Context:
Given a circle O: $x^2+y^2=1$ and a point $P(-1, \sqrt{3})$, where A and B are two moving points on circle O, find the maximum value of $\angle APB$. Express your answer in radians, using π to represent pi.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function $f(x)$ is defined on the reals such that \( f\left(\frac{1-4x}{4-x}\right) = 4-xf(x) \) for all \( x \neq 4 \). There exist two distinct real numbers \( a, b \neq 4 \) such that \( f(a) = f(b) = \frac{5}{2} \). The sum \( a+b \) can be represented as \( \frac{p}{q} \) where \( p, q \) are relatively prime positive integers. Find \( 10p + q \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ways are there to express $1,000,000$ as a product of exactly three integers greater than $1$? For the purpose of this problem, $abc$ is not considered different from $bac$, etc.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the infimum of the set $\{\lvert\lvert f\rvert\rvert_{\infty} : f \in F\}$, where $F = \{ f \in C[0,1]\: f(0)=1, \int_0^1 (1-t)f(t)dt =2 \}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral $$\iiint (x+y+z) dx\,dy\,dz$$ over the region $$D:=\left \{ (x,y,z) \in\mathbb{R}^3 :x^2+y^2+z^2 \leq 9 \wedge z > y\right \}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-1|\le1$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a standard normal random variable $N \sim \mathcal{N}(0,1)$. Determine if there exists a dependent random variable $D$ such that the random variable $M = N + D$ is still a standard normal distribution. If such a $D$ exists, construct it and find the variance of $D$. Provide your answer in the form of the variance of $D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a, b, c$ be roots of the polynomial $x^3-3x^2-4x+5$. Compute the remainder when $[\frac{a^4+b^4}{a+b} + \frac{b^4+c^4}{b+c} + \frac{c^4+a^4}{c+a}]$ is divided by $7$, where $[x]$ denotes floor function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
把半径为 1 的四个小球叠成两层放在桌面上：下层三个，上层一个，两两相切，求上层小球最高点离桌面的高度与$\frac{2 \sqrt{6}}{3}$的差。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a $100 \times 100$ grid where the cells are colored in black and white such that every column has an equal number of black cells, while each row has a different number of black cells. What is the maximum possible number of adjacent pairs of differently colored cells?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( x, y, z \) be positive real numbers. Define \( M = \max \left\{ xy + \frac{2}{z}, z + \frac{2}{y}, y + z + \frac{1}{x} \right\} \). Find the minimum value of \( M \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 g(x)=\displaystyle\int_x^{\pi}\df{(\sin t)^5}{t}\ \dd t，记 S=\displaystyle\int_0^{\pi}g(x)\ \dd x，求 [100S].
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the product  $$ \prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}. $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest positive integer $n$ whose prime factors are all greater than 18, and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a first-year class of a certain school, there are 1200 students. There are two extracurricular activities for students to choose from, and each student is required to choose at least one to participate in. A statistical survey shows that the number of students choosing one of the activities accounts for 60% to 65% of the total, while the number of students choosing the other activity accounts for 50% to 55% of the total. Calculate the range of possible values for the number of students who choose to participate in both activities. Express your answer as a range of whole numbers, representing the minimum and maximum number of students.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a parallelogram $ABCD$, let $M$ be the point on the $BC$ side such that $MC = 2BM$ and let $N$ be the point on the $CD$ side such that $NC = 2DN$. If the distance from point $B$ to the line $AM$ is $3$, calculate the distance from point $N$ to the line $AM$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A $5\times 5$ grid is to be filled with 12 red tiles and 13 green tiles. Determine the number of distinct ways to fill the grid, considering rotations and reflections of the grid as equivalent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $\left( r + \frac{1}{r} \right)^2 = 5,$ then find $r^3 + \frac{1}{r^3}.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct minimal polynomials of degree 3 exist for elements of a finite field extension $L/K$ of degree 3?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of roots of the polynomial equation \( f(z) = z^4 + 8z^3 + 3z^2 + 8z + 3 = 0 \) that lie in the right half of the complex plane.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of solutions to the equation \(6\ln(x^2+1)-e^x = 0\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. After that, at what time do the hour hand and the minute hand form a 90-degree angle for the 6th time? (12-hour format)
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 5555 children, numbered 1 to 5555, sitting around a circle in order. Each child has an integer in hand: the child numbered 1 has the integer 1, the child numbered 12 has the integer 21, the child numbered 123 has the integer 321, and the child numbered 1234 has the integer 4321. It is known that the sum of the integers held by any 2005 consecutive children is equal to 2005. What is the integer held by the child numbered 5555?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 10\}$, including the empty set, are spacy?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many 12-digit positive integers have all digits either 1 or 2, and have numbers ending in 12, but have no two consecutive 1's?
Please reason step by step, and put your final answer within \boxed{}.


Context:
将一枚骰子连续投掷五次，则事件 "五次出现的点数既不全相同，也不两两互异，且从第二次起每一次的点数都不小于前一次的点数"的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
In trapezoid \(ABCD\), a point \(X\) is taken on the base \(AD\) such that segments \(XB\) and \(XC\) divide the trapezoid into three similar, yet pairwise unequal, non-isosceles triangles. The side \(AB\) has a length of 6. Find \(AX \cdot DX\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. Given a six-digit decimal number composed of six positive integers, the digit in the units place is a multiple of 4, the digits in the tens and hundreds places are multiples of 3, and the sum of the digits of the six-digit number is 21. Then the number of six-digit numbers that satisfy the above conditions is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $I_n=\int_0^{\frac {\pi}{2}} \frac {\sin ((2n+1)x)dx}{\sin x}$, then evaluate $$\lim_{n\to \infty} I_n$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
## Task Condition

Find the derivative of the specified order.

$y=\frac{\log _{3} x}{x^{2}}, y^{IV}=?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $f(x) = (m^2 - m - 1)x^{-5m-3}$ is a power function, and it is increasing on the interval $(0, +\infty)$, determine the value of $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A casino table pays 3:2 if you get a blackjack with your first 2 cards. If you scored a blackjack and had an original bet of $40, how much were you paid?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the determinant of the following matrix A:

$$\begin{pmatrix}
5 & 3 & 3 & . & . & . & 3 \\
3 & 5 & 3 & . & . & . & 3 \\
3 & 3 & 5 & . & . & . & 3 \\
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
. & . & . & . & . & . & . \\
3 & 3 & 3 & . & . & . & 5
\end{pmatrix}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the function $y=x^2+2x+a^2-1$ reaches its maximum value of 16 on the interval $[1, 2]$, find the value of the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each of the spots in a  $8\times 8$  chessboard is occupied by either a black or white “horse”. At most how many black horses can be on the chessboard so that none of the horses attack more than one black horse?**Remark:** A black horse could attack another black horse.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a tetrahedron \(ABCD\), the sum of the angles \(BAC\) and \(BAD\) is \(180^{\circ}\). \(AK\) is the angle bisector of \(\angle CAD\). Find the measure of \(\angle BAK\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular tetrahedron $ABCD$ with edge length 1, point $M$ is the midpoint of $AC$, and point $P$ is on line segment $DM$. Find the minimum value of $AP + BP$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a geometric sequence $\{a_n\}$, the first term is $\frac{3}{2}$, and the common ratio is $-\frac{1}{2}$. The sum of the first $n$ terms is $S_n$. When $n \in N^*$, the ratio of the maximum and minimum values of $S_n - \frac{1}{S_n}$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 50 students in the class. Half of them play basketball, one-third play volleyball, one-fifth play soccer, and one-eighth play badminton. One-tenth of the students play both basketball and volleyball, one-twelfth play both basketball and soccer, one-sixteenth play both basketball and badminton, and one-eighth play both volleyball and soccer. Additionally, one-tenth play both volleyball and badminton, and one-twentieth play both soccer and badminton. Finally, 1/25 of the students play all four sports: basketball, volleyball, soccer, and badminton. How many students in this class do not engage in any of these sports?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a, b \in \mathbb{R}$ and $c \in [0, 2\pi)$, if $2\sin\left(3x - \frac{\pi}{3}\right) = a\sin(bx + c)$ holds for any real number $x$, find the number of ordered triplets of real numbers $(a, b, c)$ that satisfy the condition.
Please reason step by step, and put your final answer within \boxed{}.


Context:
若函数$f\left( x \right)$满足：对任意实数$x$，有$f\left( 2-x \right)+f\left( x \right)=0$且$f\left( x+2 \right)+f\left( x \right)=0$，当$x\in $[0,1]时，$f\left( x \right)=-{{\left( x-1 \right)}^{2}}$,则$~x\in $[2017,2018]时,$~f\left( x \right)=$______________________________．
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an eight-digit palindrome that is a multiple of three, composed of the digits 0 and 1, given that all its prime divisors only use the digits 1, 3, and %. (Palindromes read the same forwards and backwards, for example, 11011).
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a rational number $r$, its *period* is the length of the smallest repeating block in its decimal expansion. For example, the number $r = 0.123123123\ldots$ has period $3$. If $S$ denotes the set of all rational numbers of the form $r = \overline{abcdefgh}$ having period $8$, find the sum of all elements in $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given three non-collinear points \( A, B, C \) on a plane, construct a line \( l \) such that the ratio of the distances from points \( A, B, C \) to line \( l \) is \( 1: 1: 2 \) or \( 1: 2: 1 \) or \( 2: 1: 1 \). Calculate the total number of such lines. Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Sidney follows a pattern in doing jumping jacks: on Monday, he does 20, on Tuesday he does 36, and he continues to add an increasing even number of jumping jacks each day for the next five days. Brooke does four times as many jumping jacks as Sidney each day. How many jumping jacks did Brooke do during the entire eight-day period?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum sample size for which the precision of the estimate of the population mean $a$ based on the sample mean with a confidence level of 0.975 is $\delta=0.3$, given that the standard deviation $\sigma=1.2$ of the normally distributed population is known.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $V$ be a vector space with an inner product $(\cdot,\cdot)$. Consider the function $f:V\rightarrow \mathbb{R}$ defined by $f(u) = (u,u)$. Determine the Fréchet derivative of $f$ at a point $u \in V$, denoted as $f'(u)$. Provide your answer in terms of the inner product.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many natural number \( n \) less than 2015 that is divisible by \( \lfloor \sqrt[3]{n}\rfloor \) ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be a finite set of points in the plane. A linear partition of $S$ is an unordered pair $\{A,B\}$ of subsets of $S$ such that $A \cup B = S$, $A \cap B = \emptyset$, and $A$ and $B$ lie on opposite sides of some straight line disjoint from $S$ ($A$ or $B$ may be empty). Let $L_S$ be the number of linear partitions of $S$. For each positive integer $n$, find the maximum of $L_S$ over all sets $S$ of $n$ points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the function $f(x) = a|2x - 4|$ ($a > 0$, $a \neq 1$) and $f(1) = 9$, then the interval of monotonic decrease for $f(x)$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Rectangle $ABCD$ has $AB = 6$ and $BC = 3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
a group of people participate in some curriculum , 25 of them practice yoga , 15 study cooking , 8 study weaving , 2 of them study cooking only , 7 of them study both the cooking and yoga , 3 of them participate all curriculums . how many people study both cooking and weaving ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A right circular cone ($S-ABC$) has its apex $S$ perpendicular to the plane $ABC$, with $SA = 3$. The base triangle $△ABC$ is an equilateral triangle with side length $2$. Calculate the surface area of the circumscribed sphere.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function f(x) = ln((x^2 + 1)) / (x + 4), find the x-value such that the functions y = f(3-x) and y = f(3+x) are symmetric about a line x = a. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
若 $k$ 是一个正整数,且 $2^k$ 整除 $C_{4006}^0+C_{4006}^2 3^1+\cdots+C_{4006}^{2 i} 3^i+\cdots+C_{4006}^{4006} 3^{2003}$, 则 $k$ 的最大值为 $\qquad$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all integer values of \( n \) such that \( 19 \mid 2^{3n+4} + 3^{2n+1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the greatest common divisor of the numbers:
\[ 5^5 - 5, \ 7^7 - 7, \ 9^9 - 9, \ \ldots, \ 2017^{2017} - 2017 \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
在平面直角坐标系 xOy 中，\Gamma_1 是以 (2,1) 为圆心的单位圆，\Gamma_2 是以 (10,11) 为圆心的单位圆. 过原点 O 作一条直线 l，使得 l 与 \Gamma_1,\Gamma_2 各有两个交点，将 \Gamma_1, \Gamma_2 共分成四段圆弧，且这四段圆弧中有两段等长. 所有满足条件的直线 l 的斜率之和为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sides of a parallelogram are \( a \) and \( b \). What are the lengths of the diagonals of the quadrilateral formed by the angle bisectors of the parallelogram?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \((B_2 - B_1) \sim N(0, 3)\), find \(\mathbb{E}[B_1 | B_2]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $x$, $y$, and $z$ are nonzero complex numbers such that $(x+y+z)(x^2+y^2+z^2)=x^3+y^3+z^3$. Compute $$(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right).$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
The cars Niva and Toyota are driving on a ring track of a test site, a quarter of which is on a dirt road and the remaining part is on asphalt. The speed of the Niva on the dirt road is 80 km/h, and on asphalt it is 90 km/h. The speed of the Toyota on the dirt road is 40 km/h, and on asphalt it is 120 km/h. The cars start simultaneously at the beginning of the dirt road section and first drive through this dirt part. On which lap will one of the cars overtake the other for the first time?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(15^{-1} \equiv 31 \pmod{53}\), find \(38^{-1} \pmod{53}\), as a residue modulo 53.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer that can be expressed as the sum of the cubes of three positive integers in two different ways.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the square root of 4. Express your answer in the form ± followed by the absolute value of the root.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a cube $A B C D A 1 B 1 C 1 D 1$. A sphere touches the edges $A D, D D 1, C D$ and the line $B C 1$. Find the radius of the sphere, if the edge of the cube is 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given linear operators $X$ and $P$ on a complex vector space $V$, suppose the commutator $[X,P] = ciI$ for some real, positive constant $c$. What can be concluded about the dimension of $V$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Point \((x,y)\) is randomly picked from the rectangular region with vertices at \((0,0), (3000,0), (3000,3001),\) and \((0,3001)\). What is the probability that \(x > 3y\)? Express your answer as a common fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The cost of 60 copies of the first volume and 75 copies of the second volume is 2700 rubles. In reality, the total payment for all these books was only 2370 rubles because a discount was applied: 15% off the first volume and 10% off the second volume. Find the original price of these books.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability that a complete bipartite random graph $G(n,n,p)$ is disconnected when $np=\ln n + w(n)$ where $w(n)\rightarrow -\infty$ when $n\rightarrow\infty$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given three vectors $v_1, v_2, v_3 \in \mathbb{R}^2$, if the minimum eigenvalue of $v_i v_i^T + v_j v_j^T$ is less than or equal to 1 for all distinct $i, j \in \{1, 2, 3\}$, determine an upper bound for the minimum eigenvalue of $v_1 v_1^T + v_2 v_2^T + v_3 v_3^T$ under the constraint $|z_1^2 + z_2^2 + z_3^2| = 0$, where $z_j = \alpha_j + i \beta_j$ and $v_j = [\alpha_j, \beta_j]^T$. Provide your answer as an expression or a number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the convergence interval for the series $\sum_{n=1}^{\infty} \frac{(n+1) \sqrt{n+1}}{4^{n}}(x-1)^{n}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
函数 f(x)=\df{x+1}{x}+\df{x+2}{x+1}+\cdots+\df{x+2023}{x+2022} 的对称中心为 (a,b)，则 2a+b=__________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using Cauchy's integral formula, determine a tighter upper bound for \(|f^{(4)}(0)|\) for all functions \(f\) in the family \(\mathscr{F}\) as defined in the original context.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many lattice points $(v, w, x, y, z)$ does a 5-sphere centered on the origin, with radius 3, contain on its surface or in its interior?
Please reason step by step, and put your final answer within \boxed{}.


Context:
【Example 5】 There are $n$ young people and $r$ elderly people, $n>2r$, to be arranged in a row. It is required that each elderly person has a young person on both sides to support them (but each pair of young people only supports one elderly person). How many different arrangements are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define $x \star y=\frac{\sqrt{x^{2}+3 x y+y^{2}-2 x-2 y+4}}{x y+4}$. Compute $$((\cdots((2007 \star 2006) \star 2005) \star \cdots) \star 1)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the $xOy$ plane, consider two lines $A_{1}B_{1}$ and $A_{2}B_{2}$ passing through a point $P$ on the $x$-axis, where the points $A_{1}$, $B_{1}$, $A_{2}$, and $B_{2}$ lie on the parabola $\Gamma: y^{2}=x$. The lines $A_{1}B_{2}$ and $A_{2}B_{1}$ intersect the $x$-axis at points $S$ and $T$, respectively. Determine whether the magnitude of $\overrightarrow{OS} \cdot \overrightarrow{OT}$ is greater than, less than, or equal to $|OP|^{2}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Circle $A$ has radius $100$. Circle $B$ has an integer radius $r<100$ and remains internally tangent to circle $A$ as it rolls once around the circumference of circle $A$. The two circles have the same points of tangency at the beginning and end of circle $B$'s trip. How many possible values can $r$ have?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the divisor function $d(\phi(n)) = 2017$, where $d()$ counts the number of divisors and $\phi()$ is the totient function, find the smallest natural number $n$. This problem is from the 2017 Regional Mathematical Olympiad (RMO) in Maharashtra and Goa, India.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of \(x^{46}\) in the generating function \(\frac{1}{1 - x^3 - x^4 - x^{20}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the rate of population growth \( \frac{dP}{dt} = \frac{0.2P}{4(4-\frac{P}{400})} \), and an initial population of 3 at time \( t = 0 \), find the time \( t \) when the population \( P \) will reach 1200.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the prime factorization of \(1007021035035021007001\). (You should write your answer in the form \(p_{1}^{e_{1}} p_{2}^{e_{2}} \ldots p_{k}^{e_{k}}\), where \(p_{1}, \ldots, p_{k}\) are distinct prime numbers and \(e_{1}, \ldots, e_{k}\) are positive integers.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of field extensions between the fields $\mathbb{Q}(\zeta^3)$ and $\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive 12th-root of unity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct cycle subgraphs of the complete bipartite graph $K_{3,3}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number of real roots of the equation $\frac{x}{100} = \sin x$ is:
(32nd United States of America Mathematical Olympiad, 1981)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $h$ and $k$ be functions of $u$ and $v$ such that $h(2, 1) = 1$ and $k(2, 1) = 3$. At the point $(2, 1)$, the differentials are given by $dh = 3du - 2dv$ and $dk = -du + 4dv$. The function $w$ is defined implicitly by the equation $w^3h - wk + u + v = 0$, with the condition $w(2, 1) = 2$. Find the value of $\frac{\partial w}{\partial v}$ at the point $(2, 1)$. \(\boxed{\text{Answer}}\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Approximately calculate the value of the function \( f(x) = \sqrt{x^{2} + x + 3} \) at \( x = 1.97 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\omega_{1}, \omega_{2}, \ldots, \omega_{100}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set $S=\left\{\omega_{1}^{1}, \omega_{2}^{2}, \omega_{3}^{3}, \ldots, \omega_{100}^{100}\right\}$. Let $M$ be the maximum possible number of unique values in $S$, and let $N$ be the minimum possible number of unique values in $S$. Find $M-N$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the odd function $f(x)$ is a monotonically decreasing function on the interval $[-1,0]$, and $\alpha, \beta$ are acute angles of a triangle, determine the relationship between $f(\cos \alpha)$ and $f(\sin \beta)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of the image of the homomorphism \( \varphi: \mathbb{Z}^2 \to S_{15} \) given by \( \varphi(1,0)=(2,7,3)(3,11,5)(12,13) \) and \( \varphi(0,1)=(14,15)(1,4,6,8,9,10) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the flux of the vector field \( F(x,y,z)=(z^2-x,-xy,3z) \) across the surface \( S \) of the solid bounded by the equations \( z=4-y^2, x=0, x=3 \), and \( z=0 \), with the normal vector pointing outward. Compute \( \iint_S F \cdot n \,ds \) without using the Divergence Theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the largest number of white and black chips that can be placed on a chessboard so that on each horizontal and each vertical, the number of white chips is exactly twice the number of black chips?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sine function $y = \sin x$ has an arc length from $0$ to $2\pi s$ given by:

$$ L = \int_0^{2\pi s} \sqrt{1 + \cos^2 x}\, dx $$

where $s \in \mathbb{Q}$ and $s > 1$. If the period of the sine function is changed from $2\pi$ to $\frac{4\pi}{3}$, determine the new amplitude of $\sin x$ (currently $|A|=1$) such that the arc length from $0$ to $2\pi s$ remains unchanged.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, medians $\overline{AM}$ and $\overline{BN}$ are perpendicular. If $AM = 15$ and $BN = 20$, and the height from $C$ to line $AB$ is $12$, find the length of side $AB$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ordered pairs \((\mathcal{A}, \mathcal{B})\) of nonempty sets such that \(\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}\), \(\mathcal{A} \cap \mathcal{B} = \emptyset\), the number of elements in \(\mathcal{A}\) is not an element of \(\mathcal{A}\), and the number of elements in \(\mathcal{B}\) is not an element of \(\mathcal{B}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many ordered pairs of sets \((A, B)\) have the properties:

1. \( A \subseteq \{1, 2, 3, 4, 5, 6\} \)
2. \( B \subseteq \{2, 3, 4, 5, 6, 7, 8\} \)
3. \( A \cap B \) has exactly \(3\) elements.
Please reason step by step, and put your final answer within \boxed{}.


Context:
You are allowed to throw a standard die 10 times initially. Each time you roll a 6, you are allowed to throw 5 more times. Let N be the total number of throws. Calculate the expected value of \( E[N^2] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the price of Product A was set at 70 yuan per piece in the first year, with an annual sales volume of 118,000 pieces, starting from the second year, the price per piece increased by $$\frac {70\cdot x\%}{1-x\%}$$ yuan due to a management fee, and the annual sales volume decreased by $10^4x$ pieces, calculate the maximum value of x such that the management fee collected in the second year is not less than 1.4 million yuan.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The length of a rectangle is increased by $ 10\%$ and the width is decreased by $ 10\%$. What percent of the old area is the new area?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the linear function \( y = ax + b \) and the hyperbolic function \( y = \frac{k}{x} \) (where \( k > 0 \)) intersect at points \( A \) and \( B \), with \( O \) being the origin. If the triangle \( \triangle OAB \) is an equilateral triangle with an area of \( \frac{2\sqrt{3}}{3} \), find the value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a hyperbola with equation x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0), let the left and right foci be F1 and F2, respectively. A circle centered at F2 with radius |F1F2| intersects the hyperbola in the first and second quadrants at points A and B, respectively. If 3|F1B| = |F2A|, find the eccentricity of the hyperbola.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The random variable $X$ follows a normal distribution $N(1,4)$. Given that $P(X \geqslant 2) = 0.2$, calculate the probability that $0 \leqslant X \leqslant 1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If m is an integer such that a certain number squared by 2m equals 2 ^ ( 18 - m ), and m equals 6, what is that certain number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the radius of convergence of the power series \( \sum_{n=1}^{\infty}c_{n}z^{n} \) is finite and nonzero, find the radius of convergence of the series \( \sum_{n=1}^{\infty}c_{n}z^{n^{2}} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
During my workout yesterday, I did 30 squats. Each day, I plan to increase my number of squats by 5 more than the previous day. If I do my workout for four consecutive days, how many squats will I perform the day after tomorrow?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bill buys a stock for $100. On the first day, the stock decreases by $25\%$, on the second day it increases by $35\%$ from its value at the end of the first day, and on the third day, it decreases again by $15\%$. What is the overall percentage change in the stock's value over the three days?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum number of charts required to cover a torus.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sum of 99 consecutive natural numbers equals $abcd$, where $a$, $b$, $c$, and $d$ are all prime numbers. What is the minimum value of $a+b+c+d$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are perpendicular and $|\overrightarrow{OA}| = |\overrightarrow{OB}| = 24$. If $t \in [0,1]$, the minimum value of $|t \overrightarrow{AB} - \overrightarrow{AO}| + \left|\frac{5}{12} \overrightarrow{BO} - (1-t) \overrightarrow{BA}\right|$ is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
A doctor told Mikael to take a pill every 75 minutes. He took his first pill at 11:05. At what time did he take his fourth pill?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the probability of never losing an infinite game where you start with a single four-sided die numbered 1 to 4? In each round, rolling a 1 loses the die, a 2 keeps the die, a 3 or 4 gains an additional die. You lose the game when you have no dice left.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A function $f(x)$ is defined for all real numbers $x$. For all non-zero values $x$, we have
\[3f\left(x\right) + f\left(\frac{1}{x}\right) = 6x + \sin x + 3\]
Let $S$ denote the sum of all of the values of $x$ for which $f(x) = 1001$. Compute the integer nearest to $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute
$$\sum^{\infty}_{m=1} \sum^{\infty}_{n=1} \frac{m\cos^2(n) + n \sin^2(m)}{3^{m+n}(m + n)}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be the set of integers of the form $2^{x}+2^{y}+2^{z}$, where $x, y, z$ are pairwise distinct non-negative integers. Determine the 100th smallest element of $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the endomorphism $\phi: f(x) \mapsto x^2 f(x)$ has an infinite norm in the space $C(\mathbb{R})$ of continuous functions on $\mathbb{R}$, equipped with the norm $\|f\| = \sup_{x \in \mathbb{R}} |f(x)|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let real numbers \(x\) and \(y\) be such that \(x-y\), \(x^2-y^2\), and \(x^3-y^3\) are all prime numbers. Find the value of \(x-y\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). Given that \(AB = AD\) and \(CA\) is the bisector of angle \(C\), with \(\angle BAD = 140^\circ\) and \(\angle BEA = 110^\circ\), find angle \(CDB\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area enclosed between the curves \( r = \frac{1}{\phi} \) and \( r = \frac{1}{\sin\phi} \) for \( \phi \in (0, \frac{\pi}{2}) \) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Danielle Bellatrix Robinson is organizing a poker tournament with 9 people. The tournament will have 4 rounds, and in each round the 9 players are split into 3 groups of 3. During the tournament, each player plays every other player exactly once. How many different ways can Danielle divide the 9 people into three groups in each round to satisfy these requirements?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the ellipse $\frac{x^{2}}{49}+ \frac{y^{2}}{24}=1$, if the angle between the lines connecting a point $P$ on the ellipse and the two foci $F\_1$ and $F\_2$ is a right angle, then $|PF\_1| \cdot |PF\_2| =$ _____ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int_{-\infty }^{\infty } \frac{-1+e^{-i k x}}{x^2} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If both $a$ and $b$ in the fraction $\frac{a+b}{a^2+b^2}$ are enlarged by a factor of $2$, then calculate the new value of the fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each side of the square grid is 15 toothpicks long. Calculate the total number of toothpicks used to construct the square grid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( x \) and \( y \) are positive numbers, determine the minimum value of \(\left(x+\frac{1}{y}\right)^{2}+\left(y+\frac{1}{2x}\right)^{2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a$ be the greatest natural number such that $4^{2016} + 4^a + 4^{37}$ is the square of a natural number. Determine $a \pmod{37}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $A B C$ is a triangle with incircle $\omega$, and $\omega$ is tangent to $\overline{B C}$ and $\overline{C A}$ at $D$ and $E$ respectively. The bisectors of $\angle A$ and $\angle B$ intersect line $D E$ at $F$ and $G$ respectively, such that $B F=1$ and $F G=G A=6$. Compute the radius of $\omega$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Lauren sent 65 pieces of mail on Monday. On Tuesday, she sent twice as many pieces as she sent on Monday, minus 5 pieces. On Wednesday, she sent 5 fewer pieces than twice the amount she sent on Tuesday, and on Thursday, she sent the square root of the number of pieces sent on Wednesday multiplied by 10. On Friday, Lauren sent the average number of pieces she sent on the previous days plus 15 pieces. How many pieces of mail did Lauren send in total?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the maximum value of the function $f(x)=a^{x} (a > 0, a \neq 1)$ on $[-2,1]$ is $4$, and the minimum value is $m$, what is the value of $m$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int\limits_{\mathbb{R}^n} -\Delta \Phi(x) \, dx \), where \( \Phi(x) = \frac{1}{\alpha(n)}((\lvert x\rvert^2 + 1)^{\frac{-n-2}{2}}\lvert x\rvert^2 + (\lvert x\rvert^2 + 1)^{-n/2}) \) and \( x \in \mathbb{R}^n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An isosceles triangle with a base of $\sqrt{2}$ has medians intersecting at a right angle. What is the area of this triangle?
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
A line $l$ passing through the origin intersects the circle $x^2+y^2=1$ at points $P$ and $Q$. Point $A$ is the intersection of the circle and the negative semi-axis of the $x$-axis. The circle with $AQ$ as the diameter intersects line $l$ at point $N$, which is different from point $Q$. The product of the slopes of lines $AN$ and $AP$ is equal to 1. Determine the equation of line $l$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_n = \frac{1 + x_{n -1}}{x_{n - 2}}$ for $n \geq 2$. Find the number of ordered pairs of positive integers $(x_0, x_1)$ such that the sequence gives $x_{2018} = \frac{1}{1000}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the sum of  $x^2+y^2$  over all four ordered pairs  $(x,y)$  of real numbers satisfying  $x=y^2-20$  and  $y=x^2+x-21$ .

*2021 CCA Math Bonanza Lightning Round #3.4*
Please reason step by step, and put your final answer within \boxed{}.


Context:
If all $x_{k}$ ($k=1,2,3,4,5$) are positive reals, and {$a_{1},a_{2},a_{3},a_{4},a_{5}$} = {$1,2,3,4,5$}, find the maximum of
$$\frac{(\sqrt{s_{1}x_{1}}+\sqrt{s_{2}x_{2}}+\sqrt{s_{3}x_{3}}+\sqrt{s_{4}x_{4}}+\sqrt{s_{5}x_{5}})^{2}}{a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}+a_{4}x_{4}+a_{5}x_{5}}$$

($s_{k}=a_{1}+a_{2}+...+a_{k}$)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate plane \(xOy\), given that points \(A\) and \(B\) lie on the hyperbola \(C: 2x^{2} + 4x - y^{2} = 0\), and \(\triangle ABC\) is an isosceles right triangle with \(O\) as the right-angle vertex. Find the total number of such \(\triangle OAB\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given three positive real numbers $a, b, c$ satisfying $abc + a + c = b$, find the maximum value of the expression:
\[ P = \frac{2}{a^2 + 1} - \frac{2}{b^2 + 1} + \frac{3}{c^2 + 1}. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Example 5 (2005 National High School Mathematics Competition Question) Define the function
$$f(k)=\left\{\begin{array}{l}
0, \text { if } k \text { is a perfect square } \\
{\left[\frac{1}{\{\sqrt{k}\}}\right], \text { if } k \text { is not a perfect square }}
\end{array} \text {, find } \sum_{k=1}^{240} f(k)\right. \text {. }$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \(\int_{|z|=1}\sin(1/z)\cos(1/z)\,dz\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $T$ is a $C_0$-semigroup with $\omega \leq 0$ and $M \geq 0$ such that $\| T(t) \| \leq M e^{\omega t}$. Let $A$ be the generator for $T$ and let $\xi_1, ..., \xi_n > \omega$ be constants. Obtain a bound for $\| (\xi_1 \cdots \xi_n)^{-1} (\xi_1^{-1} - A)^{-1} \cdots (\xi_n^{-1} - A)^{-1} \|$ in terms of $M$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are a total of 4 lamps in the classroom. Each is controlled by a separate switch. The total number of ways to turn on the lights in the classroom is $\_\_\_\_\_\_\_\_\_\_\_$ (Answer in digits).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If there is a positive integer \( m \) such that the factorial of \( m \) has exactly \( n \) trailing zeros, then the positive integer \( n \) is called a "factorial trailing number." How many positive integers less than 1992 are non-"factorial trailing numbers"?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the nth order Taylor polynomial for cos x centered at 0 denoted T(n) (x,0). How large must we take n to guarantee that the error |cos x-T(n) (x,0) |is at most 10^-3 for x in [-pi/2,pi/2)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of squares in the sequence given by $a_0 = 91$ and $a_{n+1} = 10a_n + (-1)^n$ for $n \ge 0.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of the function \( f \) defined for integers \( k \) by

\[
f(k) = (k)_{3} + (2k)_{5} + (3k)_{7} - 6k
\]

where \( (k)_{2n+1} \) denotes the multiple of \( 2n+1 \) closest to \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a triangle with vertices A, B, and C, where random points (X, Y) are uniformly distributed along the line segments AB and AC, find the covariance of X and Y. The line segments AB and AC are defined as follows: AB consists of points (x, 0) for 0 ≤ x ≤ 1, and AC consists of points (0, y) for 0 ≤ y ≤ 1. Consider the distribution of XY first.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the geometric series $6 - \frac{12}{5} + \frac{36}{25} - \dots$, calculate the limiting sum of this series as the number of terms increases without bound.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Cauchy Principal Value of the integral \( \int_0^\infty \frac{x^{-p}}{x-1} \, dx \) for \(|p|<1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In order to obtain steel for a specific purpose, the golden section method was used to determine the optimal addition amount of a specific chemical element. After several experiments, a good point on the optimal range $[1000, m]$ is in the ratio of 1618, find $m$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sample \(X_1, X_2, \ldots, X_n\) from a random variable \(X\) with probability density function (pdf) \(f(x|\theta) = \frac{1}{\sqrt{2\pi}}\exp\left(-\frac{1}{2}(x-\theta)^2\right)\), find the 1-dimensional sufficient statistic for the parameter \(\theta\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given three iid Unif(0,1) random variables a, b, and c, what is the probability that c is the maximum if a+b < 1?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Knowing that:
$$I_n=\int_0^{\frac{\pi}{2}}\cos^n(t) \, dt$$
$$I_{2n}=\frac{1\times 3\times \ldots \times (2n-1)}{2\times 4\times \ldots\times 2n}\times\dfrac{\pi}{2}\quad \forall n\geq 1$$
$$I_{n}\sim \sqrt{\dfrac{\pi}{2n}}$$
Calculate:
$$ \lim_{n\to+\infty} \frac{1\times 3\times \ldots \times (2n+1)}{2\times 4\times \ldots\times 2n}\times\dfrac{1}{\sqrt{n}}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
设集合 $M=\{1,2, \cdots, 10\}, T$ 为 $M$ 的某些二元子集构成的集合，满足对 $T$ 中任意两个不同的元素 $\{a, b\} 、\{x, y\}$ ，均有
$11 \nmid(a x+b y)(a y+b x)$ 。
求 $T$ 的元素个数的最大值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
36. Using a protractor with only $0^{\circ}, 4^{\circ}, 10^{\circ}, 16^{\circ}, 43^{\circ}, 89^{\circ}$ and $180^{\circ}$ markings, the maximum number of angles (not greater than $180^{\circ}$) that can be directly measured is $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the correct operating point \((x_{10}, x_{20}, u_{0})\) for the dynamic system described by the ODE \(10\ddot{y}(t) + \dot{y}(t) = u^{2}(t)\) when linearizing around \(u=1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the definite integral:

$$
\int_{0}^{\frac{\pi}{2}}\left(x^{2}-5x+6\right) \sin 3x \, dx
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the constant $c$ such that the equation \( \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N \) holds true for any positive integer \( N \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the right triangle \(ABC\), the lengths of the legs are \(AC = 3\) and \(BC = 4\). Point \(A\) is moved to point \(A_1\) parallel to \(BC\). Then point \(B\) is moved to point \(B_1\) parallel to the line \(A_1C\). Finally, point \(C\) is moved to point \(C_1\) parallel to the line \(A_1B_1\) such that the resulting triangle \(A_1B_1C_1\) is a right triangle at \(B_1\) and the length of the leg \(A_1B_1\) is 1 unit. What is the length of the leg \(B_1C_1\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the smallest $k$ such that $k$ cells can be marked on a $10 \times 11$ board so that any placement of a three-cell L-shaped piece will touch at least one marked cell?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $\vartriangle ABC$, $D$ is on $AC$ such that $BD$ is an altitude, and $E$ is on $AB$ such that $CE$ is an altitude. Let F be the intersection of $BD$ and $CE$. If $EF = 2FC$, $BF = 8DF$, and $DC = 3$, then find the area of $\vartriangle CDF$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit of the following expression as $n$ approaches infinity:

$$n\left(\sum_{k=1}^n \sqrt{1+\frac{k}{n^2}} - n - \frac{1}{4} - \frac{5}{24n}\right)$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the limit $\lim_{n \to \infty} \left(\frac{(kn)!}{n!(kn-n)!}\right)^{1/n}$ using the Stirling approximation and Taylor series expansion.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac{2}{3}$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac{3}{4}$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( g(z) = z^4 + iz^3 + 1 \). Determine the number of zeros that \( g \) has in the region \( \{z \in \mathbb{C}: \text{Re}(z), \text{Im}(z) > 0\} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $g(y)$ is a differentiable function on $\mathbb{R}$ satisfying the equation \( \{g(y)\}^4 - y\{g(y)\}^3 + y^2\{g(y)\}^2 - 3y^3 + 8y^4 - 8y^5 = 0 \) for each \( y \in \mathbb{R} \). Determine the value of \( g'(1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the sequence $\{y_n\}$ defined by the equation $y_n^3 - (n^2 - 5n)y_n + 5n^2 = 0$ for all $n \in \mathbb{N}$. Determine the limit of $y_n$ as $n$ approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is $9$, spots a license plate with a $4$-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Find the age that is not one of Mr. Jones's children.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Segment $A B$ is the diameter of a circle. A second circle with center at point $B$ has a radius of 2 and intersects the first circle at points $C$ and $D$. Chord $C E$ of the second circle is part of a tangent to the first circle and is equal to 3. Find the radius of the first circle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
7. Xiao Ming and Xiao Hua are counting the picture cards in a box together. Xiao Ming is faster, and in the time it takes him to count 6 cards, Xiao Hua can only count 4. When Xiao Hua counted to 48 cards, he forgot the number and had to start over. When he counted to 112 cards, there was only 1 card left in the box. The box originally had ( ) picture cards.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic complexity \( \Theta \) of the recurrence relation: \[ t(n) = 2^n t(n/2) + n. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given 3 cities, each pair of cities is connected by a road with an exponentially distributed cost with parameter 1. Calculate the expected value of the minimum cost to connect all cities, denoted as \( E(c_3) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four, $n^{2}(n \geqslant 4)$ positive numbers are arranged in $n$ rows and $n$ columns;
$$
\begin{array}{llllll}
a_{11} & a_{12} & a_{13} & a_{14} & \cdots & a_{1 n} \\
a_{21} & a_{22} & a_{23} & a_{24} & \cdots & a_{2 n} \\
a_{31} & a_{32} & a_{33} & a_{34} & \cdots & a_{3 n} \\
a_{41} & a_{42} & a_{43} & a_{44} & \cdots & a_{4 n} \\
\cdots \cdots
\end{array}
$$

where the numbers in each row form an arithmetic sequence, and the numbers in each column form a geometric sequence, with all common ratios being equal. Given $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, find
$$
a_{11}+a_{22}+a_{33}+a_{44}+\cdots+a_{n n} .
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the functions $f(x)=x^{2}-2x+2$ and $g(x)=-x^{2}+ax+b- \frac {1}{2}$, one of their intersection points is $P$. The tangent lines $l_{1}$ and $l_{2}$ to the functions $f(x)$ and $g(x)$ at point $P$ are perpendicular. Find the maximum value of $ab$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The chord $AB$ of the ellipse $O: \frac{x^2}{12} + \frac{y^2}{6} = 1$ satisfies $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. Let the projection of the origin $O$ onto $AB$ be $M$. The number of points $M$ at a distance of $1$ from the line $x+y=1$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A motorist travels for some hours, the first half at 60 kmph and the rest at 48 kmph. The motorist traveled a distance of 324 km. How many hours did the motorist travel?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five standard six-sided dice are rolled.  We are told there is no three-of-a-kind, but there is a pair of dice that show the same number.  These two dice are set aside, and the other three dice are re-rolled.  What is the probability that after re-rolling these three dice, at least three of the five dice show the same value?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, and the altitude on side $BC$ is $\frac{a}{2}$. Determine the maximum value of $\frac{c}{b}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of $3:2:1$, respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two singular matrices $X$ and $Y$ such that $XY = Y$ and $YX = X$, express $X^2 + Y^2$ in terms of $X$ and $Y$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Shapovalov A.V.

A banker learned that among identical-looking coins, one is counterfeit (lighter). He asked an expert to identify this coin using a balance scale without weights, and required that each coin participate in weighings no more than twice. What is the maximum number of coins the banker can have so that the expert can definitely identify the counterfeit one in $n$ weighings?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic behavior of the sequence \( n - \left( {\sum\limits_{i = n+1}^\infty  {\frac{1}{{i^2 }}} } \right)^{ - 1} \) as \( n \to \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
$\left.\begin{array}{c}\text { Sphere inscribed in a pyramid } \\ \text { Regular pyramid }\end{array}\right]$

In a regular quadrilateral pyramid with a height not less than $h$, a hemisphere of radius 1 is placed such that it touches all the lateral faces of the pyramid, and the center of the hemisphere lies on the base of the pyramid. Find the smallest possible value of the total surface area of such a pyramid.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many factors of $$(20^{12})^2$$ less than $$20^{12}$$ are not factors of $$20^{12}$$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the lateral side \( CD \) of the trapezoid \( ABCD (AD \parallel BC) \), point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is dropped onto segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \) if it is known that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $$\int_{0}^{\infty}e^{-\sqrt 3 x^{\frac{2}{3}}}\sin(x^{\frac{2}{3}})\mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 (a_1,b_1),(a_2,b_2),\cdots,(a_{100},b_{100}) 是不同的非负整数对. 用 N 表示满足 1\leq i<j\leq100 且 |a_ib_j-a_jb_i|=1 的数对 (i,j) 的个数. 求 N 的最大可能值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral in closed form: \[ \mathfrak{R} \int_{0}^{\pi/2} \frac{x^2}{x^2 + \log^2(-2\cos x)} \, \mathrm{d}x \] where \(\mathfrak{R}\) denotes the real part, and \(\log(z)\) denotes the principal value of the logarithm for \(z \neq 0\) given by \[\log(z) = \ln |z| + i \arg z, \quad -\pi < \arg z \leq \pi.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( k \) be a positive integer. A sequence \( {a}_{0},{a}_{1},\ldots ,{a}_{n},n > 0 \) of positive integers satisfies the following conditions:
 (i) \( {a}_{0} = {a}_{n} = 1 \) ;
 (ii) \( 2 \leq  {a}_{i} \leq  k \) for each \( i = 1,2, \) ... \( ,n - 1 \) ;
 (iii)For each \( j = 2,3,\ldots ,k \) ,the number \( j \) appears \( \phi \left( j\right) \) times in the sequence \( {a}_{0},{a}_{1},\ldots ,{a}_{n} \) ,where \( \phi \left( j\right) \) is the number of
 positive integers that do not exceed \( j \) and are coprime to \( j \) ;
 (iv) For any \( i = 1,2,\ldots ,n - 1,\gcd \left( {{a}_{i},{a}_{i - 1}}\right)  = 1 = \gcd \left( {{a}_{i},{a}_{i + 1}}\right) \) ,and \( {a}_{i} \) divides \( {a}_{i - 1} + {a}_{i + 1} \) .
 Suppose there is another sequence \( {b}_{0},{b}_{1},\ldots ,{b}_{n} \) of integers such that \( \frac{{b}_{i + 1}}{{a}_{i + 1}} > \frac{{b}_{i}}{{a}_{i}} \) for all \( i = 0,1,\ldots ,n - 1 \) . Find the minimum
 value of \( {b}_{n} - {b}_{0} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Differentiate the following integral with respect to the function $f(t)$, where the upper bound of the integral is $t$:
\[
\frac{\partial}{\partial f(t)} \int_0^t f^3(\tau) \cdot g(\tau) \, d\tau
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Assume that $a$ and $b$ are nonnegative real numbers such that \(\sin{(ax+b)}=\sin{29x}\) for all integers \(x\). Find the smallest possible value of \(a\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the differential equation \( \frac{dy}{dx} = 1-y^2 \) and the condition that the function \( y \) is symmetric about the point \( (\ln(9),0) \), find the value of \( y(\ln(4)) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected number of Bernoulli trials required to achieve a streak of 10 consecutive successes, given that the probability of success for each trial is 0.2.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the line $y=ax$ intersects the circle $C:x^2+y^2-2ax-2y+2=0$ at points $A$ and $B$, and $\Delta ABC$ is an equilateral triangle, then the area of circle $C$ is __________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function $f_0(t)$ that is pointwise discontinuous at a finite number of points and continuous everywhere else, and a sequence of functions $f_1, f_2, \cdots$ generated by the recursive process:
$$f_n(t) = \frac{1}{2\epsilon}\int_{t-\epsilon}^{t+\epsilon}f_{n-1}(\tau)d\tau, \quad \text{for } n = 1, 2, \cdots \text{ and some } \epsilon > 0$$
Determine the maximum number of times the function $f_k(t)$ can be continuously differentiated.
Please reason step by step, and put your final answer within \boxed{}.


Context:
An insurance policy covers two employees of a company, reimbursing no more than one loss per employee per year, up to a company-wide maximum of $8000. The probability of an employee incurring a loss is $40\%$, with losses uniformly distributed between $(1000, 5000)$, independent of other employees' losses. Given that one employee has incurred a loss exceeding $2000$, what is the probability that the total losses will exceed the policy's reimbursement limit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} \left( \sin^4{x} + \frac{1}{4} \sin^4{2x} + \cdots + \frac{1}{{4}^n} \sin^4{2nx} \right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( g'(\frac{1}{2}) \) where \( g(a) = \lim_{h \to 0^+} \int_h^{1-h} t^{-a}(1-t)^{a-1} \, dt \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest value of \( T \) such that \( |y(x)| \leq 0.1 \) for all \( x > T \), where \( y(x) = e^{-\frac{x}{5}} \left[2\cos\left(\frac{\sqrt{34}}{5} x\right) + \frac{7}{\sqrt{34}}\sin\left(\frac{\sqrt{34}}{5}x\right)\right] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$ \lim_{m\to\infty} \frac{\int_0^{\pi/2}(\sin x)^{2m}dx}{\int_0^{\pi/2}(\sin x)^{2m+1}dx} $$ using integral properties or special functions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A physical system consists of four identical particles. The total energy of the system is \(4E_0 > 0\). Each particle can have an energy level equal to \(kE_0\) for \(k \in \{0, 1, 2, 3, 4\}\). A particle with energy \(kE_0\) can occupy any one of the \(k^2 + 1\) distinct energy states at that energy level. How many distinct energy configurations can the system have, considering the particles obey Bose-Einstein statistics?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \(\int \int_R |xy|\,dx\,dy\) over the region \(R\) bounded by \(x^2+4y^2=1\) and \(x^2+y^2=1\) using polar coordinates.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the particular solution $p_n$ for the nonhomogeneous recurrence relation $a_n = 2a_{n-1} + 3a_{n-2} + 8n - 4$, given the initial conditions $a_0 = 1$ and $a_1 = 3$. The non-homogeneous term is $f(n) = 8n - 4$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the formula for the number of reflection subgroups of the dihedral group $D_{2n}$, where the group has order $2n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Use the residue theorem to evaluate the integral \( \int_{0}^{2\pi}  \frac {27} {(5+4\sin\theta)^2} d\theta \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of axes of symmetry that the union of \( k \) line segments on a plane can have.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three cyclists started simultaneously: the first and second from point A, and the third towards them from point B. After 1.5 hours, the first cyclist was equidistant from the other two, and 2 hours after departure, the third cyclist was equidistant from the first and second. Determine how many hours after departure the second cyclist was equidistant from the first and third.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\mathcal{F}$ be the family of all analytic functions $f: D_1(0) \to D_1(0)$ such that $f(\frac{i}{3}) = 0$. Determine the supremum $M = \sup\{\operatorname{Im} f(0) : f \in \mathcal{F}\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine a region in the xy-plane where the differential equation \( \frac{dy}{dx} = y^{4/5} \) has a unique solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the degree of the splitting field of the polynomial $x^6 + 1$ over the field $\mathbb{Z}_3$. Express your answer as an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the tangent map of the function \( f: O(n) \rightarrow O(n) \) defined by \( f(M) = M^3 \) at the identity matrix \( I \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A tank is full of oil weighing 20 lb/ft³. The tank is an inverted right rectangular pyramid with a base width of 2 feet, a depth of 2 feet, and a height of 5 feet. Calculate the work required to pump the oil to a height of 3 feet above the top of the tank.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \oint_{C}\frac{dz}{z^2-1} \) around the unit circle \( C \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimum number of vertices in a connected graph where each vertex has a degree of at least 3, and there is no path that visits each vertex exactly once.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\rightarrow +\infty}{8x(x+1)-\sqrt{4x^2+2x}\cdot\sqrt[3]{64x^3+144x^2+90x+17}}$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Your favorite baseball team is playing against your uncle's favorite team in the World Series. At the beginning of each game, you and your uncle bet on the game's outcome. If you bet \( b \) dollars and your team wins the game, your uncle gives you an IOU for \( b \) dollars. But if they lose the game, you give him an IOU for \( b \) dollars. When the series is over, all outstanding IOUs are settled in cash. You would like to walk away with \( \$100 \) in cash if your team wins the series, and lose \( \$100 \) if your team loses the series. How much should you bet on the opening game to achieve this outcome? (The first team to win a total of four games wins the series.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
A set \( M \) of points on a line can be covered by three segments of length 1. Determine the smallest number of segments of length 1 required to cover all midpoints of segments with endpoints in the set \( M \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_{-\infty}^\infty \exp(-\sqrt{1+x^2})\,dx. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int_{-1}^1 \frac{d}{dx}\ \frac{1}{1 + 2^{1/x}}\ dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8, 4$. Define $n = (10a + b)^{10c+d}$. Calculate the probability that $1984!$ is divisible by $n$. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the remainder when \(35^{32} + 51^{24}\) is divided by 1785. Use the fact that 1785 factors as \(3 \times 5 \times 7 \times 17\), and that 35 is congruent to 0 modulo 5 and 7, while 51 is congruent to 0 modulo 3 and 17. Apply the Chinese Remainder Theorem to solve the problem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the value of \(a > 0\) such that the integral \(I = \int_1^\infty \int_1^\infty (x+y)^{-a} \, dx \, dy\) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that f(x) = log(-ax) is an odd function, find the value of the real number a that satisfies the condition f(x) + f(-x) = 0. Express your answer as a numerical value, including the ± symbol if applicable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For $0 \leq d \leq 9$, we define the numbers
\[S_{d}=1+d+d^{2}+\cdots+d^{2006}\]
Find the last digit of the number
\[S_{0}+S_{1}+\cdots+S_{9}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a parallelogram $ABCD$, a perpendicular is dropped from vertex $C$ to line $CD$, intersecting a perpendicular from vertex $A$ to diagonal $BD$ at point $F$. Another perpendicular is dropped from point $B$ to line $AB$, intersecting the perpendicular bisector of segment $AC$ at point $E$. Determine the ratio in which segment $EF$ divides side $BC$. Provide your answer as a ratio of two integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the game of projective set, each card contains some nonempty subset of six distinguishable dots. A projective set deck consists of one card for each of the 63 possible nonempty subsets of dots. How many collections of five cards have an even number of each dot? The order in which the cards appear does not matter.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( f(x)=\left(1-x^{2}\right)\left(x^{2}+b x+c\right) \) for \( x \in [-1, 1] \), find the minimum value of the maximum of \( |f(x)| \) as \( b \) and \( c \) vary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the quadratic equation \(x^2 - 4x + 7 = 0\) with roots \(x_1\) and \(x_2\), calculate the value of the expression:
\[
\frac{x_1^4 - 2x_1^3 + 3}{x_1^3 + x_1^2} + \frac{x_2^4 - 2x_2^3 + 3}{x_2^3 + x_2^2}
\]
using Vieta's formulas without manually calculating the roots.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dimension of the 2 by 2 orthogonal group, \( O(2) = \{ A \in M(2,2,\mathbb{R}): AA^T = I \} \), where \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). Given the equations \( a^2 + b^2 = 1 \), \( c^2 + d^2 = 1 \), and \( ac + bd = 0 \), find the dimension of this group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Right triangle  $ABC$  has its right angle at  $A$ . A semicircle with center  $O$  is inscribed inside triangle  $ABC$  with the diameter along  $AB$ . Let  $D$  be the point where the semicircle is tangent to  $BC$ . If  $AD=4$  and  $CO=5$ , find  $\cos\angle{ABC}$ .

[asy]
import olympiad;

pair A, B, C, D, O;
A = (1,0);
B = origin;
C = (1,1);
O = incenter(C, B, (1,-1));
draw(A--B--C--cycle);
dot(O);
draw(arc(O, 0.41421356237,0,180));
D = O+0.41421356237*dir(135);
label(" $A$ ",A,SE);
label(" $B$ ",B,SW);
label(" $C$ ",C,NE);
label(" $D$ ",D,NW);
label(" $O$ ",O,S);
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
The sides of a rectangular park are in the ratio 3 : 2 and its area is 3750 sq m. The cost of fencing it at a certain rate per meter is $200. What is the cost of fencing per meter in ps?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n \in \mathbb{N}^*$, $a_n$ be the sum of the coefficients of the expanded form of $(x+4)^n - (x+1)^n$, $c=\frac{3}{4}t-2$, $t \in \mathbb{R}$, and $b_n = \left[\frac{a_1}{5}\right] + \left[\frac{2a_2}{5^2}\right] + ... + \left[\frac{na_n}{5^n}\right]$ (where $[x]$ represents the largest integer not greater than the real number $x$). Find the minimum value of $(n-t)^2 + (b_n + c)^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $a, b>0$, 满足关于 $x$ 的方程 $\sqrt{|x|}+\sqrt{|x+a|}=b$ 恰有三个不同的实数解 $x_{1}, x_{2}, x_{3}$,且 $x_{1}<x_{2}<x_{3}=b$, 则 $a+b$ 的值为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There is a certain amount of milk and water in which milk forms 84%. 26.9 liters of water must be added to this solution to make it a solution in which milk forms 58%. What is the initial volume of the milk and water solution?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $\frac ab = \frac35$,$\frac bc=\frac{15}6$, and $\frac cd = 6$. What is the value of $\frac ad$? Express your answer in simplest form.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a cube with 8 vertices, where any two vertices define a line, calculate the probability that two randomly chosen lines from all possible lines determined by these vertices are skew lines. Express your answer as a simplified fraction in the form \(\boxed{\frac{a}{b}}\), where \(a\) and \(b\) are integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
三棱雉 $P-A B C$ 中, $P A \perp P B, P B \perp P C, P C \perp P A, M$ 为底面三角形内一点, 若 $\sin \angle A P M$ $=\frac{4}{5}, \cos \angle B P M=\frac{\sqrt{3}}{3}$, 则 $\cos \angle C P M=$ $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be defined by $f(x) = cx$ for some constant $c \neq 0$, and let $S$ be a regular surface. Express the Gaussian curvature of $f(S)$ in terms of the Gaussian curvature of $S$. Provide your answer in terms of $c$ and the Gaussian curvature of $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function $y=\sqrt{ax^2+4x+1}$ with a range of $[0,+\infty)$, determine the range of values for $a$ that satisfies this condition. Express your answer using interval notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest positive integer \( n \) such that there exist \( n \) lattice points on the coordinate plane, and any 3 points form a triangle whose centroid is not a lattice point.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the functions $f(x)=1+x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots +\frac{x^{2017}}{2017}$ and $g(x)=1-x+\frac{x^{2}}{2}-\frac{x^{3}}{3}+\frac{x^{4}}{4}+\cdots -\frac{x^{2017}}{2017}$, let $F(x)=f(x+4)\cdot g(x-4)$. If all the zeros of the function $F(x)$ are in the interval $[a-1, a]\cup[b-1,b] (a < b, b\in\mathbb{Z})$, find the value of $a+b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( f(x)=1+\frac{x^{3}}{3 !}+\frac{x^{6}}{6 !}+\cdots \), \( g(x)=\frac{x}{1 !}+\frac{x^{4}}{4 !}+\frac{x^{7}}{7 !} \cdots \), and \( h(x)=\frac{x^{2}}{2 !}+\frac{x^{5}}{5 !}+\frac{x^{8}}{8 !}+\cdots \). If \( v=f(x)^{2}+g(x)^{2}+h(x)^{2} \), find the value of \( v(\ln 3) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Lucius owns a small business and spends $10 every day on ingredients. He then makes a portion of French Fries and a portion of Poutine and sells them to the market. The price of French fries is $12 and the price of Poutine is $8. He earns $56 every week after selling all his products and buying all the ingredients. What percentage of his weekly income does he pay as tax?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the integral $\int_0^\infty \int_0^\infty \int_0^\infty e^{-(x^2+xy+xz)}\ dx\ dy\ dz$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
若点$M$，$N$在圆${{C}_{1}}$：${{x}^{2}}+{{y2}}=1$上运动，且$\left| MN \right|=\sqrt{3}$，点$P\left( {{x0}},{{y}_{0}} \right)$是圆${{C2}}$：${{x}^{2}}+{{y2}}-6x-8y+24=0$上一点，则$\left| \overrightarrow{PM}+\overrightarrow{PN} \right|$的取值范围为______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the coefficient of the term \(x^6 y^5 z\) in the expansion of \(\left( 2x + xy - z + \frac{1}{xyz} \right)^{10}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the upper bound for the integral $$\int_0^1 \frac{1}{1+x^2e^{x^2}}\,\mathrm{d}x$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly nonconsecutive) days total, and he cannot wear the same outfit (which consists of one T-shirt and one pair of pants) on two different (not necessarily consecutive) days. How many ways can he choose the outfits for these seven days?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\zeta = e^{2\pi i/99}$ and $\omega = e^{2\pi i/101}$. The polynomial \[ x^{9999} + a_{9998}x^{9998} + \cdots + a_1x + a_0 \] has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + \cdots + a_{9998}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, given that $AB = 3$, $BC = 6$, and $AC = 4$, let $D$ be a point on side $BC$ such that $AD = 3$. Find the value of $\frac{BD}{DC}$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mary typed a six-digit number, but the two $1$s she typed didn't show. What appeared was $2002.$ How many different six-digit numbers could she have typed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of divisors of the number $2^{2} \cdot 3^{3} \cdot 5^{3} \cdot 7^{5}$ that are of the form $4n+1$, where $n$ is a natural number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, respectively, and let $P$ be any point on the ellipse. The coordinates of point $M$ are $(6,4)$. Then, the maximum value of $|PM| + |PF_1|$ is ______________.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the length of the shortest sequence that contains all possible 4-digit sequences (from 0000 to 9999) as substrings?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $(1+\sqrt{2})^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers. Find the greatest common divisor of $b$ and $81$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^\infty Ai(y) \, dy \), where the Airy function \( Ai(y) \) is given by the integral representation \( Ai(y) = \frac{1}{\pi} \int_0^\infty \cos(\alpha y + \frac{\alpha^3}{3}) \, d\alpha \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知实数$x、y$满足：
$x>\max\{-3,y\},(x+3)\left(x^{2}-y^{2}\right)=8$。
则$3x+y-4\sqrt6$的最小值为_______。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Georgia's school club is selling carnations at various prices and in different bundles for a fundraising event. The pricing tiers are as follows:

Tier 1:
- A single carnation costs $0.60
- A dozen carnations cost $5.00
- A bundle of 25 carnations cost $10.00

Tier 2:
- A single carnation costs $0.50
- A dozen carnations cost $4.50
- A bundle of 50 carnations cost $16.00

Tier 3:
- A single carnation costs $0.40
- A dozen carnations cost $4.00
- A bundle of 100 carnations cost $30.00

Georgia wants to send specific quantities of carnations from different pricing tiers to each of her 8 teachers:

- Teacher 1 receives 2 dozen from Tier 1
- Teacher 2 receives 1 dozen + 5 individual carnations from Tier 2
- Teacher 3 receives 1 bundle of 25 carnations from Tier 1
- Teacher 4 receives 3 dozen + 10 individual carnations from Tier 3
- Teacher 5 receives 2 bundles of 50 carnations from Tier 2
- Teacher 6 receives 1 dozen from Tier 3 + 7 individual carnations from Tier 1
- Teacher 7 receives 1 bundle of 100 carnations from Tier 3
- Teacher 8 receives 3 dozen from Tier 1

She also wants to buy different quantities of carnations from various pricing tiers for each of her 20 friends:

- Friends 1-4 receive 3 carnations each from Tier 2
- Friend 5 receives 1 dozen carnations from Tier 1
- Friends 6-9 receive 5 carnations each from Tier 3
- Friend 10 receives 15 individual carnations from Tier 2
- Friends 11-13 receive 1 dozen + 2 individual carnations each from Tier 1
- Friend 14 receives 25 individual carnations from Tier 3
- Friends 15-18 receive 4 carnations each from Tier 1
- Friend 19 receives 1 bundle of 50 carnations from Tier 2
- Friend 20 receives 2 dozen + 5 individual carnations from Tier 1

How much money would Georgia spend on purchasing all these carnations?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(a, b, c\) are positive integers and the quadratic equation \(a x^{2}+b x+c=0\) has two real roots whose absolute values are both less than \(\frac{1}{3}\), find the minimum value of \(a + b + c\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the plane Cartesian coordinate system \( xOy \), the set \( S=\left\{(x, y) \mid (|x|+|y|-1)\left(x^{2}+y^{2}-1\right) \leqslant 0\right\} \) represents a plane region. Find the area of this region.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A batch of cookies was divided among 3 tins: 2/3 of all the cookies were placed in either the blue tin or the green tin, and the rest were placed in the red tin. If some fraction of all the cookies were placed in the blue tin and 0.5555555555555556 of the cookies that were placed in the other tins were placed in the green tin, what fraction of all the cookies were placed in the blue tin?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f:R\to R$ be a continuous odd function, which vanishes at exactly one point and $f(1)=\frac{1}{2}$. Suppose that $F(x)=\int_{-1}^{x}f(t)dt,\hspace{1cm}\forall x\in[-1,2]$, and $G(x)=\int_{-1}^{x}t|f(f(t))|dt\hspace{1cm} \forall x\in[-1,2]$. If $\lim_{x\to 1}\frac{F(x)}{G(x)}=\frac{1}{14},$ then find the value of $f(\frac{1}{2})$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vasya and Petya set off to go fishing at the lake on foot, while Kolya stayed home with a bicycle. Vasya was the first to head back home, and Kolya set off to meet him on the bicycle. Petya left the lake for home at the moment Vasya and Kolya met. Kolya brought Vasya home on the bicycle, then returned to meet Petya and brought him home as well. The time Petya spent traveling from the lake to home was \( \frac{5}{4} \) of the time Vasya took for the same journey. Calculate how much slower Vasya would have been if he walked the entire way home by himself.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain product of 936 and w, where w is a positive integer, has some factors. If 3^3 and 12^2 are also factors of the product, and the smallest possible value of w is 144, what is the other factor in the form of 2^x?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given \( [x] \) denotes the greatest integer less than or equal to the real number \( x \). The sequence \( \{a_n\} \) is defined by
\[
a_1 = a_2 = \frac{1}{2}, \quad a_{n+1} = 2a_n + a_{n-1} \quad (n \in \mathbf{Z}_{+}).
\]

Find \(\left[\sum_{k=2}^{2016} \frac{1}{a_{k-1} a_{k+1}}\right]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the fourteen numbers, $1^4, 2^4, \ldots, 14^4$. Find the smallest natural number $n$ such that these numbers leave distinct remainders when divided by $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the lengths of the sides of a regular hexagon as follows: 5 inches, 8 centimeters, 6 inches, 10 centimeters, 7 inches, and 12 centimeters. Convert all sides to one unit (either inches or centimeters) and then find the perimeter.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $\operatorname{Re}(z) \geq 1$ ，求 $\left|\frac{1}{z}+1+i\right|_{\min }$ 的值. 原始答案是 \(\frac{\sqrt{m}-n}{k}\) 的形式，请给出 m + n + k 的值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2005}\left|x_{k}-x_{k+1}\right|=2007 \). Define \( y_{k}=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+x_{k}\right) \) for \( k=1, 2, \cdots, 2007 \). Find the maximum possible value of \( \sum_{k=1}^{2006}\left|y_{k}-y_{k-1}\right| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume of the solid that lies outside the sphere defined by $r=2$ and inside the surface defined by $r=1+\cos\phi$ using a triple integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \(\lambda\) that satisfies the equation \(\binom{\lambda}{n} = n+1\) for some positive integer \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum number of natural numbers  $x_1,x_2, ... , x_m$   satisfying the conditions:
a) No  $x_i - x_j , 1 \le i < j \le m$  is divisible by  $11$ ,  and
b) The sum  $x_2x_3 ...x_m + x_1x_3  ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$  is divisible by  $11$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $f(x)$ 是定义在 $\mathbf{R}$ 上的函数, 对任意实数 $x$ 有 $f(x+3) \cdot f(x-4)=-1$. 又当 $0 \leqslant x<7$ 时, $f(x)=\log _{2}(9-x)$, 则 $f(-100)$ 的值为 \qquad. 原始答案是以-\frac{m}{n}的形式给出的，请提供m + n的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $O$ be a point outside a given circle. Two lines $OAB$, $OCD$ through $O$ meet the circle at $A$, $B$, $C$, $D$, where $A$, $C$ are the midpoints of $OB$, $OD$, respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A die with faces showing the numbers $0,1,2,3,4,5$ is rolled until the total sum of the rolled numbers exceeds 12. What is the most likely value of this sum?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected number of draws needed to obtain a sum divisible by 3 when drawing numbers from a bag containing \(1, 3, 5, 7, 9\), with replacement after each draw.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $[x]$ be the greatest integer less than or equal to the real number $x$. Given the sequence $\left\{a_{n}\right\}$ which satisfies $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1$ for $n \in N^{*}$, find the value of $\left[\sum_{k=1}^{2017} \frac{a_{k}}{a_{k}+2}\right]$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. If $\sin^2 A - \sin^2 B = 2 \sin B \sin C$ and $c = 3b$, find the value of angle $A$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a kilometer race, A can give B a 100 meters start and C a certain amount of meters start. B can give C a 111.11111111111109 meters start in a kilometer race. How much start can A give C in the race?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知 $f(x)=e^{\sin x^{2}}$, 则 $\left[-f^{(8)}(0)\right]=$？
Please reason step by step, and put your final answer within \boxed{}.


Context:
A driver travels 30 miles per hour for 3 hours and at a certain speed for 4 hours to deliver goods to a town every day from Monday to Saturday. The driver travels 1140 miles in a week. What is the speed of the driver in the last 4 hours?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the limit of the series \(\lim_{n\rightarrow \infty}\left(\frac{(n+1)(n+2)\ldots(3n)}{n^{2n}}\right).\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
The product of three consecutive integers is 384. What is their sum?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the coordinate plane, points (x, 8) and (20, y) are on line k. Line k passes through the origin. The product of x and y is 160. What is the slope of line k?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the derivative of the expression \(-a^T X^T y\) with respect to the vector \(a\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume $\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose  $a_{1} < a_{2}< \cdots < a_{2024}$  is an arithmetic sequence of positive integers, and   $b_{1} <b_{2} < \cdots <b_{2024}$  is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \iint_D xy\ dxdy \), where \( D \) is the region bounded by the curves \( y=x^2 \), \( y=2x^2 \), \( x=y^2 \), and \( x=3y^2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
One face of a tetrahedron has sides of length $3$, $4$, and $5$. The tetrahedron’s volume is $24$ and surface area is $n$. When $n$ is minimized, it can be expressed in the form $n = a\sqrt{b} + c$, where $a$, $b$, and $c$ are positive integers and b is not divisible by the square of any prime. Evaluate $a + b + c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a square on a Cartesian plane with vertices at \((0, 0), (0, 65), (65, 65),\) and \((65, 0)\). Determine the number of ways to select two grid points inside this square (excluding its boundary) such that at least one of these points lies on the line \(y = x\) or \(y = 65 - x\), and neither point lies on any line parallel to the coordinate axes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of cyclic subgroups of order 10 in the group \( \mathbb{Z}_{30} \oplus \mathbb{Z}_{120} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a school, there are three classes, each with \( n \) students, and all \( 3n \) students have different heights. These students are grouped into \( n \) groups, with each group consisting of 3 students from different classes. The tallest student in each group is called the "tall one". It is known that no matter how the groups are formed, each class has at least 10 "tall ones". What is the smallest possible value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Euler-Lagrange equation for the energy functional \[ E[u] = \frac{1}{2}\int_{\Omega} \|\nabla^3 u \|^2 \, dx \] where \( \Omega \subseteq \mathbb{R}^n \) and \( u: \Omega \to \mathbb{R} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 2004 boxes on a table, each containing one ball. Some of the balls are white, and their number is even. You can point to any two boxes and ask if there is at least one white ball among them. What is the minimum number of questions required to guarantee determining at least one box that contains a white ball?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 7, 8, 9\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the recursive sequence defined by \( a_{n+1} = \frac{a_n}{n} + \frac{n}{a_n} \) with the initial condition \( a_1 = 1 \), evaluate \( \left \lfloor a_{2015} \right \rfloor \).
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:
${\tiny ■} $She rolled a $6$ exactly three times;
${\tiny ■} $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:
Evaluate the integral \( \int_{\mathbb{C}} \frac{e^{-\frac{(x-im)^2}{2}}}{\sqrt{2\pi}} \, dx \) when \( m \) is replaced with \( im \), where \( i \) is the imaginary unit and \( m \) is a real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^4 g'(\sqrt{x}) \, dx \), where the function \( f(x) \) is defined as \( f(x) = \begin{cases} x^x \ln x + k & : \ (x \geq 1) \\ f(2-x) & : \ (x<1) \end{cases} \) and satisfies \( \int_0^1 f(x) \, dx = 5 \). The function \( g(x) \) is given by \( g(x) = \int_0^x f(t) \, dt \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A and B start a business, with A investing the total capital of Rs. 5000, on the condition that B pays A interest @ 10% per annum on his half of the capital. A is a working partner and receives Rs. 500 per month from the total profit and any profit remaining is equally shared by both of them. At the end of the year, it was found that the income of A is twice that of B. What is the total profit for the year?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the region bounded by the polar curve \( r^2 = a^2 \cos(2\theta) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
对 $\mathbb{R}^{100}$ 中的点 $x=\left(x_{1}, x_{2}, \cdots, x_{100}\right)$ 和 $y=\left(y_{1}, y_{2}, \cdots, y_{100}\right)$ 定义函数 $f(x,y)=\sqrt{\sum_{i=1}^{100}\left(x_{i}-y_{i}\right)^{2}}$. 当 $x \neq y$ 时，定义矩阵 $A_{x, y}=\left(a_{i j}\right)$ ，其中 $a_{i j}=$ $\frac{\partial}{\partial x_{i}} \frac{\partial}{\partial y_{j}} f(x, y)(i, j=1,2, \cdots, n)$. 则 $A_{x, y}$ 秩的最小可能值与最大可能值之和为 $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
If there exist positive real numbers $x$ and $y$ satisfying $x\ln\frac{y}{x}-y{e^x}+x({x+1})≥0$, then the maximum value of $y$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知四面体$D-ABC$中，$AC=BC=AD=BD=1$，求$D-ABC$体积的最大值。答案的形式为\frac{k\sqrt{m}}{n}，请给出k + m + n的值。
Please reason step by step, and put your final answer within \boxed{}.


Context:
A vessel of capacity 100 litres is fully filled with pure milk. Fifteen litres of milk is removed from the vessel and replaced with water. This process of removing 15 litres of the solution and replacing it with water is repeated three more times. Determine the quantity of pure milk remaining in the final milk solution.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( k \) be a positive integer such that for any positive numbers \( a, b, c \) satisfying the inequality \( k(ab + bc + ca) > 5(a^2 + b^2 + c^2) \), there must exist a triangle with side lengths \( a, b, \) and \( c \). Find the minimum value of \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle \(ABC\), angle \(C\) is \(60^\circ\) and the radius of the circumcircle of this triangle is \(2\sqrt{3}\).

A point \(D\) is taken on the side \(AB\) such that \(AD = 2DB\) and \(CD = 2\sqrt{2}\). Find the area of triangle \(ABC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A lady made 3 round doilies with radii of 2, 3, and 10 inches, respectively. She placed them on a round table so that each doily touches the two others and the edge of the table. What is the radius of the table?
Please reason step by step, and put your final answer within \boxed{}.