inclusionAI/Ring-lite-rl-data huggingface dataset:
From the 205 positive integers 1, 2, ..., 205, what is the maximum number of integers that can be selected so that for any three selected numbers $a$, $b$, $c$ ($a < b < c$), it holds that $ab \neq c$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{193}
The natural numbers \(a, b, c\) are chosen such that \(a < b < c\). Given that the system of equations \(2x + y = 2023\) and \(y = |x - a| + |x - b| + |x - c|\) has exactly one solution, determine the minimum possible value of \(c\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1012}
10 is 6% of a, and a certain percentage of b is 6. c equals b / a. What is the value of the percentage that represents the relationship between 6 and b?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.6\%}
If the graph of the function $f(x)=ax^{3}-\frac{1}{3}$ ($a>0$) and the function $g(x)=x^{2}-\frac{2}{3}cx$ have exactly three distinct intersection points, and the abscissas of the intersection points form an arithmetic sequence, then the range of real number $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, \frac{1}{3})}
Find the minimal value of integer n that guarantees: Among n sets, there exist at least three sets such that any of them does not include any other; or there exist at least three sets such that any two of them include the other. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given a sequence $\{a_n\}$ with the first term $a_1=1$, the function $f(x)=x^4+a_{n+1}\cos 2x-(2a_n+1)$ has a unique zero. Find the sum of the first $n$ terms of the sequence $\{n(a_n+1)\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(n-1)\cdot2^{n+1} + 2}
Masha has 2 kg of "Lastochka" candies, 3 kg of "Truffle" candies, 4 kg of "Ptichye Moloko" candies, and 5 kg of "Citron" candies. What is the maximum number of New Year gifts she can make if each gift must contain 3 different types of candies, with 100 grams of each type?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{45}
A computer component has a lifetime (in years) that follows a Gamma distribution with a mean of 6 years and a variance of 18 years squared. Calculate the probability that the component will last at least 9 years.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4e^{-3}}
We selected the vertices, the centroid, and the trisection points of the sides of an equilateral triangle. How many points can we keep among them such that no three of them form an equilateral triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A person can swim in still water at 16 km/h. If the speed of water is 8 km/h, how many hours will the man take to swim back against the current for a certain distance if it takes him 1.5 hours?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.5}
Determine the set of limit points of the set \( T = \{ m + \frac{1}{n^2} \mid m, n \in \mathbb{N} \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{N}}
Calculate \( h'(2) \) given that \( h(x) = g(f(x, x)) \), where \( g(x) = f(x, f(x, x)) \). The function \( f(x, y) \) is differentiable, and the following values are provided: \( f(2, 2) = 4 \), \( f_x(2, 2) = 3 \), and \( f_y(2, 2) = 7 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{730}
Given a function \( f(u) \le 2\sqrt{u}f(\sqrt{u}) + 1 \), determine the asymptotic growth of \( f(u) \) in terms of Big O notation.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{O(u \log u)}
Determine the number of homomorphisms from the group \( G = \langle x,y \mid x^{-1}yx = y^6 \rangle \) to the symmetric group \( S_5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{240}
Evaluate the integral \( I = \iint_R |xy|\,dx\,dy \) using polar coordinates, where \( R \) is the region bounded by \( x^2+4y^2 \ge 1 \) and \( x^2+y^2 \le 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{8}}
Determine the number of non-isomorphic semidirect products between the groups \( \mathbb{Z}_2 \) and \( SL(2,3) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the triple integral \( \int_{-1}^{-2}\int_{-1}^{-2}\int_{-1}^{-2}\frac{x^2}{x^2+y^2+z^2}dxdydz \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{3}}
Determine the Krull dimension of the algebra \( \mathbb{Q}[X,Y,Z]/(X^{2}-Y,Z^{2}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the cardinality of the set \( A = \{ f \in C'[0,1] : f(0)=0, f(1)=1, |f'(t)| \leq 1 \text{ for all } t \in [0,1] \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the area of the figure defined by the inequality \( \left|x-2 y^{2}\right|+x+2 y^{2} \leq 8-4 y \). The figure is symmetric about the Y-axis. Calculate the total area of this figure.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Find the order of a permutation $a$ in the symmetric group $S_n$ if $a^k$ is a cycle of length $n$, where $n$ is not necessarily a prime number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n}
Let \( T = {TNFTPP} \) . \( x \) and \( y \) are nonzero real numbers such that
 \( {18x} - 4{x}^{2} + 2{x}^{3} - {9y} - {10xy} - {x}^{2}y + T{y}^{2} + {2x}{y}^{2} - {y}^{3} = 0. \)
 The smallest possible value of \( \frac{y}{x} \) is equal to \( \frac{m}{n} \) 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{7}
In triangle \(KLM\), the ratio of the circumradius to the inradius is 3. The incircle touches the sides of triangle \(KLM\) at points \(A\), \(B\), and \(C\). Find the ratio of the area of triangle \(KLM\) to the area of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( M = \left\{(x, y) \mid \tan(\pi y) + \sin^2(\pi x) = 0 \right\} \) and \( N = \left\{(x, y) \mid x^2 + y^2 \leq 2 \right\} \). Calculate the number of points in the intersection of sets \(M\) and \(N\), denoted as \(|M \cap N|\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{8} \ln(2)\)}
What is the area of the fundamental domain for the Fuchsian group \(\Gamma \leq PSL(2, \mathbb{R})\) consisting of integer matrices that are congruent to the identity mod 2, given that the area of the fundamental domain for \(PSL(2,\mathbb{Z})\) is \(\frac{\pi}{3}\) and the index of \(\Gamma\) in \(PSL(2,\mathbb{Z})\) is 6?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi}
Given a pyramid \( S A B C \) with a triangular base \( A B C \) that is equilateral, and all the plane angles at the vertex \( S \) are equal to \( \alpha \). What is the smallest \( \alpha \) for which we can claim that this pyramid is regular?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Five students are lining up in a row for a group photo. There are two female students and three male students. Female student A cannot stand at either end of the row, and there are only two male students standing next to each other. Calculate the total number of different arrangements. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Given that \( S_{n} \) is the sum of the first \( n \) terms of the sequence \( \{a_{n}\} \) with \( S_{0} = 0 \), and for any \( n \in \mathbf{Z}_{+} \),
\[
\frac{a_{n}}{2017} = -\frac{2017 + S_{n-1}}{n},
\]
find \( \sum_{n=1}^{2017} 2^{n} a_{n} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4034}
The scoring rules in a football match are: 3 points for a win, 1 point for a draw, and 0 points for a loss. How many different outcomes are there for a team that plays 14 matches and gets a total of 19 points?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let $S$ be the unilateral shift operator on a separable, infinite-dimensional Hilbert space $\mathcal{H}$. Determine the distance from $S$ to the set of Hilbert-Schmidt operators on $\mathcal{H}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that the circumference of a sector is $10cm$ and its area is $4cm^{2}$, the radian measure of the central angle $\alpha$ of the sector is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{2}}
Determine whether there exists a locally integrable function $g$ on the interval $(0,1)$ such that \( \int_x^1 g(t) \, dt = \ln(x) \) for all \( x \in (0,1) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{x}}
Find all differentiable functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(x^4+y) = x^3f(x) + f(y) \) for all \( x, y \in \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = kx}
a 120 meter long train crosses a man standing on the platform in 6 sec . what is the speed of the train ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Compute the equivalence classes of extensions of $\mathbb{Z}_p$ by $\mathbb{Z}$, denoted as $E(\mathbb{Z}_p,\mathbb{Z})$. What is the structure of this set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_p}
Evaluate the limit as $x$ approaches $0$ from the positive side: \[ \lim_{{x\to 0^+}} \left(\frac{{(2^{(e^{x^{3/2}}+1)}-4)^2}}{{\sin^6(\sqrt{x})}}\right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16 (\ln 2)^2}
Between 6 and 7 o'clock, how many moments are there when, after swapping the positions of the hour and minute hands, they still represent the positions of the hour and minute hands at some actual time? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral: $$I=\int_0^{1/2}\frac{\ln(1+2x)}{1+4x^2}{\rm d}x$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{16}\ln 2\)}
Find the number of ordered 8-tuples $(a_7,a_6,\ldots,a_0) $ of nonnegative integers such that $0 \le a_i \le 4$ for all $i$, and $$a_7n^7+a_6n^6+a_5n^5+a_4n^4+a_3n^3+a_2n^2+a_1n+a_0 \equiv 0 \pmod{5}$$ for all integers $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{125}
Circle $X$ with radius $3$ is internally tangent to circle $O$ with radius $9$. Two distinct points $P_1$ and $P_2$ are chosen on $O$ such that rays $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$ are tangent to circle $X$. What is the length of line segment $P_1P_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the maximum number of subsets that can be chosen from a set of 20 elements such that no two subsets have more than 2 elements in common.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1351}
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1507509}
Suppose \( f \) is an entire function such that \( f(\sqrt{n}) = 0 \) for \( n = 1, 2, 3, \ldots \), and there exists a positive constant \( \alpha \) such that \( |f(z)| < \exp(|z|^\alpha) \) for all sufficiently large \( |z| \). Determine the values of \( \alpha \) for which it follows that \( f(z) = 0 \) for all \( z \). Consider the function \( \sin(\pi z^2) \) in your analysis.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\alpha < 2}
Calculate the intersection number of the two rational points \(P=(-1,t^3)\) and \(Q=(-t^2,1)\) on the elliptic surface defined by the equation \(E_6: y^2= x^3 + t^6 + 1\) over the field \(\mathbb{F}_5\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Find the value of \( k \) such that \( \int_0^k x^x \, dx = 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.194}
$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数, $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left\{x_{i 3}, x_{i 4}\right\}, Y=\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right.\right.$, 求 $x$ 大于 $y$ 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{4}
Given the inequality
$$
\log _{x^{2}+y^{2}}(x+y) \geqslant 1
$$
find the maximum value of \( y \) among all \( x \) and \( y \) that satisfy the inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{1}{2} + \\frac{\\sqrt{2}}{2}"]}
Edward stopped to rest at a place 1,875 feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of \( 100 \, \mathrm{ft/s} \). At the same time, Edward started running away with an acceleration of \( 1 \, \mathrm{ft/s^2} \). Assuming that air resistance causes the arrow to decelerate at \( 1 \, \mathrm{ft/s^2} \) and that it does hit Edward, how fast was the arrow moving at the moment of impact (in \( \mathrm{ft/s} \))?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{75 \, \text{ft/s}}
A 10,000-meter race involves three participants: A, B, and C. A starts last. During the race, A exchanges positions with B and C a total of 9 times. Determine the final position of A in the race.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The function g(a) is defined for integers a such that if a is even, g(a) = a / 2 and if a is a certain type of number, g(a) = a + 5. Given that g(g(g(g(g(a))))) = 19, and there are 8 possible values for a that would satisfy this equation, what type of number is a when g(a) = a + 5?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Given the complex numbers ${z_1}=1-i$ and ${z_2}=\sqrt{3}+i$, where $i$ is the imaginary unit, find the imaginary part of $\frac{\overline{z_1}}{z_2}$. Express your answer as a fraction, including 'i' to denote the imaginary part.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\sqrt{3}-1}{4}i$}
For which values of the parameter \( a \) does the equation \(\sqrt{x+\frac{1}{2}+\sqrt{x+\frac{1}{4}}}+x=a \) have a unique solution?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \in \left[ \frac{1}{4}, +\infty \right)}
Fill the numbers 1 and 10 in the blanks below. Each number can only be used once. The number in the square should be the difference between the two numbers at its top left and top right corners (the larger number minus the smaller number). The number in the circle should be the sum of the two numbers at its top left and top right corners. Find the value of A.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Determine the Euler-Lagrange equation for the functional \( \int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y'' = 0}
Calculate the second-order mixed partial derivative \(\frac{\partial^2g}{\partial u \, \partial v}(1,1)\) for \(g(u,v)=uf(u^2, 2uv)\) given that \(f(1,2)=4\), \(\nabla f(1,2)=(3,-1)\), \(\frac{\partial^2f}{\partial x^2}(1,2)= \frac{\partial^2f}{\partial y^2}(1,2)=1\), and \(\frac{\partial^2f}{\partial x \,\partial y}(1,2)=-1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4}
In a cube \( ABCD-A_1B_1C_1D_1 \) with edge length \( a \), \( X \) and \( Y \) are the centers of squares \( AA_1BB_1 \) and \( BB_1C_1C \) respectively, and \( Z \) is on the diagonal \( BD \) such that \( DZ = 3ZB \). Find the area of the cross-section that passes through points \( X \), \( Y \), and \( Z \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{2}a^2}{2}}
Two plane mirrors form the edges of a dihedral angle. A light ray perpendicular to the edge of the angle and parallel to the first mirror reflects off the second mirror, then off the first, then off the second again, then the first again, and finally reflects off the second mirror for the fifth time, returning along the same path. What is the measure of the dihedral angle between the mirrors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30^\circ}
Consider all possible broken lines that travel along the sides of the cells and connect two opposite corners of a square sheet of grid paper with dimensions $100 \times 100$ by the shortest path. What is the minimum number of such broken lines that need to be taken so that their union contains all the vertices of the cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{101}
Box 1 contains one gold marble and one black marble. Box 2 contains one gold marble and two black marbles. Box 3 contains one gold marble and three black marbles. Whenever a marble is chosen randomly from one of the boxes, each marble in that box is equally likely to be chosen. A marble is randomly chosen from Box 1 and placed in Box 2. Then a marble is randomly chosen from Box 2 and placed in Box 3. Finally, a marble is randomly chosen from Box 3. Calculate the probability that the marble chosen from Box 3 is gold. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{40}}
Let $\triangle ABC$ be an acute isosceles triangle with circumcircle $\omega$. The tangents to $\omega$ at vertices $B$ and $C$ intersect at point $T$. Let $Z$ be the projection of $T$ onto $BC$. Assume $BT = CT = 20$, $BC = 24$, and $TZ^2 + 2BZ \cdot CZ = 478$. Find $BZ \cdot CZ$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{144}
On a chessboard, there is a pawn on B8 and another on G8, and there is a knight on B1. What is the minimum number of moves in which the knight can capture both pawns? (The pawns do not move during this time.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
By placing addition signs between some of the digits of the number 123456789, we can obtain various sums. What is the minimum number of addition signs required to obtain a sum of 54? Provide your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Calculate the limit superior of the following expression as \( n \) approaches infinity:

\[ \limsup_{n\to\infty}\left(\frac{A^n}{n}+\frac{B^n}{n^2}\right)^{\frac{1}{n}} \]

where \( A \) and \( B \) are positive real numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\max\{A, B\}}
Find the derivative of the integral \( L = \int f(g(x)) \, dx \) with respect to \( g(x) \). What is \( \frac{dL}{dg(x)} \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{f(g(x))}{g'(x)}}
Find the derivative of the expression \((X \beta)^t (X \beta)\), where \(X\) is an \(N \times P\) matrix and \(\beta\) is a \(P \times 1\) vector.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 X^\top X \beta}
The center of a circle inscribed in a right trapezoid is 1 cm and 2 cm away from the ends of one of its non-parallel sides. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3.6 \, \text{cm}^2}
Billy and Bobby are located at points  $A$  and  $B$ , respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point  $A$ ; the second time they meet, they are located 10 units from point  $B$ . Find all possible values for the distance between  $A$  and  $B$ .

*Proposed by Isabella Grabski*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
In a plane, 100 points are marked. It turns out that 40 marked points lie on each of two different lines \( a \) and \( b \). What is the maximum number of marked points that can lie on a line that does not coincide with \( a \) or \( b \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{23}
Consider a scenario where \( n \) people each write their name on a card and place it in a box. Each person, one by one, draws a card from the box. If a person draws their own card, they put it back and draw again until they get a different card. After drawing a different card, they return their own card to the box. What is the probability that the last person to draw a card ends up with their own card? Assume \( n = 31 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{31}}
Let $V$ be a finite-dimensional vector space over $\mathbb{C}$. Let $T:V\to V$ be a linear transformation with the minimal polynomial $x^{13}(x-1)$. What is the minimal polynomial of $T^4$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^4(x - 1)}
Calculate the outward flux of the vector field \( V = x(x^2+y^2+z^2)^{-3/2}\mathbf{i} + y(x^2+y^2+z^2)^{-3/2}\mathbf{j} + z(x^2+y^2+z^2)^{-3/2}\mathbf{k} \) through the boundary of the rectangular box defined by \(-1 \leq x \leq 1\), \(-1 \leq y \leq 1\), and \(-1 \leq z \leq 1\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
Evaluate the residue of the function \( \frac{(z^6-1)^2}{z^5(2z^4 -5z^2 + 2)} \) at \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{21}{8}}
Determine the number of distinct ways to color the vertices of an octagon using 10 different colors, where two colorings are considered identical if one can be transformed into the other by a symmetry of the octagon.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6278140}
Given 17 points on the circumference of a circle, determine the maximum number of segments that can be drawn such that each segment intersects all others inside the circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}