inclusionAI/Ring-lite-rl-data huggingface dataset:
The number \( a \) is a root of the equation \( x^{11} + x^{7} + x^{3} = 1 \). For which natural values of \( n \) does the equality \( a^{4} + a^{3} = a^{n} + 1 \) hold?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
Calculate the residue of \( \cot^2(z) \) at each of its poles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Evaluate the limit: $$\lim_{n \to \infty} \left(\prod_{k=1}^{k=n} \prod_{r=0}^{r=k}\frac {n-k+r}{r}\right)^{1/((n)(n+1))}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
While waiting for customers, a watermelon seller sequentially weighed 20 watermelons (with masses of 1 kg, 2 kg, 3 kg, ..., up to 20 kg), balancing a watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on paper the mass of the weights he used. What is the minimum number of different numbers that could appear in his records, given that the mass of each weight is an integer in kilograms?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
## Task A-2.2.

A 2021-digit number is written. Each two-digit number formed by two consecutive digits of this number (without changing the order) is divisible by 17 or 23. The unit digit of this number is 7. What is its first digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Let \( f:[0,1] \rightarrow \mathbb{R} \) be a continuous function such that \( \int_0^1 f(x) \, dx = 0 \). Compute the limit:
\[ \lim _{t\to 0} \frac{1}{t} \int_0^1 (f(x+t)-f(x))x \, dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(1)}
Evaluate the limit: \[ \lim _{x\to \infty }\left(\frac{\left((3x)!\right)}{x^{2x}(x!)^2}\right)^{\frac{1}{x}} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Find the equation of a plane that is invariant under the transformation given by the matrix \( \begin{pmatrix}4&-23&17\\ 11&-43&30\\ 15&-54&37\end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3x - 3y + z = 0}
Given a polynomial \(P(x)\) of degree \(n\) with algebraic coefficients, where each coefficient has a minimal polynomial of degree \(\le D\) over \(\mathbb{Q}\), determine an upper bound on the degree of the minimal polynomials over \(\mathbb{Q}\) for the roots of \(P(x)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n D^{n+1}}
Let $ x$ and $ y$ be positive integers such that $ 7x^5=11y^{13}$. The minimum possible value of $ x$ has a prime factorization $ a^cb^d$. What is $ a +b+c+  d$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
31}
There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Determine the smallest possible value of the total number of intersection points of these circles.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6024}
Example 3 Let $P_{m}=\left\{A_{1}, A_{2}, \cdots, A_{m}\right\}$ be a partition of an $n$-element set $X$ (i.e., $A_{1}, A_{2}, \cdots, A_{m}$ is a partition of $X$). If one of the $A_{i}$ is further divided into two sets, this results in a partition $P_{m+1}$ of $X$, which consists of $m+1$ sets. $P_{m+1}$ is called a refinement of $P_{m}$. If
$$
P_{1}, P_{2}, \cdots, P_{n}
$$

are all partitions of the $n$-element set $X$, and each is a refinement of the previous one (clearly, in this case, $P_{n}$ consists of $n$ sets, while $P_{1}$ consists of only one set, which is $X$), then (7) is called a chain of length $n$. Find the number of chains of length $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{n!(n-1)!}{2^{n-1}}}
Determine whether the number of ordered 10-tuples of positive integers $(a_1, a_2, \ldots, a_{10})$ such that \( \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{10}} = 1 \) is even or odd.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{odd}
Point \(P\) is inside an equilateral \(\triangle ABC\) such that the measures of \(\angle APB, \angle BPC, \angle CPA\) are in the ratio 5:6:7. Determine the ratio of the measures of the angles of the triangle formed by \(PA, PB, PC\) (in increasing order).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2: 3: 4}
Solve the equation for real and positive parameters $a$, $b$, and $c$:
\[ \sqrt{a+bx}+\sqrt{b+cx}+\sqrt{c+ax}=\sqrt{b-ax}+\sqrt{c-bx}+\sqrt{a-cx} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the number of distinct countable subsets that can be formed from the set of real numbers \( \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Determine the number of non-archimedean extensions of the absolute value of $\mathbb{Q}$ for the field obtained by adjoining the $n$th root of a prime number $p$ to $\mathbb{Q}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the number of elements of order 10 in the symmetric group $S_{10}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{514080}
Determine whether the series \( a_n = \frac{1}{\sqrt{n}} \left( 1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \right) \) converges or diverges as \( n \to \infty \). If it converges, find its sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Calculate the expected value \( \mathbb{E}\left[ I \otimes \left( \operatorname{diag}(ZZ^T\mathbf{1}) - ZZ^T\right)\right] \) where \( Z \sim N(0, \sigma^2I) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Let $ABCD$ be a cyclic quadrilateral. The side lengths of $ABCD$ are distinct integers less than $15$ such that $BC \cdot CD = AB \cdot DA$. What is the largest possible value of $BD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\dfrac{425}{2}}}
Given that the graph of the function \( y = f(x) \) is symmetric with respect to the point \( (1,1) \) and the line \( x + y = 0 \), if \( f(x) = \log_{2}(x + 1) \) when \( x \in (0,1) \), find the value of \( f\left(\log_{2} 10\right) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{17}{5}}
Compute \[
\left\lfloor \frac{2017! + 2014!}{2016! + 2015!}\right\rfloor.
\] (Note that $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2016}
Find the determinant of the principal matrix solution \( \Pi(t,t_0) \) for the system of differential equations \( \dot{x} = A(t)x \) in \( \mathbb{R}^2 \), where the matrix \( A(t) \in GL(\mathbb{R}^2) \) has eigenvalues \( \alpha_1(t) = t \) and \( \alpha_2(t) = 0 \). Calculate \( \det(\Pi(1,0)) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{e}}
What is the probability that in a sequence of coin flips, there will be a point where the number of heads is at least twice the number of tails?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{5 - \sqrt{5}}{4}}
Determine the number of solutions to the equation \(x_1 + x_2 + \cdots + x_{10} = 15\) where \(x_1 \geq 1\) and \(0 \leq x_i \leq 9\) for \(i = 1\) to \(10\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{808753}
11. (5 points) Arrange the natural numbers $1,2,3,4$, in ascending order without any gaps, to get: 1234567891011121314. In this sequence of digits, when the first set of 5 consecutive even digits appears, the position of the first (even) digit from the left is the how many-th digit?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{490}
Evaluate the sum \( \sum_{n=1}^{\infty} \frac{\ln n}{n(n- \ln n)} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given the random variables \(X_1, \ldots, X_n\) that are independently and identically distributed (i.i.d.) according to the probability density function (PDF) \(f(x) = \frac{1}{\sigma^{3} \sqrt{2\pi}}x^{2}e^{-\frac{x^2}{2\sigma^2}}\), where \(\sigma > 0\), find the distribution of the sum \(\sum_{i=1}^{n}\frac{X_{i}^{2}}{\sigma^{2}}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\chi^2(3n)}
Determine the values of \( \beta \) for which a transcritical bifurcation occurs in the equation \( x_{n+1} = \beta x_{n} (2 - x_{n}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
How many onto functions exist from a set \( A \) with 6 elements to a set \( B \) with 6 elements, such that the function maps the element 1 in \( A \) to the element 2 in \( B \), and no element in \( A \) maps to itself in \( B \)?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{53}
Solve the equation \(2^x \cdot 3^y = 1 + 5^z\) in positive integers \(x, y,\) and \(z\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(1, 1, 1)}
Determine the asymptotic behavior of \( \mu(\mathcal{C} \cap \{A: \|A\|_1 \le r\}) \) as \( r \to \infty \), where \( \|A\|_1 \) denotes the 1-norm of matrix \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(r^{n^2})}
Compute the Picard group of the product space \( \mathbb{P}^n \times \mathbb{C}^* \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Find the constant $c$ such that the inequality \( \left\|\begin{pmatrix} x^2 - y^2\\2xy \end{pmatrix}\right\| \leq |c|\left\|\begin{pmatrix} x\\y \end{pmatrix}\right\| \) holds for all $x$ and $y$ in polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the cardinality of the set \( A = \bigcap_{k=1}^{\infty} A_k \), where \( A_k = \{ a \in \mathbb{N}^{\mathbb{N}} \mid \sum_{i=100k}^{100k+99} a_i = 200 \} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
A square of side length $1$ and a circle of radius $\frac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["\\frac{2\\pi}{9} - \\frac{\\sqrt{3}}{3}"]}
A right regular hexagonal prism has bases $ABCDEF$ and $A'B'C'D'E'F'$ with edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, and $FF'$, each perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. A plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained within the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{323}
Given that $x, y, z$ are positive real numbers satisfying $x + 2y + 3z = 10$, find the maximum value of the expression $x^2y^2z^6$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{256}
实数 $x, y, z, w$ 满足 $x+y+z+w=1$ ，则 $M=x w+2 y w+3 x y+3 z w+4 x z+5 y z$ 的最大值是 $\qquad$.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}
Given an increasing sequence $\{a_{n}\}$ where all terms are positive integers, the sum of the first $n$ terms is $S_{n}$. If $a_{1}=3$ and $S_{n}=2023$, calculate the value of $a_{n}$ when $n$ takes its maximum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{73}
Compute the index of the unit circle with respect to the vector field given by the system of differential equations:
\[\begin {cases}
\frac{dx}{dt} = 2x^2-1\\
\frac{dy}{dt} = 2xy
\end{cases}\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Evaluate the flux of the vector field \( \vec{F} = \langle x, y, 5 \rangle \) through the surface \( S \), which is the region enclosed by \( x^2 + z^2 = 1 \), \( y = 0 \), and \( x + y = 2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi}
The price of an item is decreased by 20%. To bring it back to its original value and then increase it by an additional 10%, the price after restoration must be increased by what percentage.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37.5\%}
Let 3 < x < 4, 4 < a < x, 6 < y < 8, and 8 < b < y. Given that a^2 + b^2 > x^2 + y^2, find the greatest possible positive integer difference of x and y.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
There are \( n \) people, and it is known that any two of them can call each other at most once. Among them, the total number of calls between any group of \( n-2 \) people is the same and is \( 3^{k} \) times, where \( k \) is a natural number. Find all possible values of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
$10 \cdot 52 \quad 1990-1980+1970-1960+\cdots-20+10$ equals:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000}
A school organizes a social practice activity during the summer vacation and needs to allocate 8 grade 10 students to company A and company B evenly. Among these students, two students with excellent English scores cannot be allocated to the same company, and neither can the three students with strong computer skills. How many different allocation schemes are there? (Answer with a number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
If the fractional equation $\frac{1-x}{x-2}=\frac{a}{2-x}-2$ has a positive root, then $a=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In a positive term geometric sequence ${a_n}$, ${a_5 a_6 =81}$, find the value of ${\log_{3}{a_1} + \log_{3}{a_5} +...+\log_{3}{a_{10}}}$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Principal Skinner is thinking of two integers $m$ and $n$ and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of $mn + 13m + 13n - m^2 - n^2$. From the value of this expression alone, he miraculously determines both $m$ and $n$. What is the value of the above expression?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{169}
Find the integer $n,$ $-180 < n < 180,$ such that $\tan n^\circ = \tan 345^\circ.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-15}
Let $A$ be the set $\{1,2,\ldots,n\}$, where $n \geq 2$. Find the least number $n$ for which there exist permutations $\alpha$, $\beta$, $\gamma$, $\delta$ of the set $A$ such that:
\[ \sum_{i=1}^n \alpha(i) \beta(i) = \frac{19}{10} \sum_{i=1}^n \gamma(i)\delta(i) . \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{28}
In the 2011 Shanghai Spring College Entrance Examination, there were 8 universities recruiting students. If exactly 3 students were admitted by exactly 2 of these universities, then the number of ways this could happen is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{168}
Given a topological space \( E \) with the topology \( \theta = \{ G \subseteq E \mid \text{card}(E \setminus G) < \infty \} \cup \{ \emptyset \} \), where \( \text{card}(E) = \infty \), find the interior \( \mathring{A} \) of a set \( A \) such that \( \text{card}(A) = \infty \) and \( \text{card}(E \setminus A) = \infty \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\emptyset}
Determine the nth derivative of the function \( f(x) = \frac{1}{x}e^x - \frac{1}{x^2}e^x + \frac{1}{x^2} \) at \( x = 0 \). Express your answer in the form \( \frac{1}{a(n)} \), where \( a(n) \) is an expression in terms of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{n + 2}}
Alex writes down some distinct integers on a blackboard. For each pair of integers, he writes the positive difference of those on a piece of paper. Find the sum of all $n \leq 2022$ such that it is possible for the numbers on the paper to contain only the positive integers between $1$ and $n$, inclusive, exactly once.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Evaluate the infinite series: \[ \sum_{n=0}^{\infty} \frac{16n^2 + 20n + 7}{(4n+2)!} \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{e + \sin 1}
Consider the sequence given by $a_0 = 3$ and such that for $i \ge 1$, we have $ai = 2^{a_{i-1}} + 1$. Let $m$ be the smallest integer such that $a^3_3$ divides $a_m$. Let $m'$ the smallest integer such that $a^3_m$ divides $a_{m'}$ . Find the value of $m'$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{35}
How many subsets of the set $\{1, 2, \ldots, 11\}$ have median 6?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{351}
Find the maximum value of \(x^2 y + y^2 z + z^2 x\) for real numbers \(x, y, z\) with the conditions that their sum is zero and the sum of their squares is six.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
One eighth of three fifths of four sevenths of five eleventh of a number minus one ninth of two thirds of three fourths of five eighths of that same number is 30. What will be 75% of that number?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1476}
Given $A = \{1, 2, 3, 5, 8, 13, 21, 34, 55\}$, find how many of the numbers between $3$ and $89$ cannot be written as the sum of two elements of set $A$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{51}
Given the functions \(f(t)=(\int_0^{\sqrt{t}}e^{-x^2} dx)^2\) and \(g(t)=\int_0^1\frac{e^{-t(1+x^2)}}{1+x^2} dx\) for \(t>0\), find \(f'(t)+g'(t)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A point is chosen randomly from within a circular region with radius $r$. A related concentric circle with radius $\sqrt{r}$ contains points that are closer to the center than to the boundary. Calculate the probability that a randomly chosen point lies closer to the center than to the boundary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4}}
Given the line $y=kx-k$ and the parabola $y^{2}=4x$ intersect at points $A$ and $B$, and the distance between $A$ and $B$ is $4$, find the distance from the midpoint of chord $AB$ to the $y$-axis. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If x, y, and z are positive integers and x = 4, y = 7, and z is some value, then the least possible value of x - y - z is 17. What is the value of z?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14}
Several students are seated at a large circular table.  They pass around a bag containing $100$ pieces of candy.  Each person receives the bag, takes one piece of candy and then passes the bag to the next person.  If Chris takes the first and last piece of candy, then the number of students at the table could be
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
For which dimensions \(d \geq 1\) does the expectation \(\mathrm{E}\left(\xi^{\top} B \xi\right)^{-1}\) exist, where \(\xi\) is a Gaussian vector in \(\mathbb{R}^{d}\) with a unit covariance matrix, and \(B\) is a symmetric positive definite matrix?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d \geq 3}
Given that $\sin A + \cos A = \frac{\sqrt{5}}{5}$, find the value of $\sin A - \cos A$. Express your answer as a fraction with a square root if necessary.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{5}}{5}}
Suppose that $X_1, X_2, \dots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S = \sum_{i=1}^k X_i/2^i$, where $k$ is the least positive integer such that $X_k < X_{k+1}$, or $k = \infty$ if there is no such integer. Find the expected value of $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2e^{1/2}-3}
Fox jeans regularly sell for $15 a pair, and Pony jeans regularly sell for $18 a pair. Additionally, there are jackets available that regularly cost $25 each. During a clearance sale, these regular unit prices are discounted at different rates so that a total of $15 is saved by purchasing 7 items - 3 pairs of Fox jeans, 2 pairs of Pony jeans, and 2 jackets. If the sum of the three discount rates for the jeans and the jackets is 35 percent, given that the jacket's discount rate is 10 percent, what is the discount rate on Pony jeans?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13.89\%}
Given the equation about $x$, $x^{2}e^{x} + t - a = 0$, where $x \in [-1, 1]$, if for any $t \in [1, 3]$, the equation always has a unique real solution, then the range of values for the real number $a$ is __________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left( \frac{1}{e} + 3, e + 1 \right]}
Given the equation of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $a>b>0$, and a point $M$ on the ellipse with a perpendicular line drawn to the $x$-axis, the foot of the perpendicular being $P$, if $\frac{|MP|^2}{|A_1P| \cdot |A_2P|}=\frac{1}{3}$, calculate the eccentricity of the ellipse. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{6}}{3}}
Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers. Please provide the value of a-b. Here $[x]$ is defined to be the integer such that $[x] \leqq x<[x]+1$ and $\log x$ is the logarithm of $x$ to base $e$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given a real number $x$ and the matrix $M_x = \begin{pmatrix} 1 & x+1 & 0 & 0 \\ 0 & x & x-1 & 0 \\ 0 & x-1 & x & 0 \\ 1 & 0 & x+1 & x \end{pmatrix}$, find the value(s) of $x$ for which $M_x$ is diagonalizable. The eigenvalues of $M_x$ are $\lambda_1 = 1$, $\lambda_2 = x$, and $\lambda_3 = 2x-1$. Determine the values of $x$ that make the matrix diagonalizable.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Arrange the numbers $1, 2, \ldots, 1962$ in the sequence $a_{1}, a_{2}, \ldots, a_{1962}$ such that the sum $\left|a_{1} - a_{2}\right| + \left|a_{2} - a_{3}\right| + \ldots + \left|a_{1961} - a_{1962}\right| + \left|a_{1962} - a_{1}\right|$ is maximized.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1924722}
Determine the maximum number of elements in a sigma-algebra generated by a collection of $N$ distinct subsets of a set $X$, given that the intersection of any two subsets in the collection is non-empty.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{2^N}}