) $$ I need to find the smallest value of $n$ such that $|S-S_n|<\epsilon=10^{-4}$. I cannot use expansion series for any functions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the measurable space \((X, \mathcal{A}) = ([-1, 1], \mathcal{B}_{[-1, 1]})\) equipped with the Lebesgue measure \(m\), consider the functional \(\phi : \mathcal{L}^1(m) \to \mathbb{R}\) defined by \(\phi(f) = \int (x - \frac{1}{2}) f(x) \, dm(x)\). Determine the norm \(||\phi||\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Determine all values of $k$ for which the series \( \sum_{n=1}^{\infty} \frac{k^n}{n^k} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
Determine the number of bijections \(f: \mathbb{N} \to \mathbb{N}\) such that \(\sum_{n=1}^{\infty} \frac{f(n)}{n^2} < \epsilon\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A sum is lent at compound interest at a certain rate per annum and amounts to Rs. 240 in 2 years. If the sum amounts to Rs. 217.68707482993196, what is the rate of interest per annum?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10.25\%}
Find the limit: \[ \lim_{n \to \infty} \frac{\gamma_{n-1}}{\gamma_{n}} \] where \( \gamma_{n} \) is the \( n \)-th Stieltjes Constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
$F_{1}, F_{2}$ 是粗圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的两个焦点， $P$ 为椭圆上的一点。如果 $\triangle P F_{1} F_{2}$ 的面积为 $1, \tan \angle P F_{1} F_{2}=\frac{1}{2}, \tan \angle P F_{2} F_{1}=-2$ 。请给出原始答案的形式 $a=\frac{\sqrt{m}}{n}$ 中的 $m + n$ 值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Determine the largest real number \( k \) such that the inequality \[ \frac{a^{2}}{b} + \frac{b^{2}}{a} \geq 2 \frac{a^{k+1} + b^{k+1}}{a^{k} + b^{k}} \] holds for all positive real numbers \( a \) and \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In a certain kingdom, the workforce consists only of a clan of dwarves and a clan of elves. Historically, in this kingdom, dwarves and elves have always worked separately, and no enterprise has ever allowed itself to hire both at the same time. The aggregate labor supply of dwarves is given by the function \( w_{\text{dwarves}}^{S} = 1 + \frac{L}{3} \), and the aggregate labor supply of elves is \( w_{\text{elves}}^{S} = 3 + L \). The inverse function of aggregate labor demand for dwarves is \( w_{\text{dwarves}}^{D} = 10 - \frac{2L}{3} \), and the inverse function of aggregate labor demand for elves is \( w_{\text{elves}}^{D} = 18 - 2L \). Recently, a newly enthroned king became very concerned that the wage rates of his subjects are different, so he issued a law stating that the wages of elves and dwarves must be equal and that workers should not be discriminated against based on their clan affiliation. The king believes that regulatory intervention in wage rates will negatively impact the kingdom's economy overall and mandates that all his subjects behave in a perfectly competitive manner. By how many times will the wage of the group of workers whose wage was lower before the king's intervention increase if the firms in the kingdom are indifferent to hiring elves or dwarves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.25}
In the spring college entrance examination of Shanghai in 2011, there were 8 universities enrolling students. If exactly 3 students were admitted by 2 of these universities, the number of ways this could happen is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
Let $n$ be a fixed natural number. Find all natural numbers $ m$ for which\n\[\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m\]\nis satisfied for every two positive numbers $ a$ and $ b$ with sum equal to $2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$m \leq n + 1$}
Given a non-constant sequence {a_i} that satisfies the recurrence relation a_{i+1}^2 - a_i a_{i+1} + a_i^2 = 0 and a_{i+1}!= a_{i-1} for i = 1, 2, 3,..., n, and the boundary conditions a_1 = a_{n+1} = 1, find the sum of the sequence from i = 1 to n. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Maria ordered a certain number of televisions for the stock of a large store, paying R\$ 1994.00 per television. She noticed that in the total amount to be paid, the digits 0, 7, 8, and 9 do not appear. What is the smallest number of televisions she could have ordered?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Consider a vector space \( V \) over \( \mathbb{Z}_2 \) with dimension 4. Determine the number of linearly independent subsets of \( V \) that contain exactly 3 elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
A rectangle is divided into three smaller congruent rectangles, and each smaller rectangle is similar to the large rectangle. Find the ratio of the length of a longer side to that of a shorter side in each of the rectangles. Express your answer as a simplified ratio (e.g., x:y).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{3} : 1}
Given a square matrix $A$ with real entries, consider a generalized eigenvector $v$ of $A$ with rank $r$, satisfying $(A - \lambda I)^r v = 0$ and $(A - \lambda I)^{r-1} v \neq 0$, where $v \neq 0$, $\lambda$ is an eigenvalue of $A$, and $I$ is the identity matrix. If $v$ is also a generalized eigenvector of $A^n$ with rank $s \leq r$, such that $(A^n - \lambda^n I)^r v = Q(A, \lambda)(A - \lambda I)^r v = 0$ for some polynomial $Q(X, \lambda)$, and $Q[A]$ is invertible on the generalized eigenspace for $\lambda$ in $A$, determine the relationship between $r$ and $s$ when $\lambda \neq 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{s = r}
A mathematics class took a test with 6 true or false questions. Each pair of students had at least three different answers. What is the maximum number of students that could be in the class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
3.50. The radius of the circle inscribed in a right trapezoid is $r$, and the acute angle of the trapezoid is $\alpha$. This trapezoid rotates around the smaller lateral side. Find the lateral surface area of the solid of revolution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\pir^2(1+\sin\alpha)}{\sin^2\alpha}}
What fraction of the area of an isosceles trapezoid $KLMN (KL \parallel MN)$ is the area of triangle $ABC$, where $A$ is the midpoint of base $KL$, $B$ is the midpoint of base $MN$, and $C$ is the midpoint of leg $KN$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Kolya Vasin walked for five hours after school. First, he walked on a horizontal road, then climbed a hill, and finally returned to the starting point along the same route. His speed was 4 km/h on the horizontal section, 3 km/h when climbing the hill, and 6 km/h when descending the hill. What distance did Kolya Vasin cover?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20 \text{ km}}
Consider all pairs of points $(a,b,c)$ and $(d,e,f)$ in the 3-D coordinate system with the condition $ad + be + cf = -2023$. What is the least positive integer that can be the distance between such a pair of points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Find the fraction that equals $0.72\overline{6}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{109}{150}}
Given an isosceles triangle \(ABC\) with \(\angle A = 30^\circ\) and \(AB = AC\). Point \(D\) is the midpoint of \(BC\). Point \(P\) is chosen on segment \(AD\), and point \(Q\) is chosen on side \(AB\) such that \(PB = PQ\). What is the measure of angle \(PQC\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
For what positive integer $A$ is the sequence $(a_k)_{k=0}^\infty$, defined by $a_0 = A$ and $a_{k+1} = a_k + S(a_k)$ for $k \ge 0$, eventually constant? Here, $S(n) = n - m^2$, where $m$ is the greatest integer such that $m^2 \le n$. Determine the positive integer $A$ for which the sequence is eventually constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the multiplicative inverse of the element $x^2 + 1$ in the ring $\Bbb{F}_3[x]/(x^3 + 2x + 2)$. Express your answer as a polynomial in $\Bbb{F}_3[x]$ of degree less than 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2x^2 + 2x + 2}
Determine the number of labeled trees on 7 vertices, where the vertex set is \( V = \{1, 2, 3, 4, 5, 6, 7\} \), and the vertices \( 1, 2, 3, \) and \( 4 \) are exactly the leaves of the tree.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150}
Calculate the expectation of the product of Brownian motion terms: \( E[B(U)B(U+V)B(U+V+W)B(U+V+W+x)] \), given that \( U + V + W > U + V > U \) and \( x > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3U^2 + 3UV + UW}
Determine the smallest cubic graph that contains a square-free 2-factor, where a square-free 2-factor is a 2-factor with no cycles of length 4.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the smallest natural number \( n \) such that it is possible to tile the plane with squares whose side lengths are from the set \( \{1, 2, \ldots, n\} \) in such a way that no two squares with the same side length share a segment of an edge. Squares with the same side length may touch at a vertex, and not all side lengths from 1 to \( n \) need to be used in the tiling.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the class number $h(G)$ for the group $G = \textbf{PGL}_{2,K}$, where $K = F_q(x)$ is the rational function field.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider the 800-digit integer
$$
234523452345 \cdots 2345 .
$$

The first \( m \) digits and the last \( n \) digits of the above integer are crossed out so that the sum of the remaining digits is 2345. Find the value of \( m+n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{130}
Compute the minimal number of generators, \( \mu(mA_m) \), of the ideal \( mA_m \), where \( A=\mathbb{C}[x,y]/(x^3-y^2) \) and \( m=(\bar{x} -1 , \bar{y} -1) \) is a maximal ideal of \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}