inclusionAI/Ring-lite-rl-data huggingface dataset:
Use the bisection method to find an approximate value of a positive root of the equation $x^2 - 2 = 0$, accurate to $0.1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.4}
An isosceles trapezoid $ABCD$ is circumscribed around a circle. The lateral side $AB$ touches the circle at point $M$, and the base $AD$ at point $N$. Segments $MN$ and $AC$ intersect at point $P$, and $NP: PM=2$. Find

the ratio $AD: BC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:1}
Tom, John, and Lily each shot six arrows at a target. Arrows hitting anywhere within the same ring scored the same number of points. Tom scored 46 points and John scored 34 points. Using the scoring information, calculate Lily's total score. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A positive integer $n$ is said to be base-able if there exist positive integers $a$ and $b$, with $b > 1$, such that $n = a^b$. How many positive integer divisors of $729000000$ are base-able?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Write digits in place of the letters in the equation

\[ \overline{abc}^2 = \overline{adeff} \]

to make it correct. (The overline indicates that the letters represent the digits of a number in the decimal system. Identical letters represent identical digits, and identical digits are labeled with the same letters.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{138^2 = 19044}
There is a question involving a polynomial added to $2a^{2}-3a+5$. Li Ming mistakenly calculated it as a subtraction and obtained $a^{2}-5a+7$. What should be the correct result?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5a^{2}-11a+17}
Mr. and Mrs. Smith have three children. They own a family van with a driver's seat, a passenger seat in the front, and two seats in the back. Either Mr. Smith or Mrs. Smith must sit in the driver's seat. How many seating arrangements are possible if one of the children insists on sitting in the front passenger seat?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Suppose there are 128 ones written on a blackboard. At each step, you can erase any two numbers \(a\) and \(b\) from the blackboard and write \(ab + 1\). After performing this operation 127 times, only one number is left. Let \(A\) be the maximum possible value of this remaining number. Find the last digit of \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Five people of heights $65,66,67,68$, and 69 inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly 1 inch taller or exactly 1 inch shorter than himself?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
In a class, 20% of the students were absent for an exam. 30% failed by 20 marks and 10% just passed. The remaining students scored an average of 65 marks. The average score of the class is 36. What was the cut off for passing the exam?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Find the number of triples of sets $(A, B, C)$ such that: (a) $A, B, C \subseteq\{1,2,3, \ldots, 8\}$. (b) $|A \cap B|=|B \cap C|=|C \cap A|=2$. (c) $|A|=|B|=|C|=4$. Here, $|S|$ denotes the number of elements in the set $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45360}
The numbers $1,2 \cdots 11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{10}{33}}
The coefficient of $x^2$ in the expansion of $\left(ax-\frac{1}{\sqrt{x}}\right)^8$ is $70$. Find the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm1}
We measured a distance at $t=26^{\circ} \mathrm{C}$ temperature using a copper rod that is exactly $a=3 \mathrm{~m}$ long at $0^{\circ} \mathrm{C}$. In this way, we found $n=7856.728 \mathrm{~m}$. What is the actual distance if the linear expansion coefficient of copper is $\lambda=0.000017$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7860.201}
If $f(x)=\log_{3a}[(a^{2}-3a)x]$ is a decreasing function on $(-\infty,0)$, then the range of the real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \dfrac{1}{3},3\right)}
Let  $ k$  be a positive integer. Suppose that the integers  $ 1,2,3,\dots,3k \plus{} 1$  are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by  $ 3$  ? Your answer should be in closed form, but may include factorials.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{(k+1)! k!}{(2k)! (3k + 1)}}
Among the first 1500 positive integers, there are n whose hexadecimal representation contains only numeric digits. What is the sum of the digits of n?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Find the smallest positive integer \( n \) such that in any two-coloring of the complete graph \( K_{n} \), there exist 3 monochromatic triangles that are pairwise edge-disjoint.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
From 2014 labeled numbers, 100 numbers are to be sampled. If systematic sampling is used, then the sampling interval should be ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
7. Given that the volume of a cylinder is $V$ (a constant), then the minimum value of the total surface area of the cylinder is

保留源文本的换行和格式，直接输出翻译结果。 

Note: The last sentence is a note in Chinese, which is not part of the translation. The translation itself is provided above.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3\sqrt[3]{2\piV^{2}}}
What is the derivative of $ \log{\left(1 - \frac{r}{r_{s}} \right)} $ with respect to $ \log{(r)} $, where $r_{s}$ is a constant?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{r}{r_{s}(1 - \frac{r}{r_{s}})}}
Find the side of an equilateral triangle that is the orthogonal projection of a triangle with sides $\sqrt{6}, 3,$ and $\sqrt{14}$ onto a certain plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{5}}
Let  $n\geq 2$  be a given integer. Initially, we write  $n$  sets on the blackboard and do a sequence of moves as follows: choose two sets  $A$  and  $B$  on the blackboard such that none of them is a subset of the other, and replace  $A$  and  $B$  by  $A\cap B$  and  $A\cup B$ . This is called a  $\textit{move}$ . 
Find the maximum number of moves in a sequence for all possible initial sets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n(n-1)}{2}}
Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$  that satisfy $$|S| +\ min(S) \cdot \max (S) = 0.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{335}
For certain ordered pairs $(a,b)\,$ of real numbers, the system of equations
$ax+by=1\,$
$x^2+y^2=50\,$
 has at least one solution, and each solution is an ordered pair $(x,y)\,$ of integers. How many such ordered pairs $(a,b)\,$ are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72}
已知抛物线$\Gamma:y^{2}=2px(p>0)$在第一象限与单位圆交于点$A,\Gamma$在$A$处的切线为$l$，过原点的$\odot M$与$l$切于点$A$。当$p$变化时，求$\odot M$的半径$R$的最小值。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given that an object is displaced by the concurrent forces $\mathbf{F}_1 = (\log_2, \log_2)$ and $\mathbf{F}_2 = (\log_5, \log_2)$ to a displacement $\mathbf{S} = (2\log_5,1)$, calculate the work $\mathbf{W}$ done by the concurrent forces on the object. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Consider $f$ is differentiable, what is the value of $$\lim_{n\to \infty} \frac{1}{n^2} \sum_{k=1}^n \frac{f(a+\frac{k}{n^2}) -f(a)}{\frac{k}{n^2}}$$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the number of even digits in the product of the two 10-digit numbers \(2222222222 \times 9999999999\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Rosa received a perfume bottle in the shape of a cylinder with a base radius of $7 \mathrm{~cm}$ and a height of $10 \mathrm{~cm}$. After two weeks of using the perfume, $0.45$ liters remained in the bottle. What fraction represents the volume of perfume that Rosa has already used?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{49\pi - 45}{49\pi}}
A hundred silver coins are laid down in a line. A wizard can convert a silver coin into a golden one in $3$ seconds. Each adjacent golden coin reduces this time by $1$ second. What is the minimal time required for the wizard to convert all coins to gold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{201}
A dishonest dealer professes to sell goods at the cost price but uses different weights for different items. For some items, he uses a weight of 900 grams per kg, while for others, he uses a weight of 850 grams per kg. Calculate his overall gain percentage, considering the total weight and price of the items.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14.29\%}
Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$, let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]
Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin.  Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2019^{-2019^{-1}}}
已知无穷数列$\left\{ {{a}_{n}} \right\}$，${{a1}}=1$，${{a}_{2}}=2$，对任意$n\in {{N}^{*}}$，有${{an+2}}={{a}_{n}}$，数列$\left\{ {{bn}} \right\}$满足${{b}_{n+1}}-{{bn}}={{a}_{n}}\left( n\in {{N*}} \right)$，若数列$\left\{ \frac{{{b2n}}}{{{a}_{n}}} \right\}$中的任意一项都在该数列中重复出现无数次，则满足要求的${{b1}}$的值为__．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
What is the value of the limit $\lim_{n\to\infty} \sqrt[n]{a_n}$ for the sequence $a_n=\sum_{k=0}^{[n/2]} C_{n-k}^k$, where $[n/2]$ denotes the largest integer less than or equal to $n/2$, and $C_{n-k}^k$ represents the binomial coefficient $\frac{(n-k)!}{k!(n-2k)!}$? Note that $a_{2m+1}-a_{2m}=a_{2m-1}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\sqrt{5}+1}{2}\)}
Vertex \(E\) of equilateral \(\triangle{ABE}\) is in the interior of unit square \(ABCD\). Let \(R\) be the region consisting of all points inside \(ABCD\) and outside \(\triangle{ABE}\) whose distance from \(\overline{AD}\) is between \(\frac{1}{3}\) and \(\frac{2}{3}\). Find the area of \(R\). The original answer is in the form \(\frac{k-m\sqrt{3}}{n}\), please provide the value of k + m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
Let $R$ be a unit square region and $n \geq 4$ an integer. A point $X$ in the interior of $R$ is called <i>n-ray</i> partitional if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are $100$-ray partitional but not $60$-ray partitional?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2320$}
If  $x=\sqrt2+\sqrt3+\sqrt6$  is a root of  $x^4+ax^3+bx^2+cx+d=0$  where  $a,b,c,d$  are integers, what is the value of  $|a+b+c+d|$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{93}
Find all integer $n$ such that the following property holds: for any positive real numbers $a,b,c,x,y,z$, with $max(a,b,c,x,y,z)=a$ , $a+b+c=x+y+z$ and $abc=xyz$, the inequality $$a^n+b^n+c^n \ge x^n+y^n+z^n$$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
For a positive integer  $n$ , let  $S_n$  be the total sum of the intervals of  $x$  such that  $\sin 4n x\geq \sin x$  in  $0\leq x\leq \frac{\pi}{2}.$ Find  $\lim_{n\to\infty} S_n.$ 60 point
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{8}}
Let \( f(n) = \sum_{k=2}^{\infty} \frac{1}{k^n \cdot k!} \). Calculate \( \sum_{n=2}^{\infty} f(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 - e}
Calculate the improper integral: $$\int_{0}^{\infty}\frac{\ln x}{(x+1)^{3}}\mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1/2}
In 2012, a telecommunications company launched a series of mobile phone numbers. The first seven digits of the card number are fixed, and the last four digits range from "0000" to "9999", totaling 10,000 numbers. The company stipulates that if the last four digits of the card number are arranged in ascending order from smallest to largest, it is called a "Auspicious Dragon Card" and enjoys certain preferential policies. The number of "Auspicious Dragon Cards" in this series is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{296}
有 20 张卡片上分别写有数字1,2,…,20, 将它们放入一个盒子内.有4个人从中不放回地各抽取一张卡片,抽到两个较小数字的两人在同一组,抽到两个较大数字的两人在同一组.现其中有两人抽到5、14,则此两人在同一组的概率等于___。原始答案为最简分数形式\frac{m}{n}，请给出m + n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{58}
Compute the number of ordered pairs \((m,n)\) of positive integers that satisfy the equation \(\text{lcm}(m,n)+\gcd(m,n)=m+n+30\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
On the radius \(AO\) of a circle with center \(O\), a point \(M\) is chosen. On one side of \(AO\) on the circle, points \(B\) and \(C\) are chosen such that \(\angle AMB = \angle OMC = \alpha\). Find the length of \(BC\) if the radius of the circle is 10 and \(\sin \alpha = \frac{\sqrt{24}}{5}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{\sin B\sin C}{\sin A}=\frac{3\sqrt{7}}{2}$, $b=4a$, and $a+c=5$, find the area of $\triangle ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{7}}{4}}
Given $b$ and $r \in \{1,2,3,4\}$, the probability that the line $y=x+b$ and the circle $x^{2}+y^{2}=r$ have common points is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{16}}
A city's education committee plans to conduct a survey in seven schools named A, B, C, D, E, F, G. The survey must be completed within a week, from Monday to Friday, and at least one school must be visited each day. School A and B are scheduled for Monday and Tuesday respectively, while School C and D must be scheduled on the same day. School E cannot be scheduled on Friday. What is the total number of different arrangements? (Answer with a numerical value)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
For how many primes \( p \), there exist unique integers \( r \) and \( s \) such that for every integer \( x \), the congruence \( x^{3} - x + 2 \equiv (x - r)^{2} (x - s) \pmod{p} \) holds? Find the number of such primes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the smallest natural number, which divides  $2^{n}+15$  for some natural number  $n$  and can be expressed in the form  $3x^2-4xy+3y^2$  for some integers  $x$  and  $y$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
A $3 \times 3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{405}
Tim drops a ball off the roof of a 96-foot tall building. The ball has an initial bounce efficiency of 50%, but with each bounce, its efficiency decreases by 5%. Additionally, air resistance causes the ball to lose 2% of its height on each bounce. How high will it bounce on the fifth bounce?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.82}
Let $E$ be a point inside the rhombus $ABCD$ such that $|AE|=|EB|$, $m(\widehat{EAB}) = 11^{\circ}$, and $m(\widehat{EBC}) = 71^{\circ}$. If the answer is $x^{\circ}$, find the value of $x$ for $m(\widehat{DCE})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{68}
Points A and B move in a clockwise direction on the sides of a square with side length 100m. Point A starts from corner A of the square and moves at a speed of 5m/s, and Point B starts from the midpoint of side CD and moves at a speed of 10m/s. After how many seconds will they first be on the same side?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 10$ and $\angle O_{1}PO_{2} = 90^{\circ}$, then $AP = \sqrt{c} + \sqrt{d}$, where $c$ and $d$ are positive integers. Find $c + d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\,\frac{1}{6}$}
What is the necessary and sufficient condition for the function \( f(x)=\frac{\sqrt{a^{2}}-x^{2}}{|x+a|-a} \) to be an odd function?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 0}
The monotonic increasing interval of the function $f(x) = -(x-5)|x|$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \frac{5}{2})}
Given an ellipse $C:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with two foci $F_{1}$ and $F_{2}$. Points $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $|PQ|=|F_{1}F_{2}|$. If the area of $\triangle PF_{2}Q$ is $S≥\frac{1}{8}|PQ|^{2}$, then the range of the eccentricity of $C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[\frac{\sqrt{2}}{2},\frac{\sqrt{6}}{3}]}
On the side \( PQ \) of triangle \( PQR \), point \( N \) is taken, and on the side \( PR \), point \( L \) is taken, such that \( NQ = LR \). The point \( A \), where segments \( QL \) and \( NR \) intersect, divides the segment \( QL \) in the ratio \( m:n \) from point \( Q \). Find the ratio \( PN : PR \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n}{m}}
Let \( n \geq 4 \) be an integer. \( A = \{a_1, a_2, \cdots, a_n\} \) is a set of integers such that \( 2023 \in A \) and the absolute difference between any two elements in \( A \) is a prime number. Determine the number of such sets \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
How many roots does the equation

$$
\overbrace{f(f(\ldots f}^{10 \text{ times } f}(x) \ldots)) + \frac{1}{2} = 0
$$

have, where \( f(x) = |x| - 1 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Problem 2. Given 10 equal segments in a plane. All intersection points are colored. Each colored point divides each of the given segments it lies on in the ratio $3: 4$. What is the maximum number of colored points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9999}
A triangle with angles \( A, B, C \) satisfies the following conditions:
\[
\frac{\sin A + \sin B + \sin C}{\cos A + \cos B + \cos C} = \frac{12}{7},
\]
and
\[
\sin A \sin B \sin C = \frac{12}{25}.
\]

Given that \( \sin C \) takes on three possible values \( s_1, s_2 \), and \( s_3 \), find the value of \( 100 s_1 s_2 s_3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1, \frac{a_{n+1}-a_{n}}{a_{n}}=\frac{a_{n+2}-a_{n+1}}{a_{n+2}}\left(n \in N^{*}\right)$, 若 $a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{6} a_{7}=3$, 则 $a_{2024}=$. 请给出原始答案为frac{m}{n}格式时m + n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2035}
At the "Economics and Law" congress, a "Best of the Best" tournament was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. During one match, participants had to ask each other questions within a limited time and record correct answers. Each participant played with each other participant exactly once. A match winner got one point, the loser got none, and in case of a draw, both participants received half a point each. By the end of the tournament, it turned out that in matches against economists, each participant gained half of all their points. How many lawyers participated in the tournament? Provide the smallest possible number as the answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Identify the algebra $H\otimes_{\mathbb{R}} \mathbb{C}$ in terms of a familiar algebraic structure, where $H$ denotes the Hamiltonian quaternions, $\mathbb{C}$ represents the complex numbers, and $\mathbb{R}$ stands for the real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{M_2(\mathbb{C})}
Find the surface area of an ideal tetrahedron in hyperbolic space $\mathbb{H}^3$ as a function of its vertices, which are parametrized by four complex numbers $z_1, z_2, z_3, z_4$ on the boundary $\mathbb{CP}^1$. Express the surface area in terms of $z_1, z_2, z_3, z_4$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Determine the ratio of the circumference to the diameter of a circle on a sphere, given that the circle is defined by an angle \( \theta \) from the pole. The radius of the sphere is \( r \). Express the ratio \( R(\theta) \) in terms of \( \theta \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi \sin \theta}{\theta}}