inclusionAI/Ring-lite-rl-data huggingface dataset:
3. Find the maximum number of rectangles of dimensions $1 \times 10 \sqrt{2}$ that can be obtained from a rectangle of dimensions $50 \times 90$, if cutting along lines parallel to the sides of the given rectangle is allowed.

(Yugoslavia)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{315}
将 $1,2,3,4,5,6,7,8$ 分别放在正方体的八个顶点上，使得每一个面上的任意三个数之和均不小于 10. 求每一个面上四个数之和的最小值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In an ellipse \(\Gamma\), \(F\) is a focus, and \(A\) and \(B\) are two vertices. Given that \(|FA|=3\) and \(|FB|=2\), determine all possible values of \(|AB|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let  $n = 2188 = 3^7+1$  and let  $A_0^{(0)}, A_1^{(0)}, ..., A_{n-1}^{(0)}$  be the vertices of a regular  $n$ -gon (in that order) with center  $O$  . For  $i = 1, 2, \dots, 7$  and  $j=0,1,\dots,n-1$ , let  $A_j^{(i)}$  denote  the centroid of the triangle \[ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)}. \] Here the subscripts are taken modulo  $n$ . If \[ \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \] for relatively prime positive integers  $p$  and  $q$ , find  $p+q$ .

*Proposed by Yang Liu*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2188}
Determine the minimum degree \( n \) of the Taylor polynomial centered at \( \pi/2 \) that approximates \( \sin(1) \) with an error of less than \( 10^{-17} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
What is the term for a structure similar to a semiring, but where associativity is not required for multiplication?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{non-associative semiring}
What is the probability of having 2 adjacent white balls or 2 adjacent blue balls in a random arrangement of 3 red, 2 white, and 2 blue balls? Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{21}}
In a football tournament with 15 teams, each team played every other team exactly once. A victory awarded 3 points, a draw awarded 1 point, and a loss awarded 0 points. After the tournament, it was found that 6 teams each scored at least $N$ points. What is the maximum integer value of $N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34}
Find the variance of the cumulative distribution function of a normal distribution, where $X \sim N(\mu,\sigma^2)$ and $\Phi(x)$ is the cumulative distribution function of this normal distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Find the volume of the region bounded by the planes $4x+2y+4z=6$, $y=x$, $x=0$, and $z=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
In triangle $A B C \angle C A B=75^{\circ}, \angle A B C=45^{\circ}$. On side $C A$ a point $K$ is taken, and on side $C B$ a point $M$, $C K: A K=3: 1$.

Find $K M: A B$, if this ratio is less than $3 / 4$, and line $M K$ cuts off from triangle $A B C$ a triangle similar to it.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3(\sqrt{3}-1)}{4}}
In acute $\triangle ABC,$ let $I$ denote the incenter and suppose that line $AI$ intersects segment $BC$ at a point $D.$ Given that $AI=3, ID=2,$ and $BI^2+CI^2=64,$ compute $BC^2.$

[i]Proposed by Kyle Lee[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{272}{3}}
Given that \( \frac{\cos x+\cos y+\cos z}{\cos(x+y+z)}=\frac{\sin x+\sin y+\sin z}{\sin(x+y+z)}=k \), find the value of \( \cos(x+y)+\cos(y+z)+\cos(z+x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k}
In triangle ABC, the magnitudes of the sides are a, b, and c, as shown, with c ≤ b ≤ a. Through interior point P and the vertices A, B, C, lines are drawn meeting the opposite sides in A', B', C', respectively. Let s = AA' + BB' + CC'. Find the upper bound for s for all positions of point P. Express your answer as a single expression involving a, b, and c.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a+b+c}
Given that $A$, $B$, and $C$ are angles of a triangle, and the equations $5\sin(A) + 12\cos(B) = 15$ and $12\sin(B) + 5\cos(A) = 2$ hold, find the measure of angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
$240$ students are participating a big performance show. They stand in a row and face to their coach. The coach askes them to count numbers from left to right, starting from $1$ . (Of course their counts be like $1,2,3,...$)The coach askes them to remember their number and do the following action:
First, if your number is divisible by $3$  then turn around.
Then, if your number is divisible by $5$  then turn around.
Finally, if your number is divisible by $7$  then turn around.
What is the number of the $66^{\text{th}}$ student counting from left who is face to coach?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{118}
Over the course of a week, Trevor was tasked to collect fresh eggs from the family's 8 chickens. Each chicken laid a different number of eggs everyday. On Monday, Gertrude laid 4 eggs, Blanche laid 3 eggs, Nancy laid 2 eggs, Martha laid 3 eggs, Ophelia laid 5 eggs, Penelope laid 1 egg, Quinny laid 3 eggs, and Rosie laid 4 eggs. On Tuesday, Gertrude laid 2 more eggs than she did on Monday, Blanche laid the same number of eggs as she did on Monday, Nancy laid 1 fewer egg than she did on Monday, Martha laid 1 more egg than she did on Monday, Ophelia laid 2 fewer eggs than she did on Monday, Penelope laid 2 more eggs than she did on Monday, Quinny laid 2 less eggs than she did on Monday, and Rosie didn't lay any eggs. On Wednesday, each chicken laid the same number of eggs as they did on Monday. Each day, Trevor ate 2 eggs for breakfast. On Monday and Wednesday, he accidentally dropped 3 eggs each day. On Tuesday, he sold half of the collected eggs. How many eggs did Trevor have by the end of Wednesday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Evaluate the limit: $$\lim_{x \to 0^{+}} \left[\frac{\exp(\sin x \log x) - \exp(x \log \sin x)}{x^3} + \frac{\log x}{6}\right]$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
A $15 \times 15$ chessboard is completely covered using $3 \times 3$ and $2 \times 2$ plates. The plates do not overlap, and they do not extend beyond the edges of the chessboard. Determine the minimum number of $3 \times 3$ plates required to achieve this coverage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the number of generalized eigenvectors required to construct the Jordan canonical form for a $4 \times 4$ matrix with a single eigenvalue and two linearly independent eigenvectors.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Among the 40 students, the average score excluding Jimin and Jungkook's scores is 79. If Jimin and Jungkook scored 98 and 100 respectively, what is the average score of Jimin's class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80}
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2$. For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_{0}, x_{1}, x_{2}, \ldots$ defined by $x_{0} = 1$ and
$$x_{n+1} = \frac{x_{n} p(x_{n})}{q(x_{n})}$$

for $n \geq 0$. Find all $n$ such that $x_{n} = 1995$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
142
}
Let $\omega (n)$ denote the number of prime divisors of the integer $n>1$.
Find the least integer $k$ such that the inequality $2^{\omega (n) } \leq k \cdot  n^{\frac 1 4}$ holds for all $n > 1.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
k=5
}
Determine the values of \( t \in \mathbb{R} \) for which the function \( f_t: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) defined by the following mappings is a linear transformation:
\[
f_t\left(\begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}\right)=\begin{pmatrix} -1 \\ 2 \\ -4-t \end{pmatrix}, \quad
f_t\left(\begin{pmatrix} 0 \\ 1 \\ t \end{pmatrix}\right)=\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}, \quad
f_t\left(\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}\right)=\begin{pmatrix} 7 \\ 2 \\ 2t+7 \end{pmatrix}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t \neq 1}
Given vector $\vec{b}=(\frac{1}{2}, \frac{\sqrt{3}}{2})$, and $\vec{a}\cdot \vec{b}=\frac{1}{2}$, calculate the projection of vector $\vec{a}$ in the direction of vector $\vec{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
A circle of radius 6 is drawn centered at the origin. How many squares of side length 1 and integer coordinate vertices intersect the interior of this circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{132}
Given points $A(0,0,2)$ and $B(3,4,1)$ in the coordinate system $Oxyz$, find the minimum value of $AX + BY$ where $X$ and $Y$ are points in the $Oxy$ plane such that the distance $XY = 1$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Calculate the value of the integral \( \int_{0}^{0.4}\sqrt{1+x^4} \, dx \) such that the absolute error is less than or equal to \( \frac{0.4^9}{72} \) using the alternating series expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.401024}
Given \( x_{i} \in \mathbf{R}, x_{i} \geq 0 \) for \( i=1,2,3,4,5 \), and \( \sum_{i=1}^{5} x_{i} = 1 \), find the minimum value of \(\max \left\{ x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, x_{4} + x_{5} \right\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Suppose $v_1, ..., v_n$ are linearly independent vectors in $\mathbb{R}^m$. Let $w$ be a linear combination of these vectors such that $w = a_1v_1 + ... + a_nv_n$. If the vectors $v_1, w, v_3, ..., v_n$ are linearly independent, what can we conclude about $a_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a_2 \neq 0}
A  $3 \times 3$  grid of unit cells is given. A *snake of length  $k$* is an animal which occupies an ordered  $k$ -tuple of cells in this grid, say  $(s_1, \dots, s_k)$ . These cells must be pairwise distinct, and  $s_i$  and  $s_{i+1}$  must share a side for  $i = 1, \dots, k-1$ . After being placed in a finite  $n \times n$  grid, if the snake is currently occupying  $(s_1, \dots, s_k)$  and  $s$  is an unoccupied cell sharing a side with  $s_1$ , the snake can *move* to occupy  $(s, s_1, \dots, s_{k-1})$  instead. The snake has *turned around* if it occupied  $(s_1, s_2, \dots, s_k)$  at the beginning, but after a finite number of moves occupies  $(s_k, s_{k-1}, \dots, s_1)$  instead.

Find the largest integer  $k$  such that one can place some snake of length  $k$  in a  $3 \times 3$  grid which can turn around.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In the equation "Xiwangbei jiushi hao $\times$ 8 = Jiushihao Xiwangbei $\times$ 5", different Chinese characters represent different digits. The six-digit even number represented by "Xiwangbei jiushi hao" is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256410}
Find the number \( S \) such that if \( a + b + c + d = S \) and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = S \) (where \( a, b, c, d \) are different from zero and one), then \( \frac{1}{a - 1} + \frac{1}{b - 1} + \frac{1}{c - 1} + \frac{1}{d - 1} = S \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of independent random variables, each uniformly distributed over the interval [0, 1]. Define $Y_n = (\prod^n_{i=1} X_i)^{\frac{1}{n}}$. Use the Central Limit Theorem to approximate $\mathbb{P}(Y_{50} \geq \mathbb{E}(Y_{50}))$. Compute this probability.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.472}
Compute the integral \( \int_C i^* \alpha \) where \( C \) is a smooth, closed curve in \( \mathbb{R}^2 \) that does not pass through the origin, \( \alpha = \frac{xdy-ydx}{x^2 + y^2} \), and \( i: C \to \mathbb{R}^2 \setminus \{(0,0)\} \) is the inclusion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi n}
There were 3 candidates in an election. Two candidates received 4136 and 7636 votes respectively. The winning candidate got a certain percentage of the total votes in that election. If the winning candidate got 49.69230769230769% of the total votes, how many votes did the winning candidate receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11628}
For a given number of digits \(n\), determine the expected number of places in the decimal expansion of \(\pi\) where the first \(n\) digits repeat.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
There are three boats, A, B, and C, in a pond. Boat A can carry 3 people, boat B can carry 2 people, and boat C can carry 1 person. Now, there are 2 adults and 2 children who need to be divided among these boats. For safety reasons, children must be accompanied by an adult to board the boat. Calculate the total number of ways they can be divided among these boats.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the smallest integer \(d\) such that every simple finite planar graph can be oriented so that the out-degree of each vertex is at most \(d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
三棱雉$P-ABC$中，三个侧面与底面所成角相等，三个侧面的面积分别为$3、4、5$，且底面面积为6。已知三棱雉$P-ABC$的外接球的表面积的原始答案形式为 $\frac{m\pi}{n}$，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{82}
Using the 0.618 method to find the optimal addition amount for an experiment, if the current optimal range is $[628, 774]$ and the better point is 718, then the value of the addition point for the experiment at this time is ___.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{684}
Given a complex number $z=a+(a-1)i$ ($a\in \mathbb{R}$, $i$ is the imaginary unit), find the value of $\int_{0}^{a}{\left( \sqrt{1-{x}^{2}}+x \right)dx}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{4}+ \dfrac{1}{2}}
Calculate the limit: $$\lim_{n\rightarrow\infty}\left(\frac{\prod\limits_{r=1}^n\left(2 +\frac{r}{n^2}\right)}{e}\right)^n$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Compute the limit of the recursive sequence defined by $x_0 = a$, $x_1 = b$, and for $n \geq 1$:

$$x_{n+1} = \left(1 - \frac{1}{2n}\right)x_n + \frac{1}{2n}x_{n-1}.$$

Find the value of $\lim_{n \to \infty} x_n$, if it exists.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a + (b-a)e^{-\frac{1}{2}}}
Find the value of the series \( \sum_{n=1}^{\infty} \left( \frac{2}{2n+1} - \frac{1}{n+1} \right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 2 - 1}
Consider a  $5 \times 5$  grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do  $\textbf{not}$  form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares he could have colored red?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Each edge of a graph with $15$ vertices is colored either red or blue in such a way that no three vertices are pairwise connected with edges of the same color. Determine the largest possible number of edges in the graph.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
90
}
Find the first four non-zero terms of the Maclaurin series for the complex function $f(z) = e^{\frac{1}{1-z}}$, centered at $z_0=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e + ez + \frac{3}{2}ez^2 + \frac{13}{6}ez^3}
What is the smallest \( k \) for which one can mark \( k \) cells on a \( 12 \times 12 \) board such that any placement of a four-cell figure \(\square \square\) on the board will touch at least one marked cell? (The figure can be rotated and flipped.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Find the residue of the function \( \frac{1}{e^{e^{\frac{1}{z}}}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{e}}
若集合 $X \subseteq\{1,2, \cdots, n\}$ 满足: 在 $X$ 的任意五元子集 $A$ 中都可以找到两个元素 $a 、 b(a<b)$ 使得 $a \mid b$, 则记 $|X|$ 的最大可能值是 $f(n)$ （ $|X|$ 表示集合 $X$ 中的元素个数）。求$f(2023)+f(3000)+f(10000)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{128}
Given the power series representation of a function $f(x)$ for $x \in (0,1)$:

$$f(x) = x + \frac{2}{1\cdot3}x^3 + \frac{2\cdot4}{1\cdot3\cdot5}x^5 + \frac{2\cdot4\cdot6}{1\cdot3\cdot5\cdot7}x^7 + \cdots$$

If $f\left(\frac{1}{2}\right)$ equals $\frac{\pi}{a\sqrt{b}}$, where $a$ and $b$ are real numbers, what is the absolute value of $a+b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the number of 20-tuples of integers $x_{1}, \ldots, x_{10}, y_{1}, \ldots, y_{10}$ with the following properties: - $1 \leq x_{i} \leq 10$ and $1 \leq y_{i} \leq 10$ for each $i$; - $x_{i} \leq x_{i+1}$ for $i=1, \ldots, 9$; - if $x_{i}=x_{i+1}$, then $y_{i} \leq y_{i+1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{109}{10}}
Determine the upper bound for the length of the shortest word that can distinguish two non-equivalent NFAs, one with $n$ states and the other with $k$ states.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n + k} - 1}
What is the total variation of the Dirac delta function \( \delta(x) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the smallest natural $k$ such that among any $k$ distinct and pairwise coprime naturals smaller than $2018$, a prime can be found.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
A pizza parlor has six different toppings. How many different one- and two-topping pizzas can you order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{21}
In how many ways can 8 distinguishable people, consisting of 2 Americans, 2 Canadians, 2 Mexicans, and 2 Jamaicans, be seated around a round table such that at least two pairs of countrymen are seated together? Consider rotations as identical.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1632}
In a school with $n$ students, each student has exactly 2023 friends. Additionally, any two students who are not friends have exactly 2022 mutual friends. Determine all possible values of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2024}
There are 4 walls in a room, each with different dimensions: the first wall is 10ft by 8ft, the second wall is 10ft by 9ft, the third wall is 15ft by 8ft, and the fourth wall is 15ft by 9ft. The room includes two windows, each with dimensions 3ft by 4ft, and a door with dimensions 3ft by 7ft. The cost of painting each square foot of area is $2. 

Now, the room is being modified, and its length, breadth and height are each increased by 50%. In the new room, an additional door of the same size as before is added, and two additional windows of the same size are also added. What is the cost to paint the new room?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1732.50$}
Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, the minimum value of the sum of the distance from point $P$ to point $Q$ and the distance from point $P$ to the directrix of the parabola is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{17}-1}
Two semicircles, each with radius \(\sqrt{2}\), are tangent to each other. If \( AB \parallel CD \), determine the length of segment \( AD \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{2}}
We say that a $2023$-tuple of nonnegative integers $(a_1,\ldots,a_{2023})$ is \textit{sweet} if the following conditions hold:

- $a_1+a_2+\ldots+a_{2023}=2023$
- $\frac{a_1}{2}+\frac{a_2}{2^2}+\ldots+\frac{a_{2023}}{2^{2023}}\le 1$

Determine the greatest positive integer $L$ so that $a_1+2a_2+\ldots+2023a_{2023}\ge L$ holds for every sweet $2023$-tuple $(a_1,\ldots,a_{2023})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22228}
Find the distance between the vector \( v_2 = \begin{pmatrix} 9 \\ 0 \\ 0 \end{pmatrix} \) and the subspace \( U = (\text{span}(v))^\perp \), where \( v = \begin{pmatrix} \frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The vertices and midpoints of the sides of a regular decagon (thus a total of 20 points marked) are noted.

How many triangles can be formed with vertices at the marked points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1130}
Find the Fourier transform of the Laplacian operator in polar coordinates, given by \( \nabla^2 = \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} \). Assume no angular dependence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-k^2}
Find the volume of the solid defined by the inequality $3x^2 + 2y^2 + z^2 \leq 6$. Express your answer in terms of $
abla$. (Note: $
abla$ is a placeholder for the correct unit or constant factor, if needed.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8\nabla}
A large cube has an edge length of \(2x^2 + 1\). A smaller cube has a volume of \(x^2 - 1\). When the large cube is filled with smaller cubes of the same size, the remaining space has a volume that is 4 cubic units less than the volume of the smaller cube. What is the edge length of the larger cube?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
设 $A$ 是一个 $2 \times 2$ 的可逆矩阵, 且方阵中每个位置的元素在模 3 同余系 $\mathrm{Z} / 3 Z$ 中取值。那么满足 $A^{50} \equiv I_{2}(\bmod 3)$ 方阵 $A$ 的个数为 ？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra, and let $\mathfrak{g}^{(i)}$ denote the derived series of $\mathfrak{g}$, with $\mathfrak{g}^{(0)} = \mathfrak{g}$. Let $\operatorname{rad}(\mathfrak{g})$ be the solvable radical of $\mathfrak{g}$. There exist minimal natural numbers $n$ and $m$ such that $\mathfrak{g}^{(n)} = \mathfrak{g}^{(n+1)}$ and $\operatorname{rad}(\mathfrak{g})^{(m)} = 0$. Determine the relationship between the numbers $n$ and $m$. \(\text{Hint: Consider the properties of solvable and nilpotent Lie algebras.}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n = m}