inclusionAI/Ring-lite-rl-data huggingface dataset:
In the geometric sequence {a_n}, the product of the first three terms is 27 (i.e., a_1 * a_2 * a_3 = 27), and the sum of the second and fourth terms is 30 (i.e., a_2 + a_4 = 30). Find the common ratio q. Express your answer as a numerical value, including the ± symbol if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±3}
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. Find the minimum possible value of $BE+DE+CD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Using $1 \times 1, 2 \times 2, 3 \times 3$ tiles to completely cover a $23 \times 23$ floor (without overlapping or leaving gaps), what is the minimum number of $1 \times 1$ tiles needed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\overline{AB}$ can be written in the form $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$312$}
Compute Tor$_1(\mathbb{Q} /\mathbb{Z}, \mathbb{Q} /\mathbb{Z})$. Provide your answer as a group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Q}/\mathbb{Z}}
Origami. Mitya plans to fold a square sheet of paper \(ABCD\). Mitya calls a fold beautiful if side \(AB\) intersects side \(CD\) and the four resulting right-angled triangles are equal.

Before this, Vanya chooses a random point \(F\) on the sheet. Find the probability that Mitya can make a beautiful fold passing through point \(F\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Determine the number of non-trivial homomorphisms from the symmetric group $S_5$ to the alternating group $A_6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Determine the number of ways to express $7^{13}$ as a product of three natural numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
Evaluate the limit: \[ \lim_{n \rightarrow \infty} n \left((n^5 +5n^4)^{1/5} - (n^2 +2n)^{1/2}\right) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{3}{2}}
Find a closed form for the following sum:

$$\sum_{k=-\infty}^{\infty} \frac{(-1)^k}{x+(2k+1)}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{2} \sec\left(\frac{x\pi}{2}\right)\)}
Determine whether the sequence defined by the nested expression \(\sqrt{1! + \sqrt{2! + \sqrt{3! + \sqrt{4! + \sqrt{5! + \sqrt{6! + \ldots}}}}}}\) converges to a finite value. If it does, find the limit.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.827}
给定正整数 $a 、 b(a \leqslant b)$. 数列 $\left\{f_{n}\right\}$ 满足: $f_{1}=a, f_{2}=b, f_{n+2}=f_{n+1}+f_{n}(n=1,2, \cdots)$. 若对任意的正整数 $n$, 都有 $\left(\sum_{k=1}^{n} f_{k}\right)^{2} \leqslant \lambda f_{n} f_{n+1}$, 求实数 $\lambda$ 的最小值。原始答案的形式为 $m + \sqrt{n}$，请给出 $m + n$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
What is the maximum number of parts the plane can be divided into by $n$ rectangles with parallel sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n^2-2n+2}
In a $3 \times 3$ table, numbers are placed such that each number is 4 times smaller than the number in the adjacent cell to the right and 3 times smaller than the number in the adjacent cell above. The sum of all the numbers in the table is 546. Find the number in the central cell.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
In \(\triangle ABC\), \(D\) and \(E\) are the midpoints of the sides \(AB\) and \(AC\) respectively. \(CD\) and \(BE\) intersect at \(P\) with \(\angle BPC = 90^\circ\). Suppose \(BD = 1829\) and \(CE = 1298\). Find \(BC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2006}
Given that $a=2^{\frac{4}{3}}$, $b=4^{\frac{2}{5}}$, and $c=25^{\frac{1}{3}}$, determine the relationship between $a$, $b$, and $c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c > a > b}
3. On a circle with center $M$ lie points $A, B$, and $C$. The reflection of $M$ in the line $A B$ lies inside triangle $A B C$ and is the intersection of the angle bisectors of angle $A$ and angle $B$. (The angle bisector of an angle is the line that divides the angle into two equal angles.)

Determine the size of angle $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
How many different ways are there to define a binary operation on a set of $n$ elements, without considering isomorphic structures?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(n^{n^2}\)}
Create a five-digit number using the digits 1, 2, 3, 4, and 5 without any repetition, with the condition that 5 cannot be in the hundred's place, and neither 2 nor 4 can be in the unit's or ten-thousand's place. Calculate the total number of such five-digit numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
There are \(10!\) permutations \( s_{0}, s_{1}, \ldots, s_{9} \) of the numbers \(0, 1, \ldots, 9\). How many of these permutations satisfy \( s_{k} \geq k - 2 \) for \( k = 0, 1, \ldots, 9 \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13122}
$a$, $b$, $c$ are the three sides of triangle $ABC$, let vectors $\overrightarrow{P}=(a+c,b)$, $\overrightarrow{q}=(b-a,c-a)$. If $\overrightarrow{P} \parallel \overrightarrow{q}$, then the size of angle $C$ is \_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Find the acute angle of a rhombus, given that the volumes of the solids obtained by rotating the rhombus around its longer diagonal and around its side are in the ratio \(1 : 2\sqrt{5}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\arccos \left(\frac{1}{9}\right)}
Given a sphere with center $O$, there are four points on the surface of the sphere: $S$, $A$, $B$, and $C$, where points $O$, $A$, $B$, and $C$ are coplanar, and $\triangle ABC$ is an equilateral triangle with side length $2$. The plane $SAB$ is perpendicular to the plane $ABC$. Find the maximum volume of the tetrahedron $S-ABC$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{3}}{3}}
Given a regular tetrahedron, how many different edge-length cubes exist such that each vertex of the cube lies on a face of the tetrahedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let \( f(x) = x^2 + px + q \). Given that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([2, 4]\), find \( f(f(\ldots f_{2017}\left(\frac{5-\sqrt{11}}{2}\right)) \ldots) \). Round your answer to the nearest hundredth if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.16}
Determine the value of $y$ for which the sum of the coefficients of the odd power terms of $x$ in the expansion of $(x+y)(x+1)^4$ is 32. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given positive integers \(a\), \(b\) (neither a multiple of 5), consider the sequence defined by \(a_1 = 5\) and \(a_{n+1} = a_{a_n} + b\). Determine the largest number of prime numbers that can appear in this sequence before the first composite number is encountered.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
In a multiplication error involving two positive integers $a$ and $b$, Ron mistakenly reversed the digits of the three-digit number $a$. The erroneous product obtained was $396$. Determine the correct value of the product $ab$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{693}
Evaluate the limit \( \lim_{x \to 0^+}\frac{4e^{x^2}-4\cos(x^3)-2\sin(x^4)-8\sqrt{1+x^2}-x^4+8}{1-\sqrt[3]{1+13x^6}} \) without using L'Hôpital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
How many days have passed from March 19, 1990, to March 23, 1996, inclusive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2197}
Find the smallest positive integer  $n$  for which  $315^2-n^2$  evenly divides  $315^3-n^3$ .

*Proposed by Kyle Lee*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
In the equation $\frac{1}{(\;\;\;)} + \frac{4}{(\;\;\;)} + \frac{9}{(\;\;\;\;)} = 1$, fill in the three brackets in the denominators with a positive integer, respectively, such that the equation holds true. The minimum value of the sum of these three positive integers is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
设$S = \{ 1 , 2 , \cdots , 9 8 \}$，求最小正整数n，使得S的任意一个n元子集中都可以选出10个数，无论怎样将10个数均分成两组，总有一组中存在一个数与另外4个数互质，而另一组中总有一个数与另外4个数不互质． （1998年CMO试题）
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
11.6. Given a tetrahedron \(ABCD\), all of whose faces are similar right triangles with acute angles at vertices \(A\) and \(B\). The edge \(AB\) is equal to 1. Find the length of the shortest edge of the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(\frac{\sqrt{5}-1}{2})^{\frac{3}{2}}}
设集合 T=\{1,2,\cdots,8\}. 对 T 的非空子集 A，用 S(A) 表示 A 中所有元素的和. 问：T 有多少个非空子集 A，使得 S(A) 是 3 的倍数，但不是 5 的倍数？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
A square is cut along a diagonal and reassembled to form a parallelogram \( PQRS \). If \( PR=90 \mathrm{~mm} \), what is the area of the original square, in \( \mathrm{mm}^{2} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1620 \mathrm{~mm}^{2}}
From the set \( M = \{1, 2, \cdots, 36\} \), remove \( n \) numbers such that the sum of any two remaining elements is not a factor of 2015. Find the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
The monotonic increasing interval of the function $y=3^{-|x-2|}$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 2]}
Maryam has a fair tetrahedral die, with the four faces of the die labeled 1 through 4.  At each step, she rolls the die and records which number is on the bottom face.  She stops when the current number is greater than or equal to the previous number. (In particular, she takes at least two steps.)  What is the expected number (average number) of steps that she takes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{625}{256}}
$已知a>0，函数f(x)=ax-xe^{x}。$
$若存在a，使得f(x) \leq a + b对任意x \in \mathbf{R}成立，求实数b的取值范围.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-e，+\infty )}
Given the graph of the power function $y=f(x)$ passes through the point $\left( \frac{1}{2}, 8 \right)$, find $f(-2) = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{8}}
已知f(x)=|1-2x|,,x∈[0,,1]，那么方程.

$f ( f ( f ( x ) ) ) = \frac { 1 } { 2 } x$的解的个数是 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let  $\triangle ABC$  with  $AB=AC$  and  $BC=14$  be inscribed in a circle  $\omega$ . Let  $D$  be the point on ray  $BC$  such that  $CD=6$ . Let the intersection of  $AD$  and  $\omega$  be  $E$ . Given that  $AE=7$ , find  $AC^2$ .

*Proposed by Ephram Chun and Euhan Kim*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
A box contains blue, red, and green flags. The total number of flags in the box is an even number. A group of children are asked to pick up two flags each. If all the flags are used up in the process such that 50% of the children have blue flags and red flags, 30% have red flags and green flags, 10% have blue flags and green flags, and 10% have flags of all three colors, what percentage of children have flags of different colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90\%}
A bicycle wheel has a certain diameter. It makes 624.4536030972898 complete revolutions in 1 km. What is the diameter of the wheel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.5097}
In the new college entrance examination reform plan, it is stipulated that the scores of each candidate are set according to the mode of $3$ (Chinese, Mathematics, English) $+2$ (Physics, History) $1+4$ (Chemistry, Biology, Geography, Politics) $2$. Then the probability that two students, A and B, have exactly two subjects in common in their elective subjects is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{12}}
There are originally 5 different books placed side by side on a bookshelf. Now, if we want to insert 3 different books in three steps, one book at a time, into the gaps formed by the existing books, calculate the total number of different insertion methods. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
A square $O P Q R$ is inscribed in $\triangle A B C$. The areas of $\triangle A O R$, $\triangle B O P$, and $\triangle C R Q$ are given as $S_{1}=1$, $S_{2}=3$, and $S_{3}=1$, respectively. Calculate the side length of the square $O P Q R$. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given $α,β∈(0,π)$, $sin(α-β)=\frac{5}{6}$, $\frac{tanα}{tanβ}=-\frac{1}{4}$, find $\alpha +\beta$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{6}π}
A function $f: \mathbb{R}\to \mathbb{R}$ is essentially increasing if $f(s)\leq f(t)$ holds whenever $s\leq t$ are real numbers such that $f(s)\neq 0$ and $f(t)\neq 0$.

Find the smallest integer $k$ such that for any 2022 real numbers $x_1,x_2,\ldots , x_{2022},$ there exist $k$ essentially increasing functions $f_1,\ldots, f_k$ such that\[f_1(n) + f_2(n) + \cdots + f_k(n) = x_n\qquad \text{for every } n= 1,2,\ldots 2022.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
An electronic clock displays time from 00:00 to 23:59, each moment consisting of four digits. Calculate the probability that the sum of the four digits at any given moment is 23. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac {1}{360}}
A computer screen shows a \( 98 \times 98 \) chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white and white becomes black). Determine the minimum number of mouse-clicks needed to make the chessboard all one color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
How many lines in a three dimensional rectangular coordinate system pass through four distinct points of the form $(i, j, k)$, where $i$, $j$, and $k$ are positive integers not exceeding four?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{76}
A Turing Machine M has an alphabet with n letters and k states. If M is restricted to only move its head to the right, how many iterations are needed in the worst case to determine if M will loop?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k * n}
What is the units (i.e., rightmost) digit of $\left[\frac{10^{20000}}{10^{100}+3}\right]$? Here $[x]$ is the greatest integer $\leqslant x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the universal set $U=\{1,2,3,4,5,6,7,8,9,10\}$, and the set $A=\{a_1, a_2, a_3\}$, find the number of sets $A$ that satisfy $a_3 \geq a_2 + 1 \geq a_1 + 4$. (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
Someone in 1860 placed 100,000 frt at 5% interest for the purpose of building and maintaining an orphanage for 100 orphans at some point. When can the construction and opening of the orphanage take place if the construction and equipment costs are 100,000 frt, the staff of the orphanage costs 3,960 frt per year, and the maintenance of one orphan costs 200 frt?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1896}
Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and let $P = AC \cap BD$, with $AB = 21$, $CD = 7$, $AD = 13$, and $[ABCD] = 168$. Let the line parallel to $AB$ through $P$ intersect the circumcircle of $\triangle BCP$ at $X$. The circumcircles of $\triangle BCP$ and $\triangle APD$ intersect at $P$ and $Y$. Let $XY \cap BC = Z$. If $\angle ADC$ is obtuse, then $BZ = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers. Compute $a + b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Given that $a_{i} \in \{0,1,2,3,4\}$ for every $0 \le i \le 9$, if $6\sum_{i=0}^{9}a_{i} 5^{i} \equiv 1 \pmod{5^{10}}$, find the value of $a_{9}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let the set $S = \{P_1, P_2, \dots, P_{12}\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called "communal" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{134}
Squares $ABCD$ and $AEFG$, each with side length $12$, overlap such that $\triangle AED$ is an equilateral triangle, as shown in the diagram. The area of the region that lies within both squares, which is shaded in the diagram, is given by $m\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Determine the value of $m + n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Square $PQRS$ has a side length of $2$ units. Points $T$ and $U$ are on sides $PQ$ and $QR$, respectively, with $PT = QU$. When the square is folded along the lines $ST$ and $SU$, sides $PS$ and $RS$ coincide and lie along diagonal $RQ$. Exprress the length of segment $PT$ in the form $\sqrt{k} - m$ units. What is the integer value of $k+m$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Given that $f(x) = \frac{1}{3^x + \sqrt{3}}$, find the value of $f(-12) + f(-11) + f(-10) + ... + f(0) + ... + f(11) + f(12) + f(13)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{13\sqrt{3}}{3}}
已知凸六边形 $P$ 的所有顶点在一个单位正方形的边上，且所有内角相等. 求 $P$ 的最短边长度的最大可能值。请给出m + n的值，其中原始答案格式为1-\frac{\sqrt{m}}{n}。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the expected number of trials until we get both outcomes (a success and a failure) in an experiment with a probability of success of 0.81 and a probability of failure of 0.19?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5.5}
A thin diverging lens with an optical power of $D_{p} = -6$ diopters is illuminated by a beam of light with a diameter $d_{1} = 10$ cm. On a screen positioned parallel to the lens, a light spot with a diameter $d_{2} = 20$ cm is observed. After replacing the thin diverging lens with a thin converging lens, the size of the spot on the screen remains unchanged. Determine the optical power $D_{c}$ of the converging lens.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18 \, 	ext{Diopters}}
During the steelmaking process, materials with specific chemical elements are added to ensure the steel meets certain required standards. Suppose for a particular use, the amount of a certain element to be added to each ton of steel should be between 500g and 1000g. If we use the 0.618 method to arrange the experiment, then the amount of the element to be added on the second experimental trial could be g.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{691\text{g}}
Let $a, b$ be positive reals with $a>b>\frac{1}{2} a$. Place two squares of side lengths $a, b$ next to each other, such that the larger square has lower left corner at $(0,0)$ and the smaller square has lower left corner at $(a, 0)$. Draw the line passing through $(0, a)$ and $(a+b, 0)$. The region in the two squares lying above the line has area 2013. If $(a, b)$ is the unique pair maximizing $a+b$, compute $\frac{a}{b}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5}{3}}
Given that Liz had no money initially, and her friends gave her one-sixth, one-fifth, and one-fourth of their respective amounts, find the fractional part of the group's total money that Liz has.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{5}}
Evaluate the integral \( \int_0^{\pi} \lfloor \pi^2 \cos^3 x \rfloor \sin x \, dx \), where \( \lfloor x \rfloor \) denotes the floor function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
We are given a matrix $A \in M_n (F)$ such that all its entries are either $1$ or $0$. What is the maximum number of $1$'s that can be in matrix $A$ so that it is still invertible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n^2-n+1}
S is the set of all ( $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$ ) where  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$  are integers such that  $a^2 + b^2 + c^2 + d^2 + e^2 = f^2$ . Find the largest  $k$  which divides abcdef for all members of  $S$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 24 }
Create a cube \( C_{1} \) with edge length 1. Take the centers of the faces and connect them to form an octahedron \( O_{1} \). Take the centers of the octahedron's faces and connect them to form a new cube \( C_{2} \). Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{54 + 9\sqrt{3}}{8}}
A fair coin is flipped until the number of heads exceeds the number of tails. What is the expected number of flips required?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Given that $2^{\aleph_0} = \aleph_3$, find the value of the expression ${\aleph_2}^{\aleph_0} + {\aleph_0}^{\aleph_2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_3}