inclusionAI/Ring-lite-rl-data huggingface dataset:
A circle touches the extensions of two sides \( AB \) and \( AD \) of the square \( ABCD \) with a side length of \( 2 \sqrt{3} \) cm. From point \( C \), two tangents are drawn to this circle. Given that the angle between the tangents is \( 30^{\circ} \), find the radius of the circle. Use the fact that \( \sin 15^{\circ} = \frac{\sqrt{3}-1}{2 \sqrt{2}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In an isosceles trapezoid \(ABCD\), \(AB\) is parallel to \(CD\) and \(AB > CD\). Define a hyperbola with foci at \(A\) and \(B\) that passes through point \(D\), and an ellipse with foci at \(C\) and \(D\) that passes through point \(A\). Let the eccentricities of the hyperbola and the ellipse be \(e_1\) and \(e_2\) respectively. Find the value of \(e_1 e_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{11}{27}$}
Consider the set $X = \mathbb{R}^2 - \{(0,0),(1,0),(2,0),(3,0)\}$. Let $V$ be the vector space of irrotational vector fields over $X$, and let $W$ be the vector space of conservative vector fields over $X$. Determine the dimension of the quotient vector space $V/W$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Find the minimum value of the sum \( \sum_{i=1}^{n} a_{i} \). Express your answer in terms of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Let $a_n$ ($n=2,3,4,\ldots$) be the coefficient of the term containing $x$ in the expansion of $(3- \sqrt{x})^n$. Calculate the value of $$\frac{1}{a_2} + \frac{3}{a_3} + \frac{9}{a_4} + \ldots + \frac{3^{2011}}{a_{2013}}$$ and express your answer as a single fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4024}{2013}}
Let $\tau = \frac{1 + \sqrt{5}}{2}.$  Find
\[\sum_{n = 0}^\infty \frac{\lfloor \tau^n \rceil}{2^n}.\]Note: For a real number $x,$ $\lfloor x \rceil$ denotes the integer closest to $x.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{2}}
Dexter's Laboratory has $2024$ robots, each with a program set up by Dexter. One day, his naughty sister Dee Dee intrudes and writes an integer in the set $\{1, 2, \dots, 113\}$ on each robot's forehead. Each robot detects the numbers on all other robots' foreheads and guesses its own number based on its program, individually and simultaneously.

Find the largest positive integer $k$ such that Dexter can set up the programs so that, no matter how the numbers are distributed, there are always at least $k$ robots who guess their numbers correctly.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
What is the smallest integer $n$ , greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\boxed{337}\)}
The numbers $1, 2,..., 2023$ are written on the board in this order. We can repeatedly perform the following operation with them: We select any odd number of consecutively written numbers and write these numbers in reverse order. How many different orders of these $2023$ numbers can we get?

Example: If we start with only the numbers $1, 2, 3, 4, 5, 6$, we can perform the following steps
$$1, 2, 3, 4, 5, 6 \to 3, 2, 1,4, 5, 6 \to 3, 6, 5, 4, 1, 2 \to 3, 6, 1, 4, 5, 2 \to ...$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1011! \times 1012!$}
In isosceles triangle \( \triangle ABC \), where \( \overline{AD} \) is a median and \( \overline{EC} \) is a cevian, it is given that \( \overline{AD} = 2\overline{EC} \). Find the value of \( \cos{B} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{4}}
What is the perimeter of the triangle formed by the points of tangency of the incircle of a 5-7-8 triangle with its sides?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9 \sqrt{21}}{7}+3}
How many pairs of positive integer solutions \((x, y)\) satisfy \(\frac{1}{x+1} + \frac{1}{y} + \frac{1}{(x+1) y} = \frac{1}{1991}\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{64}
Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{351}
Seryozha arranged the numbers from 1 to 8 in circles so that each number, except one, was used exactly once. It turned out that the sums of the numbers on each of the five lines are equal. Which number did Seryozha not use?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Find the maximum possible value of  $k$  for which there exist distinct reals  $x_1,x_2,\ldots ,x_k $  greater than  $1$  such that for all  $1 \leq i, j \leq k$ ,  $$ x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}. $$  *Proposed by Morteza Saghafian*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
My frisbee group often calls "best of five" to finish our games when it's getting dark, since we don't keep score. The game ends after one of the two teams scores three points (total, not necessarily consecutive). If every possible sequence of scores is equally likely, find the expected score of the losing team. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Evaluate the limit \(\lim_{x\to \infty} \frac{\Gamma(x+1,x)}{\Gamma(x+1)}\), where \(x\) is a positive integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Find the expected value $E(X_t)$ for the stochastic differential equation given by $dX_t = (X_t + 2t) \,dW_t + (t^2 + 1) \,dW_t$ with the initial condition $X_0=0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a collection of 20 marbles drawn from an infinite pool of pink and blue marbles, with replacement. The probability of drawing a pink marble is 0.4, and the probability of drawing a blue marble is 0.6. You draw 20 marbles and check if there are at least 12 pink marbles in the collection. If true, you note this result. Then, you randomly discard one marble from the collection and draw a new marble to replace it. You repeat this process of inspecting, noting the result, discarding, and replenishing a total of 100 times. What is the probability that you will encounter at least one instance where the collection has at least 12 pink marbles during these 100 cycles?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.997}
Determine the value of $n$ for which the space $\mathbb{R}^{2n} - \mathbb{R}^{n}$ is path connected, where $\mathbb{R}^{n}$ is embedded in the first $n$ coordinates of $\mathbb{R}^{2n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A shop owner professes to sell his articles at a certain cost price, but he uses false weights with which he cheats by 12% while buying and by 30% while selling. What is his percentage profit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60\%}
Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29\sqrt{3}}
Alice and the Cheshire Cat play a game. At each step, Alice either (1) gives the cat a penny, which causes the cat to change the number of (magic) beans that Alice has from $n$ to $5n$ or (2) gives the cat a nickel, which causes the cat to give Alice another bean. Alice wins (and the cat disappears) as soon as the number of beans Alice has is greater than 2008 and has last two digits 42. What is the minimum number of cents Alice can spend to win the game, assuming she starts with 0 beans?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
11 个黑球， 9 个红球，依次取出，剩下全是一种颜色就结束，求最后只剩下红球的概率？原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{29}
Let $A,B,C$ be angles of an acute triangle with
$\begin{aligned}
\cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\
\cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9}.
\end{aligned}$
There are positive integers $p$, $q$, $r$, and $s$ for which\[ \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, \]where $p+q$ and $s$ are relatively prime and $r$ is not divisible by the square of any prime. Find $p+q+r+s$.

Note: due to an oversight by the exam-setters, there is no acute triangle satisfying these conditions. You should instead assume $ABC$ is obtuse with $\angle B > 90^{\circ}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
222
}
Find the limit of the following sequence: $$\lim_{n\to \infty} \frac{\prod_{i = 0}^{n}\pars{i + 2/3}}{\prod_{j = 0}^{n}\pars{j + 1/3}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Find the limit of the series as \( n \to \infty \):
\[ u_n(x) = \frac{1}{x+1} + \frac{2}{(x+1)(x+2)} + \frac{3}{(x+1)(x+2)(x+3)} + \cdots + \frac{n x^{n-1}}{(x+1)(x+2)\cdots(x+n)} \]
Evaluate \( \lim_{n \to \infty} u_n(x) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
A ship travels downstream from point $A$ to point $B$ and takes 1 hour. On the return trip, the ship doubles its speed and also takes 1 hour. How many minutes will it take to travel from point $A$ to point $B$ if the ship initially uses double its speed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36 \text{ minutes}}
Find the smallest positive real number $c$ such that for all nonnegative real numbers $x, y,$ and $z$, the following inequality holds:
\[\sqrt[3]{xyz} + c |x - y + z| \ge \frac{x + y + z}{3}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
Circle $\omega$ has radius $5$ and is centered at $O$.  Point $A$ lies outside $\omega$ such that $OA=13$.  The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $BC$ is tangent to $\omega$ and $\omega$ lies outside triangle $ABC$.  Compute $AB+AC$ given that $BC=7$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
Let $g$ be a non-negative function defined on the interval $[0, \pi]$. If $\int_0^x \cos g'(t) dt = \int_0^x g(t) dt$ for $0 \leq x \leq \pi$ and $g(0) = 0$, find the possible values of $g\left(\frac{\pi}{2}\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Xiao Hua, Xiao Li, and Xiao Xia all need to travel from place A to place B. Xiao Hua and Xiao Li set off from place A together at 6 am. Xiao Hua walks at a speed of 5 kilometers per hour, and Xiao Li walks at a speed of 4 kilometers per hour. Xiao Xia does not leave place A until 8 am. By 6 pm, Xiao Hua and Xiao Xia arrive at place B simultaneously. The question is: At what time does Xiao Xia catch up with Xiao Li?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12 \text{ pm}}
Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2{x} = \log_y{16}$ and $xy = 64$. What is $(\log_2{\tfrac{x}{y}})^2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$20$}
Evaluate the limit \( \lim_{n\to\infty} \sqrt[n]{\sum_{i=1}^m |x_i|^n} \) for any vector \(x\in\mathbb{R}^m\), where \(x_p=\min_{1\le i\le m} |x_i|\) and \(x_q=\max_{1\le i\le m} |x_i|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x_q}
Let \( F = \langle a, b \rangle \) be a free group. If \( N \) is the normal subgroup of \( F \) generated by \( a^2 \) and \( b^2 \), find the index of \( N \) in \( F \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
what is the remainder when 121212 . . . . . . . ( 300 ) digits is divided by 99 .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine the quadratic variation $[X]_t$ of the process $X_t = \int_0^t B_s \, ds$, where $B_t$ is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit \( \lim_{n\to\infty} \sum_{r=0}^{p-k}\frac{1}{k(n+r)} \) using integration techniques.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Calculate the variance of the present value of a whole life annuity, $Y$, using the equation $$VarY=\frac{^2A_x-(A_x)^2}{d^2},$$ where $$A_x=\sum_{j=0}^\infty v^{j+1} {}_j| q_{x};$$ $$^2 A_x=\sum_{j=0}^\infty (v^2)^{j+1} {}_j| q_{x}$$ and $\ddot{a}_x=\frac{1-A_x}{d}$ with $d=\frac{i}{1+i}$. Given that $\ddot{a}_x=10$ for $i'=1/24=e^\delta-1$ and $\ddot{a}_x=6$ for $i=e^{2\delta}-1$, find the variance of $Y$.}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
The average age of 50 students in a class is 20 years. Out of these, the average age of 15 students is 18 years, that of another 15 students is 22 years, the average age of 10 students is 25 years, and the average age of 9 students is 24 years. Find the age of the 50th student.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Let the expression x # y # z denote the product of the consecutive multiples of 3 between x and y, and between y and z, inclusive. What is the sum of the exponents in the prime factorization of 21 # 30 # 36 with the constraint that the sum of exponents of all prime factors other than 3 must be even?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE = \angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such that $c$ is not divisible by the square of any prime. Find $a+b+c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{115}
Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to  $ 1998. $ *Gheorghe Iurea*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Given a sequence of sums involving six integers, determine the minimum number of even integers among them. The sequence is as follows: the sum of the first two integers is 26, adding two more integers results in a sum of 41, and adding the final two integers to the previous sum gives 57. Express your answer as a single integer representing the minimum number of even integers required.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
设 $A=\{x \mid-2 \leqslant x \leqslant a\}, B=\{y \mid y=2 x+3, x \in A\}, C=\left\{z \mid z=x^2, x \in A\right\}$,且 $C \subseteq B$, 则实数 $a$ 的取值集合是 $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\left[\frac{1}{2}, 3\right]$}
Let's determine

$$
\left(\frac{7 \sqrt[3]{x}}{2 \sqrt{y}}-\frac{\sqrt[3]{y}}{7 \sqrt[5]{x^{2}}}\right)^{8}
$$

the sixth term of the expansion.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{\sqrt[6]{y}}{7x}}
Determine the values of parameter \( a \) for which the inequality

$$
\sqrt{1 + x^4} > 1 - 2ax + x^2
$$

holds for all positive \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 1 - \frac{\sqrt{2}}{2}}
Let $V=\{(x,y,z)\in \mathbb{R}:\frac{1}{4}\le x^2+y^2+z^2\le1\}$ and $\vec{F}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{(x^2+y^2+z^2)^2}$ for $(x,y,z)\in V$. Find the flux of $\vec{F}$ through the entire boundary of $V$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4π}
Given sets $A=\{2, a-1\}$, $B=\{a^2-7, -1\}$, and $A\cap B=\{2\}$, find the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Let  $s_n$  be the number of solutions to  $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$ , where  $a_1,a_2,a_3$  and  $a_4$  are elements of the set  $\{2, 3, 5, 7\}$  and  $b_1$  and  $b_2$  are elements of the set  $\{ 1, 2, 3, 4\}$ . Find the number of  $n$  for which  $s_n$  is odd.

*Author: Alex Zhu*

<details><summary>Clarification</summary>$s_n$  is the number of *ordered* solutions  $(a_1, a_2, a_3, a_4, b_1, b_2)$  to the equation, where each  $a_i$  lies in  $\{2, 3, 5, 7\}$  and each  $b_i$  lies in  $\{1, 2, 3, 4\}$ .</details>
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let \( ABCD \) be a quadrilateral with an inscribed circle \( \omega \) that has center \( I \). If \( IA = 5 \), \( IB = 7 \), \( IC = 4 \), and \( ID = 9 \), find the value of \( \frac{AB}{CD} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{35}{36}}
There are four positive integers that are divisors of each number in the list $$20, 40, 100, 80, 180.$$ Find the sum of these four positive integers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let  $ABCD$  be a square with the point of intersection  $O$  of the diagonals and let  $P,\ Q,\ R,\ S$  be the points which are on the segments  $OA,\ OB,\ OC,\ OD$ , respectively such that  $OP=3,\ OQ=5,\ OR=4$ . If the points of intersection of the lines  $AB$  and  $PQ$ , the lines  $BC$  and  $QR$ , the lines  $CD$  and  $RS$  are collinear, then find the length of the segment  $OS$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{60}{23}}
Let \( n \) be the integral part of \( \frac{1}{\frac{1}{1980}+\frac{1}{1981}+\cdots+\frac{1}{2009}} \). Find the value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{66}
Find the sum of the infinite series \(\sum_{k=1}^\infty \frac{(5k+7)}{k^3+4k^2+3k}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{31}{9}}
Given that $\log_3M=a_1+b_1$ and $\log_5 M=a_2+b_2$, where $a_1,a_2$ are natural numbers and $b_1,b_2 \in [0,1)$, and $a_1a_2=6$, determine the number of integral values of $M$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
设 $A, B, C$ 是 $\triangle A B C$ 的三个内角, 则 $3 \cos A+2 \cos 2 B+\cos 3 C$ 的取值范围为 $\qquad$. 答案的标准形式为[-\frac{a}{b},c]，请给出该范围的两端点a+b+c之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47}
Determine how many roots the polynomial \( z^4 + 3z^2 + z + 1 \) has in the right-half complex plane, where \( \text{Re}(z) > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Express the infinite series \( \sum_{n=1}^{\infty} \frac{i^n(1+i)}{2} \) as a specific value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
XI OM - I - Task 3

Each side of a triangle with a given area $ S $ is divided into three equal parts, and the points of division are connected by segments, skipping one point to form two triangles. Calculate the area of the hexagon that is the common part of these triangles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{9}S}
Evaluate the integral \( \int_0^\infty \frac{\sqrt{x}}{x^3+1}\ \mathrm{d}x \) using residue theory with a keyhole contour to exclude the branch cut.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{3}}
For which values of \( b > 0 \) is the set \( \{e_n(t) = e^{int} : n \in \mathbb{Z}\} \) a basis for \( L^2([0,b]) \), meaning that the orthogonal complement \( V^\perp = 0 \) where \( V = \text{span}\{e_n : n \in \mathbb{Z}\} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0 < b \leq 2\pi}
Thomas is now 6 years younger than Roy but 4 years older than Kelly. Roy is now 8 years older than Julia and half of that amount older than Kelly. If in 2 years, Roy will be 3 times as old as Julia, and Thomas will be twice as old as Kelly, what would be the product of Roy's, Kelly's, and Thomas' ages in 2 years?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{576}
Use the Divergence Theorem to evaluate the surface integral \( \iint_S F \cdot dS \), where \( F = (x^3y + z)i + xze^yj + (xy − 3z^4)k \) and \( S \) consists of the five lower faces of the cube \([0, 1]\times[0, 1]\times[0, 1]\), excluding the face with \( z = 1 \). Hint: Close the surface to apply the theorem.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{e}{4}}
During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^{3}, z^{5}, z^{7}, \ldots, z^{2013}$, in that order; on Sunday, he begins at 1 and delivers milk to houses located at $z^{2}, z^{4}, z^{6}, \ldots, z^{2012}$, in that order. Eli always walks directly (in a straight line) between two houses. If the distance he must travel from his starting point to the last house is $\sqrt{2012}$ on both days, find the real part of $z^{2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1005}{1006}}
Evaluate the integral $\int_{-2}^{2}(x^2\sin x + \sqrt{4-x^2})dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Determine the smallest value of $k$ such that the inequality $$2A+3B+k(B+2C) \le \frac{k+3}{6} (2A+5B+12C)$$ holds for all positive real numbers $A, B, C$ where $A=x^3+y^3+z^3$, $B=x^2y+xy^2+z^2x+zx^2+y^2z+yz^2$, and $C=xyz$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find the length of the arc of the semicubical parabola defined by the equation \( y^{2}=x^{3} \) from the origin to the point where \( x=5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{335}{27}}
Determine the number of distinct intersection points formed when perpendicular lines are drawn from three non-collinear points $Q_1, Q_2, Q_3$, located on the same side of a straight line $l$, to the lines formed by joining any two of the four collinear points $P_1, P_2, P_3, P_4$, excluding the point from which the perpendicular is drawn.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}