inclusionAI/Ring-lite-rl-data huggingface dataset:
Given a regular $4n$-gon $A_{1} A_{2} \ldots A_{4n}$ with area $S$, where $n > 1$. Find the area of the quadrilateral $A_{1} A_{n} A_{n+1} A_{n+2}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{S}{2}}
Let \( f(x) \) be an increasing continuous function defined on the interval \([0, 3]\), and let \( g(x) \) be its inverse function, such that \( g(x) > f(x) \) for all positive \( x \) where both functions are defined. Additionally, \( f(0) = 0 \) and \( f(3) = 2 \).

The area under the graph of \( f(x) \) on the interval \([0, 3]\) is 2. Find the area of the figure bounded by the graphs of \( f(x) \) and \( g(x) \), as well as the segment connecting the points \((3, 2)\) and \((2, 3)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.5}
i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is:
Find all positive integers m for which for all  $\alpha,\beta \in \mathbb{Z}-\{0\}$ \[ \frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m \equiv \pm 1 \pmod{6}}
Jason went to the mall on Saturday to buy clothes for a special event. He decided to purchase a jacket, a pair of shorts, a shirt, and a pair of shoes. The prices of these items before any discounts are as follows:

- Jacket: $45.28
- Shorts: $29.99
- Shirt: $26.50
- Shoes: $84.25

Jason had a 15% off coupon for the jacket and a 20% off coupon for the shoes. The mall was also offering a "buy one, get one 50% off" sale on shirts and shorts. Assume that tax is 8.25%.

Calculate the total amount Jason spent on his shopping trip, taking into consideration the discounts and tax.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$159.54}
Let $n$ be the number of integer values of $x$ such that $P = x^4 + 6x^3 + 11x^2 + 3x + 31$ is the square of an integer. Find the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
On a purchase of $118, a store offered a payment plan consisting of a $18 down payment and 12 monthly payments of a certain amount each. The customer paid 15.254237288135593% of the purchase price in interest by using this plan. What was the amount of each monthly payment?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$9.83}
Chords \(AB\) and \(CD\) of a circle with center \(O\) both have a length of 5. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) intersect at point \(P\), where \(DP=13\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL:LC\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13/18}
Point \( A \) lies on the line \( y = \frac{5}{12} x - 7 \), and point \( B \) lies on the parabola \( y = x^2 \). What is the minimum length of segment \( AB \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4007}{624}}
Evaluate the integral \( \int_{[0,1]} f \, \, d\lambda \), where \( f(x) = e_2(x) \) and \( e_k(x) \) is the \( k \)-th digit after the decimal point of \( x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{9}{2}}
Determine the number of infinite planar graphs with the degree sequence \(D=(4^4,6^\infty)\), where the superscripts denote the number of vertices with that degree.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a $418 \times 418$ grid, black and white coins are placed on some of the squares. First, remove all black coins that are in the same row as any white coin(s). Then, remove all white coins that are in the same column as any black coin(s). Let $b$ be the number of black coins remaining and $w$ be the number of remaining white coins. Find the remainder when the maximum possible value of $bw$ is divided by $2007$. \( \text{(Express your answer as an integer.)} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{952}
A rhombus is inscribed in $\triangle ABC$ in such a way that one of its vertices is $A$ and two of its sides lie along $AB$ and $AC$. If $\overline{AC} = 6$ inches, $\overline{AB} = 12$ inches, and $\overline{BC} = 8$ inches, the side of the rhombus, in inches, is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$4$}
Given that line segments AB and CD are parallel, and angle D is twice angle B, and AD = a and CD = b, find the length of AB. Express your answer as an algebraic expression in terms of a and b.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a+b}
Determine the range of convergence for the power series \( \sum_{n=1}^{\infty}\frac{(-1)^{[\sqrt n]}}{n}x^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[-1, 1]}
Evaluate the following integral:

$$\int_{0}^{1}\left(\int_{0}^{\sin^2x}\arcsin(\sqrt{t})\, dt+\int_{0}^{\cos^2x}\arccos(\sqrt{t})\, dt\right)dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{4}\)}
Determine the points at which the function \( f(x) = \begin{cases} \cos x & x \in \mathbb{Q} \\ x & x \notin \mathbb{Q} \end{cases} \) is differentiable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
There are 10 birds pecking on the ground, and among any 5 birds, at least 4 must be on the same circle. What is the minimum number of birds that must be on the circle that contains the most birds?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
There are seven cards in a hat, and on card $k$ there is a number $2^{k-1}$, where $k=1,2,...,7$. Solarin picks the cards at random from the hat, one card at a time, until the sum of the numbers on the cards in his hand exceeds $124$. What is the most probable sum he can get?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{127}
Given the sets $P=\{x|x^2-2x-3<0\}$ and $Q=\{x|x\in\mathbb{N}\}$, then $P\cap Q=$ ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{0,1,2\}}
Twenty pairs of integers are formed using each of the integers \( 1, 2, 3, \ldots, 40 \) once. The positive difference between the integers in each pair is 1 or 3. Find the greatest possible sum of the differences.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{58}
For any real numbers $m$ and $n$, when $0 < n < m < \frac{1}{a}$, the inequality $\frac{\sqrt[m]{n}}{\sqrt[n]{m}} > \frac{n^a}{m^a}$ holds true. Find the minimum value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Let  $ABC$  be an equilateral triangle.  $A $  point  $P$  is chosen at random within this triangle. What is the probability that the sum of the distances from point  $P$  to the sides of triangle  $ABC$  are measures of the sides of a triangle?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
If you set the clock back by 10 minutes, the number of radians the minute hand has turned is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
Calculate the line integral \( \int_{|z|=R}\frac{1}{P(z)}dz \), where \( P(z) \) is a polynomial with all roots inside a circle of radius \( R \) centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
How do you evaluate the limit of the sum $$\lim_{n\to\infty}\sum\limits_{0\leqslant{k}\leqslant{2n}} {\frac{k}{k+n^{2}}}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that $\overrightarrow {a}$ and $\overrightarrow {b}$ are unit vectors, and $\overrightarrow {a}$ is perpendicular to $\overrightarrow {b}$, vector $\overrightarrow {c}$ satisfies $| \overrightarrow {c}- \overrightarrow {a}- \overrightarrow {b}|=2$. The range of values for $| \overrightarrow {c}|$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2- \sqrt {2}, 2+ \sqrt {2}]}
Points  $A$ ,  $B$ ,  $C$ , and  $D$  lie on a circle such that chords  $\overline{AC}$  and  $\overline{BD}$  intersect at a point  $E$  inside the circle.  Suppose that  $\angle ADE =\angle CBE = 75^\circ$ ,  $BE=4$ , and  $DE=8$ .  The value of  $AB^2$  can be written in the form  $a+b\sqrt{c}$  for positive integers  $a$ ,  $b$ , and  $c$  such that  $c$  is not divisible by the square of any prime. Find  $a+b+c$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{115}
A guard has detained an outsider and wants to expel him. The outsider then stated that he made a bet with his friends for 100 coins that the guard would not expel him (if the guard expels him, the outsider pays his friends 100 coins; otherwise, his friends pay him 100 coins). The outsider, deciding to buy off the guard, offered him to name a sum. What is the maximum number of coins the guard can demand so that the outsider, guided only by his own benefit, will definitely pay the guard?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{199}
Let  $\triangle ABC$  be a triangle with  $AB=85$ ,  $BC=125$ ,  $CA=140$ , and incircle  $\omega$ .  Let  $D$ ,  $E$ ,  $F$  be the points of tangency of  $\omega$  with  $\overline{BC}$ ,  $\overline{CA}$ ,  $\overline{AB}$  respectively, and furthermore denote by  $X$ ,  $Y$ ,  and  $Z$  the incenters of  $\triangle AEF$ ,  $\triangle BFD$ , and  $\triangle CDE$ , also respectively.  Find the circumradius of  $\triangle XYZ$ .

*Proposed by David Altizio*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{The circumradius of $\triangle XYZ$ is 30.}
A merchant must mix salt at a certain price per pound with 40 lbs of salt that costs 25 cents/lb so that they will get a 20% profit by selling the mixture at 48 cents/lb. The merchant needs to mix 60 lbs of salt at this price. What is the price per pound of the salt that needs to be mixed?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$0.50}
If $(X-2)^8 = a + a_1(x-1) + \ldots + a_8(x-1)^8$, then the value of $\left(a_2 + a_4 + \ldots + a_8\right)^2 - \left(a_1 + a_3 + \ldots + a_7\right)^2$ is (Answer in digits).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-255}
Marcella has some pairs of shoes. If she loses 9 individual shoes and has 15 matching pairs left, how many pairs of shoes did she have initially?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Calculate \( \lim_{n \rightarrow \infty} \int_{0}^{\infty} \left(1+\frac{x}{n}\right)^{-n}\sin\left(\frac{x}{n}\right) dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
4. A circle is inscribed with 2019 numbers. For any two adjacent numbers $x$ and $y$, the inequalities $|x-y| \geqslant 2, x+y \geqslant 6$ are satisfied. Find the smallest possible sum of the recorded numbers.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6060}
James gets bored with his game so decides to play a different one. That game promises 100 hours of gameplay but a certain percentage of that is boring grinding. However, the expansion does add another 30 hours of enjoyable gameplay. James gets 50 hours of enjoyable gameplay. What is the percentage of the gameplay that is boring grinding?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{61.54\%}
At a bus station, a bus leaves at a certain interval for 12 hours a day. For 5 days, 120 buses leave the station. What is the time interval between each bus leaving the station?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{30}
Let \( f \) be a nonnegative, continuous, and strictly increasing function on \([0,1]\). For \( p > 0 \), let \( x_p \) be the number in \((0,1)\) that satisfies \( f^p(x_p) = \int_0^1 f^p(x) \, dx \). Determine \( \lim_{p \to \infty} x_p \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^{3}-9 x^{2}+10 x+5=0$. Compute the side length of the triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \sqrt{17}}
In a diagram, several circles are connected by segments. Choose a natural number \( n \) and place different natural numbers in the circles such that the following conditions are met: If two numbers \( a \) and \( b \) are not connected by a segment, then \( a - b \) is coprime with \( n \). If they are connected, then \( a - b \) and \( n \) share a common divisor greater than 1. Determine the smallest \( n \) for which such an arrangement is possible.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
I bought a TV for $1700 excluding tax. The store offers two discounts: a 10% discount on the original price, and a store membership discount of 5%. There are three different taxes and fees to consider: a 15% value-added tax (VAT), an 8% sales tax, and a $25 recycling fee. I am eligible for the store membership discount. What is the final price of the television after applying both discounts, accounting for all taxes, and adding the recycling fee?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1812.81}
Given points $A(-2,3)$ and $B(3,2)$, if line $l$: $y=kx-2$ has no intersection with line segment $AB$, then the range of the slope $k$ of line $l$ is _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{k \in (-\frac{5}{2}, \frac{4}{3})}
已知函数$f\left( x \right)=\sin x$．若存在${{x}_{1}}$，${{x2}}$，$\cdot \cdot \cdot $，${{x}_{m}}$满足$0\le {{x1}}<{{x}_{2}}<\cdot \cdot \cdot <{{xm}}\le 6\pi $，且$\left| f\left( {{x}_{1}} \right)-f\left( {{x2}} \right) \right|+\left| f\left( {{x}_{2}} \right)-f\left( {{x3}} \right) \right|+\cdot \cdot \cdot +\left| f\left( {{x}_{n-1}} \right)-f\left( {{xn}} \right) \right|=12$（$m\ge 2$，$m\in {{\Nu }^{*}}$），则$m$的最小值为_________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product
\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{112}
Determine if there exists a constant \(C_h\) such that \(\|\partial_h x\| \leq C_h \|x\|\) for all \(x \in \ell^2(\mathbb{N})\), where \(\partial_h\) is the finite difference operator defined as \((\partial_h(x))_j = (x_{j+1}-x_j)/h\) for \(h>0\). If such a constant exists, find the optimal value of \(C_h\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2}{h}}
Determine the probability of reaching a state of 100 or more, starting from 20, assuming an optimal strategy is used. Provide your answer as a probability value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.2}
Find the value of the constant $c_1$ in the steady state solution $u(x) = c_1 - 2x$ for the following PDE problem:
\begin{align}
u_{t} &= u_{xx} \\
u_{x}(t,0) &= u_{x}(t,4) = -2 \\
u(0,x) &= \begin{cases} 0 & 0 \leq x \leq 2 \\ 2x-4 & 2 \leq x \leq 4 \end{cases}
\end{align}
Choose $c_1$ such that the total heat energy as $t \rightarrow \infty$ is the same as when $t=0$.}
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
14. Teacher $D$ placed cards with the numbers $1, 2, \cdots, 13$ in a clockwise order on a circle. Initially, all cards are face up. Each time, one card can be flipped to face down (once a card is face down, it cannot be flipped again). Teacher $D$'s rule for flipping cards is: if a face-up card has the number $A$, and the card that is 2 cards away is also face up, then Teacher $D$ can flip the card with the number $A$. For example: as long as the cards with the numbers 9 or 2 are face up, Teacher $D$ can flip the card with the number 12 (of course, provided that the card with the number 12 is still face up). In the end, as long as Teacher $D$ flips 12 cards to face down, it is considered a success. How many different sequences of flipping can Teacher $D$ use to achieve success?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26624}
Suppose $A$ is an abelian group of order $p^4$, where $p$ is a prime number. If there exists a subgroup $N$ such that $A/N \simeq \mathbb{Z}/p^3\mathbb{Z}$, how many distinct groups $A$ are there satisfying this condition? Given the classification of abelian groups of order $p^4$, which are: $\mathbb{Z}/p^4\mathbb{Z}$, $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$, $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}$, $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p^3\mathbb{Z}$, and $\mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}$, how many of these admit a subgroup $N$ with the given property?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Mr. Wong harvested 560 mangoes from the tree outside his house. He sold x mangoes at the market and gave 50 mangoes to his family. He divided the rest of the mangoes evenly among 12 of his neighbors. If x is half the number of mangoes harvested, how many mangoes does each neighbor receive?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{19}
设集合 $A=\{1,2, \cdots, 10\}, A$ 到 $A$ 的映射 $f$ 满足下列两个条件:
(1)对任意 $x \in A, f_{30}(x)=x$.
(2) 对每个 $k \in \mathbf{N}_{+}, 1 \leqslant k \leqslant 29$, 至少存在一个 $a \in A$, 使 $f_k(a) \neq a$.
求这样映射 $f$ 的个数(约定 $\left.f_1(x)=f(x), f_{k+1}(x)=f\left(f_k(x)\right), k=1,2, \cdots\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120960}
Problem 8.4. Points $D$ and $E$ are marked on sides $A C$ and $B C$ of triangle $A B C$ respectively, such that $A D=E C$. It turns out that $B D=E D, \angle B D C=\angle D E B$. Find the length of segment $A C$, given that $A B=7$ and $B E=2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
In the spring round of the 2000 Cities Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, but no two students together solved all six problems. What is the minimum possible number of high school students in country $N$ who participated in the spring round?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Let \( a, b, c, d \) be any positive real numbers. Determine the range of the sum
$$
S = \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}.
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 2)}
Given the parabola y^2 = 4x and points A(1,0), B(-1,0), if point M is on the parabola, calculate the maximum value of angle MBA in radians, using pi notation for precision.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac {\pi}{4}}
Let $\pi$ be a permutation of the numbers from 1 through 2012. What is the maximum possible number of integers $n$ with $1 \leq n \leq 2011$ such that $\pi(n)$ divides $\pi(n+1)$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1006}
Given two lines $l_{1}$: $(m+1)x+2y+2m-2=0$ and $l_{2}$: $2x+(m-2)y+2=0$, if $l_{1} \parallel l_{2}$, then $m=$ ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-2}
Given the function $f(x) = |\log_{3}x|$, if $0 < m < n$ and $f(m) = f(n)$, then the range of values for $2m+n$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2\sqrt{2}, +\infty)}
在矩形 $A B C D$ 中, $A B=2, A D=1$, 在 $D C$ 边上（包含点 $D 、 C$ ）的动点 $P$ 与 $C B$延长线上（包含 $B$ ）的动点 $Q$ 满足 $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$ ，则向量 $\overrightarrow{P A}$ 与向量 $\overrightarrow{P Q}$ 的数量积 $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ 的最小值为 $\qquad$.
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{7}
Calculate the volume of the region in the first octant bounded by the cylinder \(x^2 + z^2 = 4\) and the planes \(x = 0\), \(y = z\), and \(z = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{8}{3}}
Evaluate the integral \( \int_{-\infty}^{\infty} \frac{1}{\sqrt{(x^2+2x+2)^3}} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of ten-digit positive integers with the following properties:

- Each of the digits $0, 1, 2, \ldots, 8, 9$ is contained exactly once.
- Each digit, except $9$, has a neighboring digit that is larger than it.

(Note: For example, in the number $1230$, the digits $1$ and $3$ are the neighboring digits of $2$, while $2$ and $0$ are the neighboring digits of $3$. The digits $1$ and $0$ have only one neighboring digit.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
Determine the values of \( a \) for which the improper integral \( \int_0^{\frac{1}{2}} \frac{1}{x|\log(x)|^a} \, dx \) is finite.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a > 1}
What is the smallest known constant \(c\) such that for any integer \(n \geq 2\), there are at least \(\frac{n}{\log_2{n}}\) primes in the interval \([n, cn]\]?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Compute the volumes of the solids formed by the rotation of the figures bounded by the graphs of the functions. The axis of rotation is $O y$.

$$
y=\sqrt{x-1}, y=0, y=1, x=0.5
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{97\pi}{60}}
Evaluate the integral $$\int_{-\infty}^0 \frac{dx}{(x-2)\sqrt{1-x}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{2}\)}
Let $G$ be a finite group, and let $H$ be a normal subgroup of $G$ with order $p$, where $p$ is a prime number. Let $K$ be a subgroup of $G$ such that $\gcd(|K|, p-1) = 1$. Determine the structure of the subgroup $HK$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{H \times K}
There is a sequence: some number, 15, 17, 51, 53, 159161. What is the first number in the sequence?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Calculate the volume bounded by the surfaces $x^2+z=4$ and $y^2+z=4$ in the first octant.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the minimal polynomial of the linear transformation \( \phi: V \to V \) on the vector space \( V = \mathbb{R}^n \), where \( \phi(v_1,v_2,\dots,v_n)^t = (v_2,v_3,\dots,v_1)^t \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^n - 1}
In a casino game involving fair coin flips, the game ends when the sequence $HTH$ appears. Using the concept of martingales and the optional stopping theorem, calculate the expected number of flips required for the sequence $HTH$ to occur.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}