inclusionAI/Ring-lite-rl-data huggingface dataset:
Given the parabola $C$: ${y}^{2}=2px (p > 0)$ with focus $F$, latus rectum $l$: $x=-\frac{5}{4}$, point $M$ on parabola $C$, point $A$ on latus rectum $l$. If $MA \perp l$, and the angle of inclination of line $AF$ is $\frac{\pi}{3}$, then $|MF|=$ ________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $f'(x)$ be the derivative of the function $f(x)$ on $\mathbb{R}$, and for $\forall x \in \mathbb{R}$, $f(-x) + f(x) = x^2$, and for $\forall x \in [0, +\infty)$, $f'(x) > x$. If $f(2-a) - f(a) \geq 2 - 2a$, then the range of the real number $a$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 1]}
Determine the number of idempotents in the ring $\mathbb{Z}[\sqrt{-11}] / (1+\sqrt{-11})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
How many distinct binary codes with parameters $(n, 2, n)$ exist for $n \geq 2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
A painting $18$" X $24$" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the frame area equals that of the painting itself, the ratio of the smaller to the larger dimension of the framed painting is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2:3}
Given vectors $\overrightarrow{a}=(2,1)$ and $\overrightarrow{b}=(\sin α-\cos α,\sin α+\cos α)$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, calculate the value of $\cos 2α+\sin 2α$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{7}{5}}
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\frac{{\sin B \sin C}}{{3 \sin A}} = \frac{{\cos A}}{a} + \frac{{\cos C}}{c}$, and the area of $\triangle ABC$ is ${S_{\triangle ABC}} = \frac{{\sqrt{3}}}{4}({a^2} + {b^2} - {c^2})$, then the range of $\frac{c}{{a+b}}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[\frac{1}{2}, 1\right)}
If \( x \in \left[-\frac{5 \pi}{12}, -\frac{\pi}{3}\right] \), what is the maximum value of \( y = \tan \left( x + \frac{2 \pi}{3} \right) - \tan \left( x + \frac{\pi}{6} \right) + \cos \left( x + \frac{\pi}{6} \right) \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{6} \sqrt{3}}
Four vehicles were traveling on the highway at constant speeds: a car, a motorcycle, a scooter, and a bicycle. The car passed the scooter at 12:00, encountered the bicyclist at 14:00, and met the motorcyclist at 16:00. The motorcyclist met the scooter at 17:00 and caught up with the bicyclist at 18:00. 

At what time did the bicyclist meet the scooter?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15:20}
Suppose that $X_1 , X_2 , X_3$ are independent $U (0, 1)$-distributed random variables and let $(X_{(1)} , X_{(2)} , X _{(3)} )$ be the corresponding order statistic. Compute $\mathbb{P}\{X_{(1)}+X_{(2)} > X_{(3)}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5}
Danny collects bottle caps and wrappers. He found 46 wrappers and 50 bottle caps at the park. Now he has some bottle caps and 52 wrappers in his collection. He found 4 more bottle caps than wrappers at the park. How many bottle caps does Danny have in his collection now?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Half of the population of the island of Nevezhenia consists of hares, and all the rest are rabbits. If a resident of the island asserts something, he always sincerely believes what he is saying. At the same time, hares are mistaken in an average of every fourth case, whereas rabbits are mistaken in an average of every third case.

One day, an animal emerged in the center of the island and shouted: "I am not a hare!" Then thought for a moment and said sadly: "I am not a rabbit."

What is the probability that it actually is a hare?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{59}}
What is the cardinality of the quotient ring $\frac{\mathbb{Z}_5[X,Y]}{(X^2+4Y^4)}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
A store offers an initial discount of 25% on all its merchandise. Subsequently, an additional discount of 10% is applied to the already reduced prices. The store advertises that the final price of the goods is 35% less than the original price. What is the actual percentage difference between the store's claimed discount and the true discount from the original price?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5\%}
Define the *bigness*of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer  $N$  for which there exists a rectangular prism with integer side lengths and *bigness* $N$  and another one with integer side lengths and *bigness* $N + 1$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55}
Let $X\sim N(0,\sigma^2)$ and $C>0$. For each $r>0$ define the interval centered at $C$ with radius $r$; that is $I_r:=[C-r,C+r]$. Let $E[X|I_r]$ be the conditional expectation of $X$ on $I_r$. What is the radius $r$ which maximizes $E[X|I_r]$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the domain of the function $y=\tan ( \frac {π}{2}x+ \frac {π}{3})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{x|x\neq 2k+ \frac {1}{3},k\in\mathbb{Z}\}}
A sequence of natural numbers $\left\{x_{n}\right\}$ is constructed according to the following rules:
$$
x_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \text{ for } n \geq 1.
$$

It is known that some term in the sequence is 1000. What is the smallest possible value of $a+b$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Bob Barker decided to raise the intellectual level of The Price is Right by having contestants guess how many objects exist of a certain type, without going over. The number of points you will get is the percentage of the correct answer, divided by 10, with no points for going over (i.e., a maximum of 10 points).

The first object for our contestants is a table of shape \((5,4,3,2,1)\). This table is an arrangement of the integers 1 through 15 with five numbers in the top row, four in the next, three in the next, two in the next, and one in the last, such that each row and each column is increasing (from left to right, and top to bottom, respectively). For instance:
```
6 
13 14
15
```
is one such table. How many tables are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{292864}
Solve the indefinite integral: $$\int \frac{2}{(x^2+2)\sqrt{x^2+4}} dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\tan^{-1}\left(\frac{x}{\sqrt{x^2+4}}\right)+C}
A function f(x) defined on R satisfies f(0)=0, f(x)+f(1-x)=1, f(x/5)=1/2f(x), and f(x1) ≤ f(x2) when 0 ≤ x1 < x2 ≤ 1. Find the value of f(1/2007). Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/32}
In the sequence $\left\{a_{n}\right\}$, $a_{1}=2 \sqrt{3}$ and $a_{n-1}=\frac{8 a_{n}}{4-a_{n}^{2}}$ for $n \geq 2$, find $a_{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \tan \frac{\pi}{3 \cdot 2^{n-1}}}
A positive integer with \( n+3 \) digits, \( 144\cdots430 \) (with \( n \) number of 4s), is a multiple of 2015. What is the smallest value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Consider triangle \(ABC\) where \(BC = 7\), \(CA = 8\), and \(AB = 9\). \(D\) and \(E\) are the midpoints of \(BC\) and \(CA\), respectively, and \(AD\) and \(BE\) meet at \(G\). The reflection of \(G\) across \(D\) is \(G'\), and \(G'E\) meets \(CG\) at \(P\). Find the length \(PG\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{145}}{9}}
Given that $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$, and $\sin^2 \alpha = \cos(\alpha - \beta)$, determine the relationship between $\alpha$ and $\beta$. Express your answer as a direct comparison between $\alpha$ and $\beta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > \beta}
An ant starts out at $(0,0)$. Each second, if it is currently at the square $(x, y)$, it can move to $(x-1, y-1),(x-1, y+1),(x+1, y-1)$, or $(x+1, y+1)$. In how many ways can it end up at $(2010,2010)$ after 4020 seconds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\binom{4020}{1005}^{2}$}
What is the largest integer that must divide the product of any 5 consecutive integers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Given the sequence defined by $x_{n+1} = \frac{x_n^2 + x_n + 1}{x_n + 1}$ with $x_1 = \frac{1}{2}$, find the general expression for the sum $\sum_{n=1}^{p}\frac{1}{1+x_n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_{p+1} - \frac{1}{2}}
Find a positive integer that is divisible by 18 and has a square root between 22 and 22.5.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
The pentagon \(PQRST\) is divided into four triangles with equal perimeters. The triangle \(PQR\) is equilateral. \(PTU\), \(SUT\), and \(RSU\) are congruent isosceles triangles. What is the ratio of the perimeter of the pentagon \(PQRST\) to the perimeter of the triangle \(PQR\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5:3}
对任意实数 $a_1, a_2, \cdots, a_5$, 不等式

$$
\min _{1 \leqslant i<j \leqslant 5}\left\{\left(a_i-a_j\right)^2\right\} \leqslant k\left(a_1^2+a_2^2+\cdots+a_5^2\right)
$$


恒成立, 求 $k$ 的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$k$ 的最小值为 $1 / 10$}
Find the absolute value of the difference of single-digit integers $A$ and $B$ such that  $$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c} & & & B& B & A_6\\ & & & \mathbf{4} & \mathbf{1} & B_6\\& & + & A & \mathbf{1} & \mathbf{5_6}\\ \cline{2-6} & & A & \mathbf{1} & \mathbf{5} & \mathbf{2_6} \\ \end{array} $$Express your answer in base $6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1_6}
We have a rectangular piece of paper. We want to produce 20-sided polygons using the following procedure. In each step, we select a piece of paper (initially, only the starting piece of paper is available), and cut it into two along a straight line. Continuing this process, what is the minimum number of cuts needed to obtain at least one hundred 20-sided polygons?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1699}
The sum of the largest number and the smallest number of a triple of positive integers $(x,y,z)$ is called the power of the triple. Find the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7290}
Fix a sequence  $ a_1,a_2,a_3,... $  of integers satisfying the following condition:for all prime numbers  $ p $  and all positive integers  $ k $ , we have  $ a_{pk+1}=pa_k-3a_p+13 $ .Determine all possible values of  $ a_{2013} $ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
If the function $$f(x)=(2m+3)x^{m^2-3}$$ is a power function, determine the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Given three different numbers are randomly selected from the set $\{-2, -1, 0, 0, 3, 4, 5\}$ and multiplied together, calculate the probability that the product is $0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{7}}
Given $a>0$, let proposition $p$: the function $y=a^x$ is monotonically increasing on $\mathbb{R}$; proposition $q$: the inequality $ax^2-ax+1>0$ holds for all $x \in \mathbb{R}$. If both $p$ and $q$ are false, and either $p$ or $q$ is true, find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0,1] \cup [4,+\infty)}
Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ equals $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$122$}
A driver goes on a trip of 100 kilometers divided into three segments. The first segment is 40 kilometers at a speed of 50 kilometers per hour, the second segment is 35 kilometers at a speed of 30 kilometers per hour, and the third segment is 25 kilometers at a speed of 60 kilometers per hour. What is the average speed of the entire trip in kilometers per hour?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41.96}
In the right triangle \( \triangle ABC \) with right angle at \( C \), the altitude \( CH \) and the median \( CM \) trisect the right angle. If the area of \( \triangle CHM \) is \( k \), then what is the area of \( \triangle ABC \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4k}
What digit should be placed instead of the question mark in the number 888...88?99...999 (with the eights and nines written 50 times each) so that it is divisible by 7?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
When the Black King is asleep, he judges everything incorrectly. In other words, everything the Black King considers true in his sleep is actually false, and vice versa. On the other hand, when the Black King is awake, he judges everything correctly, considering what is true as true, and what is false as false. Yesterday evening at exactly ten o'clock, the Black King believed that he and the Black Queen were already sleeping at that time.

Was the Black Queen asleep or awake at ten o'clock yesterday evening?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Awake}
Given the sequence \(\left\{a_{n}\right\}\) defined by \(a_{n}=(\sqrt{2}+1)^{n}-(\sqrt{2}-1)^{n}(n \in \mathbb{N})\), using \([x]\) to denote the greatest integer less than or equal to the real number \(x\), what is the units digit of \(\left[a_{2017}\right]\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that the domain of the function $f(x)$ is $R$, and $f(x) > f'(x) + 1$, $f(0) = 3$, find the solution set of the inequality $f(x) > 2e^{x} + 1$. Express your answer as an interval or a description of the solution set.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 0)}
Express the area of the quadrilateral determined by the midpoints of the segments \( AB \), \( BE \), \( FC \), and \( DG \), in terms of \( a \) and \( b \), given that \( A B C D \) is a square with side length \( a \), and \( B E F G \) is a square with side length \( b \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{a+b}{2}\right)^2}
Given that ${{a_n}}$ is an arithmetic sequence with a non-zero common difference, and ${{a_9}}$, ${{a_1}}$, ${{a_5}}$ form a geometric sequence, and ${{a_1} + 3{{a_5}} + {{a_9}} = 20}$, determine the value of ${{a_{13}}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
In the diagram, square \(PQRS\) has side length 40. Points \(J, K, L,\) and \(M\) are on the sides of \(PQRS\), so that \(JQ = KR = LS = MP = 10\). Line segments \(JZ, KW, LX,\) and \(MY\) are drawn parallel to the diagonals of the square so that \(W\) is on \(JZ\), \(X\) is on \(KW\), \(Y\) is on \(LX\), and \(Z\) is on \(MY\). What is the area of quadrilateral \(WXYZ\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
Given that the asymptotic line of the hyperbola $\frac{x^2}{a}+y^2=1$ has a slope of $\frac{5π}{6}$, determine the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Evaluate the double integral $$\iint_R ye^{xy}~\mathrm{d}A$$ over the rectangular region defined by $0\le x\le 2$ and $0 \le y \le 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{1}{2}\left(e^2-3\right)$}
给定正整数k≤m≤n，求

$\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { i } \frac { 1 } { n + k + i } \cdot \frac { ( m + n + i ) ! } { i ! ( n - i ) ! ( m + i ) ! }$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How many Young diagrams exist with at most $p$ rows and $q$ columns, without any restrictions on the weight of the diagrams?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{p+q}{p}}
In a right-angled triangle $ABC\left(\angle C=90^{\circ}\right)$ with side $BC=a$, points $M$ and $N$ are the midpoints of sides $AB$ and $BC$, respectively. The angle bisector of $\angle A$ intersects the line $MN$ at point $L$. Find the radius of the circumcircle of triangle $ACL$ if the angle $\angle CBL=\alpha$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a}{2 \sin(2\alpha)}}
There are 5 tables in a classroom. Each table has 4 chairs with a child sitting on it. All the children get up and randomly sit in a seat. Two people that sat at the same table before are not allowed to sit at the same table again. Assuming tables and chairs are distinguishable, if the number of different classroom arrangements can be written as $2^a3^b5^c$, what is $a+b+c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Given the ellipse \( C: \frac{x^{2}}{4}+y^{2}=1 \) with the upper vertex \( M \) and the lower vertex \( N \), \( T(t, 2) (t \neq 0) \) is a point on the line \( y=2 \). The lines \( TM \) and \( TN \), passing through point \( T \), intersect the ellipse \( C \) at points \( E \) and \( F \), respectively. If the area of triangle \( TMN \) is \( k \) times the area of triangle \( TEF \), find the value of \( t \) that maximizes \( k \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 2\sqrt{3}}
Person 1 and Person 2 enter a queue independently. Person 1's waiting time is exponentially distributed with rate $\theta$, while Person 2's waiting time is exponentially distributed with rate $\mu$. If Person 1 enters the queue $s$ minutes before Person 2, what is the probability that Person 2 leaves the queue before Person 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1-\frac{\theta  e^{-\mu  s}}{\theta +\mu }}
In triangle $XYZ$, angle $XZY$ is $45$ degrees, and angle $ZYX$ is $80$ degrees. Let $P$ be the foot of the perpendicular from $X$ to $YZ$, $Q$ the center of the circle circumscribed around triangle $XYZ$, and $R$ the other end of the diameter which goes through $X$. Determine the measure of angle $XRP$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
The numbers 407 and 370 equal the sum of the cubes of their digits. For example, \( 4^3 = 64 \), \( 0^3 = 0 \), and \( 7^3 = 343 \). Adding 64, 0, and 343 gives you 407. Similarly, the cube of 3 (27), added to the cube of 7 (343), gives 370.

Could you find a number, not containing zero and having the same property? Of course, we exclude the trivial case of the number 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{153}
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form \frac{a\sqrt{b}- c}{d}, please provide the value of a + b + c + d.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
16. $[7]$ Let $\mathbb{R}$ be the set of real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that for all real numbers $x$ and $y$, we have
$$
f\left(x^{2}\right)+f\left(y^{2}\right)=f(x+y)^{2}-2 x y \text {. }
$$

Let $S=\sum_{n=-2019}^{2019} f(n)$. Determine the number of possible values of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2039191}
From the set \( M = \{1, 2, \cdots, 2008\} \) of the first 2008 positive integers, a \( k \)-element subset \( A \) is chosen such that the sum of any two numbers in \( A \) cannot be divisible by the difference of those two numbers. What is the maximum value of \( k \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{670}
You have a circular necklace with  $10$  beads on it, all of which are initially unpainted. You randomly select  $5$  of these beads. For each selected bead, you paint that selected bead and the two beads immediately next to it (this means we may paint a bead multiple times). Once you have finished painting, what is the probability that every bead is painted?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{42}}
In how many ways can Alice, Bob, Charlie, David, and Eve split $18$ marbles among themselves so that no two of them have the same number of marbles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
Uncle Chernomor assigns 9 or 10 of his thirty-three bogatyrs to duty every evening. What is the minimum number of days required so that all the bogatyrs have been on duty the same number of times?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In triangle $PQR$, $\angle Q$ is a right angle, and the altitude from $Q$ meets $\overline{PR}$ at $S$. The lengths of the sides of triangle $PQR$ are integers, and $PR=3^{5}, PS=3^3$. If $\tan P = m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
设 S=\{1,2,\cdots,98\}. 求最小正整数 n，使得 S 的任一 n 元子集中都可以选出 10 个数，无论怎样将这 10 个数均分成两组，总有一组中存在一个数与另外 4 个数都互质，而另一组中总有一个数与另外 4 个数都不互质.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
5. Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, a sphere is constructed with vertex $A$ as the center and $\frac{2 \sqrt{3}}{3}$ as the radius. The length of the curve formed by the intersection of the sphere's surface with the cube's surface is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\sqrt{3}\pi}{6}}
Compute the integral $$I = \int_1^\infty \log^2 \left(1-\frac{1}{x}\right) \, dx$$ using a suitable substitution and appropriate techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi^2}{3}\)}
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$20$}