inclusionAI/Ring-lite-rl-data huggingface dataset:
On the sides $\mathrm{AB}$ and $\mathrm{AC}$ of a right triangle $\mathrm{ABC}\left(\angle B C A=90^{\circ}\right)$, right triangles $\mathrm{ABT}$ and $\mathrm{ACK}$ are constructed externally such that $\angle A T B=\angle A K C=90^{\circ}$ and $\angle A B T=\angle A C K=60^{\circ}$. A point $\mathrm{M}$ is chosen on the side $\mathrm{BC}$ such that $\mathrm{BM}=\mathrm{MC}$. Determine the measure of angle $\angle KMT$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120^\circ}
The number $125$ can be expressed as a sum of pairwise coprime integers, each greater than $1$. Determine the maximum number of terms possible in such a sum.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Determine the order of the quotient group $G/G'$ for the group $G$ defined by the presentation:
\[ G = \langle a, b \mid a^{2^{n-1}} = 1, b^2 = a^{2^{n-2}}, b^{-1}ab = a^{-1} \rangle. \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Five people are arranged in a row, where person A is not next to person B, and person A is also not next to person C. Find the number of different arrangements. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
If three different numbers are selected from 2, 3, 4, 5, 6 to be $a$, $b$, $c$ such that $N = abc + ab + bc + a - b - c$ reaches its maximum value, then this maximum value is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{167}
Given a circle O: $x^2+y^2=1$ and a point $P(-1, \sqrt{3})$, where A and B are two moving points on circle O, find the maximum value of $\angle APB$. Express your answer in radians, using π to represent pi.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{3}}
The function $f(x)$ is defined on the reals such that \( f\left(\frac{1-4x}{4-x}\right) = 4-xf(x) \) for all \( x \neq 4 \). There exist two distinct real numbers \( a, b \neq 4 \) such that \( f(a) = f(b) = \frac{5}{2} \). The sum \( a+b \) can be represented as \( \frac{p}{q} \) where \( p, q \) are relatively prime positive integers. Find \( 10p + q \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{180}
How many ways are there to express $1,000,000$ as a product of exactly three integers greater than $1$? For the purpose of this problem, $abc$ is not considered different from $bac$, etc.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{114}
Find the infimum of the set $\{\lvert\lvert f\rvert\rvert_{\infty} : f \in F\}$, where $F = \{ f \in C[0,1]\: f(0)=1, \int_0^1 (1-t)f(t)dt =2 \}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Evaluate the triple integral $$\iiint (x+y+z) dx\,dy\,dz$$ over the region $$D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2+y^2+z^2 \leq 9 \wedge z > y\right \}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Consider a standard normal random variable $N \sim \mathcal{N}(0,1)$. Determine if there exists a dependent random variable $D$ such that the random variable $M = N + D$ is still a standard normal distribution. If such a $D$ exists, construct it and find the variance of $D$. Provide your answer in the form of the variance of $D$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $a, b, c$ be roots of the polynomial $x^3-3x^2-4x+5$. Compute the remainder when $[\frac{a^4+b^4}{a+b} + \frac{b^4+c^4}{b+c} + \frac{c^4+a^4}{c+a}]$ is divided by $7$, where $[x]$ denotes floor function.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
把半径为 1 的四个小球叠成两层放在桌面上：下层三个，上层一个，两两相切，求上层小球最高点离桌面的高度与$\frac{2 \sqrt{6}}{3}$的差。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given a $100 \times 100$ grid where the cells are colored in black and white such that every column has an equal number of black cells, while each row has a different number of black cells. What is the maximum possible number of adjacent pairs of differently colored cells?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14751}
Let \( x, y, z \) be positive real numbers. Define \( M = \max \left\{ xy + \frac{2}{z}, z + \frac{2}{y}, y + z + \frac{1}{x} \right\} \). Find the minimum value of \( M \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
设 g(x)=\displaystyle\int_x^{\pi}\df{(\sin t)^5}{t}\ \dd t，记 S=\displaystyle\int_0^{\pi}g(x)\ \dd x，求 [100S].
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{106}
Evaluate the product  $$ \prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}. $$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the smallest positive integer $n$ whose prime factors are all greater than 18, and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1843}
In a first-year class of a certain school, there are 1200 students. There are two extracurricular activities for students to choose from, and each student is required to choose at least one to participate in. A statistical survey shows that the number of students choosing one of the activities accounts for 60% to 65% of the total, while the number of students choosing the other activity accounts for 50% to 55% of the total. Calculate the range of possible values for the number of students who choose to participate in both activities. Express your answer as a range of whole numbers, representing the minimum and maximum number of students.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[120, 240]}
In a parallelogram $ABCD$, let $M$ be the point on the $BC$ side such that $MC = 2BM$ and let $N$ be the point on the $CD$ side such that $NC = 2DN$. If the distance from point $B$ to the line $AM$ is $3$, calculate the distance from point $N$ to the line $AM$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A $5\times 5$ grid is to be filled with 12 red tiles and 13 green tiles. Determine the number of distinct ways to fill the grid, considering rotations and reflections of the grid as equivalent.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{652048}
If $\left( r + \frac{1}{r} \right)^2 = 5,$ then find $r^3 + \frac{1}{r^3}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{5}}
How many distinct minimal polynomials of degree 3 exist for elements of a finite field extension $L/K$ of degree 3?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\((q^3 - q)/3\)}
Determine the number of roots of the polynomial equation \( f(z) = z^4 + 8z^3 + 3z^2 + 8z + 3 = 0 \) that lie in the right half of the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Determine the number of solutions to the equation \(6\ln(x^2+1)-e^x = 0\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. After that, at what time do the hour hand and the minute hand form a 90-degree angle for the 6th time? (12-hour format)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3:00}
There are 5555 children, numbered 1 to 5555, sitting around a circle in order. Each child has an integer in hand: the child numbered 1 has the integer 1, the child numbered 12 has the integer 21, the child numbered 123 has the integer 321, and the child numbered 1234 has the integer 4321. It is known that the sum of the integers held by any 2005 consecutive children is equal to 2005. What is the integer held by the child numbered 5555?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-4659}
Define a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 10\}$, including the empty set, are spacy?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60}
How many 12-digit positive integers have all digits either 1 or 2, and have numbers ending in 12, but have no two consecutive 1's?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{89}
将一枚骰子连续投掷五次，则事件 "五次出现的点数既不全相同，也不两两互异，且从第二次起每一次的点数都不小于前一次的点数"的概率为 $\qquad$.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{167}
In trapezoid \(ABCD\), a point \(X\) is taken on the base \(AD\) such that segments \(XB\) and \(XC\) divide the trapezoid into three similar, yet pairwise unequal, non-isosceles triangles. The side \(AB\) has a length of 6. Find \(AX \cdot DX\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{36}
8. Given a six-digit decimal number composed of six positive integers, the digit in the units place is a multiple of 4, the digits in the tens and hundreds places are multiples of 3, and the sum of the digits of the six-digit number is 21. Then the number of six-digit numbers that satisfy the above conditions is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{126}
If $I_n=\int_0^{\frac {\pi}{2}} \frac {\sin ((2n+1)x)dx}{\sin x}$, then evaluate $$\lim_{n\to \infty} I_n$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{2}\)}
## Task Condition

Find the derivative of the specified order.

$y=\frac{\log _{3} x}{x^{2}}, y^{IV}=?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{-154+120\lnx}{x^{6}\cdot\ln3}}
Given the function $f(x) = (m^2 - m - 1)x^{-5m-3}$ is a power function, and it is increasing on the interval $(0, +\infty)$, determine the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
A casino table pays 3:2 if you get a blackjack with your first 2 cards. If you scored a blackjack and had an original bet of $40, how much were you paid?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$100}
Calculate the determinant of the following matrix A:

$$\begin{pmatrix}
5 & 3 & 3 & . & . & . & 3 \\
3 & 5 & 3 & . & . & . & 3 \\
3 & 3 & 5 & . & . & . & 3 \\
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
3 & 3 & 3 & . & . & . & 5
\end{pmatrix}$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1472}
If the function $y=x^2+2x+a^2-1$ reaches its maximum value of 16 on the interval $[1, 2]$, find the value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pm 3}
Each of the spots in a  $8\times 8$  chessboard is occupied by either a black or white “horse”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse?**Remark:** A black horse could attack another black horse.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a tetrahedron \(ABCD\), the sum of the angles \(BAC\) and \(BAD\) is \(180^{\circ}\). \(AK\) is the angle bisector of \(\angle CAD\). Find the measure of \(\angle BAK\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{90^\circ}
Given a regular tetrahedron $ABCD$ with edge length 1, point $M$ is the midpoint of $AC$, and point $P$ is on line segment $DM$. Find the minimum value of $AP + BP$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{1+\frac{\sqrt{6}}{3}}}
For a geometric sequence $\{a_n\}$, the first term is $\frac{3}{2}$, and the common ratio is $-\frac{1}{2}$. The sum of the first $n$ terms is $S_n$. When $n \in N^*$, the ratio of the maximum and minimum values of $S_n - \frac{1}{S_n}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{10}{7}}
There are 50 students in the class. Half of them play basketball, one-third play volleyball, one-fifth play soccer, and one-eighth play badminton. One-tenth of the students play both basketball and volleyball, one-twelfth play both basketball and soccer, one-sixteenth play both basketball and badminton, and one-eighth play both volleyball and soccer. Additionally, one-tenth play both volleyball and badminton, and one-twentieth play both soccer and badminton. Finally, 1/25 of the students play all four sports: basketball, volleyball, soccer, and badminton. How many students in this class do not engage in any of these sports?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{81}
Given that $a, b \in \mathbb{R}$ and $c \in [0, 2\pi)$, if $2\sin\left(3x - \frac{\pi}{3}\right) = a\sin(bx + c)$ holds for any real number $x$, find the number of ordered triplets of real numbers $(a, b, c)$ that satisfy the condition.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
若函数$f\left( x \right)$满足：对任意实数$x$，有$f\left( 2-x \right)+f\left( x \right)=0$且$f\left( x+2 \right)+f\left( x \right)=0$，当$x\in $[0,1]时，$f\left( x \right)=-{{\left( x-1 \right)}^{2}}$,则$~x\in $[2017,2018]时,$~f\left( x \right)=$______________________________．
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{${{\left( x-2017 \right)}^{2}}$}
Find an eight-digit palindrome that is a multiple of three, composed of the digits 0 and 1, given that all its prime divisors only use the digits 1, 3, and %. (Palindromes read the same forwards and backwards, for example, 11011).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10111101}
For a rational number $r$, its *period* is the length of the smallest repeating block in its decimal expansion. For example, the number $r = 0.123123123\ldots$ has period $3$. If $S$ denotes the set of all rational numbers of the form $r = \overline{abcdefgh}$ having period $8$, find the sum of all elements in $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49995000}
Given three non-collinear points \( A, B, C \) on a plane, construct a line \( l \) such that the ratio of the distances from points \( A, B, C \) to line \( l \) is \( 1: 1: 2 \) or \( 1: 2: 1 \) or \( 2: 1: 1 \). Calculate the total number of such lines. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Sidney follows a pattern in doing jumping jacks: on Monday, he does 20, on Tuesday he does 36, and he continues to add an increasing even number of jumping jacks each day for the next five days. Brooke does four times as many jumping jacks as Sidney each day. How many jumping jacks did Brooke do during the entire eight-day period?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2880}
Find the minimum sample size for which the precision of the estimate of the population mean $a$ based on the sample mean with a confidence level of 0.975 is $\delta=0.3$, given that the standard deviation $\sigma=1.2$ of the normally distributed population is known.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{62}
Let $V$ be a vector space with an inner product $(\cdot,\cdot)$. Consider the function $f:V\rightarrow \mathbb{R}$ defined by $f(u) = (u,u)$. Determine the Fréchet derivative of $f$ at a point $u \in V$, denoted as $f'(u)$. Provide your answer in terms of the inner product.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2(u, \cdot)}
How many natural number \( n \) less than 2015 that is divisible by \( \lfloor \sqrt[3]{n}\rfloor \) ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
266
}
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A \cup B = S$, $A \cap B = \emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the number of linear partitions of $S$. For each positive integer $n$, find the maximum of $L_S$ over all sets $S$ of $n$ points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\binom{n}{2} + 1}
If the function $f(x) = a|2x - 4|$ ($a > 0$, $a \neq 1$) and $f(1) = 9$, then the interval of monotonic decrease for $f(x)$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\infty, 2]}
Rectangle $ABCD$ has $AB = 6$ and $BC = 3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$75$}
a group of people participate in some curriculum , 25 of them practice yoga , 15 study cooking , 8 study weaving , 2 of them study cooking only , 7 of them study both the cooking and yoga , 3 of them participate all curriculums . how many people study both cooking and weaving ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
A right circular cone ($S-ABC$) has its apex $S$ perpendicular to the plane $ABC$, with $SA = 3$. The base triangle $△ABC$ is an equilateral triangle with side length $2$. Calculate the surface area of the circumscribed sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{43\pi}{3}}
Given the function f(x) = ln((x^2 + 1)) / (x + 4), find the x-value such that the functions y = f(3-x) and y = f(3+x) are symmetric about a line x = a. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
若 $k$ 是一个正整数,且 $2^k$ 整除 $C_{4006}^0+C_{4006}^2 3^1+\cdots+C_{4006}^{2 i} 3^i+\cdots+C_{4006}^{4006} 3^{2003}$, 则 $k$ 的最大值为 $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2004}
Determine all integer values of \( n \) such that \( 19 \mid 2^{3n+4} + 3^{2n+1} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n \equiv 0 \pmod{18}}
Determine the greatest common divisor of the numbers:
\[ 5^5 - 5, \ 7^7 - 7, \ 9^9 - 9, \ \ldots, \ 2017^{2017} - 2017 \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}
在平面直角坐标系 xOy 中，\Gamma_1 是以 (2,1) 为圆心的单位圆，\Gamma_2 是以 (10,11) 为圆心的单位圆. 过原点 O 作一条直线 l，使得 l 与 \Gamma_1,\Gamma_2 各有两个交点，将 \Gamma_1, \Gamma_2 共分成四段圆弧，且这四段圆弧中有两段等长. 所有满足条件的直线 l 的斜率之和为__________.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{87}
The sides of a parallelogram are \( a \) and \( b \). What are the lengths of the diagonals of the quadrilateral formed by the angle bisectors of the parallelogram?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{|a - b|}
Given that \((B_2 - B_1) \sim N(0, 3)\), find \(\mathbb{E}[B_1 | B_2]\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{B_2}{2}}
Suppose $x$, $y$, and $z$ are nonzero complex numbers such that $(x+y+z)(x^2+y^2+z^2)=x^3+y^3+z^3$. Compute $$(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right).$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
The cars Niva and Toyota are driving on a ring track of a test site, a quarter of which is on a dirt road and the remaining part is on asphalt. The speed of the Niva on the dirt road is 80 km/h, and on asphalt it is 90 km/h. The speed of the Toyota on the dirt road is 40 km/h, and on asphalt it is 120 km/h. The cars start simultaneously at the beginning of the dirt road section and first drive through this dirt part. On which lap will one of the cars overtake the other for the first time?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given that \(15^{-1} \equiv 31 \pmod{53}\), find \(38^{-1} \pmod{53}\), as a residue modulo 53.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Find the smallest positive integer that can be expressed as the sum of the cubes of three positive integers in two different ways.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{251}
Find the square root of 4. Express your answer in the form ± followed by the absolute value of the root.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{±2}
Given a cube $A B C D A 1 B 1 C 1 D 1$. A sphere touches the edges $A D, D D 1, C D$ and the line $B C 1$. Find the radius of the sphere, if the edge of the cube is 1.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\sqrt{2}-\sqrt{5}}
Given linear operators $X$ and $P$ on a complex vector space $V$, suppose the commutator $[X,P] = ciI$ for some real, positive constant $c$. What can be concluded about the dimension of $V$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Point \((x,y)\) is randomly picked from the rectangular region with vertices at \((0,0), (3000,0), (3000,3001),\) and \((0,3001)\). What is the probability that \(x > 3y\)? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1500}{9003}}
The cost of 60 copies of the first volume and 75 copies of the second volume is 2700 rubles. In reality, the total payment for all these books was only 2370 rubles because a discount was applied: 15% off the first volume and 10% off the second volume. Find the original price of these books.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Calculate the probability that a complete bipartite random graph $G(n,n,p)$ is disconnected when $np=\ln n + w(n)$ where $w(n)\rightarrow -\infty$ when $n\rightarrow\infty$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Given three vectors $v_1, v_2, v_3 \in \mathbb{R}^2$, if the minimum eigenvalue of $v_i v_i^T + v_j v_j^T$ is less than or equal to 1 for all distinct $i, j \in \{1, 2, 3\}$, determine an upper bound for the minimum eigenvalue of $v_1 v_1^T + v_2 v_2^T + v_3 v_3^T$ under the constraint $|z_1^2 + z_2^2 + z_3^2| = 0$, where $z_j = \alpha_j + i \beta_j$ and $v_j = [\alpha_j, \beta_j]^T$. Provide your answer as an expression or a number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Determine the convergence interval for the series $\sum_{n=1}^{\infty} \frac{(n+1) \sqrt{n+1}}{4^{n}}(x-1)^{n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3 < x < 5}
函数 f(x)=\df{x+1}{x}+\df{x+2}{x+1}+\cdots+\df{x+2023}{x+2022} 的对称中心为 (a,b)，则 2a+b=__________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Using Cauchy's integral formula, determine a tighter upper bound for \(|f^{(4)}(0)|\) for all functions \(f\) in the family \(\mathscr{F}\) as defined in the original context.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{24}