inclusionAI/Ring-lite-rl-data huggingface dataset:
A merchant purchased a vintage item and sold it for a profit of 7%. If he had managed to negotiate the purchase price down by 8% and sold it with an additional discount of $3 during a holiday sale, he still would have gained a profit of 12%. What was the cost price of the vintage item?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$75.76}
Approximate the integral \( \int_3^4 \frac{x}{\sqrt{(7x-10-x^2)^3}}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.10}
On the New Year's table, there are 4 glasses arranged in a row: the first and third glasses contain orange juice, while the second and fourth glasses are empty. In anticipation of guests, Valya absentmindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour its entire contents into one of the two empty glasses.

Find the expected number of pourings needed to achieve the opposite situation: the first and third glasses are empty, while the second and fourth glasses are full.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Calculate the argument of the sum:
\[ e^{5\pi i/36} + e^{11\pi i/36} + e^{17\pi i/36} + e^{23\pi i/36} + e^{29\pi i/36} \]
in the form $r e^{i \theta}$, where $0 \le \theta < 2\pi$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17\pi}{36}}
Two milk processing plants must process an equal amount of a certain volume of milk together. The second plant started working $a$ working days later but processed daily $m$ liters more than the first one. After $\frac{5a}{9}$ working days since the start of their joint work, $\frac{1}{3}$ of the entire task remained incomplete. How many working days were required to complete the task if the work was finished simultaneously and each plant processed half of the specified volume of milk?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2a}
Let $T = (a,b,c)$ be a triangle with sides $a$, $b$, and $c$, and area $\triangle$. Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a$, $b' = h_b$, $c' = h_c$) and denote its area by $\triangle'$. Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from the altitudes of $T'$, and denote its area by $\triangle''$. Given that $\triangle' = 30$ and $\triangle'' = 20$, find $\triangle$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
Given the line \( L: x + y - 9 = 0 \) and the circle \( M: 2x^2 + 2y^2 - 8x - 8y - 1 = 0 \), point \( A \) is on line \( L \), and points \( B \) and \( C \) are on circle \( M \). In triangle \( \triangle ABC \), \(\angle BAC = 45^\circ\) and \( AB \) passes through the center of circle \( M \). Then, the range of the x-coordinate of point \( A \) is \(\quad\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[3, 6]}
Let $R$ be the set of all rectangles centered at the origin and with perimeter $1$. Let $S$ be a region that contains all of the rectangles in $R$. The minimum possible area of $S$ is $\pi a$. Find $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/16}
How many $3\times 2$ matrices with entries in $\mathbb{Z}/3\mathbb{Z}$ are in reduced row echelon form?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Gavrila throws a ball from behind his head at a speed of $5 \text{ m/s}$ against the wind at an angle to the horizontal. After 1 second, the ball hits his stomach 1 meter below the point of release. Given that the force exerted on the ball by the air is directed horizontally and does not depend on the position and speed, and the acceleration due to gravity, $g$, is $10 \text{ m/s}^2$, determine the maximum distance the shadow of the ball moves away from Gavrila's feet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{4}}
Determine the number of lines that can be drawn such that they are a distance of \((\sec(\theta))^2 + 2 (\csc(\theta))^2\) from the point \((1,3)\) and intersect the line \(3x+4y=5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
On a circle, there are 2009 nonnegative integers, each not greater than 100. If two adjacent numbers can be increased by 1 simultaneously, what is the minimum number of such operations required to make all the numbers on the circle equal?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100400}
Suppose I have two orthonormal bases for $\mathbb{C}^d$, denoted $\mathbf{B_1}$ and $\mathbf{B_2}$, such that for any $|e \rangle \in \mathbf{B_1}$ and $|f \rangle \in \mathbf{B_2}$, it is the case that $| \langle e| f \rangle| = 1/\sqrt{d}$ (in other words, the bases are mutually unbiased). For a given vector $| \alpha \rangle \in \mathbb{C}^d$, let $Z_{\alpha} = \{ | v \rangle \in \mathbf{B_1} \cup \mathbf{B_2} | \langle v | \alpha \rangle =0 \}$, be the set of vectors from both bases that are orthogonal to $| \alpha \rangle$. What is the maximum size of $Z_{\alpha}$ for non-zero $| \alpha \rangle$ (i.e. $max_{|\alpha \rangle \in \mathbb{C}^d, |\alpha \rangle \neq 0} |Z_{\alpha}|$)? In other words, for a suitable choice of the vector $| \alpha \rangle$ (that isn't the zero vector), what is the most number of vectors from the two bases that are simultaneously orthogonal to $| \alpha \rangle$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d-1}
1.52 In coin tossing, if $Z$ represents heads and $F$ represents tails, then the sequence of coin tosses is represented by a sequence of $Z$ and $F$. We can count the occurrences of heads followed by tails $(F Z)$, heads followed by heads $(Z Z)$, etc. For example, the sequence
ZZFFZZZZFZZFFFF

is the result of 15 coin tosses, which includes 5 $Z Z$, 3 $Z F$, 2 $F Z$, and 4 $F F$. How many sequences of 15 coin tosses have exactly 2 $Z Z$, 3 $Z F$, 4 $F Z$, and 5 $F F$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{560}
Catherine originally had an equal number of pencils and pens. She gave 12 pens and 4 pencils to her first friend, 15 pens and 7 pencils to her second friend, and 9 pens and 5 pencils to her third friend. If, after distributing these items, she was left with 36 pens and 28 pencils, how many pens and pencils did Catherine originally have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, identical letters - identical digits). The result was the word "GWATEMALA". How many different numbers could Egor have originally written if his number was divisible by 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{114240}
Let \( n \) be a positive integer. Given a real number \( x \), let \( \lfloor x \rfloor \) be the greatest integer less than or equal to \( x \). For example, \( \lfloor 2.4 \rfloor = 2 \), \( \lfloor 3 \rfloor = 3 \), and \( \lfloor \pi \rfloor = 3 \). Define a sequence \( a_1, a_2, a_3, \ldots \) where \( a_1 = n \) and
\[ 
a_m = \left\lfloor \frac{a_{m-1}}{3} \right\rfloor,
\]
for all integers \( m \geq 2 \). The sequence stops when it reaches zero. The number \( n \) is said to be lucky if 0 is the only number in the sequence that is divisible by 3. For example, 7 is lucky, since \( a_1 = 7, a_2 = 2, a_3 = 0 \), and none of 7, 2 are divisible by 3. But 10 is not lucky, since \( a_1 = 10, a_2 = 3, a_3 = 1, a_4 = 0 \), and \( a_2 = 3 \) is divisible by 3. Determine the number of lucky positive integers less than or equal to 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
Evaluate the integral \( \int_{0}^{\pi/2} (\cos^{5}x)\sin (7x)\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Evaluate the limit: $$\lim_{n\to \infty}\sqrt[n^2 +n]{\prod_{i=0}^n \frac{(n+1)!}{(i+1)!(n-i+1)!}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
In triangle $ABC$, where $AB = AC = 20$ and $BC = 14$, find the minimum value of the sum $BN + MN + MC$ for points $M$ on $\overline{AB}$ and $N$ on $\overline{AC}$. Compute $100x$, where $x$ is this minimum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3514}
Julia played tag with 4 kids on Wednesday. She had played with 6 kids on Monday and with some kids on Tuesday. She played with 2 more kids on Monday than on Wednesday. How many kids did she play with on Tuesday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The mean, median, unique mode, and range of a collection of eight integers are all equal to 8. The largest integer that can be an element of this collection is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$14$}
In how many ways can a \(3 \times 6\) rectangle be divided into L-shaped pieces of three cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given the vector $$\overrightarrow {a_{k}} = (\cos \frac {k\pi}{6}, \sin \frac {k\pi}{6} + \cos \frac {k\pi}{6})$$ for k=0, 1, 2, …, 12, find the value of $$\sum\limits_{k=0}^{11} (\overrightarrow {a_{k}} \cdot \overrightarrow {a_{k+1}})$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\sqrt{3}}
Ivan Semenovich leaves for work at the same time every day, travels at the same speed, and arrives exactly at 9:00 AM. One day, he overslept and left 40 minutes later than usual. To avoid being late, Ivan Semenovich increased his speed by 60% and arrived at 8:35 AM. By what percentage should he have increased his usual speed to arrive exactly at 9:00 AM?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30 \%}
Calculate the correlation coefficient for the random variables X and Y given the joint density function: \[ f(x, y) = \begin{cases} 4 & 0 < y \le x < 1 \\ 0 & \text{otherwise} \end{cases} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
满足下述条件的最小正实数 k：对任意不小于 k 的 4 个互不相同的实数 a,b,c,d，都存在 a,b,c,d 的一个排列 p, q, r, s，使得方程 (x^2+px+q)(x^2+rx+s)=0 有 4 个互不相同的实根.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
In the answer obtained by subtracting 101011 from 10000000000, how many times does the digit 9 appear?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Consider  $9$  points in space, no four of which are coplanar. Each pair of points is joined by an edge (that is, a line segment) and each edge is either colored blue or red or left uncolored. Find the smallest value of   $\,n\,$  such that whenever exactly  $\,n\,$  edges are colored, the set of colored edges necessarily contains a triangle all of whose edges have the same color.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 33 }
Evaluate the limit $$\lim_{x \to 0} \frac{e^{\tan x} - e^x + \ln(\sec x + \tan x) - x}{\tan x - x}$$ using L'Hopital's rule or alternative methods.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{3}{2}\)}
A function  $f$  satisfies the equation
\[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\]
for every real number  $x$  except for  $x = 0$  and  $x = 1$ . Find a closed formula for  $f$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{x^3 - x^2 - 1}{2x^2 - 2x}}
Let the even function $f(x) = \log_a |x+b|$ be monotonically increasing on the interval $(0, +\infty)$. Then, compare $f(b-2)$ and $f(a+1)$ (fill in with an equality or inequality sign).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{<}
Jerry had 7 action figures, but needed a certain number of action figures for a complete collection. If each 1 costs $8, he would need $72 to finish his collection. How many action figures are needed for a complete collection?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Five contestants $A, B, C, D, E$ participate in the "Good Voice" competition, and they stand in a line for a group appearance. Each of them has a contestant number badge on their chest. The sum of the five numbers is 35. It is known that the sum of the numbers of the contestants standing to the right of $E$ is 13; the sum of the numbers of the contestants standing to the right of $D$ is 31; the sum of the numbers of the contestants standing to the right of $A$ is 21; and the sum of the numbers of the contestants standing to the right of $C$ is 7. What is the sum of the numbers of the contestants standing at the far left and far right $\qquad$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
What is the residue at $0$ of $(1-e^{-z})^{-n}$ for $n\in\mathbb{N}^+$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
What is the sum of non-negative integer solutions of $\{(x,y,z)\mid x+y+z=k\}$, $k = 1,\ldots,N$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\binom{N+3}{3}-1\)}
Determine whether a closed form formula exists for the series \( \sum_{k=0}^{n-1} \binom{n+k}{n-k-1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{F_{2n}}
Given a program with a 20% probability of having a bug that causes it to crash, there are two testing procedures that each examine 50% of the program's code. Procedure 1 detects 50% of bugs in its coverage area, while Procedure 2 detects 30%. The overlap between the two procedures is 50% of their respective code areas, meaning they can both detect bugs in this common region, independently. Neither test generates false bug warnings, and bugs are assumed to be equally likely throughout the code (not reflecting real-world software testing). What is the probability that a program is free of major bugs if it passes both tests?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.8625}
Given the number $200\ldots 002$ (100 zeros), determine how many ways two of the zeros can be replaced with non-zero digits such that the resulting number is divisible by 66.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27100}
Given a function $f(x)=a{e^x}-\frac{1}{2}{x^2}$, where $a\in\mathbb{R}$. If the function has two critical points $x_{1}$ and $x_{2}$, and $\frac{x_{2}}{x_{1}}\geq 2$, then the range of real number $a$ is ____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(0,\frac{\ln 2}{2}\right]}
A circle is tangent to the extensions of two sides \(AB\) and \(AD\) of the square \(ABCD\), and the points of tangency cut off segments of length 4 cm from vertex \(A\). From point \(C\), two tangents are drawn to this circle. Find the side length of the square if the angle between the tangents is \(60^\circ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4(\sqrt{2}-1)}
The digits $1,2,3,4,5,6$ are randomly chosen (without replacement) to form the three-digit numbers $M=\overline{A B C}$ and $N=\overline{D E F}$. For example, we could have $M=413$ and $N=256$. Find the expected value of $M \cdot N$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{143745}
Determine the values of \( \alpha \) for which the integral \( \int_E \frac{x^\alpha}{\sqrt{x^4+y^2}} \, dx \, dy \) converges, where \( E \) is the region defined by the area of the circle \( x^2+(y-1)^2=1 \) minus the area of the circle \( x^2+(y-0.5)^2=0.5^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha > -1}
Compute the distribution $x\delta'_0$, where $\delta'_0$ is defined by $\delta'_0(u) = -u'(0)$ for $u \in D\mathbb{R}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\delta_0}
Determine the area of the region \( R \) that is the intersection of all regions \( R(\alpha) \) for \( 0 < \alpha < 1 \), where each \( R(\alpha) \) is the region in \( \mathbb{R}^2 \) bounded by the convex pentagon with vertices \((0,1-\alpha), (\alpha, 0), (1, 0), (1,1), (0, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{6}}
9.1. (13 points) The surface of a round table is divided into 9 identical sectors, in which the numbers from 1 to 9 are written sequentially clockwise. Around the table sit 9 players with numbers \(1, 2, \ldots, 9\), going clockwise. The table can rotate around its axis in both directions, while the players remain in place. The players are seated at equal distances from each other, so when the table stops rotating, exactly one player is opposite each sector and receives the number of coins written on that sector. It is known that after 11 rotations of the table, player ㄱo4 received 90 coins, and player № 8 received 35 coins. How many coins did player № 1 receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57}
Find the maximum value of $y$ given the equation $2(x^3+y)^4+(x^3+y)^2=2x^3+y$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5}{8}}
Let $a$, $b$, $c$, $d$ be complex numbers satisfying
\begin{align*}
  5 &= a+b+c+d \\
  125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\
  1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\
  25 &= a^4+b^4+c^4+d^4
\end{align*}
Compute $abcd$.

[i]Proposed by Evan Chen[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70}
Evaluate the limit: $$\lim_{x\to-1}\frac{\sqrt[3]{26-x}-\sqrt{8-x}}{3x^2+4x+1}$$ without using L'Hôpital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{7}{108}}
Let $D \subset \mathbb{R}^3$ be the set $D = \{(x, y, z) \mid x^2 + y^2 \leq 1~\text{and}~z = 0\}$, representing a disk in the $xy$-plane. Consider the linear transformation $T : \mathbb{R}^3 \to \mathbb{R}^3$ defined by $T(x, y, z) = (2x − y + z, x − y + 10z, 3x − 7y + 2z)$. Find the area of the image $T(D)$ in $\mathbb{R}^3$. \(\boxed{\text{Area}}\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \sqrt{138}}
Determine the values of $x$ for which the series \( \sum _ { n = 1 } ^ \infty \frac { x ^ n } { \sin ( n \theta ) } \) converges when \( \theta = t \pi \), where \( t \) is an irrational number, such as \( t = \sqrt{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 1)}
A set S consisting of $2019$ (different) positive integers has the following property:
the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements.
What is the maximum number of prime numbers that $S$ can contain?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1820
}
Find the residue of the function \( \sin(z + \frac{1}{z}) \) at \( z = 0 \) using the Laurent series expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{J_1(2)}
Evaluate the limit of the sequence as \( n \) approaches infinity: \[ \lim_{{n \to \infty}} \left( \sum_{i={(n+1)}/{2}}^n {n \choose i} \times 0.51^i \times 0.49^{n-i} \right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the remainder when the binary message 010100011010000 is divided by the polynomial divisor $x^{14} + x^{12} + x^8 + x^7 + x^5$ using the generator polynomial $x^5 + x^4 + x^2 + 1$ in the field $GF(2)$. What is the correct remainder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{00001}
Let $\mathcal E$ be an ellipse with foci $F_1$ and $F_2$.  Parabola $\mathcal P$, having vertex $F_1$ and focus $F_2$, intersects $\mathcal E$ at two points $X$ and $Y$.  Suppose the tangents to $\mathcal E$ at $X$ and $Y$ intersect on the directrix of $\mathcal P$.  Compute the eccentricity of $\mathcal E$.

(A [i]parabola[/i] $\mathcal P$ is the set of points which are equidistant from a point, called the [i]focus[/i] of $\mathcal P$, and a line, called the [i]directrix[/i] of $\mathcal P$.  An [i]ellipse[/i] $\mathcal E$ is the set of points $P$ such that the sum $PF_1 + PF_2$ is some constant $d$, where $F_1$ and $F_2$ are the [i]foci[/i] of $\mathcal E$.  The [i]eccentricity[/i] of $\mathcal E$ is defined to be the ratio $F_1F_2/d$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2 + \sqrt{13}}{9}}
Split the number 10 into the sum of two positive integers. Under the condition that both addends are greater than 2, calculate the probability that both addends are prime numbers. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3}{5}}
Let \(a\) and \(b\) be any positive real numbers. Determine the value of the expression

\[
\frac{\sqrt{\frac{a b}{2}}+\sqrt{8}}{\sqrt{\frac{a b+16}{8}+\sqrt{a b}}}
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the integral $$∫_{ -1 }^{ 1 }(\sqrt {4-x^{2}}+x^{3})dx$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt {3}+ \frac {2π}{3}}
Solve the system of equations for real numbers:
\[ \begin{cases} \sqrt{x^2-2x+6}\cdot \log_{3}(6-y) =x \\ \sqrt{y^2-2y+6}\cdot \log_{3}(6-z)=y \\ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x)=z \end{cases}. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(3, 3, 3)}
8. Given a non-empty set $M$ satisfying
$$
M \subseteq\{0,1, \cdots, n\}\left(n \geqslant 2, n \in \mathbf{Z}_{+}\right) \text {. }
$$

If there exists a non-negative integer $k(k \leqslant n)$, such that for any $a \in M$, we have $2 k-a \in M$, then the set $M$ is said to have property $P$. Let the number of sets $M$ with property $P$ be $f(n)$. Then the value of $f(9)-f(8)$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
Arrange 5 people to be on duty from Monday to Friday, with each person on duty for one day and one person arranged for each day. The conditions are: A and B are not on duty on adjacent days, while B and C are on duty on adjacent days. The number of different arrangements is $\boxed{\text{answer}}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z} \ | \ z \in R\right\rbrace$.  Find the area of $S.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3 \sqrt{3} + 2 \pi}
The GDP of a certain city last year was 84,300,000 ten thousand yuan. Represent this number in scientific notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.43 \times 10^7}
Find the least possible number of elements which can be deleted from the set  $\{1,2,...,20\}$  so that the sum of no two different remaining numbers is not a perfect square.

N. Sedrakian , I.Voronovich
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}