Context:
已知圆$x ^ { 2 } + y ^ { 2 } = k ^ { 2 }$至少覆盖函数$f ( x ) = \sqrt { 3 } \sin \frac { \pi x } { k }$的一个最大值点与一个最小值点.求实数k的取值范围.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The hypotenuse of a right triangle, where the legs are consecutive whole numbers, is 53 units long. What is the sum of the lengths of the two legs?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Arturo has an equal number of $\$5$ bills, of $\$10$ bills, and of $\$20$ bills. The total value of these bills is $\$700$. How many $\$5$ bills does Arturo have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Into each row of a \( 9 \times 9 \) grid, Nigel writes the digits \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \) in order, starting at one of the digits and returning to 1 after 9: for example, one row might contain \( 7, 8, 9, 1, 2, 3, 4, 5, 6 \). The grid is gorgeous if each nine-digit number read along a row or column or along the diagonal from the top-left corner to the bottom-right corner or the diagonal from the bottom-left corner to the top-right corner is divisible by 9. How many of the \( 9^{9} \) possible grids are gorgeous?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions:
- She rolled a $6$ exactly three times.
- The product of her first three rolls is the same as the product of her last three rolls.

How many distinct sequences of six rolls could Kara have rolled?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_1,a_2,\cdots,a_{22}\in [1,2],$ find the maximum value of
$$
\dfrac{\sum\limits_{i=1}^{22}a_ia_{i+1}}{\left( \sum\limits_{i=1}^{22}a_i\right) ^2}
$$
where $a_{23}=a_1.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the approximate ratio of the three cases drawn, DD, Dd, dd, is 1:2:1, calculate the probability of drawing dd when two students who have drawn cards are selected, and one card is drawn from each of these two students.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many integer pairs $(x,y)$ satisfies $x^2+y^2=9999(x-y)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four small panda bears and five bigger panda bears eat a certain amount and 40 pounds of fresh bamboo sprouts every day, respectively. The pandas eat 2100 pounds of bamboo in a week. How many pounds of bamboo do the small pandas eat each day?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Frank and Joe are playing ping pong. For each game, there is a $30 \%$ chance that Frank wins and a $70 \%$ chance Joe wins. During a match, they play games until someone wins a total of 21 games. What is the expected value of number of games played per match?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define  $m(n)$  to be the greatest proper natural divisor of  $n\in \mathbb{N}$ . Find all  $n \in \mathbb{N} $  such that  $n+m(n) $  is a power of  $10$ .

*N. Agakhanov*
Please reason step by step, and put your final answer within \boxed{}.


Context:
Felix picks four points uniformly at random inside a unit circle  $\mathcal{C}$ . He then draws the four possible triangles which can be formed using these points as vertices. Finally, he randomly chooses of the six possible pairs of the triangles he just drew.

What is the probability that the center of the circle  $\mathcal{C}$  is contained in the union of the interiors of the two triangles that Felix chose?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all values of the parameter \(a\) for which there exists a number \(b\) such that the system

$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 a(a-x-y)=64 \\
y=8 \sin (x-2 b)-6 \cos (x-2 b)
\end{array}\right.
$$

has at least one solution \((x, y)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers $1, 2, 3, 4, 5$ are placed in a circle. We refer to a placement method as a "Central Ring" placement method if no number $n$ in the range $1$ to $15$ can be formed by adding some adjacent numbers on the circle. If two placement methods can be converted into each other by rotation or reflection, we consider them the same. How many distinct "Central Ring" placement methods are there in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( n \) is a positive integer and \( n \geq 4 \). Find the largest positive integer \( k \) such that there exists a triangle with side lengths all being positive integers not greater than \( n \), and the difference between any two sides (larger minus smaller) is not less than \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest number of towns that can meet the following criteria: Each pair is directly linked by just one of air, bus, or train. At least one pair is linked by air, at least one pair by bus, and at least one pair by train. No town has an air link, a bus link, and a train link. No three towns, A, B, C, are such that the links between AB, AC, and BC are all air, all bus, or all train.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A foreman noticed an inspector checking a $3$"-hole with a $2$"-plug and a $1$"-plug and suggested that two more gauges be inserted to be sure that the fit was snug. If the new gauges are alike, then the diameter, $d$, of each, to the nearest hundredth of an inch, is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the interval of convergence for the series \(\sum_{n=1}^\infty \frac{(-1)^n n^3 x^n}{3^n \sqrt{n^5}}.\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions under which the set of integer linear combinations of the vectors \(\begin{pmatrix} a \\ b \end{pmatrix}\) and \(\begin{pmatrix} c \\ d \end{pmatrix}\) spans \(\mathbb{Z}^2\). Exclude the case where one vector is a multiple of the other.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the Cauchy problem \[ y' = \frac{4\cos^2(x) - 2}{\sin(y)}, \quad y(0) = \frac{\pi}{2}. \] What is the domain of the solution?
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. There are 167 different cards with numbers $1, 3, 11, 3^{2}, 11^{2}, \ldots, 3^{83}, 11^{83}$ (each card has exactly one number, and each number appears exactly once). In how many ways can 3 cards be chosen so that the product of the numbers on the chosen cards is a square of an integer divisible by 33?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The greatest common divisor of 30 and some number between 70 and 90 is 6. What is the number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a number $n > 1$, you repeatedly replace a number $t$ with $t - \sqrt{t}$. Determine asymptotically, in terms of $n$, how many times this operation must be performed until the number becomes less than 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the signature of the bilinear form represented by the matrix \( M = \begin{pmatrix} 1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 1 \end{pmatrix} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an equilateral triangle, consider coloring each of its three edges with one of $k \geq 1$ colors. Using Burnside's lemma, determine a formula for the number of distinct colorings modulo the symmetry group $D_6$, the dihedral group of six elements acting on the triangle. The set of all colored triangles is denoted as $X$ with $|X|=k^3$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sealed bottle, constructed by attaching a cylinder of radius 1 cm to a cylinder of radius 3 cm, contains water. When the bottle is right-side up, the height of the water inside is 20 cm. When the bottle is upside-down, the height of the liquid is 28 cm. Calculate the total height of the bottle in centimeters.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum of the square of the binomial coefficients: $C_2^2+C_3^2+C_4^2+…+C_{11}^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose that \( f(x) \) and \( g(x) \) are functions which satisfy \( f(g(x)) = x^3 \) and \( g(f(x)) = x^4 \) for all \( x \ge 1 \). If \( g(81) = 81 \), compute \( [g(3)]^4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a symmetric random walk on the integers \( \mathbb{Z} \) where \( X_i \) are independent random variables such that \( P(X_i = 1) = P(X_i = -1) = 0.5 \) for all integers \( i \) with \( 1 \leq i \leq n \). Define \( S_n = X_1 + X_2 + \cdots + X_n \). Now consider \( 2^n \) independent copies of \( S_n \) and let \( Z_n \) be their maximum. Determine \( \lim_{n \to \infty} \frac{Z_n}{n} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( I = {\int }_{0}^{\frac{\pi }{4}}\left( {{\sin }^{6}{2x} + {\cos }^{6}{2x}}\right)  \cdot  \ln \left( {1 + \tan x}\right) \mathrm{d}x \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate \( \mathbb{E}(\tau) \) where \( \tau = \inf\{t \geq 0 : B^2_t = 1-t\} \) and \( B_t \) is a standard Brownian motion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all real numbers $x$ such that the series \( \sum_{n=2}^{\infty} \frac{\sin(nx)}{\log n} \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a unitary matrix \( U \), find the unitary matrix \( V \) that maximizes \( \Re \, \text{tr}(UV) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the degree of extension of the algebraic closure over the field \( \mathbb{Q}_p^{ext} = \mathbb{Z}((X))_{conti}/(X-p) \), where \( p \) is a prime number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cube is placed inside a regular tetrahedron with an edge length of 6, and the cube can rotate freely within the tetrahedron. The maximum edge length of the cube is __________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square table consists of a tabletop and four legs. If 1 cubic meter of wood can be used to make 50 tabletops or 300 legs, and there are 5 cubic meters of wood available, please design how many cubic meters of wood should be used to make tabletops so that the number of tabletops matches the number of legs exactly, with neither leftover.
Please reason step by step, and put your final answer within \boxed{}.


Context:
$P\left(x\right)$ is a polynomial of degree at most $6$ such that such that $P\left(1\right)$, $P\left(2\right)$, $P\left(3\right)$, $P\left(4\right)$, $P\left(5\right)$, $P\left(6\right)$, and $P\left(7\right)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P\left(8\right)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the kernel of the homomorphism \( D: F[x] \to F[x] \) defined by \( D(f(x)) = f'(x) \), where \( f(x) = a_0 + a_1x + \cdots + a_nx^n \) and \( f'(x) = a_1 + 2a_2x + \cdots + na_nx^{n-1} \), given that the characteristic of the field \( F \) is \( p \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 5 teachers available to lead 3 extracurricular groups for an educational field trip. Each group can have at most 2 leaders, but neither teacher A nor teacher B can lead a group on their own. How many different leadership arrangements are possible? (Answer in numerals).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $A_1A_2A_3A_4$ be a square, and let $A_5, A_6, A_7, \ldots, A_{34}$ be distinct points inside the square. Non-intersecting segments $\overline{A_iA_j}$ are drawn for various pairs $(i,j)$ with $1 \le i,j \le 34$, such that the square is dissected into triangles. Assume each $A_i$ is an endpoint of at least one of the drawn segments. How many triangles are formed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For each positive integer $n$, let $a(n)$ be the number of zeros in the base 3 representation of $n$. For which positive real numbers $x$ does the series $$ \sum_{n=1}^{\infty} \frac{x^{a(n)}}{n^3} $$ converge? Please provide the sum of the endpoints of the interval for which the series converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two trains of different lengths are running towards each other on parallel lines at 42 kmph and 30 kmph respectively. The longer train is 280 m long. They will be clear of each other in 16.998640108791296 seconds from the moment they meet. What is the length of the shorter train?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A square is made by arranging 12 marbles of the same size without gaps in both the width and length. How many marbles are placed around the perimeter?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设函数 f:\{1,2,3\}\rightarrow\{2,3,4\} 满足 f[f(x)-1]=f(x)，则这样的函数有__________个.
Please reason step by step, and put your final answer within \boxed{}.


Context:
对任意闭区间 $I$ ，用 $M_{I}$ 表示函数 $y=\sin x$ 在 $I$ 上的最大值。若正数 $a$ 满足 $M_{[0, a]}=2 M_{[a, 2 a]}$ ，则 $a$ 的值为 $\frac{m \pi}{n}$ 或 $\frac{p \pi}{q}$ 的形式，请给出 $m + n + p + q$ 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
21 Savage has a 12-car garage, with a row of spaces numbered 1, 2, 3, \ldots, 12. How many ways can he choose 6 of them to park his 6 identical cars in, if no 3 spaces with consecutive numbers may be all occupied?
Please reason step by step, and put your final answer within \boxed{}.


Context:
To get to school, Jenny decided to ride the school's double decker bus. The bus has a total of 25 rows of seats, with 15 rows on the lower deck and 10 rows on the upper deck. On the lower deck, each row has a capacity of 5 kids, while on the upper deck, each row can seat up to 3 kids. Additionally, 4 staff members must also ride the bus. Any vacant seat can be occupied by any person who possesses a ticket, either lower or upper deck ticket, except for the reserved seats.

On the lower deck, 10 students were given reserved seats, which only they can sit in. There are 4 types of bus tickets: Premium lower deck, Regular lower deck, Premium upper deck, and Regular upper deck. Premium ticket holders have priority to sit on the first row of each deck.

Considering staff members count towards the number of ticketed occupants, reserved seats for specific students, and the priority seating policy, find the maximum number of different kids that can ride the school bus.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the plane Cartesian coordinate system \(xOy\), point \(P\) is a moving point on the line \(y = -x - 2\). Two tangents to the parabola \(y = \frac{x^2}{2}\) are drawn through point \(P\), and the points of tangency are \(A\) and \(B\). Find the minimum area of the triangle \(PAB\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the real numbers \( a \) and \( b \) satisfy \( a = x_1 + x_2 + x_3 = x_1 x_2 x_3 \) and \( ab = x_1 x_2 + x_2 x_3 + x_3 x_1 \), where \( x_1, x_2, x_3 > 0 \). Then the maximum value of \( P = \frac{a^2 + 6b + 1}{a^2 + a} \) is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the equation
\[
1 + x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots - \frac{x^{2018}}{2018} = 0
\]
have all its real roots within the interval \([a, b]\), where \(a\) and \(b\) are integers such that \(a  b\). What is the minimum value of \(b - a\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Task 1. We call a collection of lines in the plane beautiful if each line in the collection intersects an odd number of the other lines in the collection.

Determine the smallest integer $k \geq 0$ with the following property: for any 2018 different lines $\ell_{1}, \ell_{2}, \ldots, \ell_{2018}$ in the plane, there exist lines $\ell_{2018+1}, \ell_{2018+2}, \ldots$, $\ell_{2018+k}$ such that the lines $\ell_{1}, \ell_{2}, \ldots, \ell_{2018+k}$ are all different and form a beautiful collection.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two groups, Group A and Group B, each have 10 students who took a math test. The scores are as follows (unit: points):
Group A: $76$, $90$, $84$, $86$, $81$, $87$, $86$, $82$, $85$, $83$
Group B: $82$, $84$, $85$, $89$, $79$, $80$, $91$, $89$, $79$, $74$
Now, one student is randomly selected from these 20 students. Let event $A$ be "the selected student is from Group A" and event $B$ be "the selected student scored at least $85$ points on the math test." Calculate $P(A|B)$, the probability that the selected student is from Group A given that the student scored at least 85 points. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all values of the variable \( x \) for which both expressions \( f(x)=\sqrt{21-x^{2}-4x} \) and \( g(x)=|x+2| \) are defined, and such that \( \min (f(x), g(x)) > \frac{x+4}{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$A$ and $B$ play the following game: random \( k \) numbers are chosen from the first 100 positive integers. If the sum of these is even, $A$ wins; otherwise, $B$ wins. For which values of \( k \) are the winning chances of \( A \) and \( B \) equal?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A function f(x) is defined on the set of real numbers and is odd. It satisfies f(3) = 0, and when x > 0, the inequality f(x) > -xf'(x) holds. Determine the number of zeros of the function g(x) = xf(x) + log|x+1|. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
在 $\triangle A B C$ 中, 已知 $\overrightarrow{A B} \cdot \overrightarrow{A C}+2 \overrightarrow{B A} \cdot \overrightarrow{B C}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$. 求 $\sin C$ 的最大值。原始答案是 \frac{\sqrt{m}}{n} 的格式，请给出 m + n 的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sarah, an Australian political science student, loves analyzing historical election results. She decides to look at the election outcomes in her country for the past 60 years. She finds that there were 15 federal elections during this period. Each election had an average of 6 major parties competing.

Sarah is particularly interested in the number of seats won by each party and discovers the following pattern: In each election, the party that won the most seats typically secured 35% of the total available seats. She learns that on average, each election had 150 seats available.

If Sarah wants to calculate the total number of seats won by the leading parties across all 15 federal elections, how many seats did they win in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the measure of angle \( B \widehat{A} D \), given that \( D \widehat{A C}=39^{\circ} \), \( A B = A C \), and \( A D = B D \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\{a_n\}$ be a sequence of integers satisfying the recurrence relation:
\[
(n-1)a_{n+1} = (n+1)a_n - 2(n-1) \quad \forall n \ge 1.
\]
Given that $2000 \mid a_{1999}$, determine the smallest integer $n \ge 2$ such that $2000 \mid a_n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle  $ABC$  lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes  $63$ ,  $73$ ,  $97$ . Suppose the curve  $\mathcal V$  with equation  $y=(x+3)(x^2+3)$  passes through the vertices of  $ABC$ . Find the sum of the slopes of the three tangents to  $\mathcal V$  at each of  $A$ ,  $B$ ,  $C$ .

*Proposed by Akshaj*
Please reason step by step, and put your final answer within \boxed{}.


Context:
9. In a rectangular box $A B C D E F G H$ with edge lengths $A B=A D=6$ and $A E=49$, a plane slices through point $A$ and intersects edges $B F, F G, G H, H D$ at points $P, Q, R, S$ respectively. Given that $A P=A S$ and $P Q=Q R=R S$, find the area of pentagon $A P Q R S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $d_1$, $d_2$, $d_3$, $d_4$, $e_1$, $e_2$, $e_3$, and $e_4$ be real numbers such that for every real number $x$, we have
\[
  x^8 - 2x^7 + 2x^6 - 2x^5 + 2x^4 - 2x^3 + 2x^2 - 2x + 1 = (x^2 + d_1 x + e_1)(x^2 + d_2 x + e_2)(x^2 + d_3 x + e_3)(x^2 + d_4 x + e_4).
\]
Compute $d_1 e_1 + d_2 e_2 + d_3 e_3 + d_4 e_4$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \( \mathrm{ABC} \), side \( AB = 4 \) and side \( BC = 6 \). Point \( M \) lies on the perpendicular bisector of segment \( AB \), and lines \( AM \) and \( AC \) are perpendicular. Find \( MA \), given that the circumradius of triangle \( ABC \) is 9.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A company has 100 shareholders, and any 66 of them together own at least 50% of the company's shares. What is the maximum percentage of all shares that one shareholder can own?
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. Let set $A=\{1,2,3, \cdots, 1997\}$, for any 999-element subset $X$ of $A$, if there exist $x, y \in X$, such that $x<y$ and $x \mid y$, then $X$ is called a good set. Find the largest natural number $a(a \in A)$, such that any 999-element subset containing $a$ is a good set.
(《Mathematics in Middle School》1999 Issue 1 Olympiad Problems)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Emily is thinking of a number. She gives the following 3 clues. "My number has 216 as a factor. My number is a multiple of 45. My number is between 1000 and 3000." What is Emily's number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the correlation coefficient between $X_{(1)}$ and $X_{(3)}$ when they are order statistics of three independent random variables $X_1, X_2, X_3$, each uniformly distributed in the interval $[0,1]$? It is given that $X_{(k)}$ follows a Beta distribution with parameters $k$ and $4-k$. Utilize the theorem for covariance to compute the correlation coefficient, where the variance of $X_{(k)}$ is $\frac{k(4-k)}{80}$ and its mean is $\frac{k}{3}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two motorcyclists departed simultaneously from points \( A \) and \( B \) towards each other and met 50 km from point \( B \). After arriving at points \( A \) and \( B \), the motorcyclists immediately turned back and met for the second time 25 km from point \( A \). How many kilometers are there from \( A \) to \( B \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit using Taylor expansion:

$$\lim_{x\to0^+} \frac{\log(\vert x \vert (1+x^2)^{1/3}-\sin{x})}{\log(\vert x \vert)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the range q of the 6 numbers 4, 314, 710, and x is 12, what is the difference between the greatest possible value of x and the least possible value of x?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The pentagon \(A B C D E\) is circumscribed about a circle.

The angles \(\angle A B C\), \(\angle B A E\), and \(\angle C D E\) are each \(104^{\circ}\). Find \(\angle A D B\). Provide the answer in degrees (only the number, without specifying the unit of measurement).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, it is known that $\angle B=45^\circ$, $AC=\sqrt{2}BC$. Find $\angle C=$______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the integral \( \int_{C} f(z) \, dz \), where \( f(z) = \frac{\bar{z}}{z+i} \) and \( C \) is the circle \( |z+i|=3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $R^5$ with coordinates $(p_1,q_1,p_2,q_2,t)$, $db=dp_1\wedge dp_2$. Calculate the integral $\int_T db\wedge dt$, where $T=\{t\in[0,1],p_1^2+q_1^2=p_2^2+q_2^2\leq t^2\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
We place four spheres of radius $r$ on a horizontal plane so that their centers form a square and they touch each other. We then place a fifth sphere of radius $r$ on top of the four spheres, touching all four of them. How far is the highest point of the fifth sphere from the plane? (H)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Figure F represents the intersection of \( n \) circles ( \( n \geq 2 \), radii are not necessarily equal). What is the maximum number of curved "sides" that Figure F can have? (A curved side is a segment of the boundary of F that belongs to one of the circles and is bounded by intersection points with other circles.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
A courtyard is 28 meters long and 13 meters wide and is to be paved with bricks of dimensions 22 cm by 12 cm. How many bricks are required to pave the entire courtyard?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The side length of a rhombus is $a$ and its acute angle is $\alpha$. The rhombus rotates around a line passing through its vertex and parallel to the larger diagonal. Find the volume of the solid of revolution.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{2 \pi a^3 \sin \frac{\alpha}{2} \sin \alpha}


Context:
In tetrahedron \(ABCD\), \(AB = 1\), \(BC = 5\), \(CD = 7\), \(DA = 5\), \(AC = 5\), \(BD = 2\sqrt{6}\). Find the distance between skew lines \(AC\) and \(BD\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A tank holds x gallons of a saltwater solution that is 25% salt by volume. One third of the water is evaporated, leaving all of the salt. Then, 12 gallons of water and 24 gallons of salt are added, raising the salt percentage to 40%. After this, another one fourth of the water is evaporated, leaving the salt once again. In the final step, 15 gallons of water and 30 gallons of salt are added, resulting in a mixture that is 50% salt by volume. What is the value of x?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are fifteen boxes, each containing either a white ball or a black ball, with no more than 12 white balls in total. You can choose any three boxes and ask, "Does any of these three boxes contain a white ball?" and receive a truthful answer. What is the minimum number of questions you need to ask to ensure you find at least one white ball?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given vectors $\overrightarrow {a}$=(2,$m$), $\overrightarrow {b}$=(-1,2), if $\overrightarrow {a}$⊥$\overrightarrow {b}$, then the projection of $\overrightarrow {b}$ on vector $\overrightarrow {c}$=$\overrightarrow {a}$-$ \overrightarrow {b}$ is ___.
Please reason step by step, and put your final answer within \boxed{}.


Context:
One autumn day, a Scattered Scientist glanced at his antique wall clock and saw three flies asleep on the dial. The first one was exactly at the 12 o'clock mark, and the other two were precisely at the 2 o'clock and 5 o'clock marks. The Scientist measured and determined that the hour hand posed no threat to the flies, but the minute hand would sweep them all away one by one. Find the probability that exactly two of the three flies will be swept away by the minute hand precisely 40 minutes after the Scientist noticed the flies.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the area of the largest rectangle that can be inscribed within an ellipse given by the equation $\frac{(x-y)^2}{2m^2} + \frac{(x+y)^2}{2n^2} = 1$, where $0<mn\leq 1$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 3000$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 3000$ with $S(n)$ even. Find $|a-b|.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given real numbers $x$ and $y$ satisfying $2x^2 + xy - y^2 = 1$, find the maximum value of $\frac{x - 2y}{5x^2 - 2xy + 2y^2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the following expression:
\[ \sum_{i=0}^{\frac{q-1}{2}}\left\lfloor\frac{ip}{q}\right\rfloor + \sum_{j=0}^{\frac{p-1}{2}}\left\lfloor\frac{jq}{p}\right\rfloor \]
where \( p = 51 \) and \( q = 81 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A trader sells three different articles, A, B, and C, each with varying profit percentages (10%, 20%, and 30%, respectively) and individual selling prices of $100, $150, and $200, respectively. If he then sells each article for twice its initial selling price, determine his overall profit percentage for selling all three articles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
12.222. The base of the pyramid is a rhombus, one of the angles of which is $\alpha$. The lateral faces are equally inclined to the plane of the base. A plane is drawn through the midpoints of two adjacent sides of the base and the vertex of the pyramid, forming an angle $\beta$ with the plane of the base. The area of the resulting section is $S$. Find the side of the rhombus.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ronald wants to make profits by re-selling some phones he bought a week ago. He bought 200 units for just $3000, and he wants to gain a third of the initial investment in profits when all units are sold. Additionally, there is an 8% sales tax on each sale, and he must also account for $2 per unit for packaging and a $500 flat shipping fee for the entire lot. Taking all these costs into consideration, what will be the selling price for each phone, including tax?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function \( f(x) = \max \left\{\sin x, \cos x, \frac{\sin x + \cos x}{\sqrt{2}}\right\} \) (for \( x \in \mathbb{R} \)) has a maximum value and a minimum value. Find the sum of these maximum and minimum values.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find $w$, such that $5^65^w=25$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $a_1$ ,  $a_2, \dots, a_{2015}$  be a sequence of positive integers in  $[1,100]$ .
Call a nonempty contiguous subsequence of this sequence *good* if the product of the integers in it leaves a remainder of  $1$  when divided by  $101$ .
In other words, it is a pair of integers  $(x, y)$  such that  $1 \le x \le y \le 2015$  and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \]Find the minimum possible number of good subsequences across all possible  $(a_i)$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知函数$f\left( x \right)={{\left( \frac{1}{5} \right)}^{x}}-\text{1},x\in \left[ -1,0 \right]$，$g\left( x \right)={{a2}}\text{lo}{{\text{g}}_{2}}x\text{+3}a,x\in \left[ \frac{\sqrt{2}}{2},2 \right]$，若对任意的${{x0}}\in \left[ \frac{\sqrt{2}}{2},2 \right]$，总存在${{x}_{1}}\in \left[ -1,0 \right]$使得$g\left( {{x0}} \right)=f\left( {{x}_{1}} \right)$成立，则实数$\textit{a}$的取值范围是__________．
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $2022$ distinct integer points on the plane. Let $I$ be the number of pairs among these points with exactly $1$ unit apart. Find the maximum possible value of $I$.
([i]Note. An integer point is a point with integer coordinates.[/i])

[i]Proposed by CSJL.[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Arrange 6 volunteers $A$, $B$, $C$, $D$, $E$, $F$ to take care of 3 elderly people $X$, $Y$, $Z$. Each pair of volunteers takes care of one elderly person. Considering the distance between the volunteers' and the elderly people's residences, volunteer $A$ cannot be assigned to take care of elderly person $X$, and volunteer $B$ cannot be assigned to take care of elderly person $Y$. Calculate the total number of different arrangements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
We are downloading a 1.5 MB file to our computer from the internet. During this process, the program continuously estimates the remaining time based on the average speed of the download so far. At one moment, we observe that exactly half of the file has already been downloaded, and at this moment, the program estimates that the remaining time is precisely 2 minutes. After this time, regardless of any $t$ time that passes (due to network congestion), the program still indicates that the remaining time to download the file is 2 minutes. Determine the size of the file that has already been downloaded as a function of $t$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of the smallest disc in the complex plane that contains all eigenvalues of the matrix A, given that the characteristic polynomial of A - I is x^4 + 15x^2 - 14x + 16, where I is the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $N = 99999$. Then $N^3 = \underline{\quad\quad}$.
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
M. Murraikin

On an island, there live 100 knights and 100 liars, each of whom has at least one friend. Knights always tell the truth, while liars always lie. One morning, each resident either said the phrase "All my friends are knights" or the phrase "All my friends are liars," and exactly 100 people said each phrase. Find the smallest possible number of pairs of friends, one of whom is a knight and the other a liar.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{2} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]
Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Determine $\cos \theta.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Iggy is training for a marathon. He runs different distances at varying paces due to changes in elevation, weather conditions, and additional factors like carrying a weighted backpack.

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

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

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

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

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

Calculate the total time Iggy spends running from Monday to Friday, taking into account the changes in elevation, weather conditions, and the added resistance from the weighted backpacks.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given four balls with colors red, yellow, blue, and green, they are to be placed into four boxes of colors red, yellow, blue, and green, with each box containing only one ball. Calculate the probability that exactly one ball's color matches the box's color. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a college, 50% of total 400 arts students are locals. 25% of students from 100 science students are locals and a certain percentage of total 120 commerce students are locals. The total percentage of locals from arts, science and commerce is 327. What is the percentage of local commerce students?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let triangle $\triangle ABC$ have side lengths $AB = 7$, $BC = 8$, and $CA = 9$. Let $M$ be the midpoint of $\overline{BC}$, and $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $m\angle AEM = m\angle AFM = 90^{\circ}$. Let $P$ be the intersection of the angle bisectors of $\angle AED$ and $\angle AFD$. If $MP$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, then find $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The product $55 \cdot 60 \cdot 65$ is written as a product of 5 distinct numbers. Find the least possible value of the largest number among these 5 numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC = 60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a cubic polynomial \( q(x) \) that satisfies \( q(3) = 2, \) \( q(8) = 20, \) \( q(16) = 12, \) and \( q(21) = 30, \) find the sum \( q(2) + q(3) + \ldots + q(22) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a house, a hall is 20 m long, of a certain width, and 5 m high. Its interior has to be covered with mat. The total expenditure for covering the hall with mat at a cost of Rs. 50 per square meter is Rs. 47,500. What is the width of the hall?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer $k \geqslant 3$ is called fibby if there exists a positive integer $n$ and positive integers $d_{1}<d_{2}<\ldots<d_{k}$ with the following properties:  - $d_{j+2}=d_{j+1}+d_{j}$ for every $j$ satisfying $1 \leqslant j \leqslant k-2$,  - $d_{1}, d_{2}, \ldots, d_{k}$ are divisors of $n$,  - any other divisor of $n$ is either less than $d_{1}$ or greater than $d_{k}$.  Find the largest fibby number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The student locker numbers at Olympic High are numbered consecutively beginning with locker number $1$. The plastic digits used to number the lockers cost two cents apiece. Thus, it costs two cents to label locker number $9$ and four cents to label locker number $10$. If it costs $137.94$ to label all the lockers, how many lockers are there at the school? Provide your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Nina wants to give presents to her children. She buys three toys at $10 each, two packs of basketball cards at $5 each, and some shirts at a certain price each. She spends $70 in all. What is the price of each shirt?
Please reason step by step, and put your final answer within \boxed{}.


Context:
\(ABCD\) is a cyclic quadrilateral such that \(|DA| = |BC| = 2\), and \(|AB| = 4\). If \(|CD| > |AB|\) and the lines \(DA\) and \(BC\) intersect at an angle of \(60^\circ\), find the radius of the circumscribing circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A die is rolled consecutively three times. Calculate the probability that the sequence of numbers facing upwards forms an arithmetic progression. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $y=\cos({2x+\frac{π}{3}})$, determine the horizontal shift of the graph of the function $y=\sin 2x$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
As shown in the diagram, in the tetrahedron \(A B C D\), the face \(A B C\) intersects the face \(B C D\) at a dihedral angle of \(60^{\circ}\). The projection of vertex \(A\) onto the plane \(B C D\) is \(H\), which is the orthocenter of \(\triangle B C D\). \(G\) is the centroid of \(\triangle A B C\). Given that \(A H = 4\) and \(A B = A C\), find \(G H\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the multiplication problem shown in the figure, \( A, B, C, D, E, F, G, H, I \) each represent a different single-digit number. What is the five-digit number represented by "FIGAA"?

```
   ABC
×   DC
-----
   BEA
------
 FIGAA
```
Please reason step by step, and put your final answer within \boxed{}.


Context:
71. A sheep is tied to a vertex of an equilateral triangular fence with a side length of 5 meters, the rope is 7 meters long. If the sheep can only walk outside the fence, the area that the sheep can reach is $\qquad$ square meters. ( $\pi$ is taken as 3.14)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\{x_n\}_{n=1}^{\infty}$ be a convergent sequence in a normed space $(S, \|\cdot\|)$. Determine the limit of the sequence
$$\left\{\frac{x_1 + 2x_2 + \cdots + nx_n}{n^2}\right\}_{n=1}^{\infty}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知存在正整数 a, b, c 满足 $a+b+c=407,10^{n} \mid a b c$ ，求 n 的最大值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the inverse Laplace transform of the function $ \ln \! \left( 1 + \dfrac{1}{s^{2}} \right) $?
Please reason step by step, and put your final answer within \boxed{}.


Context:
It is given that  $x = -2272$ ,  $y = 10^3+10^2c+10b+a$ , and  $z = 1$  satisfy the equation  $ax + by + cz = 1$ , where  $a, b, c$  are positive integers with  $a < b < c$ . Find  $y.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(a\) and \(b\) be real numbers such that \(\sin{a} + \sin{b} = \frac{\sqrt{2}}{2}\) and \(\cos {a} + \cos {b} = \frac{\sqrt{6}}{2}.\) The original answer is in the form \(\frac{\sqrt{k}}{m}\). Please find the value of \(k + m\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of positive integer solutions $(a,b,c,d)$ to the equation $(a^2+b^2)(c^2-d^2)=2020$. Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\mathcal{P}$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $\mathcal{P}$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB=20$ and $CD=14$, compute the area of $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
3- |

In the convex pentagon $A B C D E$, the diagonals $B E$ and $C E$ are the angle bisectors of the angles at vertices $B$ and $C$ respectively, $\angle A=35^{\circ}, \angle D=145^{\circ}$, and the area of triangle $B C E$ is 11. Find the area of the pentagon $A B C D E$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The 83 trainees from the Animath workshop each choose an activity for the free afternoon from 5 activities offered. We know that:

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

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

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

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

When writing the list of activities in alphabetical order, the number of students corresponding to each activity is written. How many different lists of numbers can be written?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The four hydrogen atoms in the methane molecule $\mathrm{CH}_{4}$ are located at the vertices of a regular tetrahedron with edge length 1. The carbon atom $C$ is located at the center of the tetrahedron $C_{0}$. Let the four hydrogen atoms be $H_{1}, H_{2}, H_{3}, H_{4}$. Then $\sum_{1 \leq i < j \leq 4} \overrightarrow{C_{0} \vec{H}_{i}} \cdot \overrightarrow{C_{0} \vec{H}_{j}} = \quad$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain microbrewery increased its beer output by some percent. At the same time, it decreased its total working hours by 30 percent. The factory increased its output per hour by 171.43 percent (approximately). By what percent did the microbrewery increase its beer output?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n\to \infty} \frac{1}{\sqrt{n}} \left(\frac{1}{\sqrt{n+1}}+\dotsm +\frac{1}{\sqrt{n+n}} \right) \) using a Riemann sum.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triline is a line with the property that three times its slope is equal to the sum of its \(x\)-intercept and its \(y\)-intercept. For how many integers \(q\) with \(1 \leq q \leq 10000\) is there at least one positive integer \(p\) so that there is exactly one triline through \((p, q)\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Yasha writes in the cells of a $99 \times 99$ table all positive integers from 1 to $99^2$, with each number appearing once. Grisha looks at the table and selects several cells such that no two selected cells share a common side. Grisha then sums up the numbers in all the selected cells. Determine the largest sum that Grisha can guarantee to achieve.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ordered quintuples  $(a,b,c,d,e)$  of integers with  $0\leq a<$   $b<$   $c<$   $d<$   $e\leq 30$  for which there exist polynomials  $Q(x)$  and  $R(x)$  with integer coefficients such that \[x^a+x^b+x^c+x^d+x^e=Q(x)(x^5+x^4+x^2+x+1)+2R(x).\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
An alarm clock runs 4 minutes slow every hour. It was set right  $ 3 \frac{1}{2}$  hours ago. Now another clock which is correct shows noon. In how many minutes, to the nearest minute, will the alarm clock show noon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cube is rotated around the specified axis so that the marked face is rotated in the indicated manner. To which vertex number did point \(A\) move?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a basketball season, John has a varying success rate for hitting his free throws. In 20% of the games, he hits 60% of his free throws, in another 50% of the games, he hits 70% of his free throws, and in the remaining 30% of the games, he hits 80% of his free throws. Additionally, the number of fouls he gets each game also varies. In 40% of the games, he gets 4 fouls, in 35% of the games, he gets 5 fouls, and in 25% of the games, he gets 6 fouls. In each foul, he gets 2 shots. Out of 20 games the team plays, John plays in 80% of them. Calculate the total number of free throws John gets in the season, considering the varying percentages and the number of fouls he gets in each game.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \(0<\theta<\pi\), a complex number \(z_{1}=1-\cos \theta+i \sin \theta\) and \(z_{2}=a^{2}+a i\), where \(a \in \mathbb{R}\), it is known that \(z_{1} z_{2}\) is a pure imaginary number, and \(\bar{a}=z_{1}^{2}+z_{2}^{2}-2 z_{1} z_{2}\). Determine the value of \(\theta\) when \(\bar{a}\) is a negative real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The circle passing through the vertices $A, B$, and $D$ of a rectangular trapezoid $ABCD$ ($\angle A = \angle B = 90^\circ$) intersects the extension of the base $BC$ and the extension of the lateral side $CD$ at points $M$ and $N$ respectively, such that $CM: CB = CN: CD = 1: 2$. Find the ratio of the diagonals $BD$ and $AC$ of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of zeroes of the polynomial \( p(z) = z^8 - 20z^4 + 7z^3 + 1 \) that lie within the disk \( D(0,2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
甲、乙两人从 $0,1, \cdots, 81$ 中轮流挑选互不重复的数，甲先选，每次每人从剩下的数中选一个数。当这 82 个数被选完之后，记 $A$ 、 $B$ 分别为甲、乙选择的所有数之和。在挑选数
的过程中, 甲希望 $A$ 与 $B$ 的最大公约数越大越好，而乙希望 $A$ 与 $B$ 的最大公约数越小越好。在甲、乙各自的最佳策略下, 求挑选完毕之后 $A$ 与 $B$ 的最大公约数.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The natural number solution set of the equation $(x^2 - 6x + c_1)(x^2 - 6x + c_2)(x^2 - 6x + c_3) = 0$ is $\{x_1, x_2, x_3, x_4, x_5, x_6\}$. What is the maximum value of $c_1 - c_2$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum value of $\int_0^{\pi} (x-y)^2 (\sin x)|\cos x|dx$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $V$ be an $n$-dimensional vector space over $F$, with basis $\mathcal{B} = \{\mathbf{v_1, \cdots, v_n}\}$. Let $\mathcal{B}^{*} = \{\phi_1, \cdots, \phi_n\}$ be the dual basis for $V^{*}$. Let $\psi : V \to V$ and $A$ be the matrix representing $\psi$ with respect to $\mathcal{B}$. Let $\psi^t : V^{*} \to V^{*}$ and $B$ be the matrix representing $\psi^t$ with respect to $\mathcal{B}^{*}$. Determine the relationship between $A$ and $B$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jonathan eats 2500 calories every day except for Saturday, when he consumes an extra amount of calories. He burns 3000 calories every day. His weekly caloric deficit is 2500 calories. How many extra calories does Jonathan consume on Saturday?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute $\left\lfloor \sum_{k=0}^{10}\left(3+2\cos\left(\frac{2k\pi}{11}\right)\right)^{10}\right\rfloor \pmod{100}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int^1_0 \frac{1}{\sqrt{x}} \left( \int^\sqrt{x}_1 e^{-t^2} \, dt \right) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A farmer claims that with four rods, he can fence a square plot of land sufficient for one sheep. If this is true, what is the minimum number of rods needed to fence an area for ten sheep? The answer depends on the shape of your fence.

How many rods are required for 10 sheep?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $y(x)$ is a solution of the differential equation $$ x^2 y^{\prime \prime}+x y^{\prime}-4 y=x^2 $$ on the interval $(0, \infty)$ such that $\lim _{x \rightarrow 0^{+}} y(x)$ exists and $y(1)=1$, find the value of $y^{\prime}(1)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $P\left( \frac{3}{2}, -\frac{3\sqrt{3}}{2}\right)$ is the lowest point on the graph of the function $y=A\sin(\omega x+\varphi)$ ($\omega > 0$), and $M$, $N$ are the two highest points adjacent to $P$. If $\angle MPN=60^{\circ}$, calculate the smallest positive period of this function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given:
Proposition $p$: The function $y=a^{x} (a > 0, a \neq 1)$ is a monotonically decreasing function on $\mathbb{R}$.
Proposition $q$: The range of the function $y=\lg (ax^{2}-x+a)$ is $\mathbb{R}$.
If the proposition "$p$ and $q$" is false, find the range of values for $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Janice gets paid $10 an hour for the first 40 hours she works each week, and $15 each hour of overtime after that. On weekends, her hourly wage increases to $12 an hour for the first 40 hours and $18 for each hour of overtime. Additionally, if she works on a holiday, her hourly rate doubles the usual wage for that day. In one week, Janice works 30 hours on weekdays, 25 hours on weekends, and 5 hours on a holiday that falls on a weekend. How much money does she make that week?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Seventy percent of a train's passengers are men, and fifteen percent of those men are in the business class. What is the number of men in the business class if the train is carrying 300 passengers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a,$ $b,$ and $c$ be the roots of the equation $x^3 - 24x^2 + 50x - 42 = 0.$ Find the value of $\frac{a}{\frac{1}{a}+bc} + \frac{b}{\frac{1}{b}+ca} + \frac{c}{\frac{1}{c}+ab}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n \to \infty}n\left(\frac{\cos\left(\frac{2n+1}{n^2+n}\right)\sin\left(\frac{1}{n^2+n}\right)}{\cos\left(\frac{2n+3}{(n+1)^2+n+1}\right)\sin\left(\frac{1}{(n+1)^2+n+1}\right)}-1\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a right triangle \( \triangle ABC \) with \( \angle B = 90^\circ \), the median \( AM \) is drawn. If the measure of angle \( \angle AMB \) is twice the measure of angle \( \angle A \), calculate the ratio \( \frac{AM}{BC} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of $x^{20}$ in the expansion of $(x^3+2x^2+5x+3)^5(x^2+4x+5)^4$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Krull dimension of the local ring \( \mathbb{C}[x,y]_p/(y^2-x^7,y^5-x^3) \), where \( p \neq (0,0) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{-\infty}^{\infty} A x^2 e^{-x^2/(2\sigma^2)} \, dx \), given that \( \int_{-\infty}^{\infty} A e^{-x^2/(2\sigma^2)} \, dx = 1 \). Here, \( A \) is a constant, and the expression inside the second integral represents the Gaussian or Normal distribution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{n \to \infty} \frac{\big((n+1)(n+1)(n+3)\ldots3n\big)^{1/n}}{(n^{2n})^{1/n}}.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the fundamental group \( \pi_1(X / \sim, x_0) \) for a point \( x_0 \in X / \sim \), where \( X \) is the subset of \( \mathbb{R}^3 \) defined as \( X=\left \{x \in \mathbb{R}^3 \vert 1 \leq \lvert x \rvert \leq 2 \right \} \) with boundary components \( S_1 = \left \{x \in \mathbb{R}^3 \vert \lvert x \rvert = 1 \right \} \) and \( S_2 = \left \{x \in \mathbb{R}^3 \vert \lvert x \rvert = 2 \right \} \). The equivalence relation \( \sim \) is defined such that \( x \sim y \) if \( x \in S_1 \), \( y \in S_2 \), and \( y = 2x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the convergence rate of the Newton iteration near the root \( r = 1 \) for the polynomial \( g(x) = 4x^{3} - 6x^{2} + 4x - 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an irreducible polynomial $g(x) \in \mathbb{Q}[x]$ of prime degree $p$, if $g(x)$ has $p-2$ real roots, determine the structure of the Galois group of $g(x)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when the sum \( S = \sum_{n=0}^{2015} (n! + n^3 - n^2 + n - 1) \) is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest natural number \( n \) such that in any set of \( n \) different natural numbers, each not exceeding 1000, there exist two numbers where the larger number is not divisible by the smaller one.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A quadratic polynomial with positive integer coefficients and rational roots
 can be written as \( {196}{x}^{2} + {Bx} + {135} \) for some integer \( B \) . What is the sun
 of all possible values of \( B \) such that \( \gcd \left( {B,{196} \cdot  {135}}\right)  = 1 \) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cube with edge length 1 is denoted as AC1. A moving point P is on its surface, and the distance from point A is $\frac {2 \sqrt {3}}{3}$. The set of points P forms a curve. The length of this curve is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the positive real numbers \(a_{1}, a_{2}, a_{3}, a_{4}\) satisfy the conditions \(a_{1} \geqslant a_{2} a_{3}^{2}, a_{2} \geqslant a_{3} a_{4}^{2}, a_{3} \geqslant a_{4} a_{1}^{2}, a_{4} \geqslant a_{1} a_{2}^{2}\), find the maximum value of \(a_{1} a_{2} a_{3} a_{4}\left(a_{1}-a_{2} a_{3}^{2}\right)\left(a_{2}-a_{3} a_{4}^{2}\right)\left(a_{3}-a_{4} a_{1}^{2}\right)\left(a_{4}-a_{1} a_{2}^{2}\right)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ordered 17-tuples $(a_1, a_2, a_3, \dots, a_{17})$ of integers, such that the square of any number in the 17-tuple is equal to the sum of the other 16 numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Now a ball is launched from a vertex of an equilateral triangle with side length 5. It strikes the opposite side after traveling a distance of $\sqrt{19}$. How many times does the ball bounce before it returns to a vertex? (The final contact with a vertex does not count as a bounce.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In Class 3 (1), consisting of 45 students, all students participate in the tug-of-war. For the other three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking competition and 28 students participate in the basketball shooting competition. How many students participate in all three events?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sets $A=\{y\in \mathbb{N}|y=-x^{2}+4x,x\in \mathbb{Z}\}$ and $B=\{x|\ln x > 1\}$, find the intersection of $A$ and the complement of $B$ in $\mathbb{R}$, denoted as $A\cap (\complement_{\mathbb{R}}B)$. Express your answer as a set of numbers using standard set notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the Sturm-Picone comparison theorem, determine the maximum number of zeroes of the solution to the differential equation $(1+x^2)y''+(1+4x^2)y=0$ in the interval $[0, 2\pi]$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ABCDEF$  be a regular hexagon with sidelength  $6$ , and construct squares  $ABGH$ ,  $BCIJ$ ,  $CDKL$ ,  $DEMN$ ,  $EFOP$ , and  $FAQR$  outside the hexagon. Find the perimeter of dodecagon  $HGJILKNMPORQ$ .

*Proposed by Andrew Wu*
Please reason step by step, and put your final answer within \boxed{}.


Context:
A set $D$ of positive integers is called indifferent if there are at least two integers in the set, and for any two distinct elements $x, y \in D$, their positive difference $|x-y|$ is also in $D$. Let $M(x)$ be the smallest size of an indifferent set whose largest element is $x$. Compute the sum $M(2) + M(3) + \dots + M(100)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use Stoke's Theorem to evaluate the line integral \( \int_{C} \mathbf{F} \cdot \mathrm{d}\mathbf{r} \), where \( \mathbf{F}(x,y,z) = -x^2\mathbf{i} + xz\mathbf{j} + yx\mathbf{k} \) and \( C \) is the boundary of the surface \( S \), which is the graph of \( f(x,y) = 4x - 8y + 30 \) over the rectangle \( R = \{(x,y) \mid -2 < x < 3, 0 < y < 2\} \). The curve \( C \) is oriented counterclockwise when viewed from above.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A horse stands at the corner of a chessboard, on a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally, like a knight moves. The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to the rabbit who owns the chessboard for every move it makes. When the horse reaches the square on which it began, it can leave. What is the maximum number of carrots the horse can earn without touching any square more than twice?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bicycle costs 389 yuan, and an electric fan costs 189 yuan. Dad wants to buy a bicycle and an electric fan. He will need approximately \_\_\_\_\_\_ yuan.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Martha gets prize points every time she shops at her local grocery store. She gets 50 points per $10 spent, plus a 250 point bonus if she spends more than $100. Additionally, Martha is a member of the store's loyalty program which allows her to earn 100 bonus points for every 5th visit to the store, and today is her 5th visit.

During her shopping trip, Martha buys the following items:
- 3 pounds of beef for $11 each, with an item-specific 10% discount applicable on each;
- 8 pounds of fruits and vegetables for $4/pound, with a distinct category-wise 8% discount on the total amount;
- 3 jars of spices for $6 each, with a "buy 2, get the 3rd one free" offer and a separate 5% discount on the total price;
- Other groceries totaling $37 before a $3 off coupon is applied.

Calculate the number of points Martha earns during this shopping trip, taking into account all the individual discounts, the "buy 2, get 1 free" offer, and the loyalty program bonus points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the partial derivative of the function \( f = \text{tr}((ABC)(ABC)^T) \) with respect to the matrix \( B \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Percival's castle had a square shape. One day Percival decided to expand his estate and added a square annex to the castle. As a result, the perimeter of the castle increased by $10\%$. By what percentage did the area of the castle increase?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum possible area of a triangle with side lengths at most 2, 3, and 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( S = \{1, 2, 3, 4\} \), and a sequence of \( n \) terms \( a_1, a_2, \ldots, a_n \) has the following property: For any non-empty subset \( B \) of \( S \), there exist \( |B| \) consecutive elements in the sequence that exactly form the set \( B \). Find the minimum value of \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n \to \infty} I_n^n \), where \( I_n = \int_1^{1+\frac{1}{n}}\left\{[(x+1)\ln x +1] e^{x (e^x \ln x +1)}+n \right\}\, dx \) for \( n=1,2,\ldots \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of: $$\lim_{x\to 0}\left\{\dfrac 2{x^3}(\tan x- \sin x )\right\}^{2/x^2 }$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the equation of the tangent line to the function \( f(x) \) at \( x = 1 \), where:

\[ f(x) = \int_0^{g(x)} \frac{1}{1+\sin(t) + e^t} \, dt \]

and \( g(x) \) is given by:

\[ g(x) = \int_{-x}^{x^2} h(t) \, dt \]

with \( h(t) \) being an odd function and \( h(1) = 2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Pete's bank account contains 500 dollars. The bank allows only two types of transactions: withdrawing 300 dollars or adding 198 dollars. What is the maximum amount Pete can withdraw from the account if he has no other money?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many times will a clock strike over the course of 12 hours if it chimes on the half-hours as well?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Galois group of the polynomial \( g(x) = x^6 + 2x^3 - 3 \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many different permutations $(\alpha_1 \alpha_2\alpha_3\alpha_4\alpha_5)$ of the set $\{1,2,3,4,5\}$ are there such that $(\alpha_1\dots \alpha_k)$ is not a permutation of the set $\{1,\dots ,k\}$, for every $1\leq k \leq 4$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Indiana Jones reached an abandoned temple in the jungle and entered the treasury. There were 5 chests, with the knowledge that only one contains the treasure, and the others trigger a stone slab to fall on the head of anyone who tries to open them. The chests are numbered from left to right.

- The first, fourth, and fifth chests are made of cedar.
- The second and third chests are made of sandalwood.
- The inscriptions on the chests are as follows:
  - On the first chest: "The treasure is in me or in the 4th chest."
  - On the second chest: "The treasure is in the chest to the left of me."
  - On the third chest: "The treasure is in me or in the chest at the far right."
  - On the fourth chest: "There is no treasure in the chests to the left of me."
  - On the fifth chest: "All the inscriptions on other chests are false."

The last guardian of the temple, before dying, revealed to Indiana a secret: an equal number of false statements are written on both cedar and sandalwood chests.

In which chest is the treasure?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Choose a permutation of $ \{1,2, ..., 20\}$  at random. Let  $m$  be the amount of numbers in the permutation larger than all numbers before it. Find the expected value of  $2^m$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of digits in the largest natural number $N$ that does not contain the digit $0$, has a digit sum of $2021$, and where the digit sum of $2N$ does not exceed $1202$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A professional pet groomer who specializes in the care of long-haired cats, like the Siberian, has 3 grooming stations in her shop. On a busy Saturday, she has appointments to groom 12 Siberian cats. Each cat takes 45 minutes to groom. If the groomer works 9 hours that day, how many grooming stations does she need to use at the same time to finish all the appointments within her working hours?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \sum_{d\mid N}\Lambda(d) \) for a positive integer \( N \), where \( \Lambda(n) \) is defined as follows:

\[\Lambda(n) = \begin{cases} \log p & \text{if } n = p^r, \ p \text{ a prime and } r \in \mathbb{N}, \\ 0 & \text{otherwise.} \end{cases}\]

Here, the sum ranges over all divisors \( d \) of \( N \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f \) be the function defined for all \((x,y)\) as follows: \[ f(x,y)= \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2}, &\text{if }(x,y)\ne(0,0)\\ 0, &\text{if }(x,y)=(0,0) \end{cases} \] Determine the value of \( \frac{\partial^2 f}{\partial x \partial y} \) at \((0,0)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of hyperreal numbers that have a standard part equal to 0.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the boundary value of \(\alpha\) such that \(0 < \alpha < 1\) and \(n^{\alpha}\) never surpasses \(\log(n)\) for sufficiently large \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the expansion of (mx-1)^n, where n is a positive integer, and the sum of the coefficients of all terms is equal to the sum of the binomial coefficients, and the binomial coefficient of the 6th term is the largest, find the number of different ordered real number pairs (m,n) that satisfy these conditions. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
9. (10 points) The Fifth Plenary Session of the 18th Central Committee of the Communist Party of China, which concluded on October 29, 2015, determined the policy allowing universal second children. Xiao Xiao's father, after seeing the news on that day, told Xiao Xiao: The total age of our family this year is 7 times your age. If we add a younger brother or sister next year, the total age of our family in 2020 will be 6 times your age at that time. How old is Xiao Xiao this year? $\qquad$ years old.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F_1=1, F_2=1,$ and $F_{n+2}=F_{n+1}+F_n.$ Then,$$S = \sum\limits_{n=1}^{\infty} \arctan\left(\frac{1}{F_n}\right)\arctan\left(\frac{1}{F_{n+1}}\right)$$Find $\lfloor 80S \rfloor.$

(Hint: it may be useful to note that $\arctan(\tfrac{1}{1}) = \arctan(\tfrac{1}{2})+\arctan(\tfrac{1}{3}).$)
Please reason step by step, and put your final answer within \boxed{}.


Context:
8,9

Points $K, L, M$ divide the sides of the convex quadrilateral $A B C D$ in the ratio $A K: B K=C L: B L=C M: D M=1$ : 2. The radius of the circle circumscribed around triangle $K L M$ is $\frac{5}{2}, K L=4, L M=3$. What is the area of quadrilateral $A B C D$, given that $K M<K L$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a circle with center \(O\) and radius \(\sqrt{5} + 1\), a secant \(PBA\) and a tangent \(PE\) are drawn from an external point \(P\). Given that \(PE = AB = 2\), calculate the measure of \(\angle POB\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number 2019 is expressed as a sum of different odd natural numbers. What is the maximum possible number of terms in this sum?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim\limits_{a\to 0}\frac{\delta_{a}+\delta_{-a}-2\delta}{a^{2}} \), where \( \delta_{a} \) denotes the Dirac distribution at point \( a \). Express the result of the limit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Arrange the numbers 1, 2, 3, 4, 5, 6 in the form of rows where the first row contains 1 number, the second row contains 2 numbers, and the third row contains 3 numbers, randomly. Let $N_i$ ($i=1,2,3$) denote the largest number in the $i$-th row, then the number of all permutations satisfying $N_1<N_2<N_3$ is _______ (answer in numerals).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a cinema hall with $N$ numbered seats, each seat is assigned to a specific spectator. The first $N-1$ spectators arrive early and occupy seats randomly. The last spectator, who has a designated seat, attempts to occupy it. If her seat is occupied, she asks the person sitting there to move to their assigned seat. This process continues until all spectators are in their correct seats. What is the expected number of seat changes (displacements) that occur during this process?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three stations are distributed uniformly over a distance of 10 units. Calculate the probability that the maximum distance between any two stations is at most 1 unit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least possible degree of the polynomial \( f(x) \) in the expression \( \frac{d^{100}}{dx^{100}}\left(\frac{p(x)}{x^3-x}\right) = \frac{f(x)}{g(x)} \), where \( p(x) \) is a polynomial that does not have the factor \( x^3-x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two circles $\odot M$ and $\odot N$ intersecting at points $A$ and $B$, and a line $CD$ passing through point $B$ and intersecting circles $\odot M$ and $\odot N$ at points $C$ and $D$ respectively, with $\angle ABD = 60^{\circ}$, find the ratio of the area of quadrilateral $AMBN$ to the area of triangle $ACD$. Express your answer as a ratio in simplest form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of real variables required to describe all 300 by 300 matrices that are their own inverse.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the integer \( n \) such that \( \mathbb{Z}[i]/(-2+2i) \cong \mathbb{Z}/n\mathbb{Z} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Minh enters the numbers from 1 to 81 in a $9\times9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by 3?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a restaurant, men and women arrive at the bathroom at an average rate of one person every 20 minutes. On average, men take 60 seconds to use the facilities, while women take 10 minutes. Determine the long-term ratio of the average queuing time at the women's bathroom to the average queuing time at the men's bathroom.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the function $f(z)$ given that $f = u + iv$ and $if = -v + iu$. Additionally, it is given that $\Re[(1+i)f(z)] = e^{x} (\cos y - \sin y)$. Determine $f(z)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of consecutive positive integers that are singular, where a singular number is defined as a number whose prime factorization contains no even powers other than 0. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this edge.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the derivative of a function f(x) is f'(x), if for any x in R, f'(x) - f(x) < 1, and f(0) = 2022, find the solution set of the inequality f(x) + 1 > 2023e^x. Express your answer as an interval or a set of real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the rectangular coordinate system $(xOy)$, the equations of the ellipses $(C_{1})$ and $(C_{2})$ are given by $( \frac{x^{2}}{4}+y^{2}=1)$ and $( \frac{y^{2}}{16}+ \frac{x^{2}}{4}=1)$ respectively. The ray $(OA)$ intersects $(C_{1})$ and $(C_{2})$ at points $A$ and $B$ respectively, with $( \overrightarrow{OB}=2 \overrightarrow{OA})$. Find the slope of the ray $(OA)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Out of the 21 numbers $8, 9, \cdots, 28$, randomly select three different numbers. What is the probability that their sum is exactly divisible by 9?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an axis of symmetry for the function $f(x) = \cos(2x + \frac{\pi}{6})$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the last two digits of $\binom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g., if the last two digits are $05$, just write $5$).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there of using diagonals to divide a regular 6-sided polygon into triangles such that at least one side of each triangle is a side of the original polygon and that each vertex of each triangle is a vertex of the original polygon?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, $c$, respectively, and $C=\frac{π}{3}$, $c=2$. Then find the maximum value of $\overrightarrow{AC}•\overrightarrow{AB}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the value of the integer $n$ in the expression for the area of the intersection of the surface $x^2 + y^2 + z^2 = 4$ (a sphere with radius 2) and the surface $x^2 + y^2 = 2x$ (a cylinder with radius 1), given that the area can be represented as $n(\pi - 2)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the greatest real number $a$ such that the inequality

$$
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2} \geq a\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}\right)
$$

holds for every five real numbers $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How do you evaluate the line integral $\int_C xdy - ydx$, where C is the curve composed of a straight line segment from $(−2, 0)$ to $(0, 0)$, a straight line segment from $(0, 0)$ to $(0, −2)$, and the part of the circle of radius 2, centered at the origin, traversed counterclockwise starting from $(0,−2)$ and ending at $(−2, 0)$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the random variable ξ follows a normal distribution N(4, 6^2), and P(ξ ≤ 5) = 0.89, find the probability P(ξ ≤ 3). Express your answer as a decimal value between 0 and 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an infinite sequence of natural numbers, the product of any fifteen consecutive terms is equal to one million, and the sum of any ten consecutive terms is equal to \(S\). Find the maximum possible value of \(S\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two players play a game on a pile of \( n \) beans. On each player's turn, they may take exactly \( 1 \), \( 4 \), or \( 7 \) beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many \( n \) between \( 2014 \) and \( 2050 \) (inclusive) does the second player win?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The area of the region bounded by the graph of \[x^2+y^2 = 3|x-y| + 3|x+y|\] is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the even numbers arranged in the pattern shown below, determine in which column the number 2008 will appear.

```
2  4  6  8
16 14 12 10
18 20 22 24
32 30 28 26
...
```
Please reason step by step, and put your final answer within \boxed{}.


Context:
A company is lining a 900-foot path with a fence. Every 6 feet, a fence pole is placed. The entire path is lined with a fence except for a certain length of a bridge. There are 286 fence poles placed along the whole path on both sides. What is the length of the bridge?
Please reason step by step, and put your final answer within \boxed{}.


Context:
在世界杯足球赛前，F 国教练为了考察 A_1, A_2, \cdots, A_7 这七名队员，准备让他们在三场训练比赛（每场 90 分钟）中都上场. 假设在比赛的任何时刻，这些队员中有且仅有一人在场上，且 A_1,A_2,A_3,A_4 每人上场的总时间（以分钟为单位）均能被 7 整除，A_5,A_6,A_7 每人上场的总时间（以分钟为单位）均能被 13 整除. 如果每场换人次数不限，那么，按每名队员上场的总时间计算，共有多少种不同的情况？
Please reason step by step, and put your final answer within \boxed{}.


Context:
The three sides of a triangle are $3$, $5$, and $1-2a$. Find the range of $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A kite-shaped field is planted uniformly with wheat. The sides of the kite are 120 m and 80 m, with angles between the unequal sides being \(120^\circ\) and the other two angles being \(60^\circ\) each. At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. Determine the fraction of the crop that is brought to the longest side of 120 m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of the constant $λ > 0$ such that the equation $2\lfloor{\lambda n}\rfloor = 1 - n + \lfloor\lambda \lfloor\lambda n \rfloor\rfloor$ holds true for all positive integers $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A printing company manufactures a board game that requires two types of figures: dogs and cats, cut from cardboard. In each set, the ratio of cats to dogs is \(5:3\). To minimize waste, they print two types of cardboard: one type contains 28 dogs and 28 cats, and the other type has 42 cats. In what proportion should they print the two types of cardboard so that after cutting, the ratio of the two figures is exactly as needed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties:
[list]
[*]The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and
[*]It is possible to colour the edges of $\mathcal{P}$ with $k{}$ colours such that for every colour $c{}$, and every pair of vertices $(v_1, v_2)$ of $\mathcal{P}$, there is a monochromatic path between $v_1$ and $v_2$ in the colour $c{}$.
[/list]
[i]Viktor Simjanoski, Macedonia[/i]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n$ be a positive natural number. A sequence of $n$ positive integers (not necessarily distinct) is called a "four-group" sequence if it satisfies the following requirements: for any natural $k \geq 2$, if $k$ occurs in this sequence, then $k − 1$ occurs, and the first position at which the number $k − 1$ occurs is earlier than the last position where the number $k$ occurs. How many "four-group" sequences of length $n$ are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Petya conceived a natural number and wrote the sums of each pair of its digits on the board. Then he erased some sums, and the numbers $2,0,2,2$ remained on the board. What is the smallest number Petya could have conceived?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many 1000-digit positive integers have all digits odd, and are such that any two adjacent digits differ by 2?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 2 (17th All-Soviet Union Mathematical Olympiad, 9th Grade) One side of a rectangle is $1 \mathrm{~cm}$. It is known that it is divided into four smaller rectangles by two perpendicular lines, with the area of three of them being no less than $1 \mathrm{~cm}^{2}$, and the area of the fourth one being no less than $2 \mathrm{~cm}^{2}$. How long must the other side of the original rectangle be at least?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \(g:\mathbb{R}^3 \to \mathbb{R}^3\) defined by \(g(x, y, z) = (3x + 4z, 2x - 3z, x + 3y)\) and the region \(S = \{(x, y, z) \in \mathbb{R}^3 : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1\}\), find the value of \(\alpha\) such that the following integral equality holds:

\[\iiint_{g(S)} (2x + y - 2z) \,dxdydz = \alpha \iiint_S z \,dxdydz\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a Markov chain \(\{X_n\}\) with the following transition probability matrix:

\[
P=\left( \begin{array}{ccccccc}
1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 \\
0 & 1/2 & 1/2 & 0 & 0 & 0 & 0 \\
1/4 & 0 & 0 & 1/2 & 1/4 & 0 & 0 \\
0 & 0 & 0& 1/2 & 1/2 & 0 & 0 \\
0 & 0 & 0 & 0 & 1/2 & 1/4 & 1/4 \\
0 & 0 & 0 & 0 & 1/4 & 1/2 & 1/4 \\
0 & 0 & 0 & 0 & 1/2 & 1/2 & 0 \\
\end{array} \right).
\]

Given that \(X_0=1\), calculate the expected number of times the chain visits state 3 before reaching any of the states 5, 6, or 7. Formally, find:

\[
\mathbb{E}\bigg[\sum_{k=1}^{T}\mathbb{I}_{\{X_k=3\}}\bigg],
\]

where \(T=\inf\{k\geq 0: ~X_k\in \{5,6,7\}\}\). Here, \(\mathbb{I}_A\) denotes the indicator function on set \(A\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the surface integral \(\int_S (\nabla \times \vec{F}) \cdot d\vec{S}\) for the vector field \(\vec{F}(x, y, z) = (z, x, y)\) and the surface \(S\), which is the portion of the paraboloid \(z = x^2 + y^2\) lying below the plane \(z = 4\), oriented downward.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a group $G$ with $|G| = 40$ and a set $X$ with $|X| = 67$, where $G$ acts on $X$ resulting in 5 orbits, determine the number of possible lists of orbit lengths. Each orbit length must divide the order of the group $G$, and the sum of the orbit lengths must equal the size of the set $X$. How many such lists are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( \lambda > 0 \) for which the set \( I_{\lambda} \), consisting of all irrational numbers \( \rho = [a_{1}, a_{2}, \ldots, a_{n}, \ldots] \in (0,1) \) with continued fraction partial quotients satisfying \( a_{n} \leq \text{const} \cdot n^{\lambda} \), has a positive Lebesgue measure.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the differential of the multiplication map \( m: G \times G \rightarrow G \) at the identity element \((e,e)\) for a Lie group \( G \). Specifically, find \( dm_{(e,e)}: T_{e}G \oplus T_{e}G \rightarrow T_{e}G \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the simplified form of the expression $E[E(ZX \mid Y) \mid Z,Y]$. Provide your answer in terms of $Z$ and $Y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For discrete probability distributions \( P \) and \( Q \) defined on the same probability space, \( \mathcal{X} \), the relative entropy from \( Q \) to \( P \) is defined as
\[
D_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \log \left(\frac{P(x)}{Q(x)}\right)
\]
Assume the two distributions are \( P=[p, 0, \ldots,0, 1-p] \) and \( Q=\text{binomial}(n,q)=\binom{n}{k} q^{k}(1-q)^{n-k}, \forall k \in \{0,1,2,...n\} \).
Both distributions have \( n+1 \) mass points. Find the parameter \( q \) that minimizes \( D_{\mathrm{KL}}(P \| Q) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the positive integer values of \( k \) for which the expression \( N = 3^{6n-1} - k \cdot 2^{3n-2} + 1 \) is divisible by 7 for every positive integer \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Neville's Algorithm is used to approximate $f(0)$ using the values $f(-2)$, $f(-1)$, $f(1)$, and $f(2)$. If $f(-1)$ is understated by 2 and $f(1)$ is overstated by 3, determine the error in the original calculation of the interpolating polynomial used to approximate $f(0)$. What is the error in the approximation?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the set of values for which the series \( \sum_{n=1}^\infty \left(\frac{n^2+1}{3n^2+4}\right)^n(x-2)^n \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a list of $9$ positive integers that are not necessarily all different, the mean, median, and (unique) mode are all $9$. What is largest possible positive difference between the largest element and the smallest element of the set?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A polygon has interior angles such that the difference between any two consecutive angles is 5 degrees. If the smallest angle is 120 degrees, determine the number of sides of the polygon.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For $u=2+2i$ and $A =\{z \in \mathbb{C}:|z-1|\leq|z-i|\text{ and }|z-u|\leq 1 \}$, determine the value of $|z|$ for which $z \in A$ and the argument of $z$, $\arg(z)$, is minimized.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of possible rays that satisfy Snell's law in the lifeguard problem, given only the angle between the line connecting the two people and the normal.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $b + \frac{1}{b} + b^2 + \frac{1}{b^2} + 1 = 0$, where $b$ is a real number, find the value of $b^{4m} + b^{2m} + \frac{1}{b^{2m}} + \frac{1}{b^{4m}}$ for any positive integer $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the quotient ring \( \mathbb{C}[x,y]/I \) over \( \mathbb{C} \), where \( I=\langle(x+2)^2,(x+2)(y+1),(y+1)^3\rangle \) and \( \mathbb{C}[x,y] \) is the polynomial ring over \( \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F_X(x)$ be the cumulative distribution function (CDF) of a continuous random variable $X$, and let $F_X^{-1}(y)$ be its inverse function. If $V$ is a continuous random variable with a uniform distribution on the interval $[0, 1]$, determine the CDF of the random variable $Z = F_X^{-1}(V)$. Provide your answer in terms of $F_X(x)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression: \[ 3.444 A = \cos^6\left(\frac{3\pi}{2} - \frac{1}{2} \arcsin \frac{3}{5}\right) - \cos^6\left(\frac{5\pi}{2} + \frac{1}{2} \arcsin \frac{4}{5}\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which values of \( p > 0 \) does the integral \( \int_0^{\infty} x \sin(x^p) \, dx \) converge?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the condition under which the function \( \max\{|p(a_j)| : 0 \leq j \leq k\} \), defined on the vector space \( V \) of real polynomials with degree less than or equal to \( n \), constitutes a norm. Assume \( a_j \) are distinct real numbers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a convex quadrilateral with $\angle ABC = 120^{\circ}$ and $\angle BCD = 90^{\circ}$, and let $M$ and $N$ denote the midpoints of $\overline{BC}$ and $\overline{CD}$.  Suppose there exists a point $P$ on the circumcircle of $\triangle CMN$ such that ray $MP$ bisects $\overline{AD}$ and ray $NP$ bisects $\overline{AB}$.  If $AB + BC = 444$, $CD = 256$ and $BC = \frac mn$ for some relatively prime positive integers $m$ and $n$, compute $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
At his craftwork store, Howard has a collection of wooden bowls where he rewards two to his customers for every 10 they buy. If he had 20 customers that day, half of whom bought 20 bowls each, there were 30 bowls remaining in the rewards collection. How many wooden bowls were in the collection initially?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all quadruples $(x,y,z,t)$ of positive integers such that
\[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fifty gallons of tea were poured into 200 containers. Geraldo drank 44 containers. If Geraldo had consumed only the tea's 75% liquid content, how many pints of liquid did Geraldo actually drink, given that 1 gallon is approximately 8 pints and 1 container is 5% foam?
Please reason step by step, and put your final answer within \boxed{}.


Context:
52. On the side $AB$ of the rectangular trapezoid $ABCD (AB \perp BC)$, a semicircle (with the side as the diameter) is constructed, which touches the side $CD$ at point $K$. The diagonals of the trapezoid intersect at point $O$. Find the length of the segment $OK$, if the lengths of the bases of the trapezoid $ABCD$ are 2 and 3.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum number of points that need to be marked inside a convex $n$-gon so that any triangle with vertices at the vertices of the $n$-gon contains at least one marked point?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the polynomial \(P(x) = 5x^6 - ax^4 - bx^3 - cx^2 - dx - 9\), where \(a, b, c, d\) are real numbers. If the roots of \(P(x)\) are in arithmetic progression, determine the value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
3. A $4 \times 4 \times h$ rectangular prism can hold 8 small spheres with a radius of 1 and 1 large sphere with a radius of 2. Then the minimum value of $h$ is $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
The last 5 digits of $99 \times 10101 \times 111 \times 1001001$ are _____.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the following limit using convergence theorems, such as Dominated Convergence:

$$\lim_{n\rightarrow\infty}{\int\int}_{(0,\infty)\times (0,\infty)} \frac{\sin^2(nxy)xy}{e^{(x^2+y^2)}}\,dx\,dy.$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of different connected graphs that can be formed with 6 vertices and 7 edges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the exponential equation for \(x\): $$1000.00116=\frac{1000}{(1+x)^{16}}+\frac{1-(1+x)^{-16}}{x}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Coffee is poured into a truncated cone-shaped cup at a rate of $2 \text{ cc}/\text{sec}$. The cup has an upper radius of $4 \text{ cm}$, a lower radius of $2 \text{ cm}$, and a height of $6 \text{ cm}$. Determine the rate at which the coffee level is rising when it reaches the halfway point. The volume of the coffee in the cup is given by $V = \frac{h\pi}{3}(R^2 + Rr + r^2)$, where $r < R$ and $r = 2 \text{ cm}$. Note that $R$ varies linearly with $h$ as the coffee level rises.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the point of intersection of the functions \( y = e^{-x} \) and \( y = x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the triple integral \( \int_D f(x, y, z) \, dV \) where \( D \) is the region in the first octant \((0, \infty)^3\) between the cone \( z^2 = (x - 1)^2 + y^2 \) and the sphere \( x^2 + y^2 + z^2 = 1 \). The function \( f : D \to \mathbb{R} \) is given by \( f(x, y, z) = z \sqrt{x^2 + y^2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{|z|=1} |z-1|^2 \, dz \), where \( z \) is a complex number and the integration is performed along the circle of radius 1 centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{2\pi} \frac{\cos(t)}{4+\sin(t)} \, dt \) using complex analysis techniques.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 10 Let $a, b, c$ be positive real numbers, find the minimum value of $\frac{a+3 c}{a+2 b+c}+\frac{4 b}{a+b+2 c}-\frac{8 c}{a+b+3 c}$. (3rd China Girls Mathematical Olympiad)
Please reason step by step, and put your final answer within \boxed{}.


Context:
The locus of the centers of all circles of given radius $a$, in the same plane, passing through a fixed point, is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \((x_1, x_2, \ldots, x_n)\) be \(n\) independent random variables, each uniformly distributed as \(x_i \sim U(\theta, 3\theta)\) with an unknown parameter \(0.5 \le \theta \le 1\). Determine the domain of the minimal number of samples \(\min_n\) such that \(n \ge \min_n\) ensures \[ P(2\theta - 0.05 \le \hat{X} \le 2\theta + 0.05) \ge 0.97 \] where \(\hat{X} = \frac{1}{n} \sum_{i=1}^n x_i\).
Please reason step by step, and put your final answer within \boxed{}.