inclusionAI/Ring-lite-rl-data huggingface dataset:
Calculate the correlation coefficient between $X_{(1)}$ and $X_{(3)}$, where $X_{(1)}$ and $X_{(3)}$ are the order statistics of three independent random variables $X_1, X_2, X_3$, each uniformly distributed over $[0,1]$. Given that $X_{(k)}$ follows a Beta distribution with parameters $k$ and $4-k$, and the variance of $X_{(k)}$ is $\frac{k(4-k)}{80}$ with mean $\frac{k}{3}$, use the theorem for covariance to find the correlation coefficient.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
James decides to replace his car and buy a new car for his wife. He sold his $20,000 car for 80% of its value and then was able to haggle to buy a $30,000 sticker price car for 90% of its value. Additionally, he sold his wife's $15,000 car for 70% of its value and bought another car with a $25,000 sticker price at an 85% discounted price.

During these transactions, a 7% sales tax was applied to the purchase of each new car, and James had to pay a 2% processing fee for each car sale. Considering all these transactions, how much total amount was James out of pocket?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$25,657.50}
Calculate the volume of the solid enclosed by the surface \((x^2+y^2+z^2)^2 = x\) using spherical coordinates \((r, \theta, \phi)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
The length of a circular race track is $1 \mathrm{~km}$. Two motorcyclists start simultaneously from a given point $A$ on the track, traveling in opposite directions. One of them travels at a constant speed, while the other accelerates uniformly. They first meet at point $B$ on the track, and then for the second time at point $A$. How far did the first motorcyclist travel from the starting point to point $B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{-1 + \sqrt{5}}{2} \text{ km}}
James visits 20 houses to try and make a sale. He manages to sell something in every house. The next day he visits twice as many houses. He sold two things at each house in each of the two days and sold 104 items in total. What percentage of houses did he sell to on the second day?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{80\%}
An  $8\times 8$  chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
$\begin{aligned} & \text{ Let }ABC\text{ be an acute triangle, and let }D\mathrm{~and~}E\text{ be the feet of the altitudes to }BC\mathrm{~and~}CA,\text{respectively. Suppose that} \\ & \mathrm{segments~}AD\mathrm{~and~}BE\text{ intersect at point }H\mathrm{~with~}AH=20\mathrm{~and~}HD=13.\mathrm{Compute~}BD\cdot CD. \end{aligned}$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{429}
Calculate the value of the sum: \[ \sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi i}{2000}\right) \] where \( \gcd \) denotes the greatest common divisor.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{800}
Let $x_0,x_1,x_2,\dotsc$ be a sequence of numbers, where each $x_k$ is either $0$ or $1$. For each positive integer $n$, define \[S_n = \sum_{k=0}^{n-1} x_k 2^k\] Suppose $7S_n \equiv 1 \pmod{2^n}$ for all $n \geq 1$. What is the value of the sum \[x_{2019} + 2x_{2020} + 4x_{2021} + 8x_{2022}?\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$6$}
Determine the infimum of \( \mu(P) = \sup_{|z| = 1} |P(z)| \) over all complex polynomials \( P \) of degree \( n \) in the form \( P = X^n + \sum_{k=0}^{n-1} a_kX^k \), denoted as \( \mu_n \), for \( n \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Draw the lines parallel to two sides of a triangle that bisect the area of the triangle. In what ratio does the line parallel to the third side of the triangle, passing through their intersection point, divide the area of the triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{2}-2}{7}}
Given the pattern of positive odd numbers shown below, find the 6th number from the left in the 21st row.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{811}
During one hour, Tom can read 12 pages of a book. How many pages would he be able to read during 2 hours if he could increase his reading speed by a certain factor, given that he reads 72 pages in that time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Let $n$ be a positive integer. Compute the number of words $w$ that satisfy the following three properties.

1. $w$ consists of $n$ letters from the alphabet $\{a,b,c,d\}.$

2. $w$ contains an even number of $a$'s

3. $w$ contains an even number of $b$'s.

For example, for $n=2$ there are $6$ such words: $aa, bb, cc, dd, cd, dc.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4^n-2\cdot3^n+2^n}
给定整数 $n \geqslant 3$, 设 $a_{1}, a_{2}, \cdots, a_{2 n}, b_{1}, b_{2}, \cdots, b_{2 n}$ 是 $4 n$ 个非负实数, 满足$$a_{1}+a_{2}+\cdots+a_{2 n}=b_{1}+b_{2}+\cdots+b_{2 n}>0$$且对任意 $i=1,2, \cdots, 2 n$, 有 $a_{i} a_{i+2} \geqslant b_{i}+b_{i+1}$ （这里 $a_{2 n+1}=a_{1}, a_{2 n+2}=a_{2}$, $b_{2 n+1}=b_{1}$ ), 求 $a_{1}+a_{2}+\cdots+a_{2 n}$ 的最小值。请提供当 $n=3$ 和 $n\geqslant 4$ 时最小值的和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
N是由5个不同的非零数码组成的5位数，且N等于这5个数码中取3个不同的数码构成的所有三位数的和，求所有的这种5位数N．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35964}
Determine the Galois group \( Gal \left(\frac{\mathbb{Q}(\sqrt[3]{3},\sqrt{-3})}{\mathbb{Q}}\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_3}
In the cartesian coordinate system xOy, point P is located at (1, $\sqrt {3}$). If $|$$\overrightarrow {OA}$$|=|$$\overrightarrow {OB}$$|=|$$\overrightarrow {OC}$$|=1, and $$\overrightarrow {OA}$$+ $$\overrightarrow {OB}$$+ $$\overrightarrow {OC}$$\= $$\overrightarrow {0}$$, determine the range of values for $$\overrightarrow {PB}$$• $$\overrightarrow {OA}$$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-$$\frac {5}{2}$, $ \frac {3}{2}$]}
There are 25 individuals numbered $1, 2, \ldots, 25$. Each individual randomly and independently chooses 12 others from this group to call. Let $X$ be the number of calls received by Individual 1, and $Y$ be the number of calls between Individual 1 and Individual 3. Calculate $E(X|Y=1)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In a certain game, two "electronic dogs," one black and one yellow, start from vertex A of a cube with edge length 1, labeled ABCD-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub>D<sub>1</sub>, and crawl along the edges. Completing one edge is referred to as "completing one segment." The black "electronic dog" follows the path AA<sub>1</sub>→A<sub>1</sub>D<sub>1</sub>→…, while the yellow "electronic dog" follows the path AB→BB<sub>1</sub>→…, both adhering to the rule that the (i+2)<sup>th</sup> segment must be on a skew line with the i<sup>th</sup> segment (where i is a positive integer). Assuming the black "electronic dog" completes 2006 segments and the yellow "electronic dog" completes 2007 segments, and they both stop at some vertex of the cube, calculate the distance between the black and yellow "electronic dogs" at that time. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let $a_0=5/2$ and $a_k=a^2_{k-1}-2$ for $k\geq1.$ Compute $$\prod\limits_{k=0}^{\infty}\left(1-\frac{1}{a_k}\right)$$ in closed form. The original answer is in \frac{m}{n} format, please give the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Consider a Brownian motion $X_t$, defined for $t \geqslant 0$. Let $T_a$ denote the stopping time at which $X_t$ reaches level $a$, i.e., $T_a = \inf \{t \mid X_t = a\}$. Compute the probability $\mathbb{P}\{T_{2} < T_{-1} < T_{3}\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{12}}
Helen the hippo and her friends are preparing for Thanksgiving at Helen's house. Helen baked 31 cookies three days ago, 270 cookies two days ago, and 419 cookies the day before yesterday. Unfortunately, some cookies were lost due to various reasons. On the day she baked 31 cookies, her mischievous bird friend, Beaky, ate 5 of them. On the second day, 15% of the cookies crumbled during a cookie stacking competition. And on the third day, Helen generously gave 30 cookies away to her neighbors. 

In addition to these losses, Helen received a surprise gift of 45 cookies from her friend Lucy. Considering all these events, how many cookies are available at Helen's house for Thanksgiving as of last night?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{690}
某单位有A，B两个餐厅为员工提供午餐与晚餐服务，甲、乙两位员工每个工作日午餐和晚餐都在单位就餐，近100个工作日选择餐厅就餐情况统计如下：

|选择餐厅情况(午餐，晚餐)|(A，A)|(A，B)|(B，A)|(B，B)|
|---|---|---|---|---|
|甲员工|30天|20天|40天|10天|
|乙员工|20天|25天|15天|40天|

假设甲、乙员工选择餐厅相互独立，用频率估计概率.
$记X为甲、乙两员工在一天中就餐餐厅的个数，求X的数学期望E(X)；$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1.9$}
The base of the quadrilateral pyramid \( S A B C D \) is a quadrilateral \( A B C D \) with \( A B = A D = 6 \), \( C B = C D = 2 \), and the sides \( A B \) and \( B C \) are perpendicular to each other. All the lateral faces are inclined to the plane of the base at an angle of \( 60^{\circ} \). Find the volume \( V \) of the pyramid, given that \( V > 12 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12\sqrt{3}}
Find all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the equation \( f(x) = f(x^2 + C) \) for all \( x \in \mathbb{R} \), where \( C > 0 \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = k}
Consider the set $L$ of binary strings of length less than or equal to $9$. For a string $w$, define $w^{+}$ to be the set $\{w, w^2, w^3, \ldots\}$ where $w^k$ represents $w$ concatenated to itself $k$ times. How many ways are there to pick an ordered pair of (not necessarily distinct) elements $x, y \in L$ such that $x^{+} \cap y^{+} \neq \varnothing$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1250}
Given the integers \( 1, 2, 3, \ldots, 40 \), find the greatest possible sum of the positive differences between the integers in twenty pairs, where the positive difference is either 1 or 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{58}
$
\begin{aligned}
 & \text{Given a positive integer }n,\text{find the proportion of the subsets of }\{1,2,\ldots,2n\}\mathrm{such~that} \\
 & \text{their smallest element is odd.}
\end{aligned}
$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\(\frac{2}{3}\).

}
A rectangle is cut at one corner to form a pentagon with side lengths of $13, 19, 20, 25,$ and $31$ (the order of lengths is not necessarily this). What is the area of the pentagon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{745}
For a positive integer $n$, denote by $\tau(n)$ the number of positive integer divisors of $n$, and denote by $\phi(n)$ the number of positive integers that are less than or equal to $n$ and relatively prime to $n$. Call a positive integer $n$ $\emph{good}$ if $\varphi (n) + 4\tau (n) = n$. For example, the number $44$ is good because $\varphi (44) + 4\tau (44)= 44$.

Find the sum of all good positive integers $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{172}
Ann, Bohan, Che, Devi, and Eden each ordered a different meal at a restaurant and asked for their own bill. Although the correct five bills arrived at the table, none of them were given to the correct person. If the following four statements are true, who was given the bill for Bohan's meal?

- Che and Eden ordered meals with the same cost.
- Devi's bill was given to the same person whose bill was given to Devi.
- Ann, Bohan, and Eden were each given a bill for less money than the meal they ordered.
- Ann's bill was given to Che.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Ann}
Evaluate the limit:

\[ \lim_{x \to 0} \frac{1}{x} \left[ \sqrt[3]{\frac{1 - \sqrt{1 - x}}{\sqrt{1 + x} - 1}} - 1 \right] \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{6}}
Consider the equation \( {x}^{4} = a{x}^{3} + b{x}^{2} + {cx} + {2007} \) , where \( a,b,c \) are real numbers. Determine the largest value of \( b \) for
 which this equation has exactly three distinct solutions, all of which are integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
3338
}
For a differential operator $P$ of order $p$ and a multi-index $\beta$, determine the maximum possible order of the commutator $[P, \partial_x^\beta]$. Provide your answer as an expression in terms of $p$ and $|\beta|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p + |\beta| - 1}
Calculate the probability that the minimum of a Brownian motion \((W_t)_{t \in [0, T]}\) is zero, given that \(W_0 = 0\) almost surely. Express this probability as \(\mathbb{P} \left(\min_{t \in [0, T]} W_t = 0\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a set of numbers \([1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\), calculate the expected value of the largest number when selecting 3 numbers at random from the set. Assume each number is selected with equal probability and selections are independent. What is the expected value of the largest number in such a selection?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{319}{40}}
Calculate the flux of the vector field \( \bar{V} = z\bar{i} + x\bar{j} + y\bar{k} \) through the surface defined by \( x^2 + y^2 + z^2 = 1 \), \( x = 0 \), \( y = 0 \), \( z = 0 \) with \( x > 0 \), \( y > 0 \), \( z > 0 \) using Gauss's theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that the derivative of the even function f(x) (x ≠ 0) is f'(x), and it satisfies f(1) = 0, when x > 0, xf'(x) < 2f(x). Find the range of x that makes f(x) > 0 true. Express your answer as an interval or union of intervals.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, 0) ∪ (0, 1)}
Find the least number of balanced quadruples \((a, b, c, d)\) in a set \(\mathcal{S}\) where \(1 \leq a < b < d < c \leq 20\) and the cardinality of \(\mathcal{S}\) is 4411. A quadruple is balanced if \(a + c = b + d\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
设 P_1,P_2,\cdots,P_{12} 是平面上的 12 个点（允许重合），其中任意两点之间的距离不超过 1. 求 \ds{1\leq i<j\leq 12}{}|P_iP_j|^2 的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
In the polar coordinate system, the maximum distance from a point on the circle $ρ=8\sin θ$ to the line $θ=\frac{π}{3}\ (ρ∈R)$ is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the moment generating function of a random variable $X$:
\[ M_{X}(t) = \frac{6}{\pi^{2}}\sum_{n\ge1}\frac{e^{\frac{t^2}{2n}}}{n^2}\,,\;t\in \mathbb{R} \]
Determine the probability that $X$ is a rational number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate :
 \( \sqrt[8]{{2207} - \frac{1}{{2207} - \frac{1}{{2207} - \cdots }}} \)
 Express your expression in the form \( \frac{a + b\sqrt{c}}{d} \) where \( a,b,c,d \in  \mathbb{Z} \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
\[ \frac{3 + \sqrt{5}}{2} \]}
If $f$ is a quadratic function such that $f(0)=1$ and $\int\frac{f(x)}{x^2(x+1)^3}dx $ is a rational function, find the value of $f'(0)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given the linear programming problem: Minimize \(x+y+z-\alpha t\) subject to \(2x - y + z = 6\), \(-x + y + t = 3\), and \(x, y, z, t \geq 0\). If \((3, 0, 0, \beta)\) is an optimal solution, determine the maximum value of \(\beta - \alpha\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
The average weight of some pigs is 15 kg. Weight less than 16 kg is considered as underweight. The maximum number of pigs that can be underweight is 4. How many pigs are there in total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the total length of the intervals on the number line where the inequalities \( x < 1 \) and \( \operatorname{tg} \log _{4} x > 0 \) hold.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{1+2^{\pi}}}
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1)=1,c(2n)=c(n),$ and $c(2n+1)=(-1)^nc(n)$. Find the value of $$S = \sum_{n=1}^{2013}c(n)c(n+2).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Evaluate the line integral \( \oint_C \mathbf{G} \cdot d\mathbf{r} \), where \( C \) is the boundary of the surface \( T \) defined by \( z = 10x - 20y + 60 \) over the rectangle \( R = \{ (x,y) \mid -2 < x < 3, 0 < y < 2 \} \) and \( \mathbf{G}(x,y,z) = -y^2 \mathbf{i} + xy \mathbf{j} + zx \mathbf{k} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-8970}
Three integers form a geometric progression. If 8 is added to the second number, an arithmetic progression is formed. If 64 is added to the third term of this arithmetic progression, a geometric progression is formed again. What are the original numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4, 12, 36}
$\underline{\text { Folklore }}$

On a chessboard, $n$ white and $n$ black rooks are placed such that rooks of different colors do not attack each other. Find the maximum possible value of $n$.

#
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Find the smallest prime divisor of the sum \(1^{60} + 2^{60} + \cdots + 33^{60}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
9.1. Find the number of roots of the equation

$$
\operatorname{arctg}\left(\operatorname{tg}\left(\sqrt{13 \pi^{2}+12 \pi x-12 x^{2}}\right)\right)=\arcsin \left(\sin \sqrt{\frac{13 \pi^{2}}{4}+3 \pi x-3 x^{2}}\right)
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the coefficient of  $x^2$  after expansion and collecting the terms of the following expression (there are  $k$  pairs of parentheses):  $((... (((x - 2)^2 - 2)^2 -2)^2 -... -2)^2 - 2)^2$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4^k - 1}{3} \cdot 4^{k-1}}
A train 220 m long is running with a certain speed. It passes a man who is running at 6 km/hr in the direction opposite to that in which the train is going in 11.999040076793857 seconds. What is the speed of the train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
Determine the minimum number of elements that a sample space must have to ensure the existence of $n$ independent events, each with a probability strictly between 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^n}
Chinese shooter Huang Yuting, 17 years old, won three gold medals for the Chinese delegation at the 2023 Hangzhou Asian Games with her tenacious style and superb skills, demonstrating a spirit of striving for progress and climbing to new heights. Below are the scores (in rings) she achieved in the final elimination stage of the women's 10-meter air rifle individual event:

| 1   | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   | 10  | 11  | 12  | 13  | 14  |
|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| 10.3| 10.3| 10.4| 10.4| 10.8| 10.8| 10.5| 10.4| 10.7| 10.5| 10.7| 10.7| 10.3| 10.6|

Find the variance of this data set. (Round to $0.001$)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.032}
Find all common points of the graphs

$$
y=8 \cos \pi x \cdot \cos ^{2} 2 \pi x \cdot \cos 4 \pi x \quad \text{and} \quad y=\cos 9 \pi x
$$

with abscissae belonging to the segment \( x \in [0, 1] \). In the answer, indicate the sum of the abscissae of the found points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.5}
Two runners started simultaneously in the same direction from the same point on a circular track. The first runner, moving ahead, caught up with the second runner at the moment when the second runner had only run half a lap. From that moment, the second runner doubled their speed. Will the first runner catch up with the second runner again? If so, how many laps will the second runner complete by that time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5}
In the geometric sequence {a_n}, given that a_2a_5 = -3/4 and a_2 + a_3 + a_4 + a_5 = 5/4, find the sum of the reciprocals 1/a_2 + 1/a_3 + 1/a_4 + 1/a_5. Express your answer as a single fraction or decimal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{5}{3}}
Evaluate the integral \( \frac{1}{2\pi} \int_{0}^{2\pi} e^{\cos\theta} \cos(n\theta) \,d\theta \) using the residue theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{I_n(1)}
Evaluate the integral \( \int_{0}^{\infty} f(x) g(x) \, dx \), where \( f(x) = x - \frac{x^3}{2} + \frac{x^5}{2 \cdot 4} - \frac{x^7}{2 \cdot 4 \cdot 6} + \cdots \) and \( g(x) = 1 + \frac{x^2}{2^2} + \frac{x^4}{2^2 \cdot 4^2} + \frac{x^6}{2^2 \cdot 4^2 \cdot 6^2} + \cdots \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
At 15:00 there were 20 students in the computer lab. At 15:03 and every three minutes after that, some students entered the lab. If at 15:10 and every ten minutes after that 8 students left the lab, there were 27 students in the computer lab at 15:44. How many students entered the lab every three minutes?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Consider the set  $$ \mathcal{S}=\{(a, b, c, d, e): 0<a<b<c<d<e<100\} $$  where  $a, b, c, d, e$  are integers. If  $D$  is the average value of the fourth element of such a tuple in the set, taken over all the elements of  $\mathcal{S}$ , find the largest integer less than or equal to  $D$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Convert the polar equation \(\rho^{2} \sin \theta - \rho \cos^{2} \theta - \sin \theta = 0\) for \(\theta \in (0, \pi)\) into its Cartesian coordinate system representation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = 1}
Find the number of elements that a set $B$ can have, contained in $(1, 2, ... , n)$, according to the following property: For any elements $a$ and $b$ on $B$ ($a \ne b$), $(a-b) \not| (a+b)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \left\lceil \frac{n}{3} \right\rceil }