inclusionAI/Ring-lite-rl-data huggingface dataset:
Evaluate the integral \( \int_{0}^{\infty} u^{n}e^{-u} \, du \) for \( n \geq 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Gamma(n+1)}
Find an example of a constructible angle that cannot be quintsected using an unmarked straightedge and compass.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72^\circ}
The numbers \( 1,2,\ldots ,9 \) are randomly placed into the9squares of a \( 3 \times  3 \)
 grid. Each square gets one number, and each of the numbers is used once.
 What is the probability that the sum of the numbers in each row and each
 column is odd?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\( 1/{14} \)}
Given that $A$ is a non-singular square matrix of order $2$ and satisfies the equation $|A + |A|\operatorname{adj}A| = 0$, where $\operatorname{adj}A$ denotes the adjoint of matrix $A$ and $|A|$ denotes the determinant of $A$, evaluate $|A - |A|\operatorname{adj}A|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the possible rank values for the matrix product $AB$ if $A$ and $B$ are $3 \times 3$ real matrices and $\operatorname{rank}(BA) = 2$. Provide your answer as a set of possible rank values.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{1, 2\}}
Determine the values of the parameter \( t \) for which the linear transformation \( \psi: \mathbb{R}^4 \rightarrow \mathbb{R}^3 \) defined by \( \psi([x_1,x_2,x_3,x_4]) = [x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4] \) is an epimorphism.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{t \neq 2}
The sum of $\lfloor x \rfloor$ for all real numbers $x$ satisfying the equation $16 + 15x + 15x^2 = \lfloor x \rfloor ^3$ is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33}
In triangle $ABC$ with $AC \neq BC$, a triangle $A'B'C$ is formed by rotating $\triangle ABC$ around point $C$. Let $M$, $E$, and $F$ be the midpoints of segments $BA'$, $AC$, and $CB'$ respectively. Given that $EM = FM$, determine the measure of angle $\angle EMF$. \( \text{Provide your answer in degrees.} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
You are given that $17! = 355687ab8096000$ for some digits $a$ and $b$. Find the two-digit number $\overline{ab}$ that is missing above.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Calculate the dot product of the bivectors \(\gamma_{23}\) and \(\gamma_{01}\) using the metric \(\eta=(1, -1, -1, -1)\) and the formula for the dot product of two k-blades.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of non-trivial ring homomorphisms from \( \mathbb{Z}_{12} \) to \( \mathbb{Z}_{28} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the definite integral \(\int_{-1}^{+1} \frac{2 u^{332} + u^{998} + 4 u^{1664} \sin u^{691}}{1 + u^{666}} \, \mathrm{d} \, u\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{333}\left(1 + \frac{\pi}{4}\right)}
Let $f (n)$ be a function satisfying the following three conditions for all positive integers $n$:
(a) $f (n)$ is a positive integer,
(b) $f (n + 1) > f (n)$,
(c) $f ( f (n)) = 3n$.
Find $f (2001)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
 3816

}
Maria found a store that sells handmade sweaters for $35, handmade scarves for $25, handmade mittens for $15, and handmade hats for $12. She wants to buy two sweaters, one scarf, one pair of mittens, and one hat for each of her 15 family members. The store offers a 10% discount if she spends more than $800 on her purchase. If Maria has a spending limit of $1500, how much will be left in her savings after buying all these items, considering the discount and a 5% sales tax?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$229.35}
Determine the smallest integer $N$ such that the remainder $R_N$ of the series \( \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^2} \) after summing the first $N$ terms is less than 0.1. That is, find $N$ such that:
\[
R_N = \sum_{n=N+1}^{\infty} \frac{1}{n(\ln n)^2} < 0.1.
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22027}
A $7 \times 7$ table is filled with non-zero integers. Initially, the border of the table is filled with negative numbers. Each subsequent cell is filled in any order, with the next number being equal to the product of the previously placed numbers closest to it either in the row or column. Determine the maximum number of positive numbers that can be in the table.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
For $i=1,2,$ let $T_i$ be a triangle with side length $a_i,b_i,c_i,$ and area $A_i.$ Suppose that $a_1\le a_2, b_1\le b_2, c_1\le c_2,$ and that $T_2$ is an acute triangle. Does it follow that $A_1\le A_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ A_1 \le A_2 }
Find the limit (if it exists) of $$\lim_{x\longrightarrow 0^+}\sum_{n=1}^{\infty}\frac{x}{1+x^2n^2}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi}{2}$}
There are 11 rings arranged in a circle, numbered from 1 to 11. Exactly 9 of these rings are fake, and exactly 2 are real. You can choose any 5 consecutive rings and ask how many real rings are in that selection. What is the minimum number of questions needed to identify at least one real ring?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A "fifty percent mirror" is a mirror that reflects half the light shined on it back and passes the other half of the light onward. Two "fifty percent mirrors" are placed side by side in parallel, and a light is shined from the left of the two mirrors. How much of the light is reflected back to the left of the two mirrors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{3}}
Determine the dimension of the smallest field extension of \( \mathbb{Q} \) that contains a transcendental number, such as \( e \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
In triangle $ABC$, an interior point $P$ is defined as "good" if there are exactly 27 rays emanating from $P$ that intersect the sides of the triangle, dividing it into 27 smaller triangles of equal area. Determine the number of good points for a given triangle $ABC$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{325}
10. Person A and Person B start from points $A$ and $B$ respectively (Person A starts from $A$), walking towards each other and continuously moving back and forth between the two points. Person A's speed is 4 times that of Person B. It is known that the distance between $A$ and $B$ is $S$ kilometers, where $S$ is a positive integer and has 8 factors. The first time they meet at point $C$ (note: meeting here can refer to meeting face-to-face or catching up from behind), the length of $AC$ is an integer; the second time they meet at point $D$, the length of $AD$ is also an integer; after the second meeting, Person B feels that his speed is too slow, so he borrows a motorcycle from a villager near $D$. By the time he borrows the motorcycle and returns to $D$, Person A has already reached point $E$ (Person A has not yet reached $A$), and the length of $AE$ is also an integer; finally, Person B rides the motorcycle to catch up with Person A, with the motorcycle's speed being 14 times that of Person A, and they both reach point $A$ at the same time. Therefore, the distance between $A$ and $B$ is $\qquad$ kilometers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{105}
A boat covers a certain distance downstream in 1 hour, while it comes back in 1.30 hours. The speed of the stream is some value, and the speed of the boat in still water is 15 kmph. What is the speed of the stream?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?

Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
A right rectangular prism $P$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P$ cuts $P$ into two prisms, one of which is similar to $P,$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40}
Given a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), let $M$ and $N$ be two points on the hyperbola that are symmetric about the origin, and let $P$ be any point on the hyperbola. The slopes of lines $PM$ and $PN$ are denoted as $k_{PM}$ and $k_{PN}$, respectively. If $k_{PM} < 0$ and $k_{PN} < 0$, find the maximum value of $k_{PM} + k_{PN}$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{- \frac{2b}{a}}
Given a Hidden Markov Model (HMM) with a transition matrix for temperature states (WARM and COLD) and an observation matrix relating tree ring sizes (SMALL, MEDIUM, and LARGE) to temperature, calculate the probability of the sequence of tree ring sizes SSSMMLLL.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.000329}
Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that $(8-p)(8-q)(8-r)(8-s)(8-t) = -120$. Calculate the sum $p+q+r+s+t$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{27}
Determine the number of possible ordered triplets \((a, b, c)\) such that \(\text{LCM}(a, b) = 5^2 \times 3^4\), \(\text{LCM}(b, c) = 5^3 \times 3^4\), and \(\text{LCM}(a, c) = 5^3 \times 3^4\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65}
A point \( D \) is marked on the altitude \( BH \) of triangle \( ABC \). Line \( AD \) intersects side \( BC \) at point \( E \), and line \( CD \) intersects side \( AB \) at point \( F \). It is known that \( BH \) divides segment \( FE \) in the ratio \( 1:3 \), starting from point \( F \). Find the ratio \( FH:HE \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1:3}
Each of the $5$ sides and the $5$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{253}{256}}
Laura’s House is a 20-mile round trip from her school. The supermarket is 10 miles farther away from the school. If Laura drives to school every morning and two afternoons a week drives to the supermarket. How many miles does Laura driver per week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{220 \text{ miles}}
Iveta sequentially wrote natural numbers consisting of the digits 1, 3, 5, and 7. She used no other digits, proceeded in ascending order from the smallest number, and did not omit any number. She wrote the numbers consecutively to form an exceptionally long number:

$$
1357111315173133 \ldots
$$

What digit is at the 1286th position in this number?

(E. Novotná)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given that a > 0, b < 0, and |a| < |b|, determine the correct ordering of the variables a, b, -a, and -b based on their magnitude and sign. Express your answer using the greater than (>) symbol to denote the ordering.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-b > a > -a > b}
若复数 $z_{1}, z_{2}$ 分别满足 $\left|z_{1}-3 \mathrm{i}\right|=2,\left|z_{2}-8\right|=1$ ，则在复平面上所有由复数 $z_{1}-z_{2}$ 对应的点围成的面积是 $x\pi$，求 $x$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
The surface of a 3 x 3 x 3 Rubik's Cube consists of 54 cells. What is the maximum number of cells you can mark such that the marked cells do not share any vertices?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
What is the largest divisor by which $29 \cdot 14$ leaves the same remainder when divided by $13511, 13903,$ and $14589$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
设正实数 $c_{1}, c_{2}, c_{3}$ 使得对任意定义在 $[-\pi, \pi]$ 上的三次多项式 $f(x)$ 均有 $\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) {d} x=$ $c_{1} f(-\pi)+c_{2} f(0)+c_{3} f(\pi)$, 则 $\left[\frac{c_{2}+c_{3}}{c_{1}}\right]$ 的最大可能值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the smallest \( n \) such that the sequence of positive integers \( a_1, a_2, \ldots, a_n \) has each term \( \leq 15 \) and \( a_1! + a_2! + \ldots + a_n! \) has last four digits 2001.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
已知双曲线$\frac{x^2}9-\frac{y^2}{16}=1$ 的左、右焦点分别为$F_1$、$F_2.$ 过$F_1$ 的直线与左支交于 $A$、$B$ 两点，且 $\frac{|AF_1|}{|AF_2|}=\frac12$ .则 $\triangle AF_1F_2$ 与$\triangle ABF_2$ 的内切圆面积之比为 $\qquad.$ 请给出比值两数之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
Given the parabola $C: y^2 = 4x$ with focus $F$, and a line $l$ passing through point $P(-1, 0)$ with slope $k$ ($k > 0$) intersects the parabola $C$ at points $A$ and $B$. The lines $AF$ and $BF$ intersect the parabola $C$ again at points $M$ and $N$ respectively. If $\frac {|AF|}{|FM|} + \frac {|BF|}{|FN|} = 18$, then $k = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac { \sqrt {5}}{5}}
Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\frac{3}{4}$ of the probability of rolling a sum of $10,$ and the probability of rolling a sum of $12$ is $\frac{1}{12}$. What is the least possible number of faces on the two dice combined?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{17}
An airline company operates passenger transportation between four cities A, B, C, and D. The prices for some of the direct flights between these cities are as follows: AB is 2000 yuan; AC is 1600 yuan; AD is 2500 yuan; CD is 900 yuan; BC is 1200 yuan. If the company's pricing for flights is directly proportional to the straight-line distance between the cities (assuming all four cities are on the same plane), calculate the price for the straight-line distance between B and D. Express your answer in yuan as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1500}
Marvin had a birthday on Tuesday, May 27 in the leap year $2008$. In what year will his birthday next fall on a Saturday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2017}
For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have
\[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2n}}
Ben initially had 18 marbles and John had 17 marbles. Ben gave half of his marbles to John, and another quarter of his marbles to his friend Lisa. Additionally, John gave 1/3 of the marbles he received from Ben to Lisa as well. After these transactions, how many more marbles did John have than Ben?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Two circles touch in  $M$ , and lie inside a rectangle  $ABCD$ . One of them touches the sides  $AB$  and  $AD$ , and the other one touches  $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in  $M$  divides  $AB$  and  $CD$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ 1:1 }
Given eight small squares, in how many different ways can six of these squares be shaded so that the shaded squares form the net of a cube? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
请写出一个开口向上，与y轴交点纵坐标为-2，且经过点(1,2)的抛物线关系式。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y=4x^2-2}
Let $X_1, X_2, \ldots, X_n$ be i.i.d. random variables with a uniform distribution on $[0,1]$. Define $Z = \max\{X_1, X_2, \ldots, X_n\}$. Calculate the probability $P(X_2 < a \mid Z = z, X_2 < z)$. 
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\min(a, z)}{z}}
Determine the Galois group of the polynomial \( f(x) = x^5 - x + 1 \) over the finite field \( \mathbb{F}_7 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}/6\mathbb{Z}}
How many cyclic subgroups are there in the symmetric group $S_5$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{67}
A circle is circumscribed around quadrilateral $EFGH$ as follows: $EF$ subtends a central angle of $100^\circ$, and $GH$ subtends a central angle of $120^\circ$. Find the sum of the angles $\angle EFG + \angle EHG$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{70^\circ}
Given the line $l: (2\lambda+1)x + (\lambda+2)y + 2\lambda + 2 = 0$ ($\lambda \in \mathbb{R}$) has equal intercepts on the x-axis and y-axis, then $\lambda = \_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 1}
In a drawer Sandy has 5 pairs of socks, each pair a different
 color. On Monday Sandy selects two individual socks at
 random from the 10 socks in the drawer. On Tuesday Sandy
 selects 2 of the remaining 8 socks at random and on
 Wednesday two of the reaining 6 socks at random. The
 probability that Wednesday is the first day Sandy selects
 matching socks is \( \frac{m}{n} \) ,where \( m \) and \( n \) are relatively prime
 positive integers. Find \( m + n \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{341}
Determine the distribution that is the limit of the sequence \( \langle f_n, \varphi \rangle = n(\varphi(\frac{1}{n})-\varphi(\frac{-1}{n})) \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2\delta'}
Among the 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of a cube (totaling 27 points), how many groups of three collinear points are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Determine the values of $p \in \mathbb{R}$ for which the series \( \sum^{\infty}_{n=4}{\frac{1}{n\log (n)\log( \log(n))^p}} \) converges.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 1}
In a rectangle \(ABCD\) with an area of 1 (including its boundary), there are 5 points such that no three points are collinear. Determine the minimum number of triangles, formed by these 5 points as vertices, that have an area of at most \(\frac{1}{4}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the equation of the hyperplane in \(\mathbb{R}^4\) that passes through the points \(P_1 = (1, 2, 1, 4)\), \(P_2 = (0, 1, 2, 1)\), \(P_3 = (-2, 1, 1, 0)\), and \(P_4 = (1, 1, 1, 0)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4y + z - w = 5}
Compute the distance from the constant function $g(x) = 0$ to the set $A = \{f \in \mathcal{C}([0,1]) \mid f(0) = 1 \text{ and } \int_0^1 f^2 = 2\}$ in the normed space $(\mathcal{C}([0,1]), \|\cdot\|_\infty)$ with the supremum norm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{2}}
There are 35 even integers between a certain number and 95. What is the starting number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find the value of $X$ that satisfies the equation involving recursive $n$-th roots:
\[ \sqrt[n]{X+\sqrt[n]{X+\sqrt[n]{X+ \dots}}}  =\sqrt[n]{X\sqrt[n]{X\sqrt[n]{X \dots}}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the set of values for $x$ that makes the fraction $\frac{{x-1}}{{x+1}}$ meaningful. Express your answer using mathematical notation to describe the condition for $x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x \neq -1}
Determine the order of the best constant \(\lambda = \lambda(n)\) such that for any \(n \times n\) complex matrix \(A\) with trace zero, there exist \(n \times n\) matrices \(B\) and \(C\) satisfying \(A = BC - CB\) and \(\|B\| \cdot \|C\| \le \lambda \|A\|\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(n)}
When the two diagonals of an $8 \times 10$ grid are drawn, calculate the number of the $1 \times 1$ squares that are not intersected by either diagonal.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Let \( B = (B_t)_{t \geq 0} \) be a standard Brownian motion and define \( M_t = 4B_t^2 + e^{4B_t -8t} -4t \). Find \( \mathbb{E}(M_\sigma) \) for \( \sigma = \inf\{t \geq 0 : |B_t| = 1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the minimum value of \( P = \dfrac{1}{(a-b)^2} + \dfrac{1}{(b-c)^2} + \dfrac{1}{(c-a)^2} \) given that \( a, b, c \in \mathbb{R}^+ \) and \( a^2 + b^2 + c^2 = 3 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{15}{4}}
Evaluate the integral \[ I = \frac{1}{2\pi i}\oint_{|z|=1}\frac{2(1-\cos(z))e^z}{z^4}dz \] using the identity of Laurent's expansion \[ a_{n} = \frac{1}{2\pi i}\oint_{C}\frac{f(z)}{(z-z_{0})^{n+1}}dz. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Compute the differential of the i-th coordinate map \(x_i: \mathbb{R}^n \to \mathbb{R}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{dx_i}
6. Given that the base of the tetrahedron $S-ABC$ is an equilateral triangle, the projection of point $A$ on the side face $SBC$ is $H$, which is the orthocenter of $\triangle SBC$, the dihedral angle $H-AB-C$ is $30^{\circ}$, and $SA=2\sqrt{3}$. Then the volume of the tetrahedron $S-ABC$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{9}{4}\sqrt{3}}
Determine the maximum number of distinct subsets that can be formed from a set of 10 subsets using the operations of union, intersection, complement, and symmetric difference, including the possibility of empty subsets.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{1024}}
Determine the domain of the function \( f(x) = \int_0^1 \frac{t^x - 1}{\ln(t)} \, dt \), where \( x \) is a real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-1, \infty)}