inclusionAI/Ring-lite-rl-data huggingface dataset:
A deck of cards contains 52 cards, where each of the suits "diamonds", "clubs", "hearts", and "spades" has 13 cards, marked $2,3, \cdots, 10, J, Q, K, A$. Any two cards of the same suit and adjacent ranks are called "straight flush cards," and $A$ with 2 is also considered a consecutive pair (i.e., $A$ can be used as 1). Determine the number of ways to draw 13 cards from this deck so that each rank appears exactly once and no "straight flush cards" are included.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3^{13} - 3}
M. Murraikin

On an island, there live 100 knights and 100 liars, each of whom has at least one friend. Knights always tell the truth, while liars always lie. One morning, each resident either said the phrase "All my friends are knights" or the phrase "All my friends are liars," and exactly 100 people said each phrase. Find the smallest possible number of pairs of friends, one of whom is a knight and the other a liar.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Let $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{2} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]
Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Determine $\cos \theta.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{2}}
Iggy is training for a marathon. He runs different distances at varying paces due to changes in elevation, weather conditions, and additional factors like carrying a weighted backpack.

On Monday, he runs 3 miles at a pace of 1 mile in 10 minutes. On this day, there is an elevation gain of 300 feet, which adds an additional 1 minute per mile to his pace. Iggy also carries a 10-pound backpack, adding another 30 seconds per mile.

On Tuesday, he runs 5 miles at a pace of 1 mile in 9 minutes. The weather becomes hotter, impacting his speed by slowing him down by 1 extra minute per mile. He also runs with a backpack of 15 pounds, adding an additional minute for each mile.

On Wednesday, he runs 7 miles with a 2 minutes per mile reduction due to wind, resulting in an overall pace of 1 mile in 12 minutes. He carries a 20-pound backpack, adding an extra 2 minutes to his pace per mile.

On Thursday, Iggy runs 10 miles at a pace of 1 mile in 8 minutes. However, due to steady rain on this day, his pace is slowed down by an extra 2 minutes per mile. He also carries a 25-pound backpack, which adds 4 minutes per mile to his pace.

On Friday, he runs 4 miles at a pace of 1 mile in 10 minutes. The weather is perfect and there are no added impacts on his pace, but he runs with a 5-pound backpack, adding 15 seconds per mile.

Calculate the total time Iggy spends running from Monday to Friday, taking into account the changes in elevation, weather conditions, and the added resistance from the weighted backpacks.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{354.5}
Given four balls with colors red, yellow, blue, and green, they are to be placed into four boxes of colors red, yellow, blue, and green, with each box containing only one ball. Calculate the probability that exactly one ball's color matches the box's color. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
In a college, 50% of total 400 arts students are locals. 25% of students from 100 science students are locals and a certain percentage of total 120 commerce students are locals. The total percentage of locals from arts, science and commerce is 327. What is the percentage of local commerce students?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{85\%}
Let triangle $\triangle ABC$ have side lengths $AB = 7$, $BC = 8$, and $CA = 9$. Let $M$ be the midpoint of $\overline{BC}$, and $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $m\angle AEM = m\angle AFM = 90^{\circ}$. Let $P$ be the intersection of the angle bisectors of $\angle AED$ and $\angle AFD$. If $MP$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, then find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
The product $55 \cdot 60 \cdot 65$ is written as a product of 5 distinct numbers. Find the least possible value of the largest number among these 5 numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC = 60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{408}
Given a cubic polynomial \( q(x) \) that satisfies \( q(3) = 2, \) \( q(8) = 20, \) \( q(16) = 12, \) and \( q(21) = 30, \) find the sum \( q(2) + q(3) + \ldots + q(22) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{336}
In a house, a hall is 20 m long, of a certain width, and 5 m high. Its interior has to be covered with mat. The total expenditure for covering the hall with mat at a cost of Rs. 50 per square meter is Rs. 47,500. What is the width of the hall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
A positive integer $k \geqslant 3$ is called fibby if there exists a positive integer $n$ and positive integers $d_{1}<d_{2}<\ldots<d_{k}$ with the following properties:  - $d_{j+2}=d_{j+1}+d_{j}$ for every $j$ satisfying $1 \leqslant j \leqslant k-2$,  - $d_{1}, d_{2}, \ldots, d_{k}$ are divisors of $n$,  - any other divisor of $n$ is either less than $d_{1}$ or greater than $d_{k}$.  Find the largest fibby number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$. If it costs $137.94$ to label all the lockers, how many lockers are there at the school? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2001}
Nina wants to give presents to her children. She buys three toys at $10 each, two packs of basketball cards at $5 each, and some shirts at a certain price each. She spends $70 in all. What is the price of each shirt?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$6}
\(ABCD\) is a cyclic quadrilateral such that \(|DA| = |BC| = 2\), and \(|AB| = 4\). If \(|CD| > |AB|\) and the lines \(DA\) and \(BC\) intersect at an angle of \(60^\circ\), find the radius of the circumscribing circle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\sqrt{21}}{3}}
A die is rolled consecutively three times. Calculate the probability that the sequence of numbers facing upwards forms an arithmetic progression. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{12}}
Given the function $y=\cos({2x+\frac{π}{3}})$, determine the horizontal shift of the graph of the function $y=\sin 2x$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{12}}
As shown in the diagram, in the tetrahedron \(A B C D\), the face \(A B C\) intersects the face \(B C D\) at a dihedral angle of \(60^{\circ}\). The projection of vertex \(A\) onto the plane \(B C D\) is \(H\), which is the orthocenter of \(\triangle B C D\). \(G\) is the centroid of \(\triangle A B C\). Given that \(A H = 4\) and \(A B = A C\), find \(G H\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4\sqrt{21}}{9}}
In the multiplication problem shown in the figure, \( A, B, C, D, E, F, G, H, I \) each represent a different single-digit number. What is the five-digit number represented by "FIGAA"?

```
   ABC
×   DC
-----
   BEA
------
 FIGAA
```
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15744}
71. A sheep is tied to a vertex of an equilateral triangular fence with a side length of 5 meters, the rope is 7 meters long. If the sheep can only walk outside the fence, the area that the sheep can reach is $\qquad$ square meters. ( $\pi$ is taken as 3.14)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{136.59}
Let $\{x_n\}_{n=1}^{\infty}$ be a convergent sequence in a normed space $(S, \|\cdot\|)$. Determine the limit of the sequence
$$\left\{\frac{x_1 + 2x_2 + \cdots + nx_n}{n^2}\right\}_{n=1}^{\infty}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{x}{2}\)}
已知存在正整数 a, b, c 满足 $a+b+c=407,10^{n} \mid a b c$ ，求 n 的最大值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
What is the inverse Laplace transform of the function $ \ln \! \left( 1 + \dfrac{1}{s^{2}} \right) $?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{2}{t}(1-\cos t)\)}
It is given that  $x = -2272$ ,  $y = 10^3+10^2c+10b+a$ , and  $z = 1$  satisfy the equation  $ax + by + cz = 1$ , where  $a, b, c$  are positive integers with  $a < b < c$ . Find  $y.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1987}
Let \(a\) and \(b\) be real numbers such that \(\sin{a} + \sin{b} = \frac{\sqrt{2}}{2}\) and \(\cos {a} + \cos {b} = \frac{\sqrt{6}}{2}.\) The original answer is in the form \(\frac{\sqrt{k}}{m}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the number of positive integer solutions $(a,b,c,d)$ to the equation $(a^2+b^2)(c^2-d^2)=2020$. Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $\mathcal{P}$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $\mathcal{P}$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB=20$ and $CD=14$, compute the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{140}
3- |

In the convex pentagon $A B C D E$, the diagonals $B E$ and $C E$ are the angle bisectors of the angles at vertices $B$ and $C$ respectively, $\angle A=35^{\circ}, \angle D=145^{\circ}$, and the area of triangle $B C E$ is 11. Find the area of the pentagon $A B C D E$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
The 83 trainees from the Animath workshop each choose an activity for the free afternoon from 5 activities offered. We know that:

$\triangleright$ Shopping was at least as popular as Laser Tag;

$\Delta$ Movie tickets are sold in batches of 6;

$\Delta$ At most 5 students go to the beach;

$\triangleright$ At most one student visited a Romanesque church.

When writing the list of activities in alphabetical order, the number of students corresponding to each activity is written. How many different lists of numbers can be written?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3570}
The four hydrogen atoms in the methane molecule $\mathrm{CH}_{4}$ are located at the vertices of a regular tetrahedron with edge length 1. The carbon atom $C$ is located at the center of the tetrahedron $C_{0}$. Let the four hydrogen atoms be $H_{1}, H_{2}, H_{3}, H_{4}$. Then $\sum_{1 \leq i < j \leq 4} \overrightarrow{C_{0} \vec{H}_{i}} \cdot \overrightarrow{C_{0} \vec{H}_{j}} = \quad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{3}{4}}
A certain microbrewery increased its beer output by some percent. At the same time, it decreased its total working hours by 30 percent. The factory increased its output per hour by 171.43 percent (approximately). By what percent did the microbrewery increase its beer output?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Evaluate the limit \( \lim_{n\to \infty} \frac{1}{\sqrt{n}} \left(\frac{1}{\sqrt{n+1}}+\dotsm +\frac{1}{\sqrt{n+n}} \right) \) using a Riemann sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2} - 2}
A triline is a line with the property that three times its slope is equal to the sum of its \(x\)-intercept and its \(y\)-intercept. For how many integers \(q\) with \(1 \leq q \leq 10000\) is there at least one positive integer \(p\) so that there is exactly one triline through \((p, q)\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{57}
Yasha writes in the cells of a $99 \times 99$ table all positive integers from 1 to $99^2$, with each number appearing once. Grisha looks at the table and selects several cells such that no two selected cells share a common side. Grisha then sums up the numbers in all the selected cells. Determine the largest sum that Grisha can guarantee to achieve.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24017351}
Determine the number of ordered quintuples  $(a,b,c,d,e)$  of integers with  $0\leq a<$   $b<$   $c<$   $d<$   $e\leq 30$  for which there exist polynomials  $Q(x)$  and  $R(x)$  with integer coefficients such that \[x^a+x^b+x^c+x^d+x^e=Q(x)(x^5+x^4+x^2+x+1)+2R(x).\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5208}
An alarm clock runs 4 minutes slow every hour. It was set right  $ 3 \frac{1}{2}$  hours ago. Now another clock which is correct shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
A cube is rotated around the specified axis so that the marked face is rotated in the indicated manner. To which vertex number did point \(A\) move?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a basketball season, John has a varying success rate for hitting his free throws. In 20% of the games, he hits 60% of his free throws, in another 50% of the games, he hits 70% of his free throws, and in the remaining 30% of the games, he hits 80% of his free throws. Additionally, the number of fouls he gets each game also varies. In 40% of the games, he gets 4 fouls, in 35% of the games, he gets 5 fouls, and in 25% of the games, he gets 6 fouls. In each foul, he gets 2 shots. Out of 20 games the team plays, John plays in 80% of them. Calculate the total number of free throws John gets in the season, considering the varying percentages and the number of fouls he gets in each game.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{155}
Given \(0<\theta<\pi\), a complex number \(z_{1}=1-\cos \theta+i \sin \theta\) and \(z_{2}=a^{2}+a i\), where \(a \in \mathbb{R}\), it is known that \(z_{1} z_{2}\) is a pure imaginary number, and \(\bar{a}=z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}\). Determine the value of \(\theta\) when \(\bar{a}\) is a negative real number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{2}}
The circle passing through the vertices $A, B$, and $D$ of a rectangular trapezoid $ABCD$ ($\angle A = \angle B = 90^\circ$) intersects the extension of the base $BC$ and the extension of the lateral side $CD$ at points $M$ and $N$ respectively, such that $CM: CB = CN: CD = 1: 2$. Find the ratio of the diagonals $BD$ and $AC$ of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{3}:\sqrt{7}}
Determine the number of zeroes of the polynomial \( p(z) = z^8 - 20z^4 + 7z^3 + 1 \) that lie within the disk \( D(0,2) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
甲、乙两人从 $0,1, \cdots, 81$ 中轮流挑选互不重复的数，甲先选，每次每人从剩下的数中选一个数。当这 82 个数被选完之后，记 $A$ 、 $B$ 分别为甲、乙选择的所有数之和。在挑选数
的过程中, 甲希望 $A$ 与 $B$ 的最大公约数越大越好，而乙希望 $A$ 与 $B$ 的最大公约数越小越好。在甲、乙各自的最佳策略下, 求挑选完毕之后 $A$ 与 $B$ 的最大公约数.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{41}
The natural number solution set of the equation $(x^2 - 6x + c_1)(x^2 - 6x + c_2)(x^2 - 6x + c_3) = 0$ is $\{x_1, x_2, x_3, x_4, x_5, x_6\}$. What is the maximum value of $c_1 - c_2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Find the minimum value of $\int_0^{\pi} (x-y)^2 (\sin x)|\cos x|dx$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{\pi^2}{8}-\frac{1}{2}$
}
Let $V$ be an $n$-dimensional vector space over $F$, with basis $\mathcal{B} = \{\mathbf{v_1, \cdots, v_n}\}$. Let $\mathcal{B}^{*} = \{\phi_1, \cdots, \phi_n\}$ be the dual basis for $V^{*}$. Let $\psi : V \to V$ and $A$ be the matrix representing $\psi$ with respect to $\mathcal{B}$. Let $\psi^t : V^{*} \to V^{*}$ and $B$ be the matrix representing $\psi^t$ with respect to $\mathcal{B}^{*}$. Determine the relationship between $A$ and $B$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{B = A^\top}
Jonathan eats 2500 calories every day except for Saturday, when he consumes an extra amount of calories. He burns 3000 calories every day. His weekly caloric deficit is 2500 calories. How many extra calories does Jonathan consume on Saturday?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
Compute $\left\lfloor \sum_{k=0}^{10}\left(3+2\cos\left(\frac{2k\pi}{11}\right)\right)^{10}\right\rfloor \pmod{100}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{91}
Evaluate the integral \( \int^1_0 \frac{1}{\sqrt{x}} \left( \int^\sqrt{x}_1 e^{-t^2} \, dt \right) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{e} - 1}
A farmer claims that with four rods, he can fence a square plot of land sufficient for one sheep. If this is true, what is the minimum number of rods needed to fence an area for ten sheep? The answer depends on the shape of your fence. 

How many rods are required for 10 sheep?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Given that $y(x)$ is a solution of the differential equation $$ x^2 y^{\prime \prime}+x y^{\prime}-4 y=x^2 $$ on the interval $(0, \infty)$ such that $\lim _{x \rightarrow 0^{+}} y(x)$ exists and $y(1)=1$, find the value of $y^{\prime}(1)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{4}}
Given that $P\left( \frac{3}{2}, -\frac{3\sqrt{3}}{2}\right)$ is the lowest point on the graph of the function $y=A\sin(\omega x+\varphi)$ ($\omega > 0$), and $M$, $N$ are the two highest points adjacent to $P$. If $\angle MPN=60^{\circ}$, calculate the smallest positive period of this function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given:
Proposition $p$: The function $y=a^{x} (a > 0, a \neq 1)$ is a monotonically decreasing function on $\mathbb{R}$.
Proposition $q$: The range of the function $y=\lg (ax^{2}-x+a)$ is $\mathbb{R}$.
If the proposition "$p$ and $q$" is false, find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 1}
Janice gets paid $10 an hour for the first 40 hours she works each week, and $15 each hour of overtime after that. On weekends, her hourly wage increases to $12 an hour for the first 40 hours and $18 for each hour of overtime. Additionally, if she works on a holiday, her hourly rate doubles the usual wage for that day. In one week, Janice works 30 hours on weekdays, 25 hours on weekends, and 5 hours on a holiday that falls on a weekend. How much money does she make that week?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$720}
Seventy percent of a train's passengers are men, and fifteen percent of those men are in the business class. What is the number of men in the business class if the train is carrying 300 passengers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 24x^2 + 50x - 42 = 0.$ Find the value of $\frac{a}{\frac{1}{a}+bc} + \frac{b}{\frac{1}{b}+ca} + \frac{c}{\frac{1}{c}+ab}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{476}{43}}
Evaluate the limit: $$\lim_{n \to \infty}n\left(\frac{\cos\left(\frac{2n+1}{n^2+n}\right)\sin\left(\frac{1}{n^2+n}\right)}{\cos\left(\frac{2n+3}{(n+1)^2+n+1}\right)\sin\left(\frac{1}{(n+1)^2+n+1}\right)}-1\right)$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In a right triangle \( \triangle ABC \) with \( \angle B = 90^\circ \), the median \( AM \) is drawn. If the measure of angle \( \angle AMB \) is twice the measure of angle \( \angle A \), calculate the ratio \( \frac{AM}{BC} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{2}}
Find the coefficient of $x^{20}$ in the expansion of $(x^3+2x^2+5x+3)^5(x^2+4x+5)^4$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2991}
Determine the Krull dimension of the local ring \( \mathbb{C}[x,y]_p/(y^2-x^7,y^5-x^3) \), where \( p \neq (0,0) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the integral \( \int_{-\infty}^{\infty} A x^2 e^{-x^2/(2\sigma^2)} \, dx \), given that \( \int_{-\infty}^{\infty} A e^{-x^2/(2\sigma^2)} \, dx = 1 \). Here, \( A \) is a constant, and the expression inside the second integral represents the Gaussian or Normal distribution.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sigma^2}
Evaluate the limit: $$\lim_{n \to \infty} \frac{\big((n+1)(n+1)(n+3)\ldots3n\big)^{1/n}}{(n^{2n})^{1/n}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{27}{e^2}}
Determine the fundamental group \( \pi_1(X / \sim, x_0) \) for a point \( x_0 \in X / \sim \), where \( X \) is the subset of \( \mathbb{R}^3 \) defined as \( X=\left \{x \in \mathbb{R}^3 \vert 1 \leq \lvert x \rvert \leq 2 \right \} \) with boundary components \( S_1 = \left \{x \in \mathbb{R}^3 \vert \lvert x \rvert = 1 \right \} \) and \( S_2 = \left \{x \in \mathbb{R}^3 \vert \lvert x \rvert = 2 \right \} \). The equivalence relation \( \sim \) is defined such that \( x \sim y \) if \( x \in S_1 \), \( y \in S_2 \), and \( y = 2x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Determine the convergence rate of the Newton iteration near the root \( r = 1 \) for the polynomial \( g(x) = 4x^{3} - 6x^{2} + 4x - 1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given an irreducible polynomial $g(x) \in \mathbb{Q}[x]$ of prime degree $p$, if $g(x)$ has $p-2$ real roots, determine the structure of the Galois group of $g(x)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{S_p}
Find the remainder when the sum \( S = \sum_{n=0}^{2015} (n! + n^3 - n^2 + n - 1) \) is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{578}