inclusionAI/Ring-lite-rl-data huggingface dataset:
Some people know each other in a group of people, where \u0022knowing\u0022 is a symmetric relation. For a person, we say that it is $social$ if it knows at least $20$ other persons and at least $2$ of those $20$ know each other. For a person, we say that it is $shy$ if it doesn\u0027t know at least $20$ other persons and at least $2$ of those $20$ don\u0027t know each other. Find the maximal number of people in that group, if we know that group doesn\u0027t have any $social$ nor $shy$ persons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
40
}
Given a line $l$ with a slope of $2$ intersects the hyperbola $C$: $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a > 0, b > 0$) at points $A$ and $B$. If point $P(3,1)$ is the midpoint of $AB$, calculate the eccentricity of the hyperbola $C$. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\sqrt{15}}{3}}
Triangle  $ABC$  with  $\measuredangle A = 90^\circ$  has incenter  $I$ . A circle passing through  $A$  with center  $I$  is drawn, intersecting  $\overline{BC}$  at  $E$  and  $F$  such that  $BE < BF$ . If  $\tfrac{BE}{EF} = \tfrac{2}{3}$ , then  $\tfrac{CF}{FE} = \tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

*Proposed by Lewis Chen*
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
In the expansion of $\left(x+1\right)\left(2x+1\right)\cdots \left(nx+1\right)$ (where $n$ is a positive integer), find the coefficient of the linear term in $x$. Express your answer as a combination formula (e.g., C_n^k).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C_{n+1}^2}
Compute\[\log_2\left(\prod_{a=1}^{2015}\prod_{b=1}^{2015}\left(1+e^{2\pi iab/2015}\right)\right)\]Here $i$ is the imaginary unit (that is, $i^2=-1$).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
13725}
A piece of rope is randomly cut into two parts. Calculate the probability that the longer segment is at least x times the shorter segment. Express your answer as a fraction or ratio involving x.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2}{x+1}}
Calculate the phase margin \( \phi \) of the second-order system given by the transfer function:
\[ F(j\omega) = \frac{1}{(1-\frac{\omega^2}{\omega_f^2}) + j\omega\frac{2\xi}{\omega_f}} \]
where \( \xi < 1 \). Find \( \phi = \text{Arg}(F(j\omega)) \) for \( \omega = 2\omega_f \) and \( \xi = 0.25 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-161.57^\circ}
10. (20 points) Point $A$ lies on the line $y=\frac{15}{8} x-8$, and point $B$ lies on the parabola $y=x^{2}$. What is the minimum length of segment $A B$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1823}{544}}
Find the value of \( n \) that is the closest solution to the equation:
\[
\frac{(4.554 \times 10^{9})!(4.6 \times 10^9 - n)!}{(4.554 \times 10^{9} -n)!(4.6 \times 10^9)!} \approx 0.997
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
In the pyramid \( SABC \), the product of the edge lengths for each of the four faces is equal to the same number. The length of the height of the pyramid, dropped from \( S \) to the face \( ABC \), is \( 2 \sqrt{\frac{102}{55}} \), and the angle \( CAB \) is \( \arccos \left(\frac{1}{6} \sqrt{\frac{17}{2}}\right) \). Find the volume of the pyramid \( SABC \) given that

\[
|SA|^{2} + |SB|^{2} - 5|SC|^{2} = 60
\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{ \frac{34 \sqrt{6}}{3}}
Determine the quotient ring $T_2(\mathbb{Z_4})/I$, where $I = \left\{ \begin{bmatrix}0&b\\0&c\end{bmatrix}: b,c \in \mathbb{Z_{4}}\right\}$ is an ideal of the ring of upper triangular matrices $T_2(\mathbb{Z_4})$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}_4}
How many distinct bracelets can be created using 12 beads, with 3 red, 4 green, and 5 blue beads, considering rotation and reflection symmetry? Use the Dihedral group formula to determine this number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1170}
Consider the function $$f(x) =1+3x \int^{x^2}_{2x} e^{-(t)^{1/2}}dt$$ What is the first order Taylor polynomial around $x=2$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 + 12e^{-2} (x-2)}
Given the parabola $y^{2}=2x$, and $A$, $B$ are two points on the parabola, the perpendicular bisector of segment $AB$ intersects the $x$-axis at point $P(x_{0},0)$. The range of $x_{0}$ is _______. (Express the range as an interval)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1,+\infty)}
Given the first-order partial differential equation (PDE) $xu_x + (x^2 + y)u_y = u$ for all $x, y \in \mathbb{R}$, where $u(2, y) = y - 4$, find the value of $u(1, 2)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given that the random variable $\xi$ follows the normal distribution $N(1, 4)$, if $P(\xi > 4) = 0.1$, then $P(-2 \leq \xi \leq 4)$ equals _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["0.8"]}
Let \(a\) and \(b\) be two natural numbers. There are \(ab\) pieces of candy, which are to be packaged into several bags in such a way that the candies can be divided into \(a\) parts, each with \(b\) pieces, and also can be divided into \(b\) parts, each with \(a\) pieces. Find the minimum number of bags needed.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a + b - \gcd(a, b)}
Marge had a winning lottery ticket for $20000. She paid 55% of her winnings towards taxes. She then used a half of the remaining amount to pay off a part of her mortgage. Wanting to secure her future, she put a third of what was left into a retirement fund. Afterwards, she decided to contribute a quarter of the remaining amount to her children’s college fund. Eventually, she also put $1200 in savings and invested half of her savings in the stock market. How many dollars does she have left for fun?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1050}
In the land of Draconia, there live red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Moreover, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each one said:

- 1st head: "The dragon to my left is green."
- 2nd head: "The dragon to my right is blue."
- 3rd head: "There is no red dragon next to me."

What is the maximum number of red dragons that could have been at the table?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{176}
Find all real functions \( f(x) \) that are defined and continuous on the interval \((-1, 1)\) and satisfy the functional equation:
\[ f(x+y) = \frac{f(x) + f(y)}{1 - f(x) f(y)} \quad \left( x, y, x+y \in (-1,1) \right). \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = \tan(kx)}
Given \( x = -2272 \), \( y = 10^3 + 10^2 c + 10 b + a \), and \( z = 1 \), which satisfy the equation \( a x + b y + c z = 1 \), where \( a \), \( b \), \( c \) are positive integers and \( a < b < c \). Find \( y \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1987}
Find the area of the region bounded by the curves $xy=4$, $xy=8$, $xy^3=5$, and $xy^3=15$. Express your answer in terms of natural logarithms.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2 \ln 3}
Let \( f:\mathbb{R} \to \mathbb{R} \) be a differentiable function defined by \( f(x) = x + \int_0^1 (xy + x^2)f(y) \, dy \). Find the value of \( \int_0^1 f(x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{36}{23}}
Below is a "right triangular number array":
\[
\frac{1}{4},
\]
\[
\frac{1}{2}, \quad \frac{1}{4}
\]
\[
\frac{3}{4}, \quad \frac{3}{8}, \quad \frac{3}{16}
\]
Each column forms an arithmetic sequence, starting from the third row, each row forms a geometric sequence with a constant common ratio. Let the number in the \(i^\text{th}\) row and \(j^\text{th}\) column be \(a_{i j}\) (\(i \geqslant j, i, j \in \mathbf{N}_{+}\)), find \(a_{n 3}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n}{16}}
设 $P-A B C D$ 与 $Q-A B C D$ 为两个正四棱雉, 且 $\angle P A Q=90^{\circ}$, 点 $M$ 在线段 $A C$ 上, 且 $C M=3 A M$. 将异面直线 $P M, Q B$ 所成的角记为 $\theta$ ，则 $\cos \theta$ 的最大可能值为 $\qquad$.  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{5}
Find the value of \( \sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{14-6\sqrt{5}}}} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the asymptote of the function \( g(x) = \frac{x^2 + f(x)}{x + 1 + 2f(x)} \), given that \( \lim_{x \to \infty} x^n f(x) = 0 \) for any \( n \in \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{y = x - 1}
Find a value of $R > 0$ such that for all complex numbers $z$ with $|z| > R$, the inequality $|z^{10} + 100z + 9| > 1$ holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the dual group of the multiplicative group of positive real numbers, denoted by $\mathbb{R}_{>0}^*$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{R}}
Evaluate the limit: \[ \lim_{n \rightarrow \infty} \int^{n}_{0} \left(1+\frac{x}{n}\right)^{-n} \log\left(2+ \cos\left(\frac{x}{n}\right)\right)\ \mathrm{d}x. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\log 3}
Find the locus of the centers of circles with a constant radius that move while touching the faces of a trihedral angle, where all plane angles are right angles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(r, r, r)}
When the integer $({\sqrt{3} + 5})^{103} - ({\sqrt{3} - 5})^{103}$ is divided by 9, what is the remainder?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
The number $2.29^{\star \star} N$ is an integer. Its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a perfect fourth power.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Determine the number of points on the curve defined by the equation \(x^3 + x + 1 = y^2\) over the finite field \(\mathbb{F}_{125}\). Given that the curve has 9 points over \(\mathbb{F}_5\), and using Hasse's theorem, the number of points over \(\mathbb{F}_{125}\) must be a multiple of 9. The possible orders are 108, 117, 126, 135, and 144. Find the correct number of points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{108}
Quadrilateral $ABCD$ is inscribed in a circle with the following properties:
- $\angle BAC = 70^{\circ}$
- $\angle ADB = 40^{\circ}$
- $AD = 4$
- $BC = 6$

What is the length of $AC$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Given the cubic equation $2x^3+8x^2-42x-188$, with roots $a, b, c$, evaluate $|a^2b+a^2c+ab^2+b^2c+c^2a+bc^2|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{198}
A finite set of integers is called bad if its elements add up to 2010. A finite set of integers is a Benelux-set if none of its subsets is bad. Determine the smallest integer \( n \) such that the set \( \{502, 503, 504, \ldots, 2009\} \) can be partitioned into \( n \) Benelux-sets. 

(A partition of a set \( S \) into \( n \) subsets is a collection of \( n \) pairwise disjoint subsets of \( S \), the union of which equals \( S \).)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{182}
In triangle ABC, where \( \angle A = 70^\circ \), the angle bisector AE intersects side AC at point D and side BD at point H. Given the ratios \( \frac{AH}{HE} = \frac{3}{1} \) and \( \frac{BH}{HD} = \frac{5}{3} \), find the measure of \( \angle C \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{55^\circ}
A car traveling at a certain constant speed takes 5 seconds longer to travel 1 km than it would take to travel 1 km at a certain speed. The car is traveling at 80 km/hour. What is the speed it is being compared to?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90}
Example 10. Solve the equation

$$
x \int_{0}^{x} y(t) d t=(x+1) \int_{0}^{x} t y(t) d t, \quad x>0
$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{C\frac{1}{x^{3}}e^{-1/x}}
9.5. If there are several piles of stones on the table, it is considered that there are many stones on the table if one can find 50 piles and number them from 1 to 50 such that the first pile has at least one stone, the second pile has at least two stones, ..., the fiftieth pile has at least fifty stones. Suppose initially there are 100 piles, each with 100 stones. Find the largest \( n \leqslant 10000 \) such that after removing any \( n \) stones from the initial piles, there will still be many stones on the table. (When removing stones, a pile does not split into several.) (D. Khramov)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5099}
设a,b,c,d&gt;0，且a+b+c+d=3，求

$\frac { a ^ { 3 } } { b + c + d } + \frac { b ^ { 3 } } { a + c + d } + \frac { c ^ { 3 } } { a + b + d } + \frac { d ^ { 3 } } { a + b + c }$的最小值.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac { 3 } { 4 }$}
Determine the image of the set \( \mathcal{G} \) under the map that takes a function \( g(x) \) in \( \mathcal{G} \) to \( \int_{0}^{2}g(x) \, dx \), where \( \mathcal{G} \) is the subset of real-valued continuous functions on \([0, 2]\) such that \( 0 \leq g(x) \leq 1 \) for all \( x \) in \([0, 2]\) and \( \liminf_{n\rightarrow \infty}\int_{0}^{2}g(x)^{n}dx >0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(0, 2]}
Evaluate the integral: $$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{611\pi}}
A particle is located on the coordinate plane at $(5,0)$. Define a ''move'' for the particle as a counterclockwise rotation of $\frac{\pi}{4}$ radians about the origin followed by a translation of $10$ units in the positive $x$-direction.  Find the particle's position after $150$ moves.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-5 \sqrt{2}, 5 + 5 \sqrt{2})}
Determine the value of \( p \) that minimizes the expression \( S = \frac{p-1}{p^n-1}\sum_{i=1}^n(n-i) p^{i-1} \) for the maximum possible value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A subset of $\left\{1,2,3,\ldots,2017,2018\right\}$ has the property that none of its members are $5$ times another. What is the maximum number of elements that such a subset could have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
1682
}
Find the sum of all roots of the equation \((\sin x + \cos x)^{(1 + \sin 2x)} = 2\) for \(x \in [-2\pi, 4\pi]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3\pi}{4}}
1. Given that $x$ satisfies $\log _{5 x} 2 x=\log _{625 x} 8 x$, then the value of $\log _{2} x$ is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\ln5}{2\ln2-3\ln5}}
An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h$, For every pair of people at the party, they are either friends or enemies. If every MIT student has 16 MIT friends and 8 Harvard friends, and every Harvard student has 7 MIT enemies and 10 Harvard enemies, compute how many pairs of friends there are at the party.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{342}
In the sequence \{a_n\}, a_1 = 0, a_2 = 2, and for any k in the set of positive integers, a_{2k-1}, a_{2k}, a_{2k+1} form an arithmetic sequence, and a_{2k}, a_{2k+1}, a_{2k+2} form a geometric sequence with common ratio q_k. Calculate q_{10}. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{11}{10}}
Determine the fundamental group of the complement of a circle in 3D space, \( \Bbb{R}^3 \setminus S^1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
A company stipulates that a retired employee can receive a pension each year, which is directly proportional to the square root of the number of years they have worked. If they work $a$ more years, their pension increases by $p$ dollars, and if they work $b$ more years ($b \neq a$), their pension increases by $q$ dollars. The annual pension of the employee is (expressed in terms of $a$, $b$, $p$, $q$) ______ dollars.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{a q^2 - b p^2}{2(bp - aq)}}
A number is upwards if its digits in base 10 are nondecreasing when read from left to right. Compute the number of positive integers less than $10^6$ that are both upwards and multiples of 11.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
219
}
Given that \( A^4 = -1.6A^2 - 0.64I \), find \( \lim_{m \to \infty} A^m \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the smallest integer $n$ such that the error in approximating $\pi$ using the formula $\pi = 4\arctan(1/2) + 4\arctan(1/3)$ with $n$ terms of the series expansion is less than $10^{-8}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Determine the minimum number of straight lines required to construct exactly 100 squares.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Determine the order of $25+3017\mathbb{Z}$ in $(\mathbb{Z}/3017\mathbb{Z})^\times$. Note that $3017 = 7 \cdot 431$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1290}
Let \( x < {0.1} \) be a positive real number. Let the foury series be
 \( 4 + {4x} + 4{x}^{2} + 4{x}^{3} + \ldots \) , and let the fourier series be
 \( 4 + {44x} + {444}{x}^{2} + {4444}{x}^{3} + \ldots \) . Suppose that the sum of the fourier
 series is four times the sum of the foury series. Compute \( x \) .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{3}{40}}
Find $ \lim\limits_{x\rightarrow \infty} \left(\int_0^x \sqrt{1+e^{2t}}\ dt-e^x\right)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$ \ln(1 + \sqrt{2}) - \sqrt{2} $}
The director of the school decided to take a photo of the 2008 graduates. He arranged the students in parallel rows, all with the same number of students, but this arrangement was too wide for the camera's field of view. To solve this problem, the director noticed that it was enough to remove one student from each row and place them in a new row. This arrangement did not please the director because the new row had 4 students fewer than the others. He then decided to remove one more student from each row and place them in the new row he created, and found that this way all the rows had the same number of students, and finally took the photo. How many students appeared in the photo?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
Find the first three 7-adic digits of a root of the polynomial \(x^3 - 1\) in \(\mathbb{Z}_7\) that is not equal to 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{246}
There are  $110$  guinea pigs for each of the  $110$  species, arranging as a  $110\times 110$  array. Find the maximum integer  $n$  such that, no matter how the guinea pigs align, we can always find a column or a row of  $110$  guinea pigs containing at least  $n$  different species.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Determine the order of $k$ such that the expression \( \sum_{i=0}^{k} \frac{i}{2n - i} \) converges to a limit independent of $n$ as $n \to \infty$, given that $k = cn$ does not work.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{n}}
Let $p(x)=2+4x+3x^{2}+5x^{3}+3x^{4}+4x^{5}+2x^{6}$. For $k$ with $0<k<5$, define $$I_{k}=\int_{0}^{\infty} \frac{x^{k}}{p(x)} d x$$ For which $k$ is $I_{k}$ smallest?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the number of subgroups of $A = \mathbb{Z}_{360} \oplus \mathbb{Z}_{150} \oplus \mathbb{Z}_{75} \oplus \mathbb{Z}_{3}$ that are isomorphic to $\mathbb{Z}_{5} \oplus \mathbb{Z}_{5}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
For a positive integer $n$, let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$. For example, $f(96) = 1 + 2 + 3 + 6 + 9 = 21$. Find the largest positive integer $n$ such that for all positive integers $k$, there is some positive integer $a$ such that $f^k(a) = n$, where $f^k(a)$ denotes $f$ applied $k$ times to $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
What is the remainder when \(2,468,135,790\) is divided by \(99\)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{54}
在 \(\triangle A B C\) 中, 已知 \(\angle C=90^{\circ}, \angle B=30^{\circ}\), \(A C=2, M\) 是 \(A B\) 的中点，将 \(\triangle A C M\) 沿 \(C M\) 翻折，使 \(A 、 B\) 两点间的距离为 \(2 \sqrt{2}\)。求三棱锥 \(A-B C M\) 的体积 \(V\)。答案的形式为\(\frac{k \sqrt{m}}{n}\)，请给出k+m+n的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Find the minimal polynomial of \(\beta^7\) over \(\mathbb{F}_2\), where \(\beta\) is an element of \(\mathbb{F}_{16}\) with minimal polynomial \(x^4 + x + 1\) over \(\mathbb{F}_2\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{x^4 + x^3 + 1}
The odometer of a family car shows \(15,951\) miles. The driver noticed that this number is palindromic: it reads the same backward as forwards. "Curious," the driver said to himself, "it will be a long time before that happens again." Surprised, he saw his third palindromic odometer reading (not counting \(15,951\)) exactly five hours later. How many miles per hour was the car traveling in those 5 hours (assuming speed was constant)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
The integers  $ 1,2,...,n$  are placed in order so that each value is either strictly bigger than all the preceding values or is strictly smaller than all preceding values. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{n-1}}