inclusionAI/Ring-lite-rl-data huggingface dataset:
Two trains of different lengths run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. One train is 500 m long, and they take 44.99640028797697 seconds to cross each other. How long is the other train?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{750}
In triangle $A B C, \angle A B C$ is obtuse. Point $D$ lies on side $A C$ such that \angle A B D$ is right, and point $E$ lies on side $A C$ between $A$ and $D$ such that $B D$ bisects \angle E B C$. Find $C E$, given that $A C=35, B C=7$, and $B E=5$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Determine the Hausdorff dimension of the graph of the function \( y = \log(1+x)\sin\frac{1}{x} \) for \( x > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let f(x) = x^3 + 7x^2 + 9x + 10. Find the value of p that satisfies the statement f(a) ≡ f(b) (mod p) ⇒ a ≡ b (mod p) for every integer a, b. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
In triangle \( \triangle ABC \), points \( K \) and \( M \) are on sides \( AB \) and \( BC \) respectively, such that \( \frac{AK}{KB} = \frac{1}{4} \) and \( \frac{BM}{MC} = \frac{4}{5} \). On line segment \( KM \), point \( O \) is chosen such that \( \frac{KO}{OM} = 3 \). Let \( N \) be the intersection of ray \( BO \) with \( AC \), where \( AC = a \). The distance from point \( O \) to side \( AC \) is \( OD = d \). Find the area of \( \triangle KMN \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{a d}{3}}
Evaluate the integral $$\int_0^1 \frac{x^b-x^a}{\ln x} dx$$ for $a > b > 0$. Initial attempts to simplify the integral using substitutions like $t = \ln x$ were unsuccessful.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\ln\frac{b+1}{a+1}\)}
Determine the number of elements of order \( p \) in the general linear group \( \text{GL}(2, \mathbb{Z}_p) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2 - 1}
Determine the growth order of the function \(g(z) = \prod_{k = 1}^\infty \left(1 + \frac{z}{k^3}\right)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
The stem and leaf plot represents the heights, in inches, of the players on the Pine Ridge Middle School boys' basketball team. Calculate the mean height of the players on the team. (Note: $5|3$ represents 53 inches.)

Height of the Players on the Basketball Team (inches)

$4|8$

$5|0\;1\;4\;6\;7\;7\;9$

$6|0\;3\;4\;5\;7\;9\;9$

$7|1\;2\;4$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61.44}
In a regular quadrilateral pyramid $P-ABCD$, if $\angle APC = 60^{\circ}$, find the cosine of the dihedral angle between planes $A-PB-C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$-\frac{1}{7}$}
Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. Call a partition $(S, T)$ of $X$ $f$-balanced if for all $s \in S$, we have $f(s) \in T$, and for all $t \in T$, we have $f(t) \in S$. 

A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S \cap T = \emptyset$ and $S \cup T = X$. Note that $(S, T)$ and $(T, S)$ are considered the same partition.

Let $g(f)$ be the number of $f$-balanced partitions, and let $m$ equal the maximum value of $g$ over all functions $f$ from $X$ to itself. If there are $k$ functions satisfying $g(f) = m$, determine $m+k$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{961}
An engineer undertakes a project to build a road 10 km long in 300 days and employs 30 men for the purpose. After a certain number of days, he finds only 2 km of the road has been completed. He must employ approximately 30 extra men to finish the work in time. How many days had passed when he realized the progress?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
在正方体的八个顶点中任取 3 个构成三角形，求所构成的三角形是等腰三角形的概率。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Let $p=2^{24036583}-1$, the largest prime currently known. For how many positive integers $c$ do the quadratics \pm x^{2} \pm p x \pm c all have rational roots?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Given a sequence of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ whose sum is 2009, and which can be partitioned into 41 groups with equal sums as well as 49 groups with equal sums, find the smallest value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
Two circles with radii \( R \) and \( r \) intersect at point \( A \). Let \( BC \) be a common tangent to both circles at points \( B \) and \( C \). Determine the radius of the circumcircle of triangle \( ABC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{Rr}}
Given is a set of $2n$ cards numbered $1,2, \cdots, n$, each number appears twice. The cards are put on a table with the face down. A set of cards is called good if no card appears twice. Baron Munchausen claims that he can specify $80$ sets of $n$ cards, of which at least one is sure to be good.  What is the maximal $n$ for which the Baron's words could be true?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
7
}
Find the value of $a$ such that the $y$-coordinate of the point on the curve $f(x) = e^x$ that is closest to the line $y = \frac{x}{a} - 1$ is $\frac{1}{e}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e}
Let $W$ be the hypercube $\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}\right) \mid 0 \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 1\right\}$. The intersection of $W$ and a hyperplane parallel to $x_{1}+x_{2}+x_{3}+x_{4}=0$ is a non-degenerate 3-dimensional polyhedron. What is the maximum number of faces of this polyhedron?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let \( y = g(x) \) be the inverse of the bijective mapping \( f: \mathbb{R} \to \mathbb{R} \) where \( f(x) = 3x^3 + 2x \). Find the area bounded by the graph of \( g(x) \), the x-axis, and the ordinate at \( x = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{13}{4}}
Given a right triangular prism ABC-A$_{
m 1}$B$_{
m 1}$C$_{
m 1}$ with a base edge length of $2\sqrt {3}$ and a height of 3, circle O is the inscribed circle of triangle ABC, and point P is any point on circle O. The volume of the circumscribed sphere of the triangular pyramid P-A$_{
m 1}$B$_{
m 1}$C$_{
m 1}$ is $\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{20 \sqrt {5}}{3}π}
Evaluate \( \lim_{n \rightarrow \infty} \int_{0}^{x} \frac{(-\ln{z})^n}{n!} \, dz \) for \( x \in (0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that the even function $f(x)$ is decreasing on $[0,2]$, compare the sizes of $a=f(1)$, $b=f(\log _{ \frac {1}{2}} \frac {1}{4})$, and $c=f(\log _{2} \frac { \sqrt {2}}{2})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c > a > b}
7.5. There are 100 boxes numbered from 1 to 100. One of the boxes contains a prize, and the host knows where it is. The audience can send the host a batch of notes with questions that require a "yes" or "no" answer. The host shuffles the notes in the batch and, without reading the questions aloud, honestly answers all of them. What is the minimum number of notes that need to be sent to definitely find out where the prize is?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{99}
The sides of a rectangular park are in the ratio 3 : 2 and its area is 5766 sq m. The cost of fencing it at a certain rate per meter is 155. What is the cost of fencing per meter in ps?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
An arithmetic sequence has the following properties:

- When the first, third, fifth, and so on terms are added, up to and including the last term, the sum is 320.
- When the first, fourth, seventh, and so on, terms are added, up to and including the last term, the sum is 224.

Find the sum of the entire sequence. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{608}
Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the class group of the complement of the union of the three lines $L_1: x=0$, $L_2: y=0$, and $L_3: z=0$ in the projective plane $\mathbb{P}^2$. Let $U = \mathbb{P}^2 \setminus (L_1 \cup L_2 \cup L_3)$. What is the class group of $U$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the greatest common divisor in \( \mathbb{Z}[i] \) of \( 11+7i \) and \( 18-i \) using the Euclidean algorithm or ideal membership tests.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that $a^{1008} = b^2 = e$ and $a^jb = ba^{-j}$ holds for all $j$, find a positive value for $x$ such that $ba^{-x} = e$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{504}
in the coordinate plane a slope of the line k is 4 times the x - intercept of the line k . what is the y - intercept of the line k ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4a^2}
Find the torsion of the helix defined by the parametric equations: \(x(t) = \cos(t), \quad y(t) = \sin(t), \quad z(t) = t.\)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Evaluate the sum of the series: \( \sum_{n=1}^\infty \frac{n}{\sqrt{n^3+2}} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
If rose bushes are spaced about 1 foot apart, approximately how many bushes are needed to surround a circular patio whose radius is 12 feet? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75}
Given point $M(x_{0},1)$, suppose there exists a point $N$ on the circle $O$: $x^{2}+y^{2}=1$ such that $\angle OMN=30^{\circ}$. Determine the range of values for the real number $x_{0}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[-\sqrt{3}, \sqrt{3}\right]}
Given the polynomial \(0 + x + 12x^2 + 11x^3\) in the quotient ring \(\mathbb{Z}_{13} / (x^4 + 1)\), determine the smallest positive integer \(r\) such that the sequence of rows generated by multiplying the polynomial by \(x^i\) for \(i = 0, 1, 2, \ldots, r\) starts repeating. Assume the width of the matrix is 4. What is the value of \(r\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A month has 5 Saturdays and 5 Sundays, but only 4 Mondays and 4 Fridays. If the month has 30 days and ends on a Sunday, what day of the week must occur 5 times in the next month? Express your answer as a specific day of the week.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{Wednesday}
The tangent lines from the point $A$ to the circle $C$ touches the circle at $M$ and $N$. Let $P$ a point on $[AN]$. Let $MP$ meet $C$ at $Q$. Let $MN$ meet the line through $P$ and parallel to $MA$ at $R$. If $|MA|=2$, $|MN|=\sqrt 3$, and $QR \parallel AN$, what is $|PN|$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Approximate the definite integral \( \int_{0}^{1} x^{2} \sin(x^{4}) \, dx \) using a power series such that the error is less than 0.001.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.132}
Find the number of unordered pairs {a, b}, where \(a, b \in \{0, 1, 2, \ldots, 108\}\) such that 109 divides \(a^3 + b^3 - ab\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
In the country of PUMaC-land, there are $5$ villages and $3$ cities. Vedant is building roads
between the $8$ settlements according to the following rules:
a) There is at most one road between any two settlements;
b) Any city has exactly three roads connected to it;
c) Any village has exactly one road connected to it;
d) Any two settlements are connected by a path of roads.
In how many ways can Vedant build the roads?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
90
}
The relationship between the velocity $v$ of object A and time $t$ is given by $v=2t-1$ (with $v$ in meters per second and $t$ in seconds), while the relationship between the velocity $v$ of object B and time $t$ is $v=1+8t$. Both objects start simultaneously from points 405 meters apart on the same line and move towards each other. Calculate the distance traveled by object A when they meet.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{72 \ \text{meters}}
Find the smallest positive integer that is both an integer power of 7 and is not a palindrome.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Find the smallest positive $\alpha$ (in degrees) for which all the numbers $\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots$ are negative.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120}
10.204. From a point on the circumference, two chords of lengths 10 and 12 cm are drawn. Find the radius of the circle if the distance from the midpoint of the shorter chord to the longer chord is 4 cm.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["6.25"]}
Evaluate the integral \( \int_S x\,dy\,dz+y\,dz\,dx+z\,dx\,dy \), where \( S=\{(x,y,z)\in \mathbb{R}^{3}: x^2+y^2+z^2=1\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Determine the holonomy group of the 1-dimensional octonionic projective space, denoted as $\mathbb{O}P^1$. Provide your answer in terms of a well-known mathematical group.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{SO(8)}
A function $f(x)$ defined on $\mathbb{R}$ satisfies $f(-x) + f(x) = 2x^2$, and its graph is continuous. When $x \leq 0$, $f'(x) < 2x$. Determine the solution set of the inequality $f(x) + 1 \geq f(1-x) + 2x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, \frac{1}{2}]}
有限项等差数列的公差为 4 ，第二项起各项之和+首项的平方 <100 ，则该数列可能有几项？请给出可能项数的最大值与最小值之和。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Calculate the expected number of dice throws needed to roll a 6, given that all previous rolls resulted in even numbers (2 or 4). Use the concept of fixed point updating to determine the expected value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Divers extracted a certain number of pearls, not exceeding 1000. The distribution of the pearls happens as follows: each diver in turn approaches the heap of pearls and takes either exactly half or exactly one-third of the remaining pearls. After all divers have taken their share, the remainder of the pearls is offered to the sea god. What is the maximum number of divers that could have participated in the pearl extraction?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
A particle is placed at the center of an interval on a 1-D axis. The particle moves according to the following rules: it chooses its direction freely and randomly, and the span of each step is chosen with equal probability from the range (0, r]. The total number of steps is unlimited. Determine the probability that the particle will eventually exit the interval as a function of r.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Suppose 2012 reals are selected independently and at random from the unit interval  $[0,1]$ , and then written in nondecreasing order as  $x_1\le x_2\le\cdots\le x_{2012}$ . If the probability that  $x_{i+1} - x_i \le \frac{1}{2011}$  for  $i=1,2,\ldots,2011$  can be expressed in the form  $\frac{m}{n}$  for relatively prime positive integers  $m,n$ , find the remainder when  $m+n$  is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find all values of the parameter \( b \) for which there exists a value of \( a \) such that the system

$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 b(b-x+y)=4 \\
y=\frac{9}{(x+a)^{2}+1}
\end{array}\right.
$$

has at least one solution \((x, y)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{b \in [-11, 2)}
Find the value of \( \dfrac{d z^*}{dz} \), where \( z^* \) denotes the complex conjugate of \( z \) and \( z \in \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A large container is 30% full with water. If 27 liters of water are added, the container becomes 3/4 full. Now, assume there is another smaller container which can hold half the volume of the larger container when it is 30% full. What is the capacity of this smaller container?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Suppose that P(z), Q(z), and R(z) are polynomials with real coefficients, having degrees 2, 3, and 6, respectively, and constant terms 1, 2, and 3, respectively. Let N be the number of distinct complex numbers z that satisfy the equation P(z) * Q(z) = R(z). What is the minimum possible value of N? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
During a certain season, a team won 65 percent of its first 100 games and 50 percent of its remaining games. The team played a total of 75 games. What percentage of its games did the team win for the entire season?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{65\%}
Adva van három egyenló sugarú, egymást érintő kör. Mekkora a körök között fekvő idom területe, ha mindegyik kör sugara $r$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{r^2 \left( \sqrt{3} - \frac{\pi}{2} \right)}
Given that \( \triangle ABC \) has a circumcenter \( O \), and \( 2 \overrightarrow{OA} + 3 \overrightarrow{OB} + 4 \overrightarrow{OC} = 0 \), find \( \cos \angle BAC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Given the Lie algebra \(\mathfrak{g} = \mathfrak{sl}_{n+1}\) with its canonical basis \(\{x^\pm_i, h_i : i=1,\ldots, n\}\), where \(x_i^+ = e_{i,i+1}\), \(x_i^- = e_{i+1,i}\), and \(h_i = e_{i,i} - e_{i+1,i+1}\), and \(e_{ij}\) denotes the standard basis of \((n+1) \times (n+1)\) matrices. Let \(\alpha_i\) for \(i=1,\ldots, n\) be the simple positive roots. Determine the root space \(\mathfrak{g}_\theta\), where \(\theta = \alpha_1 + \cdots + \alpha_n\), and \(\mathfrak{g}_\theta = \{x\in \mathfrak{g}: [h,x] = \theta(h) x \; \forall \; h\in \mathfrak{h} \}\). Find an element \(x_\theta \in \mathfrak{g}\) such that \([h,x_\theta] = \theta(h) x_\theta\) for all \(h\in \mathfrak{h}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e_{1,n+1}}
Color three faces of an $8 \times 8 \times 8$ cube red and three faces blue (ensuring that any three faces sharing the same vertex are not all the same color), and then cut it into 512 smaller $1 \times 1 \times 1$ cubes. How many of these 512 smaller cubes have at least one red face and one blue face?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{56}
The points \( K, L, M, N \) are the centers of the circles inscribed in the faces \( S A B, S A C, S B C, \) and \( A B C \) of the tetrahedron \( S A B C \). Given that \( A B = S C = 5 \), \( A C = S B = 7 \), and \( B C = S A = 8 \), find the volume of the tetrahedron \( K L M N \). Round your answer to two decimal places if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.66}
Find the $L_\infty$ distance between the essential class $(a,\infty)$ and the non-essential class $(b,c)$ on the extended real line $\bar{\mathbb{R}}$, where $a, b, c < \infty$. Provide your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
There are 21 students in Class 3(1), with student numbers ranging from 1 to 21. They are arranged in a circle. The differences between the student numbers of each adjacent pair are recorded as positive values, calculated as the larger number minus the smaller number. What is the maximum possible sum of these 21 differences?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{220}
In the coordinate plane $xOy$, the angle $\angle MON$ is formed by the ray $y = x$ $(x \geq 0)$ and the positive $x$-axis. Points $A(6, 5)$ and $B(10, 2)$ are two fixed points inside the angle $\angle MON$. Points $P$ and $Q$ are movable points located on the two sides of $\angle MON$. Determine the minimum possible value of the perimeter $l$ of the quadrilateral $ABQP$. \( \boxed{} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5 + \sqrt{241}}
Given the sequence defined by $$a_{i,j} = \frac{1}{i^2 j^2},\quad i,j=1,2,\dots,$$ what is the asymptotic behavior of the $n$-th largest number when the sequence is arranged in increasing order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\log^2(n)}{n^2}\)}
The common difference of the arithmetic sequence $\{a\_n\}$ is $d$, and the solution set of the quadratic inequality about $x$, $\frac{d}{2}x^2 + (a\_1 - \frac{d}{2})x + c \geqslant 0$, is $[0, 22]$. Determine the maximum positive integer value of $n$ that makes the sum of the first $n$ terms of the sequence, $S\_n$, the greatest.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Triangle $ABC$ has side lengths $AB = 14$, $BC = 15$, and $CA = 13$. We draw a circle with diameter $AB$ such that it intersects $BC$ again at $D$ and intersects $CA$ again at $E$. If the circumradius of $\triangle CDE$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are coprime positive integers, determine $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3308}
Let \( f(x) = e^{x+1} - 1 \) and \( g(x) = 100|f(x)| - \sum_{k=1}^n |f(x^k)| \), where \( n \in \mathbb{N} \). Given that \( g(x) \) is differentiable over the entire set of real numbers, find the sum of all possible values of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{39}
In triangle $ABC$, if $\sin^2A + \sin^2B = 2\sin^2C$, then the maximum value of $\angle C$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^{\circ}}
Jesse has ten squares, labeled $1, 2, \dots, 10$. In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$, if $i$ divides $j$, then the $i$-th and $j$-th squares have different colors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2304}