inclusionAI/Ring-lite-rl-data huggingface dataset:
We say that some positive integer  $m$  covers the number  $1998$ , if  $1,9,9,8$  appear in this order as digits of  $m$ . (For instance  $1998$  is covered by  $2\textbf{1}59\textbf{9}36\textbf{98}$  but not by  $213326798$ .) Let  $k(n)$  be the number of positive integers that cover  $1998$  and have exactly  $n$  digits ( $n\ge 5$ ), all different from  $0$ . What is the remainder of  $k(n)$  on division by  $8$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Marisa has a collection of $2^{8}-1=255$ distinct nonempty subsets of $\{1,2,3,4,5,6,7,8\}$. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process $2^{8}-2=254$ times until there is only one set left in the collection. What is the expected size of this set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1024}{255}}
Lily bought a Russian nesting doll as a souvenir. The biggest doll is 243 cm tall, and each doll is 2/3rd the size of the doll that contains it. How big is a certain numbered doll if the height of that doll is 32 cm?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively so that $AK = CK$, and $\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$, and let $M$ be the point on line $BK$ for which $K$ is the midpoint of $\overline{PM}$. If $AM = 180$, find $LP$.
Diagram
[asy] import markers; defaultpen(fontsize(8)); size(300); pair A=(0,0), B=(30*sqrt(331),0), C, K, L, M, P; C = intersectionpoints(Circle(A,450), Circle(B,300))[0]; K = midpoint(A--C); L = (3*B+2*A)/5; P = extension(B,K,C,L); M = 2*K-P; draw(A--B--C--cycle); draw(C--L);draw(B--M--A); markangle(n=1,radius=15,A,C,L,marker(markinterval(stickframe(n=1),true))); markangle(n=1,radius=15,L,C,B,marker(markinterval(stickframe(n=1),true))); dot(A^^B^^C^^K^^L^^M^^P); label("$A$",A,(-1,-1));label("$B$",B,(1,-1));label("$C$",C,(1,1)); label("$K$",K,(0,2));label("$L$",L,(0,-2));label("$M$",M,(-1,1)); label("$P$",P,(1,1)); label("$180$",(A+M)/2,(-1,0));label("$180$",(P+C)/2,(-1,0));label("$225$",(A+K)/2,(0,2));label("$225$",(K+C)/2,(0,2)); label("$300$",(B+C)/2,(1,1)); [/asy]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7 . Play continues until either player wins, or else indefinitely. If Nathaniel goes first, determine the probability that he ends up winning.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{11}}
A palindrome is a positive integer that reads the same backwards as forwards, such as 82328. What is the smallest 5 -digit palindrome that is a multiple of 99 ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54945}
In the drawing, there is a grid consisting of 25 small equilateral triangles.

How many rhombuses can be formed from two adjacent small triangles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
For each positive integer $k$, let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$. Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$. Determine the number of digits in $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
设集合 $T=\left\{(a, b, c) \mid a, b, c \in \mathbf{N}^{*}\right.$ ，且 $a, b, c$ 可构成某个三角形的边长 $\}$ 。则 $\sum_{(a, b, c) \in T} \frac{2^{a}}{3^{b} \cdot 5^{c}}$ 的值为 \frac{m}{n} 格式，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
Esmeralda writes  $2n$  real numbers  $x_1, x_2, \dots , x_{2n}$ , all belonging to the interval  $[0, 1]$ , around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products  $p_1 = x_1x_2$ ,  $p_2 = x_2x_3$ ,  $\dots$  ,  $p_{2n} = x_{2n}x_1$ . She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left\lfloor \frac{n}{2} \right\rfloor}
Let $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
将方程 $(1+2+\cdots+k)-3(1+2+\cdots+t)=1$ 的全体正整数解 $(k, t)$ 按照从小到大的方式排列，得到序列 $\left\{\left(k_{n}, t_{n}\right)\right\}$ ，则 $k_{6}=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3196}
Given a polynomial equation \(z^{4} + a z^{3} + b z^{2} + c z + d = 0\) with real coefficients and all its roots lying on the unit circle in the complex plane centered at the origin, calculate the sum of the reciprocals of these roots. Express your answer as a single value in terms of the coefficients of the polynomial.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-a}
对 $\mathbb{R}^{100}$ 中的点 $x=\left(x_{1}, x_{2}, \cdots, x_{100}\right)$ 和 $y=\left(y_{1}, y_{2}, \cdots, y_{100}\right)$ 定义函数 $f(x, y)=\sqrt{\sum_{i=1}^{100}\left(x_{i}-y_{i}\right)^{2}}$. 当 $x \neq y$ 时，定义矩阵 $A_{x, y}=\left(a_{i j}\right)$ ，其中 $a_{i j}=$ $\frac{\partial}{\partial x_{i}} \frac{\partial}{\partial y_{j}} f(x, y)(i, j=1,2, \cdots, n)$. 则 $A_{x, y}$ 秩的最小可能值与最大可能值之和为 $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{198}
Let  $A,B,C,D$ , be four different points on a line  $\ell$ , so that  $AB=BC=CD$ . In one of the semiplanes determined by the line  $\ell$ , the points  $P$  and  $Q$  are chosen in such a way that the triangle  $CPQ$  is equilateral with its vertices named clockwise. Let  $M$  and  $N$  be two points of the plane be such that the triangles  $MAP$  and  $NQD$  are equilateral (the vertices are also named clockwise). Find the angle  $\angle MBN$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Five people are doing some work. The first, second, and third, working together, complete the work in an hour, just like the second, fourth, and fifth do the work in an hour. The first, working with the fifth, and the third, working with the fourth, complete the work in 2 hours. How long will it take for all five people to complete the work together? (15 points)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3} \text{ hours (40 minutes)}}
Let  $ABCDEF$  be a regular hexagon with sidelength  $6$ , and construct squares  $ABGH$ ,  $BCIJ$ ,  $CDKL$ ,  $DEMN$ ,  $EFOP$ , and  $FAQR$  outside the hexagon. Find the perimeter of dodecagon  $HGJILKNMPORQ$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
In a park, there are paths laid out as shown in the diagram. Two workers started paving the paths simultaneously from point $A$. They pave the asphalt at constant speeds: the first one on the segment $A-B-C$, and the second one on the segment $A-D-E-F-C$. They finished their work at the same time, spending 9 hours on it. It is known that the second worker is 1.2 times faster than the first worker. How many minutes did the second worker spend paving the segment $D-E$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
用4个红色珠子与3个蓝色珠子镶成项链，问有多少种不同的项链？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
You flip a fair coin which results in heads ( $\text{H}$ ) or tails ( $\text{T}$ ) with equal probability. What is the probability that you see the consecutive sequence  $\text{THH}$  before the sequence  $\text{HHH}$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{8}}
Let $n$ be the product of the first 10 primes, and let $$S=\sum_{x y \mid n} \varphi(x) \cdot y$$ where $\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$, and the sum is taken over ordered pairs $(x, y)$ of positive integers for which $x y$ divides $n$. Compute $\frac{S}{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1024}
Let  $\mathcal{P}_1$  and  $\mathcal{P}_2$  be two parabolas with distinct directrices  $\ell_1$  and  $\ell_2$  and distinct foci  $F_1$  and  $F_2$  respectively. It is known that  $F_1F_2||\ell_1||\ell_2$ ,  $F_1$  lies on  $\mathcal{P}_2$ , and  $F_2$  lies on  $\mathcal{P}_1$ . The two parabolas intersect at distinct points  $A$  and  $B$ . Given that  $F_1F_2=1$ , the value of  $AB^2$  can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ .

[i]Proposed by Yannick Yao
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1504}
Let  $S = \{1, 2,..., 8\}$ . How many ways are there to select two disjoint subsets of  $S$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6561}
Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{160}
Harvard has recently built a new house for its students consisting of $n$ levels, where the $k$ th level from the top can be modeled as a 1-meter-tall cylinder with radius $k$ meters. Given that the area of all the lateral surfaces (i.e. the surfaces of the external vertical walls) of the building is 35 percent of the total surface area of the building (including the bottom), compute $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Evaluate $$\lim_{x \to 0+}\left[\frac{x^{\sin x}-(\sin x)^{x}}{x^3}+\frac{\ln x}{6}\right].$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/6}
We color some cells in  $10000 \times 10000$  square, such that every  $10 \times 10$  square and every  $1 \times 100$  line have at least one coloring cell. What minimum number of cells we should color ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10000}
Reimu and Sanae play a game using 4 fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then neither of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{16}}
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85}
A alone can do a piece of work in 6 days and B alone in 8 days. A and B undertook to do it for Rs. 6000. With the help of C, they completed the work in 3 days. How much is to be paid to C?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{750}
During the Shanghai World Expo, a store sold 11 types of souvenirs, 8 types for 10 yuan each, and 3 types for 5 yuan each. Xiao Zhang spent exactly 50 yuan to buy souvenirs (at most one of each type, and the 50 yuan was exactly used up). Calculate the total number of different ways to buy the souvenirs.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{266}
How do we compute the $3$-adic order of $\beta_1$, one of the roots of the equation $X^2-252X+3^{11}=0$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Two samovars - one large and one small - are filled with very hot water at the same temperature. Both samovars have the same shape and are made of the same material. Which one will cool down to room temperature first?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{small}
You are given a positive integer  $k$  and not necessarily distinct positive integers  $a_1, a_2 , a_3 , \ldots,
a_k$ . It turned out that for any coloring of all positive integers from  $1$  to  $2021$  in one of the  $k$  colors so that there are exactly  $a_1$  numbers of the first color,  $a_2$  numbers of the second color,  $\ldots$ , and  $a_k$  numbers of the  $k$ -th color, there is always a number  $x \in \{1, 2, \ldots, 2021\}$ , such that the total number of numbers colored in the same color as  $x$  is exactly  $x$ . What are the possible values of  $k$ ?

*Proposed by Arsenii Nikolaiev*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2021}
设复数列 \{z_n\} 满足 |z_1|=1,|z_n+z_{n-1}|=n, n\geq2. 记 |z_{20}-z_{23}| 和 |z_{20}+z_{23}| 的最大值分别为 m 和 n，则 m-n 的值为__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{420}
Find $k$ such that $k\pi$ is the area of the region of points in the plane satisfying $$\frac{x^2+y^2+1}{11} \le x \le \frac{x^2+y^2+1}{7}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
用1,2,3，···，9这九个数字可组成多少个数码不重复且8、9不相邻的七位数？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{151200}
The polynomial \[x^4 + bx^3 + cx^2 + 7dx + e = 0\] has integer coefficients, where $d$ is nonzero. Let $m$ be the exact number of integer roots of the polynomial, counting multiplicities. Determine all possible values for $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0, 1, 2, 4}
A point  $E$  on side  $CD$  of a rectangle  $ABCD$  is such that  $\triangle DBE$  is isosceles and  $\triangle ABE$  is right-angled. Find the ratio between the side lengths of the rectangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\sqrt{5} - 2}}
In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is the value of $k+m+n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Triangle \( ABC \) is a right-angled triangle with the vertex \( C \) at the right angle. Let \( P \) be the point of reflection of \( C \) about \( AB \). It is known that \( P \) and two midpoints of two sides of \( ABC \) lie on a line. Find the angles of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ, 60^\circ, 90^\circ}
Suppose  $m$  and  $n$  are relatively prime positive integers. A regular  $m$ -gon and a regular $n$ -gon are inscribed in a circle. Let  $d$  be the minimum distance in degrees (of the arc along
the circle) between a vertex of the  $m$ -gon and a vertex of the  $n$ -gon. What is the maximum
possible value of  $d$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{180}{mn}}
Given that \( AB \parallel DE \), \(\angle A = 120^\circ\), \(\angle B = 50^\circ\), \(\angle D = 60^\circ\), \(\angle E = 140^\circ\), calculate the difference between \(\angle BCD\) and \(\angle AFE\). Express your answer in degrees (°).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10^\circ}
In the complex plane, the point corresponding to the complex number $z = -1 + 2i$ is located in which quadrant?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{second quadrant}
Let  $k$  be a positive integer such that  $1 +\frac12+\frac13+ ... +\frac{1}{13}=\frac{k}{13!}$ . Find the remainder when  $k$  is divided by  $7$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
某学校高一学生有720人，现从高一、高二、高三这三个年级学生中采用分层抽样方法，抽取180人进行英语水平测试，已知抽取高一学生人数是抽取高二学生人数和高三学生人数的等差中项，且高二年级抽取65人，则该校高三年级学生人数是__________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{660}
Let \( M = \{1, 2, \cdots, 1995\} \). Suppose \( A \) is a subset of \( M \) that satisfies the condition: if \( x \in A \), then \( 15x \notin A \). What is the maximum number of elements in \( A \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1870}
What is the median of the following list of $4040$ numbers?  
$1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2$
Calculate the median of this list and provide the value of the median multiplied by 10.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19765}
地面上有10只小鸟在啄食，并且任意5只中至少有4只在同一个圆周上，问：有鸟最多的圆圈上至少有几只鸟？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Cape May and Daytona Beach have 40 shark sightings a year. Cape May has a certain relationship to the number of shark sightings of Daytona Beach, specifically 8 less than this relationship. If Cape May has 24 shark sightings, what is the relationship between the number of shark sightings in Cape May and Daytona Beach?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $x_1=1/20$, $x_2=1/13$, and \[x_{n+2}=\dfrac{2x_nx_{n+1}(x_n+x_{n+1})}{x_n^2+x_{n+1}^2}\] for all integers $n\geq 1$.  Evaluate $\textstyle\sum_{n=1}^\infty(1/(x_n+x_{n+1}))$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
On the dart board shown in the figure below, the outer circle has radius $6$ and the inner circle has radius $3$. Three radii divide each circle into three congruent regions, with point values shown. The probability that a dart will hit a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values of the regions hit. What is the probability that the score is odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{35}{72}}
A satellite of mass 'm' is orbiting Earth in a circular path with radius $r_0$ and speed $v_0$. At a point in its orbit, the satellite's velocity direction is changed by an angle $\theta = \cos^{-1}(\frac{3}{5})$, while maintaining the same speed. This change results in an elliptical orbit. Determine the ratio of the speed at perigee to the speed at apogee of the satellite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
$2016$  numbers written on the board:  $\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016},  ..., \frac{2016}{2016}$ . In one move, it is allowed to choose any two numbers  $a$  and  $b$  written on the board, delete them, and write the number  $3ab - 2a - 2b + 2$  instead. Determine what number will remain written on the board after  $2015$  moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{3}}
In chess, there are two types of minor pieces, the bishop and the knight. A bishop may move along a diagonal, as long as there are no pieces obstructing its path. A knight may jump to any lattice square  $\sqrt{5}$  away as long as it isn't occupied.

One day, a bishop and a knight were on squares in the same row of an infinite chessboard, when a huge meteor storm occurred, placing a meteor in each square on the chessboard independently and randomly with probability  $p$ . Neither the bishop nor the knight were hit, but their movement may have been obstructed by the meteors.

The value of  $p$  that would make the expected number of valid squares that the bishop can move to and the number of squares that the knight can move to equal can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a, b$ . Compute  $100a + b$ .

*Proposed by Lewis Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{102}
Find the fundamental period of the function \( f: \mathbb{R} \to \mathbb{R} \) given that \( f(x+13) + f(x+630) = 0 \) for all \( x \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1234}
There are weights of 3 grams, 7 grams, and 14 grams, totaling 11 weights with a combined weight of 108 grams. How many of each type of weight are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1, 5, 5}
Find the mass percentage of O in Acetone
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27.55\%}
In  $10\times 10$  square we choose  $n$  cells. In every chosen cell we draw one arrow from the angle to opposite angle.  It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or
the distance between their ends is at least 2. What is the maximum possible value of  $n$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Compute the probability that a standard Brownian motion $W_t$ hits 1 before hitting -3, and then hits -3 before hitting 3. That is, find $P(T_{1} < T_{-3} < T_3)$, where $T_a$ is the hitting time of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
Given the function $Z(x,y)$ defined by the equation $$yz + \sin(z) + xyz - y\cos (z) = -1$$ around the point $(1,1,0)$, calculate $\frac{\partial Z}{\partial y}(1,1)$. Provide your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Suppose $15\%$ of $x$ equals $20\%$ of $y.$ What percentage of $x$ is $y?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$75$}
Given the matrix \(A=\begin{bmatrix} 0 & 0 & 1\\ 1 & 0 &0\\ 0& 1& 0\end{bmatrix}\) and the vector \(B=\begin{bmatrix}1\\ 1\\ 1\end{bmatrix}\), determine the angle of rotation about the axis defined by vector \(B\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{3}}
Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$12+10\sqrt3$}
B4 In the following diagram, two lines that meet at a point $A$ are tangent to a circle at points $B$ and $C$. The line parallel to $A C$ passing through $B$ meets the circle again at $D$. Join the segments $C D$ and $A D$. Suppose $A B=49$ and $C D=28$. Determine the length of $A D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{63}