inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $ABCDE$ be a convex pentagon, and let $H_A,$ $H_B,$ $H_C,$ $H_D$ denote the centroids of triangles $BCD,$ $ACE,$ $ABD,$ and $ABC,$ respectively. Determine the ratio $\frac{[H_A H_B H_C H_D]}{[ABCDE]}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{9}}
A trapezoid is divided into three smaller trapezoids by lines parallel to the bases. It is known that each of the three resulting trapezoids can have an inscribed circle. Find the radius of the circle inscribed in the middle trapezoid if the radii of the circles inscribed in the other two trapezoids are $R$ and $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{rR}}
In the diagram, $D$ is on side $A C$ of $\triangle A B C$ so that $B D$ is perpendicular to $A C$. Also, $\angle B A C=60^{\circ}$ and $\angle B C A=45^{\circ}$. If the area of $\triangle A B C$ is $72+72 \sqrt{3}$, what is the length of $B D$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 \sqrt[4]{3}}
What is the cardinality of the set of all subsets of the real numbers $\mathbb{R}$ that are neither open nor closed and have no intersection with the set of integers $\mathbb{Z}$ (i.e., if $A$ is such a set, then $A \cap \mathbb{Z} = \emptyset$)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$2^{\aleph_0}$}
For positive integers $n$, let $L(n)$ be the largest factor of $n$ other than $n$ itself. Determine the number of ordered pairs of composite positive integers $(m, n)$ for which $L(m) L(n)=80$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Two different digits from 1 to 9 are chosen. One digit is placed in each box to complete the two 2-digit numbers shown. The result of subtracting the bottom number from the top number is calculated. How many of the possible results are positive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
How many positive integers which divide  $5n^{11}-2n^5-3n$  for all positive integers  $n$  are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
If k^4 is divisible by 1680, and the prime factorization of k is in the form 2^a * 3^b * 5^c, where a, b, and c are positive integers, what is the least possible value of integer k?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{210}
For  $i=1,2,\ldots,7$ , Zadam Heng chooses a positive integer  $m_i$  at random such that each positive integer  $k$  is chosen with probability  $\frac{2^i-1}{2^{ik}}$ . If  $m_1+2m_2+\ldots+7m_7\neq35$ , Zadam keeps rechoosing the  $m_i$  until this equality holds. Given that he eventually stops, what is the probability that  $m_4=1$  when Zadam stops?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{5}}
The least common multiple of three consecutive natural numbers is 660. Find these three numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10, 11, 12}
11.2. What is the minimum number of factors that need to be crossed out from the number $99!=1 \cdot 2 \cdot \ldots \cdot 99$ so that the product of the remaining factors ends in $2?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
8. There is an unlimited number of test tubes of three types - A, B, and C. Each test tube contains one gram of a solution of the same substance. Test tubes of type A contain a $10\%$ solution of this substance, type B $-20\%$ solution, and type C $-90\%$ solution. Sequentially, one after another, the contents of the test tubes are poured into a certain container. In this process, two consecutive pourings cannot use test tubes of the same type. It is known that a $20.17\%$ solution was obtained in the container, performing the minimum number of pourings. What is the maximum number of test tubes of type C that can be used in this process?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
In a table \( A \) of size \( 10 \times 10 \), some numbers are written. Let the sum of all numbers in the first row be denoted as \( s_{1} \), in the second row as \( s_{2} \), and so on. Similarly, the sum of the numbers in the first column is denoted as \( t_{1} \), in the second column as \( t_{2} \), and so on.

A new table \( B \) of size \( 10 \times 10 \) is constructed such that in the first cell of the first row is written the smallest of the numbers \( s_{1} \) and \( t_{1} \), in the third cell of the fifth row is written the smallest of the numbers \( s_{5} \) and \( t_{3} \), and so on for the whole table. It turns out that it is possible to number the cells of table \( B \) with numbers from 1 to 100 such that in the cell with number \( k \) there will be a number less than or equal to \( k \). What is the maximum value that the sum of all the numbers in table \( A \) can take under these conditions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{955}
Given integers \( a, b, c \) from the set \( \{1, 2, 3, \ldots, 9\} \), where \( \overline{ab} = b^2 \) and \( \overline{acbc} = (\overline{ba})^2 \), find the value of \( \overline{abc} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{369}
Find the number(s) that satisfy the equation x = 1/x, where x is the original number. Express your answer using the ± symbol if there are multiple solutions.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±1}
Find an integer $1 < a < 9991$ such that:
\[
a^{4995} \not\equiv 1 \pmod{9991}, \\
a^{4995} \not\equiv -1 \pmod{9991}, \\
a^{2 \cdot 4995} \not\equiv -1 \pmod{9991}.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine a function that, when applied to a triangular number \( M \), always results in an odd square number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8M + 1}
Determine the asymptotic growth rate, in big O notation, of the divisor function \( \sigma_{-1}(n) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(\log \log n)}
What is the maximum area of a triangle that can be covered by a square if each vertex of the square is on one of the triangle's sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
If a sphere $O$ with a radius of $2$ has a point $A$ on its surface, and $A$ creates a section of the sphere $O$, and the angle formed by $OA$ and this section is $60^{\circ}$, then the area of this section is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Compute the measure of the set \( A = \{ (x, x): x \in [0, 1] \} \) under the product measure \( \mu_1 \times \mu_2 \), where \( \mu_1 \) is the Lebesgue measure and \( \mu_2 \) is the counting measure on \([0, 1]\). Express \( A \) as a Cartesian product to facilitate the computation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Consider a homomorphism $f: D_{18} \to GL(2,\Bbb R)$, where $D_{18}$ is the dihedral group with 18 elements and $f(r) = R := \begin{pmatrix}1& 1 \\ -1& 0\end{pmatrix}$. Given that the reflection $s$ maps to $f(s) = S := \begin{pmatrix}a &b \\ b-a &-a\end{pmatrix}$ with $a^2 -ab + b^2 = 1$, determine the order of $\ker(f)$, denoted as $\lvert \ker(f) \rvert$. What is the order of the kernel?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. The area of $ABCDE$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$7 \sqrt{3}$}
In triangle \(ABC\), median \(BM\) and height \(AH\) are drawn. It is known that \(BM = AH\). Find the angle \(\angle MBC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Given that \( x_{1}, \cdots, x_{n} \) are all real numbers within the interval \(\left(\frac{1}{4}, 1\right)\), find the minimum value of the sum \( \sum_{k=1}^{n} \log _{x_{4}}\left(x_{k+1} - \frac{1}{4}\right) \), where \( x_{n+1} = x_{1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2n}
Simplify the expression: \(\frac{a^{1 / 2}+a b^{-1}}{a^{-1 / 3}-a^{-1 / 6} b^{-1 / 3}+b^{-2 / 3}}-\frac{a}{\sqrt[3]{b}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a^{5/6}}
Tokorev S.i.

In each cell of a $4 \times 4$ table, there is a sign "+" or "-". It is allowed to simultaneously change the signs to the opposite in any cell and in all cells that share a side with it. How many different tables can be obtained by repeatedly applying such operations?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{12}}
Calculate the degree of the field extension \( \mathbb{Q}(\sqrt[80]{43},\sqrt[84]{43})/\mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1680}
Given the function $f(x) = A \sin(\omega x + \varphi)$, where $A$, $\omega$, and $\varphi$ are constants with $A > 0$ and $\omega > 0$. If $f(x)$ is monotonic in the interval $[\frac{\pi}{6}, \frac{\pi}{2}]$ and $f(\frac{\pi}{2}) = f(\frac{2\pi}{3}) = -f(\frac{\pi}{6})$, determine the smallest positive period of $f(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi}
Calculate the volume of the region enclosed by the cone $z^2 = x^2 + y^2$ and the sphere $x^2 + y^2 + z^2 = z$ using cylindrical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{8}}
On June 1, a bicycle dealer noted that the number of bicycles in stock had decreased by 4 for each of the past few months. If the stock continues to decrease at the same rate for the rest of the year, there will be 36 fewer bicycles in stock on October 1 than were in stock on January 1. For how many months has the stock been decreasing?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the norm of the linear functional \( \phi(x_n) = \sum_{n\ge 1} (-2)^{1-n}x_n \), where \( \phi: c_0 \to \mathbb{R} \) and \( c_0 \) is the set of sequences converging to 0.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the expected value of the function \( \phi(x) = x + \frac{F(x) - 1}{f(x)} \), where \( x \in [\underline{x}, \bar{x}] \) and \( f(x) \) is a density function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\underline{x}}
A, B, and C are standing on a staircase with 7 steps, where each step can accommodate up to 2 people. If the positions of people on the same step are not distinguished, calculate the number of different ways they can stand.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
The area of the figure enclosed by the curve $y= \sqrt{x}$, the line $y=2-x$, and the x-axis is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{7}{6}}
Which of the following numbers is an odd integer, contains the digit 5, is divisible by 3, and lies between $12^2$ and $13^2$?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
165
}
In a directed graph, there are two cycles: one cycle consists of 2 vertices, and the other consists of 3 vertices. Determine the number of 12-step paths from a vertex 'w' back to itself, denoted as \( f(12) \), using the edges of these two cycles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
The largest prime factor of the number $520$, $302$, $325$ has $5$ digits. What is this prime factor?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10613}
In the figure, \(O\) is the center of the circle and \(AB = 5 \text{ cm}\). What is the diameter of this circle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10 \text{ cm}}
Let $X_1, X_2, \ldots$ be an i.i.d. sequence of random variables with a distribution that has a continuous density function on $[0,1]$. Determine whether the following limit holds almost surely:

$$\lim_{n\to \infty}\frac{\sum_{1\le i<j \le n}{\bf{1}}_{(-1/n,1/n)}{(X_i-X_j)}}{n}=1$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
李阿姨配制一种洗洁水，由洗洁液和水按10mL：1L的浓度配成．现有30mL的洗洁液，可以配制多少升的洗洁水？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a
 disphenoid whose faces are scalene triangles with integer side lengths?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\( {15}\sqrt{7} \)
}
Let $z_1,$ $z_2,$ $z_3$ be complex numbers such that $|z_1| = 2,$ $|z_2| = 3,$ and $|z_3| = 4.$  Find the largest possible value of
\[|z_1 - z_2|^2 + |z_1 - z_3|^2 + |z_2 - z_3|^2.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{87}
Consider the set \( M = \{x \in \mathbb{R}^n : x^{⊤} A x = r\} \subset \mathbb{R}^n \), where \( r > 0 \) and \( A \) is a symmetric, positive semidefinite matrix \( A \in \mathbb{R}^{n \times n} \). Determine if \( M \) is a \( C^l \)-submanifold of \( \mathbb{R}^n \). If it is, find its dimension.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n - 1}
Evaluate the limit of the integral \( I(r) = \int_{\gamma_r} \frac{e^{iz}}{z^2} \, dz \) as \( r \to \infty \), where \( \gamma_r \) is a specified contour.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a Poisson random variable $Z$ with parameter $\lambda = \mathbb{E}[Z]$, and another random variable $G$ that converges in distribution to $Z$ with a total variation distance bound $d_{TV}(G,Z) \leq c$, where $c$ is a constant, find an upper bound for the probability $P(|G-Z| > t)$ for a fixed real number $t > 0$. Use the convergence in distribution to derive the bound.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c}
Let \( g(x) \) be a continuous and bijective function on \( \mathbb{R} \). Suppose that for every \( t \in \mathbb{R} \), the area bounded by \( y=g(x) \), \( x=b-t \), \( x=b \), and the \( x \)-axis is equal to the area bounded by \( y=g(x) \), \( x=b+t \), \( x=b \), and the \( x \)-axis. If \( g(b)=0 \) and \( \int_{\alpha}^{\beta}g^{-1}(x)~\mathrm{d}x = (\beta-\alpha) mb \), find the value of \( m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral \( \int_{-\infty}^\infty \frac{e^{ix}}{x^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\pi}
Determine the values of \( p \in (1, +\infty) \) for which the Banach space \( l_p \) is a Fréchet differentiable space (FD-space).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, +\infty)}
Determine the range of values for $a$ such that the function \( f(x) = ax + \cos(2x) + \sin(x) + \cos(x) \) is strictly increasing for all real numbers $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \dfrac{17}{8}, \infty \right)}
Determine whether the infinite product \( \prod_{k=1}^n \cos(bk) \) converges to zero as \( n \to \infty \), where \( b \) is an irrational number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In a measure space $(X, \mathcal F, \mu)$ with $\mu(X)=1$, there are $n$ measurable sets $F_1, F_2, \ldots, F_n$ such that $\mu(F_j) \geq 1/2$ for all $j$. Determine the smallest integer $k$ such that there exists a subset of indices $\{i_1, i_2, \ldots, i_k\}$ for which $F_{i_1} \cap F_{i_2} \cap \cdots \cap F_{i_k} \neq \varnothing$. What is the smallest possible value of $k$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\lceil n/2 \rceil}
Evaluate the limit of the integral \( \int_{0}^{\pi}\frac {\sin x}{1+\cos^2(nx)}\mathrm{d}x \) as \( n \to +\infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
Determine the minimum number of terms required in the Maclaurin series to approximate the square root of $e$ with an error less than $10^{-3}$. Use the inequality \( \left| \frac{e^c}{(n+1)!} \cdot 0.5^{(n+1)} \right| \leq 10^{-3} \), where $0 < c < 0.5$, to find your answer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Calculate the integral \( \int_{-2}^2 \left(x^3\cos\frac{x}{2}+\frac{1}{2}\right)\sqrt{4-x^2}\ dx \) and provide the first 10 digits of the result.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.141592653}
Consider a bin containing 1047 distinct numbers. You extract 10 elements without replacement, note them, and then reinsert them into the bin. This process is repeated 50 times. What is the expected number of different elements you will have seen after these 50 extractions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{399}
某船上的人开始看见灯塔在南偏东30°方向，后来船沿南偏东60°方向航行45n mile后，看见灯塔在正西方向，则这时船与灯塔的距离是多少？
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15\sqrt{3}}
Let \( V \) be an \( n \)-dimensional real inner product space. Define the operator \( M: L(V) \to L(V) \) by \( M(S) = S + S^* \), where \( S^* \) is the adjoint of \( S \). Determine the trace of the operator \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n(n + 1)}
In a shop, two employees are each serving a customer. The service time for each customer is an independent random variable following an exponential distribution with parameter \( \lambda \). After finishing with their current customer, each employee will serve you. What is the probability that you complete your transaction and leave the shop before both of the initial customers finish their transactions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
In a vector space of dimension $m$, determine the maximum total number of dimensions that two non-intersecting affine subspaces can have.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2m - 2}
Let $ \triangle ABC $ be a non-degenerate triangle with perimeter 4 such that $ a = bc \sin^2 A $. If $ M $ is the maximum possible area of $ \triangle ABC $, and $ m $ is the minimum possible area of $ \triangle ABC $, then $ M^2 + m^2 $ can be expressed in the form $ \frac{a}{b} $ for relatively prime positive integers $ a $ and $ b $. Compute $ a + b $.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
59
}
The non-terminating periodic decimal $0.124124 \ldots = 0.\overline{124}$ has period three and is abbreviated by placing a bar over the shortest repeating block.

 If all digits $0$ through $9$ are allowed, how many distinct periodic decimals $0.\overline{d_1d_2 \ldots d_6}$ have period exactly six? Do not include patterns like $0.323$ and $0.17$ that have shorter periods.


Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{998,910 
}
Determine whether the equation \( x^{2}+y^{2}+z^{2}=2 x y z \) has a solution in non-negative integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 0, 0)}
The distance from the point $P$ to two vertices $A$ and $B$ of an equilateral triangle are $|P A| = 2$ and $|P B| = 3$. Find the greatest possible value of $|P C|$.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of (positive) integers such that $k$ divides $\gcd(a_{k-1},a_k)$ for all $k\geq 2$.  Compute the smallest possible value of $a_1+a_2+\cdots+a_{10}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{440}
A polygon $A_1A_2A_3\dots A_n$ is called beautiful if there exist indices $i$, $j$, and $k$ such that $\measuredangle A_iA_jA_k = 144^\circ$. Compute the number of integers $3 \le n \le 2012$ for which a regular $n$-gon is beautiful.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{401}
Evaluate the infinite series: $$\sum_{k=0}^{\infty} \sum_{j=0, j\neq k}^{\infty} \sum_{i=0, i\neq j, i\neq k}^{\infty} \frac{1}{3^i3^j3^k}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{81}{208}}
In triangle $\triangle ABC$, $\angle C = 120^\circ$. Point $M$ is located on side $AB$ such that $\angle CMB = 60^\circ$ and the ratio $BM:AB = 1:3$. Determine the measure of $\angle B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45^\circ}
In the expansion of \((2x+3y-4z+w)^9\), determine the coefficient of the term \(x^3y^2z^3\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-23224320}
Evaluate the integral \( \int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)} \) using residue theory.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \sqrt{2}}
Calculate the limit \( \lim_{n \to \infty} x_n \) for the sequence defined by:
\[ x_n = \frac{n}{2^{n+1}} \sum_{i=1}^{n} \frac{2^i}{i} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Identify the second nonabelian simple group and determine its order.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
Determine whether the following limit diverges:
\[ \lim_{n \to +\infty} \frac{n+o(n)}{\sqrt{n+o(n)} - \sqrt{n+o(n)}} \cdot \cos(1) \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Determine the complexity class for subsumption in the description logic \(\mathcal{AL}(\circ, ^{-})\). Is it in \(\mathrm{P}\) or does it belong to a different complexity class?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathrm{P}}
Let \( n \in \mathbb{N}_{\geq 2} \). For any real numbers \( a_1, a_2, \ldots, a_n \), denote \( S_0 = 1 \) and for \( 1 \leq k \leq n \), denote \[ S_k = \sum_{1 \leq i_1 < i_2 < \ldots < i_k \leq n} a_{i_1} a_{i_2} \ldots a_{i_k}. \] Find the number of \( n \)-tuples \((a_1, a_2, \ldots, a_n)\) such that \[ (S_n - S_{n-2} + S_{n-4} - \ldots)^2 + (S_{n-1} - S_{n-3} + S_{n-5} - \ldots)^2 = 2^n S_n. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}
The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation. 
Find the smallest possible number of independent deputies after the vacation.

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Determine whether the limit of the summation diverges: $$\lim\limits_{n \to +\infty} \sum_{k=1}^n \frac{n\cdot \ln (k)}{n^2+k^2}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In the given geometric figure, calculate the ratio \( \frac{\text{area}(\triangle CDF)}{\text{area}(\triangle CEF)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}