Context:
How many multiplications and subtractions are required in the Cholesky decomposition of an $n \times n$ matrix, excluding square roots and divisions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $M$ and $N$ be two closed subspaces of a Hilbert space $H$ such that $N \subset M$. If $P_M$ and $P_N$ are the orthogonal projections onto $M$ and $N$, respectively, determine the subspace of $H$ on which $P_M - P_N$ is an orthogonal projection.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Dave had 15 apps and some files on his phone. After deleting some apps and files he had 21 apps and 4 files left. Dave has 17 more apps than files left on his phone. How many files did Dave initially have on his phone?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field \( \bar{V}=z\bar{i}+x\bar{j}+y\bar{k} \) through the portion of the sphere \( x^2+y^2+z^2=1 \) that lies in the first octant using Gauss's theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
To combat the mice, Leopold the Cat needs to drink a potion called "ozverin" daily. He has a bottle of ozverin solution in water (a colorless, clear liquid) in the closet with a volume of \( V = 0.5 \) liters and an initial ozverin concentration of \( C_{0} = 40\% \). Each day, Leopold drinks \( v = 50 \) ml of the solution, and to avoid detection by the mice, he refills the bottle with the same volume of water. Find the volume concentration of ozverin in the bottle after the cat has drunk \( n = 5 \) times \( v = 50 \) ml of the solution and refilled the bottle with water each time.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are $25$ towns in a country. Find the smallest $k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) From each town, there are exactly $k$ direct routes to $k$ other towns.
2) If two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A jar contains quarters (worth $0.25$ each), nickels (worth $0.10$ each), and pennies (worth $0.02$ each). The value of the quarters is $12.50$, the value of the nickels is $15.00$, and the value of the pennies is $8.00$. If Lena randomly chooses one coin from the jar, what is the probability that it is a quarter?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a regular tetrahedron \(P-ABC\) with edge length 1, let \(E\), \(F\), \(G\), and \(H\) be points on edges \(PA\), \(PB\), \(CA\), and \(CB\) respectively. If \(\overrightarrow{EF}+\overrightarrow{GH}=\overrightarrow{AB}\) and \(\overrightarrow{EH} \cdot \overrightarrow{FG}=\frac{1}{18}\), then find \(\overrightarrow{EG} \cdot \overrightarrow{FH}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
When a cube is cut into two pieces with a single plane cut, two polyhedra are obtained. Determine the number of faces of a polyhedron that cannot be obtained in this way. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A cubic polynomial $p(x)$ satisfies
\[p(n) = \frac{1}{n^2}\]for $n = 1, 2, 3,$ and $4.$  Find $p(5).$
Please reason step by step, and put your final answer within \boxed{}.


Context:
What could be the minimum length of a cycle in a graph where no single vertex is connected to all others, any two non-adjacent vertices have a common neighbor, and if the number of vertices is denoted by $n$, the sum of the squares of the degrees is $n^2 - n$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \frac{1}{2 \pi} \int_0^{2 \pi} \frac{1 - r^2}{1 - 2r \cos(\theta) +r^2}d\theta \) for \(0 < r < 1\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the representing object of the functor $D_n : \mathsf{Cat} \to \mathsf{Set}$, which maps a category to the set of all composable $n$-tuples of morphisms.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $r$ for which the Joukowsky transform is injective on the circle of radius $r$ centered at the origin.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A frog is at the point \((0,0)\). Every second, he can jump one unit either up or right. He can only move to points \((x, y)\) where \(x\) and \(y\) are not both odd. How many ways can he get to the point \((8,14)\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Bob is building raised beds for his vegetable garden. Each bed has a certain height, 2 feet wide, and 8 feet long. The sides are going to be built of 1-foot wide planks. Bob needs 50 8-foot long planks to construct 10 raised beds. What is the height of each raised bed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the directional derivative of the function \( f(x,y) = \begin{cases} \frac{x^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} \) at the point \((0,0)\) in the direction of the unit vector \( u = (u_1, u_2)^T \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the Galois group of the polynomial \( f(x) = x^2 + 2ix + 1 \) over \( \mathbb{C} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the integral $\int_0^\infty \frac{\ln ^2z} {1+z^2}{d}z$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \displaystyle\int_{C(1,2)}\frac{\cos z}{z^3}\,\mathrm{d}z \), where \( C(1,2) \) is the circle centered at 1 with radius 2, using the Cauchy integral formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A hotel consists of a $2 \times 8$ square grid of rooms, each occupied by one guest. All the guests are uncomfortable, so each guest would like to move to one of the adjoining rooms (horizontally or vertically). Of course, they should do this simultaneously, in such a way that each room will again have one guest. In how many different ways can they collectively move?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate \( \lim_{x \to 0^+} \sum_{n=1}^\infty \sin(\sqrt{n}) e^{-nx} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a circle with a circumference of 2013 units, there are 2013 equally spaced points, each with a token. The distance between two points is the length of the shorter arc between them. Determine the largest integer $n$ such that the tokens can be rearranged so that each point still has a token, and the distance between any two tokens, originally separated by no more than $n$ points, increases.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the triple integral: $$\int_{0}^{2}\int_{6}^{1}\int_{0}^{1} xyz\sqrt{2-y^2-x^2}\,dx\,dy\,dz$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$, she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$, and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. Find the value of $a+b+c+d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In \(\triangle ABC\), if \(\tan A\), \(\tan B\), \(\tan C\) are in harmonic progression, find the minimum value of \(\cot \frac{B}{2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Vasya plays the strings of a 6-string guitar from 1 to 6 and then back. Each subsequent hit lands on an adjacent string. On which string number will the 2000th hit land? (The order of string hits is: $1-2-3-4-5-6-5-4-3-2-1-2-\ldots)$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Pål has more chickens than he can manage to keep track of. Therefore, he keeps an index card for each chicken. He keeps the cards in ten boxes, each of which has room for $2021$ cards.

Unfortunately, Pål is quite disorganized, so he may lose some of his boxes. Therefore, he makes several copies of each card and distributes them among different boxes, so that even if he can only find seven boxes, no matter which seven, these seven boxes taken together will contain at least one card for each of his chickens.

What is the largest number of chickens Pål can keep track of using this system?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the value of \( \int_{-\infty}^{\infty} y(t) \, dt \), where \( y(t) = \int_{-\infty}^{t} g(\tau) \, d\tau \) and for a function \( g(t) \), it is given that \( \int_{-\infty}^{\infty} g(t)e^{-j\omega t} \, dt = \omega e^{-2\omega^2} \) for any real value \( \omega \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many people are needed in a room to have a 50% chance that three people share the same birthday, assuming birthdays are equally distributed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
An equilateral triangle and a square are inscribed in a circle as shown. $ABC$ is isosceles. The triangle and square share a common vertex. What is the number of degrees in the measure of the angle indicated by the question mark?

[asy]

import markers; defaultpen(linewidth(0.8));

pair A,B,C,D,E,F,G;

draw(unitcircle);

A=(-1,0);
B=(0,-1);
C=(1,0);
D=(0,1);

draw(A--B--C--D--A);

E=(-0.9,-0.45);
F=(0.9,-0.45);

draw(D--E--F--D);

G=(-0.76,-0.23);

markangle(Label("?"),n=1,radius=10,D,G,A,marker(stickframe(n=0),true));

label("$A$",(-0.5,-0.34));
label("$B$",(0.5,-0.34));
label("$C$",B,S);

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


Context:
Given that \(7^{-1} \equiv 55 \pmod{102}\), find \(49^{-1} \pmod{102}\), as a residue modulo 102. (Give an answer between 0 and 101, inclusive.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two non-negative integers $m>n$, let's say that $m$ ends in $n$ if we can get $n$ by erasing some digits (from left to right) in the decimal representation of $m$. For example, 329 ends in 29, and also in 9.
Determine how many three-digit numbers end in the product of their digits.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of digits in the duodecimal (base 12) representation of \(\tau\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an upper bound, independent of \( p \), for the sum \( S_p = \sum_{j=1}^p \frac{p+1}{p-j+1} \frac{1}{2^j} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the board game Tsuro, each tile has 8 entry points, with each point connecting to exactly one other point. Given that there are 35 unique tiles, determine the number of distinct configurations of these tiles, considering rotations as equivalent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A partially silvered mirror covers a square area with vertices at $(\pm 1, \pm 1)$. The fraction of incident light it reflects at a point $(x, y)$ is given by $\frac{(x-y)^2}{4}$. Assuming a uniform intensity of incident light, calculate the total fraction of light reflected by the mirror.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Provide an example of a sequence of open, bounded intervals \( J_n \) such that \( J_{n+1} \subset J_n \) for all \( n \), and \( \bigcap_{n=1}^{\infty} J_n \) contains exactly one element.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( -\frac{1}{n}, \frac{1}{n} \right)}


Context:
Each face of a cube is painted either red or blue, each with probability \( \frac{1}{2} \). The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? The original answer is in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please provide the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The terms of a geometric sequence $\{a_{n}\}$ are all real numbers, and the sum of the first $n$ terms is $S_{n}$. If $S_{10}=10$ and $S_{30}=70$, then what is $S_{4n}$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the periods of the functions \(x(t) = (a - b) \cdot \cos(t) + b \cdot \cos(t \cdot (k - 1))\) and \(y(t) = (a - b) \cdot \sin(t) - b \cdot \sin(t \cdot (k - 1))\), where \(k\) is a rational number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The probability that a light bulb lasts more than 1000 hours is 0.2. Determine the probability that 1 out of 3 light bulbs fails after 1000 hours of use.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the location of the set of all numbers \( e \) such that \( \phi_e \) has a primitive recursive extension in the arithmetical hierarchy, where \( \phi_e \) is the \( e \)th partial recursive function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate plane $xOy$, a circle with center $C(1,1)$ is tangent to the $x$-axis and $y$-axis at points $A$ and $B$, respectively. Points $M$ and $N$ lie on the line segments $OA$ and $OB$, respectively. If $MN$ is tangent to circle $C$, find the minimum value of $|MN|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the geometric sequence $\{a_n\}$, it is given that $a_{13}=1$ and $a_{12} > a_{13}$. Find the largest integer $n$ for which $(a_1-\frac{1}{a_1})+(a_2-\frac{1}{a_2})+(a_3-\frac{1}{a_3})+\cdots+(a_n-\frac{1}{a_n}) > 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the 2-form $dxdy + 3dxdz$ on the oriented triangle with vertices $(0,0,0)$, $(1,2,3)$, and $(1,4,0)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a, b$ be positive integers, and $1 = d_1 < d_2 < d_3 < ... < d_n = a$ be the divisors of $a$, and $1 = e_1 < e_2 <e_3 < ... < e_m = b$ be the divisors of b. Given $gcd(a, b) = d_2 = e_6$, find the smallest possible value of $a + b$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the residue of the function \( \frac{z}{e^z - 1} \) at its singularity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the total surface area of the remaining part of a solid sphere with radius 11 after a cylindrical hole with radius 8 is drilled through the center of the sphere.
Please reason step by step, and put your final answer within \boxed{}.


Context:
7. In how many different ways can we fill the 9 fields of a $3 \times 3$ table with the first 9 natural numbers (one in each field) so that the sums of the numbers written in each row of the table are equal?

The use of a pocket calculator or any reference materials is not allowed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the integer \(D\) such that the quadratic subfield of the cyclotomic field \(\mathbb{Q}(\zeta_7)\) can be expressed as \(\mathbb{Q}(\sqrt{D})\), where \(\zeta_7 = e^{i\frac{2\pi}{7}}\) is a 7th root of unity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Oliver is learning words starting with an alphabet, focusing on vowels first, before moving to others. He takes 5 days to learn words of one alphabet thoroughly but decides to set a unique challenge for himself. For each vowel, the number of rest days Oliver takes before starting to learn the next vowel is equal to the index of the vowel in the alphabet. Also, Oliver plans to learn the last two vowels simultaneously, while incorporating the appropriate rest days between them. Considering these factors, how many days does Oliver need to finish learning all vowels?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the range of values of \(\lambda\) for which the transformation \(T\) defined by \((Tf)(x) = x + \lambda \int_0^x (x-t)f(t)\) is a contraction with respect to the supremum norm on \(C([0,1])\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the expected value of the maximum distance of three jumps, where each jump has a probability of \( \frac{1}{3} \) of being disqualified (resulting in a distance of 0) and a probability of \( \frac{2}{3} \) of having a distance uniformly distributed over \([2, 3]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the integral $$I(\zeta) = \int_{-\infty}^{\infty} \frac{1}{u^{4} + \left(4 \zeta^{2} - 2\right)u^{2} + 1}\, du$$ for positive real $\zeta$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tara is saving up to buy a new clarinet. She already has $10 saved. She plans to sell her old books to make the money. She sells each book of hers for $5. However, when she is halfway towards her goal, she loses all her savings and has to start over. She sells a total of 25 books by the time she reaches her goal. What is the cost of the clarinet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Eight classes in the first year of high school negotiate to form a grade basketball team, which needs a total of 10 players. Each class must contribute at least 1 player. Calculate the total number of different composition methods to form the basketball team.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many positive five-digit integers exist such that the product of their five digits is 900?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{1}{x^{a - 1} \over \left(1 - x\right)^{a + 1}} \,\exp\left(x \over x - 1\right)\,\mathrm{d}x \) where \( a \in \mathbb{R} \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Fox jeans regularly sell for $10 a pair and Pony jeans regularly sell for $18 a pair. During a sale, these regular unit prices are discounted at different rates so that a total of $9 is saved by purchasing 5 pairs of jeans: 3 pairs of Fox jeans and 2 pairs of Pony jeans. If the sum of the two discount rates is 22 percent, what is the discount rate on Pony jeans?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the values of $a \in \mathbb{R}$ for which the system of differential equations \( \begin{cases} x'(t) = x(t) + ay(t) \\ y'(t) = ay(t) \end{cases} \) has exactly one non-trivial solution \((x(t), y(t))\) that approaches \((0, 0)\) as \(t \rightarrow \infty\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate
\[\prod_{n = 1}^{13} \frac{n(n + 2)}{(n + 4)^2}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of \( p \geq 0 \) for which the integral \( \int_0^\infty x \sin(x^p) \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A positive integer $n$ is magical if $\lfloor \sqrt{\lceil \sqrt{n} \rceil} \rfloor=\lceil \sqrt{\lfloor \sqrt{n} \rfloor} \rceil$. Find the number of magical integers between $1$ and $10,000$ inclusive.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\lim_{(h,k)\to (0,0)} \frac{|e^k-1-h|(e^k-1)}{\sqrt{h^2+k^2}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find an element in the group \((\mathbb{Z}/100\mathbb{Z})^\ast\) that has order 20.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{x \to 1} \frac{x^x - x^{x^x}}{(x-1)^2} \) using L'Hôpital's rule or alternative methods.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the largest integer K such that for integers a > 0, b > 0, and c > 0 satisfying the inequality:
\[ 126 > 2^{a+1}(2^b(2^c-1) - 3) > 0 \]
the sum a + b + c is less than K.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(x)$ be an odd function defined on $\mathbb{R}$, and it satisfies $f(x+3) \cdot f(x) = -1$ and $f(-1) = 2$. Find the value of $f(2018)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number of geese in a flock increases so that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$. If the populations in the years $1994$, $1995$, and $1997$ were $39$, $60$, and $123$, respectively, then the population in $1996$ was
Please reason step by step, and put your final answer within \boxed{}.


Context:
850 students are going to the zoo. They have to be divided into groups so that each teacher has 1 group. There are 23 teachers. There will be _____ students in each group.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Complex numbers $z_{1}$ and $z_{2}$ correspond to points in the complex plane that are symmetric with respect to the imaginary axis. If $z_{1}=1-2i$, where $i$ is the imaginary unit, find the imaginary part of $\dfrac{z_{2}}{z_{1}}$. Express your answer as a fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let real numbers \( a_1, a_2, \cdots, a_n \) be in the interval \([0,2]\), with \( a_{n+1} = a_1 \). Find the maximum value of \(\frac{\sum_{i=1}^{n} a_i^2 a_{i+1} + 8n}{\sum_{i=1}^{n} a_i^2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \int_{-2}^{0} \int_{-2x}^{x+6} \frac{y+2x}{(y-x)^3} \cos(y-x) \, dy \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a configuration of two right triangles, $PQR$ and $PRS$, squares are constructed on three sides of the triangles. The areas of three of the squares are 25, 4, and 49 square units. Determine the area of the fourth square built on side $PS$.

[asy]
defaultpen(linewidth(0.7));
draw((0,0)--(7,0)--(7,7)--(0,7)--cycle);
draw((1,7)--(1,7.7)--(0,7.7));
draw((0,7)--(0,8.5)--(7,7));
draw((0,8.5)--(2.6,16)--(7,7));
draw((2.6,16)--(12.5,19.5)--(14,10.5)--(7,7));
draw((0,8.5)--(-5.6,11.4)--(-3.28,14.76)--(2.6,16));
draw((0,7)--(-2,7)--(-2,8.5)--(0,8.5));
draw((0.48,8.35)--(0.68,8.85)--(-0.15,9.08));
label("$P$",(7,7),SE);
label("$Q$",(0,7),SW);
label("$R$",(0,8.5),N);
label("$S$",(2.6,16),N);
label("25",(-2.8/2,7+1.5/2));
label("4",(-2.8/2+7,7+1.5/2));
label("49",(3.8,11.75));
[/asy]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find \( \alpha \in \mathbb{Z}[i] \) such that the ideal \((3+5i,1+3i)\) is equal to the ideal \((\alpha)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given $\sin 2α - 2 = 2\cos 2α$, find the value of $\sin^{2}α + \sin 2α$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( N \) be the smallest positive integer such that \( 2008N \) is a perfect square and \( 2007N \) is a perfect cube. Find the remainder when \( N \) is divided by 25.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the supremum \( \sup_{x \in \mathbb{R}} \sum_{k \in \mathbb{Z}} \frac{|k|}{(1+|x-k|)^{100}} \) is finite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a queuing system denoted as $(M | M | 2 | 5)$, the input flow rate is $240$ requests per hour, and the average service time for one request is $30$ seconds. Calculate the average waiting time for a request in the queue.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of equivalence relations \(S\) on \(\mathcal P(\mathbb N)\) such that the equivalence class \([\{8\}]_S\) is given by \(\{A \in \mathcal P(\mathbb N) \mid A \neq \{1\} \wedge A \neq \{2\}\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation \( h = g^a \mod p \), where \( p = 541 \), \( g = 2 \), and \( h = 419 \), find the integer \( a \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
在 9 阶置换群 S_9 中，称两个元素 \sigma_1,\sigma_2 是共轭的，如果存在 \tau\in S_9，使得 \tau^{-1}\sigma_1\tau=\sigma_2. 记 \sigma=(12345)(4567)\in S_9，求 S_9 中与 \sigma 共轭的元素个数.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral for a natural number $n$:
$$\int_{1}^{n+1} \frac {\{x\}^{[x]}}{[x]} \, dx$$
where $[x]$ is the greatest integer less than or equal to $x$ and $\{x\}=x-[x]$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a plane, 36 points are marked, with no three points being collinear. Some pairs of these points are connected by segments such that no more than 3 segments originate from any point. Determine the maximum number of distinct closed 4-segment broken lines that can be formed, where the vertices are the marked points and the segments are the drawn segments.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A submarine has three navigational devices, labeled $A$, $B$, and $C$, with mean lifetimes of 1, 1.5, and 3 years, respectively. The submarine can continue operating if at least two devices are functioning. Assuming the failure times for these devices follow an exponential distribution, calculate the expected average length of time the submarine can remain at sea using the conditional expectation approach. Let $T$ represent the length of time the submarine can remain operational.
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Context:
There are 5 pots of chrysanthemums: 2 pots of yellow chrysanthemums, 2 pots of white chrysanthemums, and 1 pot of red chrysanthemums. When arranging them in a row, the 2 pots of yellow chrysanthemums must be adjacent to each other, and the 2 pots of white chrysanthemums must not be next to each other. Calculate the number of different arrangements for these 5 pots of flowers. Express your answer as a single integer.
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Context:
Determine the number of homomorphisms from the symmetric group $S_4$ to the dihedral group $D_4$. The group $D_4$ has 8 elements, and $S_4$ is generated by the transpositions $(12), (13), (23), (14), (24), (34)$. Consider that the order of the image of a generator under a homomorphism must divide the order of the generator.
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Context:
若 $\triangle A B C$ 满足 $\frac{\angle A}{3}=\frac{\angle B}{4}=\frac{\angle C}{5}$, 则 $\frac{\overrightarrow{A B} \cdot \overrightarrow{A C}}{|B C|^{2}}$ 的值为$\frac{a+\sqrt{b}}{c}$，求$a+b+c$的值。
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Context:
Determine the number of solutions for the equation \( \sum_{i=1}^{n} i \ a_i = k \), where \( a_i \in \mathbb{N} \cup \{0\} \) for \( 1 \leq i \leq n \) and \( k \) is a fixed natural number.
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Context:
Let  $a,\ b$  be real constants. If real numbers  $x,\ y$  satisfy  $x^2+y^2\leq 25,\ \ 2x+y\leq 5$ , then find the minimum value of  $z=x^2+y^2-2ax-2by$ .
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Context:
Calculate the contour integral \(\int_{\gamma}\frac{1}{z}dz\) where \(\gamma\) is a circle that excludes the origin, using the definition of the integral along a curve.
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Context:
Two people, A and B, each roll a six-sided die. When a "6" is rolled, they stop rolling. The probability that the number of rolls between the two people differs by at most one is denoted as $\mathrm{P}$. Find $\mathrm{P}$.
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Context:
Calculate the volume of the region defined by the inequalities: \( x^2 + y^2 + z^2 \le 2x \) and \( z \ge \sqrt{x^2 + y^2} \).
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Context:
In $\triangle ABC$, given that $b=2$, $B=45^{\circ}$, if using the Law of Sines to solve the triangle yields two solutions, then the range of values for the side length $a$ is _______.
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Context:
6. Given that the three sides of $\triangle A B C$ are $13,14,15$, there are 4 circles $O, O_{1}, O_{2}, O_{3}$ with the same radius $r$ placed inside $\triangle A B C$, and circle $O_{1}$ is tangent to sides $A B$ and $A C$, circle $O_{2}$ is tangent to sides $B A$ and $B C$, circle $O_{3}$ is tangent to sides $C B$ and $C A$, and circle $O$ is tangent to circles $O_{1}, O_{2}, O_{3}$. Then $r=$ $\qquad$ .
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Context:
Let the sequence $\{a_n\}$ for $n \ge 0$ be defined as $a_0 = c$, and for $n \ge 0$,
$$a_n = \frac{2a_{n-1}}{4a^2_{n-1} - 1}.$$
Compute the sum of all values of $c$ such that $a_{2020}$ exists but $a_{2021}$ does not exist.
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Context:
Determine the condition under which the points \((x_1,0)\) and \((x_2,0)\) divide the segment between \((x_3,0)\) and \((x_4,0)\) harmonically. Here, \(x_1\) and \(x_2\) are the roots of the quadratic equation \(ax^2+bx+c=0\), and \(x_3\) and \(x_4\) are the roots of the quadratic equation \(Ax^2+Bx+C=0\).
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Context:
Find the operator norm \( \|T\|_{\ell^\infty \rightarrow \ell^2} \) for the linear operator \( T: \ell^\infty \rightarrow \ell^2 \) defined by \( T(a_1, a_2, \ldots) = (v_1a_1, v_2a_2, \ldots) \), where \((v_n)_{n \geq 1} \in \ell^2\) is a fixed bounded sequence of real numbers. Use the fact that \( \|Tx\|_2 \leq \|(v_n)\|_2 \cdot \|x\|_\infty \) for all \( x \in \ell^\infty \).
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Context:
Let \( a_{1}, a_{2}, \cdots, a_{2006} \) be 2006 positive integers such that the ratios \( \frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}} \) are all distinct. Determine the minimum number of distinct integers in the sequence \( a_{1}, a_{2}, \cdots, a_{2006} \).
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Context:
A cyclist has 4 miles left to go for every mile he has already completed in a 40-mile race. Write an equation for the total distance, letting $m$ stand for the distance he has left to cycle.
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Context:
3. On the side $B C$, resp. $C D$ of the parallelogram $A B C D$, determine points $E$, resp. $F$ such that segments $E F, B D$ are parallel and triangles $A B E, A E F$ and $A F D$ have the same areas.
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Context:
Let's determine the multiples of 5 that, when divided by 11, leave a remainder of 4.
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Context:
There is colored paper that is 25 centimeters (cm) wide and 30 centimeters (cm) long. When 15 pieces of this colored paper are attached horizontally to the wall, overlapping each other by 0.5 cm, find the horizontal length in meters (m) of the glued colored paper.
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Context:
Valerie needs to put stamps on the envelopes she is about to mail. She has thank you cards for each of her grandmother, uncle, and aunt for the birthday presents they sent. She also has to pay the water bill, electric bill, and the internet bill separately. The electric bill requires 2 stamps, while the internet bill requires 3 stamps. She wants to send three more mail-in rebates than the sum of the three types of bills, and she has twice as many job applications as rebates to mail. For each rebate, she needs 2 stamps, and for each job application, she needs 1 stamp. How many stamps does she need in total for all the envelopes?
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Context:
Find a lower bound for the quadratic form \( \langle M \, y , y \rangle \), where \( y \) is a vector in \( \ell(\mathbb{Z}) \) and \( M \) is the pentadiagonal matrix given by:

\[ M= \begin{pmatrix} \ddots & \ddots & & & & \\ \ddots & 25 & -6 & 0 & 0 & \\ \ddots & -6 & 25 & -6 & 0 & \ddots \\ & 0 & -6 & 25 & -6 & \ddots \\ & 0 & 0 & -6 & 25 & \ddots \\ & & \ddots & \ddots &\ddots \end{pmatrix}. \]
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Context:
Determine the number of points at which the function \( f(x) = \lim_{n\to\infty}\left(4^n+x^{2n}+\frac{1}{x^{2n}}\right)^{\frac{1}{n}} \) is non-derivable.
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Context:
Calculate the covariance \( \operatorname{Cov}(X_0, w_0) \) for an ARMA(1,1) process given by \( X_t = a_1X_{t-1} + w_t + b_1w_{t-1} \), where \( w_t \sim \mathcal{N}(0, \sigma^2) \) independently.
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Context:
Given three integers \(a, b,\) and \(c\) such that \(a, b, c, a + b - c, a + c - b, b + c - a,\) and \(a + b + c\) are seven distinct prime numbers, and \(800\) is an element in the set \(\{a + b, b + c, c + a\}\). Determine the maximum possible value of \(d\), where \(d\) is the difference between the largest and smallest of these seven primes.
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Context:
Determine the radius of convergence for the series \(\sum\limits_{n=1}^\infty (-1)^{n+1} \left(\left|\frac{1-x}{1+x}\right|-1\right)^n \frac{1}{n}\), which represents \(\ln\left|\frac{1-x}{1+x}\right|\). Given that the ratio test yields the inequality \((x<-3) \cup (-\frac{1}{3}<x)\), find the radius of convergence.
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Context:
Evaluate the area of the surface of the cylinder given by the equation \(x^2 + y^2 = 2x\), bounded by \(z = 0\) and \(z = \sqrt{x^2 + y^2}\), using a double integral. Express your answer in terms of polar coordinates if preferred.
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Context:
A point $P(\frac{\pi}{12}, m)$ on the graph of the function $y = \sin 2x$ can be obtained by shifting a point $Q$ on the graph of the function $y = \cos (2x - \frac{\pi}{4})$ to the left by $n (n > 0)$ units. Determine the minimum value of $mn$. Express your answer as a fraction of π.
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Context:
Janice works at a convenience store where her schedule fluctuates. She has to work 5 to 6 days per week with varying hours per day and can earn $30 per day if she works 5 hours. When she works more than 5 hours, she earns an additional hourly rate: $10 per additional hour on weekdays, and $15 per additional hour on weekends. This week, she worked 5 days, including one weekend day. Her total work hours are as follows: Monday - 5 hours, Tuesday - 6 hours, Wednesday - 7 hours, Thursday - 5 hours, and Sunday - 6 hours. Janice earned $15 in tips this week, but she needs to pay 10% tax on her basic earnings and a 5% tax on her overtime and tips. How much will Janice earn this week after deductions?
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Context:
Let $\vec a, \vec b,\vec c$ be three unit vectors such that $\vec a \cdot \vec b+\vec b \cdot \vec c-\vec a \cdot \vec c=\frac{3}{2}$. What is the value of $\vec a \cdot \vec b+\vec b \cdot \vec c+\vec a \cdot \vec c$?
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Context:
Place 5 points in a rectangle of size $17 \times 32$ such that the minimum distance between any two points is maximized. What is the highest possible minimum distance between any two points?
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Context:
How many functions \( f : \{1,2, \ldots, 2013\} \rightarrow \{1,2, \ldots, 2013\} \) satisfy \( f(j) < f(i) + j - i \) for all integers \( i, j \) such that \( 1 \leq i < j \leq 2013 \) ?
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Context:
Consider the initial value problem for the classical Burgers equation given by \( u_t + uu_x = 0 \) with the initial condition:
\[
 u(x,0) = \phi(x) = \begin{cases}
 2, & x \leq \pi/2 \\
 \sin x + 1, & \pi/2 < x \leq 3\pi/2 \\
 0, & x > 3\pi/2
\end{cases}
\]
Determine the breaking time \( t_B \) for this problem.
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Context:
Let \( S \) be the set of functions \( f \) defined on the interval \([0, 1]\) with non-negative real values such that \( f(1) = 1 \) and \( f(x) + f(y) \leq f(x + y) \) for all \( x, y \) with \( x + y \leq 1 \). Determine the smallest constant \( k \) such that \( f(x) \leq kx \) for all functions \( f \) in \( S \) and for all \( x \) in \([0, 1]\).
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Context:
What is the cardinality of \(|\mathbb{R}| + |\mathbb{N}|\)?
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Context:
Amy has divided a square into finitely many white and red rectangles, each with sides parallel to the sides of the square. Within each white rectangle, she writes down its width divided by its height. Within each red rectangle, she writes down its height divided by its width. Finally, she calculates $x$, the sum of these numbers. If the total area of white equals the total area of red, determine the minimum of $x$.
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Context:
Egor wrote a number on the board and encrypted it using a letter puzzle, where different letters correspond to different digits, and the same letters correspond to the same digits. The word "GVATEMALA" was obtained. How many different numbers could Egor have originally written if his number was divisible by 8?
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Context:
Find the maximum distance between two parallel normals of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), where \(a > b\). Express your answer in terms of \(a\) and \(b\).
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Context:
Let $ABC$ be a triangle with inradius $r$ and circumradius $R$. Construct triangle $A'B'C'$ such that $A'B'$ is perpendicular to $OC$ through $C$, and similarly for the other sides. Let $r_1$, $r_2$, and $r_3$ be the inradii of triangles $A'BC$, $AB'C$, and $ABC'$ respectively. Find the sum $r_1 + r_2 + r_3$ in terms of $r$ and $R$. Express your answer as a single expression.
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Context:
Find the form of the polynomial $Q(x)$ that satisfies the equation $Q(x+y) + Q(x-y) = 2Q(x) + 2Q(y)$. Provide your answer in terms of the degree and coefficients of $Q(x)$.
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Context:
Find the sum of the shifted sinc function given by:
\[ g(y) = \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \]
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Context:
Find a closed formula for the power series $A(x) = \sum_{n\geq0} a_nx^n$ that satisfies the differential equation $A'(x) = 2xA(x)$. Express your answer in terms of $x$.
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Context:
For a Riemannian 3-manifold diffeomorphic to \(\mathbb{R}^3\), determine the number of conditions of the spherical coordinate metric that can be satisfied by choosing a suitable coordinate system, considering the curvature of the manifold.
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Context:
Given independent random samples \((X_{1},X_{2},X_{3},...,X_{8}) \sim N(8,16)\), \((Y_{1},Y_{2},Y_{3}) \sim N(1,9)\), and \((Z_{1},Z_{2},Z_{3}) \sim N(6,10)\), find the probability \(P(2\bar{X}-\bar{Z}>3\bar{Y})\). The sample means are distributed as \(\bar{X} \sim N(8,2)\), \(\bar{Y} \sim N(1,3)\), and \(\bar{Z} \sim N(6,\frac{10}{3})\). Calculate \(P(2\bar{X}-\bar{Z}>3\bar{Y})\).
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Context:
if \( a,b \in  \mathrm{R} + ,{a}^{\log b} = 2,{a}^{\log a}{b}^{\log b} = 5 \) ,find out \( {\left( ab\right) }^{\log {ab}} \)
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Context:
Determine the number of three-digit numbers with the following property:
The number formed by the first two digits is prime and the number formed by the last two digits is prime.
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Context:
In a Poisson process with intensity \( \lambda = 2 \), let \( S_{n} \) denote the time of the \( n^{th} \) event. Given that \( N(2) = 4 \), find \( E(S_{6}|N(2)=4) \).
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Context:
Evaluate the limit: \[ \lim_{n\to\infty}\sum_{k=0}^n\binom{n}{k}\frac{3k}{2^n(n+3k)} \]
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Context:
Evaluate the limit: \[ \lim_{x\to 0}\frac{10^x-2^x-5^x+1}{x\tan x} \]
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Context:
Determine the value of \( \theta \) in the interval \( 0 < \theta < \pi \) that satisfies the equation \( \int_{\pi/2}^\theta \sin x\,dx = \sin 2\theta \).
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Context:
Find the number of ways to arrange 6 people at three identical round tables such that each table is occupied.
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Context:
Several students are seated at a large circular table. They pass
 around a bag containing 100 pieces of candy. Each person
 receives the bag, takes one piece of candy and then passes the
 bag to the next person. If Chris takes the first and last piece of
 candy, then the number of students at the table could be
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Context:
In a country with 110 cities, each pair of cities may or may not have a road connecting them. A driver starts in a city with exactly one road leading out of it. He travels along this road to a second city, which has exactly two roads leading out of it. Continuing this pattern, he travels to a third city with exactly three roads, and so on. Eventually, he reaches the $N$-th city, which has exactly $N$ roads leading out of it, and stops his journey. What is the maximum possible value of $N$?
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Context:
Solve in real numbers the system
$$
\begin{cases} x^3 + y = 3x + 4 \\ 2y^3 + z = 6y + 6 \\ 3z^3 + x = 9z + 8\end{cases}
$$
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Context:
Determine the values of $i$ for which the isomorphism $H_i(K^{(p)}) \simeq H_i(K)$ holds, where $K$ is a topological space.
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Context:
Determine the value of $p$ for which the series \( \sum\limits_{n=1}^\infty\left(\dfrac1{n^p}\sum\limits_{k=1}^nk^{3/2}\right) \) converges.
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Context:
Determine the number of zero elements in the inverse of the $n \times n$ matrix $A=\begin{bmatrix} 1&1&1&1&\cdots&1\\ 1&2&2&2&\cdots&2\\ 1&2&1&1&\cdots&1\\ 1&2&1&2&\cdots&2\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ 1&2&1&2&\cdots&\cdots \end{bmatrix}$. \(n \geq 2\)
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Context:
Given a set $\mathcal{A}$ of $K$ unique integers, $\mathcal{A} = (X_{1}, \ldots, X_{K})$, we randomly pick an integer $X_{j}$ from the set. We then draw a random subset $\mathcal{B} \subset \mathcal{A} \setminus (X_{j})$ of $M < K-1$ integers (excluding $X_{j}$ from the draw). What is the probability that $X_{j}$ is less than the minimum integer in subset $\mathcal{B}$? That is, find/estimate the probability: $\mathbb{P}(X_{j} < \min(X_{i}): X_{i} \in \mathcal{B})$.

Example: Let $\mathcal{A} = (2,3,4,5,6,7)$ and $X_{j} = 4$. Let also $\mathcal{B1} = (3,6,7)$ and $\mathcal{B2} = (6,7,5)$. In $\mathcal{B1}$, $X_{j} = 4$ is not less than 3 = min(3,6,7), thus, does not satisfy the criterion, while in $\mathcal{B2}$, $X_{j} = 4$ is less than 5 = min(5,6,7), which satisfies the criterion.
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Context:
Let \( \{U_n\} \) be a sequence of i.i.d Uniform\((0,1)\) random variables and \( \{a_k\} \) be a sequence of i.i.d random variables such that \( P(a_k = \pm1) = \frac{1}{2} \) and are independent of the \( \{U_n\} \). Define the quantities:
\[ X_n = \sum\limits_{k=1}^nU_ka_k, \quad Y_n = \sum\limits_{k=1}^n(U_k^2 - 1/2)a_k \]
Find the limit:
\[ \lim\limits_{n\rightarrow\infty}P\left(|X_n| > Y_n\sqrt{\frac{20}{7}}\right) \]
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Context:
In three-dimensional space, determine the maximum number of points that can be arranged such that the Euclidean distance between every pair of points is an integer, and no four points are coplanar.
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Context:
Determine the smallest natural number \( n \) such that for any finite subset \( A \) of \( \mathbb{R}^2 \), if for every \( n \) points in \( A \) there exist two lines that include all these \( n \) points, then all points in \( A \) lie on two lines.
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Context:
Consider two independent random variables, $X \sim \mathrm{Exp}(4)$ and $Y \sim \mathrm{Exp}(12)$. Calculate the expected value of the minimum of $4X$ and $Y$, i.e., find $E[\min(4X, Y)]$. Provide your answer as a single value.
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Context:
Given the set $S = \{1, 2, 3, 4, 5\}$ and non-empty subsets $A, B, C \subseteq S$, consider functions $f: A \rightarrow B$ and $g: B \rightarrow C$ with their composite function $h: A \rightarrow C$ defined as $h(x) = g(f(x))$. If $h$ is a bijective function, determine the number of possible functions $h(x)$. Note that $n(X)$ denotes the number of elements in set $X$.
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Context:
Identify the finite abelian group of order 8 that has three elements of order two and four elements of order four, up to isomorphism.
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Context:
Determine the range of values of \( k \) for which the function \( f(x) = \frac{x^2 - 1}{(x-2)(x+k)} \) takes all real values, where \( x \) is a real number.
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Context:
Determine the theoretical number of iterations required for the BFGS method to minimize a quadratic function.
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Context:
Determine the smallest integer \( n \geq 3 \) such that for any partition of the set \( I = \{1, 2, \ldots, n\} \) into two non-empty disjoint subsets \( A \) and \( B \), there exist two numbers in either \( A \) or \( B \) whose sum is a perfect square.
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Context:
What is the maximum amount a player with an initial wealth of 2 would pay to participate in the St. Petersburg paradox, assuming a logarithmic utility function?
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Context:
Find the area under the curve $x^2 + y^2 = 3\sqrt{2} x - 5\sqrt{2} y + 2xy$ and above the x-axis between $x = 0$ and $x = 3\sqrt{2}$. Provide your answer as a numerical value.
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Context:
Given the graph $G$ represented by the adjacency matrix:

$$A=\begin{pmatrix} 0 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 &1 &0 \\ 0 &1 &1 &0 & 1\\ 1&1 &0&1&0 \end{pmatrix}$$

Determine the number of minimum spanning trees of the graph $G$. Provide your answer as a single integer.
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Context:
Determine the condition on \( \alpha \) such that the function \( f(x) = |x|^\alpha \) with \( f(0) = 0 \) is weakly differentiable on the unit ball \( A = B_1(0) \subset \mathbb{R}^n \). Assume \( \alpha \neq 0 \) and \( n \geq 2 \).
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Context:
In a regular triangular prism, a plane passes through a side of the lower base and the opposite vertex of the upper base, forming a $45^\circ$ angle with the plane of the lower base. The area of the cross-section is $S$. Find the volume of the prism.
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\boxed{\frac{S \sqrt{S} \cdot \sqrt[4]{6}}{2}}


Context:
Let $S$ be a regular oriented surface such that for any simple, closed, and positively oriented curve in $S$, the integral of the geodesic curvature along this curve is always the same. What can be concluded about the Gaussian curvature of $S$?
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Context:
Calculate the residue of the function \( f(z) = z \sin\left(\frac{1}{1-z}\right) \) at \( z = 1 \).
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Context:
Let $X$ be as in problem 13. Let $Y$ be the number of ways to order $X$ crimson flowers, $X$ scarlet flowers, and $X$ vermillion flowers in a row so that no two flowers of the same hue are adjacent. (Flowers of the same hue are mutually indistinguishable.) Find $Y$.
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Context:
Find the rank of the matrix \( A \) given by:
\[
A=\begin{bmatrix}
a & b & c& d& e& f& g\\
a+1& b-1& 0 & 0 & e& 0 & g \\
a+1& 0 & c-1& 0 & e& f& 0\\
a+1& 0 & 0& d-1& 0& f & g\\
a+1& b& c& 0&  e-1& 0& 0\\
a+1& 0 & c& d& 0& f-1& 0\\
a+1& b& 0 & d& 0 &0 & g-1
\end{bmatrix}.
\]
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Context:
A sequence $\{a_n\}$ is defined such that $a_i=i$ for $i=1,2,3\ldots,2020$ and for $i>2020$, $a_i$ is the average of the previous $2020$ terms. What is the largest integer less than or equal to $\displaystyle\lim_{n\to\infty}a_n$?

2020 CCA Math Bonanza Lightning Round #4.4
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Context:
Let $X = (x_1, x_2, \ldots, x_9)$ be a permutation of the set $\{1, 2, \ldots, 9\}$. Let $A$ be the set of all such permutations $X$. For any $X \in A$, define the function $f(X) = x_1 + 2x_2 + \cdots + 9x_9$. Let $M = \{f(X) \mid X \in A\}$. Find $|M|$, where $|S|$ denotes the number of elements in the set $S$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a focal chord PQ in the parabola \( y^2 = 4ax \), where \( a > 0 \). The tangents to the parabola at points P and Q intersect at a point lying on the line \( y = 2x + a \). If chord PQ makes an angle \( \theta \) with the vertex of the parabola, find the value of \( \tan\theta \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the following limit superior:
\[ \limsup_{n\to\infty}\left(\frac{A^n}{n}+\frac{B^n}{n^2}\right)^{\frac{1}{n}} \]
where \( A, B > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers 123456789 and 999999999 are multiplied. How many of the digits in the final result are 9's?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of distinct monic quadratic polynomials with integer coefficients such that they have at least one root, all roots are powers of 5 with non-negative integer exponents, and all coefficients in absolute value do not exceed $122^{85}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A triangle is inscribed in a circle with a unit radius. Determine the possible distance between the centers of two excircles of the triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum number of cells that need to be marked on a 5 by 5 grid so that among the marked cells, there are no neighbors (sharing a common side or a common vertex), and adding any one cell to these cells would violate this condition?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given rectangle ABCD with AD = 5 and CD = 3, draw three circles with centers at A, B, and C and radii of 1, 2, and 3, respectively. Calculate the approximate area of the shaded region. Express your answer as a single number with one decimal place.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the linear transformation \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) defined by \( T\left(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\right) = \begin{bmatrix} xy \\ xz \end{bmatrix} \). Determine the dimension of the kernel of \( T \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\xi, \eta$ be independent random variables, both following the same exponential distribution. Find $\mathrm{P}\left(\log _{2}(\eta /(\xi-1))<0\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the derivative of the cross-entropy loss function \( J = -\sum_i p_i \log{q_i} \) with respect to \( z_j \), where \( q_j \) is defined using the softmax function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Ali Baba arrived at a cave where there is gold, diamonds, and a chest in which they can be carried. A full chest of gold weighs 200 kg, a full chest of diamonds weighs 40 kg, and an empty chest weighs nothing. A kilogram of gold costs 20 dinars in the market, and a kilogram of diamonds costs 60 dinars. Ali Baba can lift and carry no more than 100 kg. What is the maximum amount of money he can get for the treasures he brings from the cave in one trip?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit \(\lim\limits_{n\to\infty}\left(\frac{a_1}{a_0S_1}+\frac{a_2}{S_1S_2}+\ldots+\frac{a_n}{S_{n-1}S_n}\right)\) where the sequence is defined by \(a_n=2015^n\) and \(S_n=\frac{2015^{n+1}-1}{2014}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Xiao Ming needs to do the following things after school in the evening: review lessons for 30 minutes, rest for 30 minutes, boil water for 15 minutes, and do homework for 25 minutes. The minimum time Xiao Ming needs to spend to complete these tasks is ___ minutes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
We write the equation on the board:

$$
(x-1)(x-2) \ldots (x-2016) = (x-1)(x-2) \ldots (x-2016) .
$$

We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimal number of factors that need to be erased to achieve this?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the sum of the first $n$ terms of a geometric sequence ${a_{n}}$ is $S_{n}$, if $a_{3}=4$, $S_{3}=12$, find the common ratio.
Please reason step by step, and put your final answer within \boxed{}.


Context:
求最大的正实数 $\alpha$, 对于满足 $a+b+c \mid a^{2}+b^{2}+c^{2}$ 的不全相等的正整数 $a, b, c$, 都有方程
$$
(x+b)^{2}=(x+c)(x+a)
$$
在区间 $(0, \alpha)$ 内没有解。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:
Calculate the integral \( \int_{-\infty}^\infty |f(x)|^2 \mathrm{d}x \) using Plancherel's formula, where \( f(\xi) = \int_0^1 \sqrt{x} \sin(\xi x) \mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{\beta\to+\infty}\frac{\int_{0}^2 I_0(\sqrt{t(2-t)})e^{-\beta t}(t-1)dt}{\int_{0}^2 I_0(\sqrt{t(2-t)})e^{-\beta t}dt} \] where \( I_0(x) \) is the modified Bessel function of the first kind.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a Markov chain with the following transition matrix and time weights for each state:

\[ \begin{bmatrix} \frac{3}{4} & \frac{1}{4} & 0 & 0 \\
\frac{4}{5} & 0 & \frac{1}{5} & 0 \\
\frac{6}{7} & 0 & 0 & \frac{1}{7} \\
0 & 0 & 0 & 1 \end{bmatrix} \]

and time weights:
- \( t_0 = 5 \)
- \( t_1 = 10 \)
- \( t_2 = 15 \)
- \( t_3 = 20 \)

Calculate the expected time to reach the absorbing state \( t_3 \) starting from \( t_0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Arnel had ten boxes of pencils with the same number of pencils in each box. He kept some pencils and shared the remaining pencils equally with his five friends. His friends got eight pencils each, and there were 5 pencils in each box. How many pencils did Arnel keep?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value of the maximum distance among three jumps. Each jump has a \( \frac{1}{3} \) probability of being disqualified, resulting in a distance of 0. If a jump is qualified, its distance is uniformly distributed on the interval \([2, 3]\].
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the minimal polynomial of \( \beta \) over \( \mathbb{F}_5 \), where \( \beta \) is a root of the polynomial \( q(Y) = Y^2 - \alpha \) in the extension \( \mathbb{L} \) of \( \mathbb{K} = \mathbb{F}_5(\alpha) \), with \( \alpha \) being a root of the irreducible polynomial \( p(X) = X^2 - 2 \) over \( \mathbb{F}_5 \). The minimal polynomial \( s(Z) \) should be of degree 4.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( n \) be a positive integer. Determine the first digit after the decimal point of the number

$$
\sum_{k=1}^{n} \frac{k(k+1)}{n}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $V$ be a finite-dimensional $
Q$-linear space, and $T: V \rightarrow V$ a linear operator such that $T^2 = -Id$. If $V$ has a $T$-invariant proper subspace $W$ with $\dim(W) \ge 1$, find the smallest possible dimension of $V$. \( \text{Provide your answer as a single integer.} \)
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Context:
A positive integer is called *Tiranian* if it can be written as $x^2 + 6xy + y^2$, where $x$ and $y$ are (not necessarily distinct) positive integers. The integer $36^{2023}$ is written as the sum of $k$ Tiranian integers. What is the smallest possible value of $k$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the variables $a$ and $b$ satisfying $b=-\frac{1}{2}a^2+3\ln(a)$ (with $a>0$), if point $Q(m,n)$ is on the line $y=2x+\frac{1}{2}$, calculate the minimum value of $(a-m)^2+(b-n)^2$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $p(n) = n^4 - 6n^2 - 160$. If $a_n$ is the least odd prime dividing $q(n) = |p(n-30) \cdot p(n+30)|$, find $\sum_{n=1}^{2017} a_n$. ($a_n = 3$ if $q(n) = 0$.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using the 24-hour clock, the time \(\ 16:11:22 \) on the date \(16 / 11 / 22\) (i.e. the 16th of November 2022) has hours, minutes, and seconds equal to the date, month, and (two-digit) year respectively. Let \(S\) be the number of seconds which will elapse after that date and time until this phenomenon next occurs. What is the value of \(\sqrt{S}\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given independent random variables $X, Y \sim \exp(1)$ and $U \sim U[0,1]$, find the distribution of the random variable $U(X+Y)$. Provide your answer in terms of a probability density function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy
$1-3\sin\theta+5\cos3\theta = 0?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the projective closure of the complement of the locus defined by $x_0x_1x_2$ in the projective space $\mathbb{P}^2$. Provide your answer as a geometric description or set.
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Context:
Find the asymptotes for the curve given by the equation \(x_1=\frac{C}{x_{1}^{2}-3x_1x_2+3x_{2}^{2}}\), where \(C\in \mathbb{R}\), in the \((x_1,x_2)\)-plane.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit: \[ \lim_{n \to \infty} \int_{0}^{\pi} \frac{\sin x}{1 + 3\cos^{2}(nx)} \, \mathrm{d}x. \]
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Context:
Calculate the expected value of the random variable Y with the cumulative distribution function given by:
\[ F_Y(y) = 1 - 0.3e^{-0.5y} + 0.6e^{-0.25y} \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
A polygon is said to be friendly if it is regular and it also has angles that, when measured in degrees, are either integers or half-integers (i.e., have a decimal part of exactly 0.5). How many different friendly polygons are there?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions on the initial value $x_0$ such that the differential equation $x' = \sqrt{|x|} + x^2$ has a global solution defined for all $t > 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $F(1,0)$ is the focus of the ellipse $\frac{x^2}{9} + \frac{y^2}{m} = 1$, $P$ is a moving point on the ellipse, and $A(1,1)$, find the minimum value of $|PA| + |PF|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a matrix $A$ with a null trace, determine the minimum number of linear measurements, defined as $A\vec{x}_i \cdot \vec{h}_j$ for specified $\vec{x}_i$ and $\vec{h}_j$, necessary to uniquely determine the matrix $A$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value of $\exp(Y)$, where $Y$ is a binomial random variable with parameters $x$ (number of trials) and $p$ (probability of success).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( P \) and \( Q \) are points on the parabola \( y^{2} = x \) and the circle \( (x-3)^{2} + y^{2} = 1 \) respectively, find \(|PQ|_{\operatorname{mnn}}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the projective space $\mathbb{P}^2$, consider the curves defined by the polynomials $f = x^2 - yz$ and $g = (x+z)^2 - yz$. Compute the intersection multiplicity of $V(f)$ and $V(g)$ at the point $p = [-2:1:4]$. Use the fact that the intersection multiplicity at $p$ is equal to the Hilbert polynomial of the $I(p)$-primary component of $(f)+(g)$, where $I(p) = (X+2, Y-1, Z-4)$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the radius of convergence for the power series $$\sum \left(\frac{n^3}{2^n}\right)^{n^2} x^{2n^2+1}$$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $P$ be the set of positive integers that are prime numbers. Find the number of subsets of $P$ that have the property that the sum of their elements is $34$ such as $\{3, 31\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the dimension of the tangent space of the submanifold $G = \{M \in M_4(\mathbb{R}) \mid M^{tr}PM = P\}$, where $P = \operatorname{diag}(1,1,1,-1)$, at the identity matrix.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the square matrix $A$ of minimal order such that the vector function \( \vec{X}_0(t) = \vec{W}(t) + \vec{V}e^{2t} \) satisfies the differential equation \( \vec{X}_0'(t) = A \cdot \vec{X}_0(t) \), where:

\[ \vec{W}(t) = \begin{pmatrix} \sin(t) + \cos(t) \\ \cos(t) - \sin(t) \\ 0 \\ \vdots \\ 0 \end{pmatrix} \quad \text{and} \quad \vec{V} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \\ \vdots \\ v_n \end{pmatrix} \]

Determine the smallest possible value of \( n \) for which such a matrix \( A \) exists.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a nice functional $F: \mathcal B \to \mathbb R$ on a Banach space $\mathcal B$, define the map $\phi_x: \mathbb R \to \mathbb R$ for a fixed $x \in \mathcal B$ and $c \in \mathbb R$ by $\phi_x(c) = F(cx)$. Find an expression for $\frac{d}{dc}\phi_x(c)$. Assume that $F$ is differentiable in a suitable sense.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A barn with a flat roof is rectangular in shape, $ 10$ yd. wide, $ 13$ yd. long and 5 yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the Cartesian coordinate system $xOy$, the figure formed by the line $y=x$, the line $x=1$, and the x-axis is rotated around the x-axis to form a cone. The volume $V_{\text{cone}}$ of the cone is $\int_{0}^{1} \pi x^{2} dx = \frac{\pi}{3}x^{3}\bigg|_{0}^{1} = \frac{\pi}{3}$. By analogy: rotating the figure formed by the curve $y=2\ln x$, the line $y=1$, and the x and y axes around the y-axis once, we obtain a solid of revolution. The volume $V$ of this solid is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the length of the primary component of the ideal \((xy, y^2)\) at the origin in the polynomial ring \(\mathbb{R}[x, y]\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer \( N \) such that all elements of the symmetric group \( S_{11} \) have orders dividing \( N \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of distinct $2 \times 2$ matrices with complex entries that satisfy the equation $A^3 = A$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field \( F = y\hat{i} + z\hat{k} \) across the boundary of the solid cone defined by \( 0 \le z \le 1 - \sqrt{x^2 + y^2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral: \[ \int_{1}^{\infty} \int_{1}^{\infty} \frac{1}{x^4y^2\ln (xy)} \, dx \, dy \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest integer \( n \) such that the remainder estimate \( |R_n| \le \frac{M}{(n+1)!} (x-a)^{n+1} \), where \( M \) is the maximum value of \( |f^{(n+1)}(z)| \) on the interval between \( a \) and the indicated point, satisfies \( |R_n| \le \frac{1}{1000} \) for the function \( f(x) = e^{-2x} \) on the interval \([-1,1]\), with \( a = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In James' aquarium, there's a Bobbit worm, a predatory fish, and 60 other fish. The Bobbit worm and the predatory fish are nocturnal, feeding only at night. Every night, the Bobbit worm eats 2 fish, and the predatory fish eats 3 fish. The other fish in the aquarium reproduce at a rate of 2 new fish every 3 days, with reproduction occurring at noon. After two weeks, James adds 8 more fish to the aquarium. A week later, he discovers the Bobbit worm and the predatory fish. How many fish does the aquarium have when James discovers the Bobbit worm and the predatory fish?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sample $X_1, X_2, \ldots, X_{15}$ from a distribution with probability density $\exp(-(x-\theta))$ for $x \ge \theta$, test the hypothesis $H_0: \theta=0$ against $H_1: \theta>0$. The test statistic is $T = \min\{x_1, x_2, \ldots, x_{15}\}$. If the observed value of $T$ is $t=0.1$, what is the p-value of this test? (Hint: If $X_1, X_2,\ldots,X_n$ is a sample from an $\operatorname{Exp}(\lambda)$ distribution, then $\min\{X_1, X_2,\ldots,X_n\}$ has an $\operatorname{Exp}(n\lambda)$ distribution.)
Please reason step by step, and put your final answer within \boxed{}.


Context:
The Kovács couple was visited by four other couples. After the introductions, Mr. Kovács observed that, besides himself, each of the remaining nine attendees had met a different number of people.

How many people did Mrs. Kovács meet?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in the 3-by-3 array of boxes, then Carl places an O
 in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all 9 boxes are filled or
 one of the players has 3 of their symbols in a row - horizontal, vertical, or diagonal - whichever comes first, in which case that
 player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and tha
 Carl wins the game when he places his third O. How many ways can the board look after the game is over?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the limit: \[ \lim_{z\rightarrow \infty}\dfrac{\Gamma(k,iz)}{\Gamma(k)} \] where $z$ is a real number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the residue of the function \( \frac{(z^6+1)^2}{(z^5)(z^2-2)(z^2-\frac{1}{2})} \) at \( z = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance,
$S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}$
 .Find the least positive integer $k$ making the number $k!\cdot  S_{2016}$ an integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \(\lim_{x \to 0} \frac{e^{\tan x} - e^x + \ln(\sec x + \tan x) - x}{\tan x - x}\) using L'Hopital's rule or alternative methods.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a diagonalizable square matrix $A$ with eigendecomposition $A = S_A D_A S_A^{-1}$, express the matrix $S_{A \oplus A}$ that diagonalizes the Kronecker sum $A \oplus A = A \otimes I + I \otimes A$ in terms of $S_A$. Here, $I$ is the identity matrix of the same size as $A$. Assume that $A \otimes I$ and $I \otimes A$ are simultaneously diagonalizable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the maximum value of \( x_1^2 + x_2^2 + \cdots + x_{2014}^2 \) given that \( x_i \in [-11, 5] \) for \( i = 1, 2, \ldots, 2014 \) and \( x_1 + x_2 + \cdots + x_{2014} = 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the largest power of 7 that divides the binomial coefficient \( \binom{2016}{1003} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $M$ be a $7\times6$ real matrix, and let $X$ be the matrix defined as:
$$
X = \pmatrix{
0&0&x_{13}&x_{14}&0&0\\
0&0&0&0&0&0\\
0&0&x_{33}&x_{34}&0&0\\
0&0&0&0&0&0\\
0&0&0&x_{54}&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0}.
$$
If $N = M + X$ has rank $4$, what is the maximum possible rank of $X$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x \to 0} \frac{(x-\sqrt{2})^4}{\sin^3(x)\cos^2(x)}\int_{0}^x \frac{t^2}{t^4+1}dt$$ using L'Hopital's rule or other appropriate methods.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $H$ be a Hilbert space representing single-particle states, and let $\mathfrak{F}$ be the fermionic Fock space. Determine whether the creation operator $c^*(f)$ is a bounded linear operator on $\mathfrak{F}$ for $f \in H$. If it is bounded, find its norm.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tej is typing a string of $L$s and $O$s that consists of exactly $7$ $L$s and $4$ $O$s. How many different strings can he type that do not contain the substring ‘$LOL$’ anywhere? A substring is a sequence of consecutive letters contained within the original string.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of conjugacy classes in the Dihedral group \( D_{11} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the matrix representation of $\exp(tE_{a,b})$ given that $E_{a,b}^2 = 0$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the values of $a$ for which the improper integral \( \int_{0}^{\infty} \frac{\sin x}{x^a} \, dx \) converges.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Define the sequence \(\{a_n\}\) as \(a_n = n^3 + 4\), where \(n \in \mathbb{N}_+\). Let \(d_n = \gcd(a_n, a_{n+1})\), the greatest common divisor of \(a_n\) and \(a_{n+1}\). Determine the maximum value of \(d_n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $a_1, \dots, a_n$ be a set of vectors in a vector space $V$. It is given that $a_1, \dots, a_n$ are linearly independent. Find all values of $\lambda$ such that the following set is linearly independent:

$$a_1 + a_2, \dots, a_{n-1} + a_n, a_n + \lambda a_1$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a client-server scenario, a client masks a value $b \in \mathbb{F}_q$ as $M=(r_1\cdot b+r_2)\bmod q$, where $r_1$ and $r_2$ are chosen uniformly at random from the field such that $r_1, r_2 > \frac{q}{2}$. The client sends $M$ to the server and deletes $b$. The client can later check the integrity of $M$ by computing $s_1=(M- r_2)\bmod q$ and $s_2=(s_1\cdot (r_1)^{-1})\bmod q$. If $s_2$ is a uniformly random value, the client can detect a modification. The server can modify $b$ by finding $k=c\cdot r_1$ with $k<q$ and computing $M+k$, resulting in $s_2=b+c$. If $k \neq c\cdot r_1$, $s_2$ is uniformly random. What is the probability that the server modifies $M$ without being detected?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-isomorphic well-orderings of the set of natural numbers \( \mathbb{N} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimal degree $d$ such that there exists a map of degree $d$ from $S^2 \times S^2$ to $\mathbb{CP}^2$. Consider the cohomology ring of $\mathbb{CP}^2$ and the classes in $H^2(S^2 \times S^2)$ to find the minimal $d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given two independent and identically distributed random variables $X$ and $Y$ with moment generating function \( M(t) = \left( \frac{1}{1-1.5t}\right)^2 \) for \( t < \frac{2}{3} \), calculate the standard deviation of the sum $X + Y$. Assume $X$ and $Y$ are Erlang distributed.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $\overline{X_n}$ denote the mean of a random sample of size $n$ from a Poisson distribution with mean $\lambda = 2$. Use the delta method to find the limiting distribution of $\sqrt{n}\left( (\overline{X_n})^2 -4 \right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the expression \( 2^{2010}\frac{\int^{1}_{0}x^{1004}(1-x)^{1004}dx}{\int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the class to which the language \(L_3=\{\langle M\rangle\mid|\langle M\rangle|\le 2016\text{ and M is a TM that accepts }\varepsilon \}\) belongs.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an initial boundary value problem (IBVP) for the heat equation with non-homogeneous Neumann boundary conditions, where the boundary conditions satisfy $A = B$, determine the equation that governs the function $v(x)$. Provide the resulting equation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \(a+b+c=10\), \(abc=2015\), and \(ab+bc+ca=1\), find the value of the expression \((bc-a^2)(ac-b^2)+(ac-b^2)(ba-c^2)+(ba-c^2)(bc-a^2)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \displaystyle\int_0^{2\pi} \frac{\cos2x}{5+ 4\cos x}\mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the square of the perpendicular distance from the origin to the line L, which intersects perpendicularly both the lines \( \frac{x+2}{2}=\frac{y+6}{3}=\frac{z-34}{-10} \) and \( \frac{x+6}{4}=\frac{y-7}{-3}=\frac{z-7}{-2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest integer $n$ such that the $n$th Taylor polynomial $P_n(x)$ of $f(x) = \tan^{-1}(x)$ about $x_0 = 0$ approximates $f(x)$ within $10^{-6}$ on the interval $[-0.5, 0.5]$. Provide the value of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \int_\gamma \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy + \ln{(z^4+z^2+1)}dz \) where \( \gamma \) is the curve formed by the intersection of the cone \( x^2+y^2=z^2 \) and the plane \( x+y+10z=10 \). The curve is oriented to make one clockwise revolution around the \( z \)-axis when viewed from above.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A computer science laboratory has 15 workstations and 10 servers. A cable can be used to directly connect a workstation to a server, with only one direct connection to each server active at any time. Determine the minimum number of direct connections required to ensure that any set of 10 or fewer workstations can simultaneously access different servers via direct connections.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose 20 real numbers are chosen independently from the interval [0, 20] with uniform probability. Find a lower bound for the probability that their average lies between 8 and 12.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ordered pairs \((F_1, F_2)\) such that \(F_1 \not\subseteq F_2'\), where \(F_1, F_2 \subseteq \{a, b, c, d\}\) and \(F_2' = F_2 \cup \{c \mid d \in F_2\}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^\frac{\pi}{2} \frac{\cos x}{(1+\sqrt{\sin (2x)})^2}\,dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A vehicle is equipped with four regular tires and one spare tire. Each regular tire has an independent exponential lifetime with a mean of 4000 miles, and the spare tire has an exponential lifetime with a mean of 2000 miles. Calculate the expected mileage the vehicle can travel before a tire replacement service is required, assuming the spare tire is used when one of the regular tires fails.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of $r$ such that a Möbius transformation maps the annulus $\{z:r<|z|<1\}$ to the region between the circles $\{z:|z-1/4|=1/4\}$ and $\{z:|z|=1\}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the divergence of the vector field \(r^n \vec{r}\), where \(r\) is the magnitude of the position vector \(\vec{r}\) and \(n\) is a constant, using tensor notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the asymptotic behavior of the function \( \left(\frac{c+\sqrt{x}}{-c+\sqrt{x}}\right)^x \exp(-2c\sqrt{x}) \) as \( x \to \infty \), where \( c \) is a constant.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Estimate the number of digits of accuracy that can be expected in the solution \( x \) of the linear system \( Ax = b \), given that the entries of \( A \) have about 6 digits of accuracy, \( b \) has about 4 digits of accuracy, and the condition number of \( A \) is about \( 10^3 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the infimum of the normalized distances between the zeros of the Riemann zeta function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine all values of \(x\) for which the power series \(\sum_{n=0}^\infty \frac{(x-2)^n}{4^n\sqrt{n}}\) is convergent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{n \to \infty} (-1)^n \sin\left(\pi(n^2+0.5n+1)^{\frac{1}{2}}\right)\sin\left(\frac{(n+1)\pi}{4n}\right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence of identically distributed continuous random variables $\xi_1, \xi_2, \ldots, \xi_n, \ldots$, where each variable has the same distribution, let $\nu$ denote the minimum index $k$ such that $\xi_{k-1} > \xi_k$. Find the expected value of $\nu$, denoted as $\mathbb{E}(\nu)$. \( \text{Express your answer in terms of known constants or functions.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in some order, compute the height of the cave.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the dual space of the sum $X=L^2(\mathbb{R})+L^3(\mathbb{R})$ using the norm defined in the problem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{\frac{3\pi}{2}}^{\frac{7\pi}{2}} \frac{e^{\tan^{-1}(\cos(x))}}{e^{\tan^{-1}(\sin(x))} + e^{\tan^{-1}(\cos(x))}} \,dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of partial orders on a set of 4 elements, given that the number of partial orders on a set of 3 elements is 19.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the inverse Z-transform of \( \frac{z}{z+1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of three-digit numbers such that:
- The number formed by the first two digits is prime.
- The number formed by the last two digits is prime.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In an acute triangle \( ABC \), let \( M \) be the midpoint of \( AC \). A circle \( \omega \) passing through \( B \) and \( M \) intersects the sides \( AB \) and \( BC \) again at points \( P \) and \( Q \), respectively. Let \( T \) be the point such that the quadrilateral \( BPTQ \) forms a parallelogram. If \( T \) lies on the circumcircle of triangle \( ABC \), find all possible values of the ratio \( \frac{BT}{BM} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A tetrahedron inscribed in a parallelepiped cuts off four tetrahedra from the parallelepiped with its face planes. What fraction of the volume of the parallelepiped is the volume of the tetrahedron whose vertices are the centroids of the cut-off tetrahedra?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For four distinct integers, all pairwise sums and pairwise products were calculated. The obtained sums and products were written on the board. What is the smallest number of distinct numbers that could be on the board?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Say that a 4-digit positive integer is *mixed* if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits.  For example, 2013 is mixed.  How many 4-digit positive integers are mixed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all pairs of prime numbers whose difference of squares is a prime number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A random walk is initiated at a vertex of a tetrahedron, with each of the three edges connected to this vertex being equally likely to be traversed. Determine the expected number of steps required to reach the opposite vertex.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A computer company has a chance to get four different types of contracts: hardware, software, network services, and maintenance. The probability that the company will get the hardware contract is 3/4, the probability it will not get the software contract is 3/5, the probability of getting a network services contract is 2/3, and the probability of getting a maintenance contract is 1/2. If the probability of getting at least one contract is 7/8, what is the probability that the company will get all four contracts?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a sphere, there are four points A, B, C, and D. If AB, AC, and AD are mutually orthogonal and AB = AC = AD = 4, then calculate the surface area of the sphere. Express your answer in terms of pi (π).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two numbers following it in a clockwise direction, and the sum of all numbers is 278, how many different values can $n$ take? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
某班准备购买班服,确定从A, B两种款式中选出一种统一购买，现在全班50位同学赞成购买A, B款式的人数分别为20,30位，为了尽量统一意见，准备在全班进行三轮宣传,每轮宣传从全班同学中随机选出一位，介绍他赞成款式的理由，假设每轮宣传后，赞成该同学所选款式的不会改变意见，不赞成该同学所选款式的会有5位改变意见，赞成该同学所选款式.
设经过三轮宣传后赞成A款式的人数为X，求随机变量X的期望.
Please reason step by step, and put your final answer within \boxed{}.


Context:
According to the rules of a chess match, the winner is the one who gains two victories over the opponent. Draws do not count. The probabilities of winning for the opponents are equal. The number of decisive games in such a match is a random variable. Find its expected value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given point $M(3,t)$, if there exist two points $A$ and $B$ on circle $O$: $x^{2}+y^{2}=6$ such that $\angle AMB=90^{\circ}$, then the range of $t$ is _______.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{- \sqrt {3}\leqslant t\leqslant \sqrt {3}}


Context:
Let $\phi (x, u)$ be the smallest positive integer $n$ so that $2^u$ divides $x^n + 95$ if it exists, or $0$ if no such positive integer exists. Determine$ \sum_{i=0}^{255} \phi(i, 8)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A sum of Rs. 5060 is divided among 4 persons, p, q, r, and s, in such a way that the ratios p : q : r : s = 3 : 5 : 7 : 9 and q : r : s = 10 : 11 : 13. Find the amount that person r receives.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function $f(x) = x^n + 1$ always passes through a fixed point. What are the coordinates of this fixed point?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $T$ be a trapezoid with two right angles and side lengths $4,4,5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into 4 regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240 d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the smallest positive integer \( n \) such that, if there are initially \( 2n \) townspeople and 1 goon, then the probability that the townspeople win is greater than 50%.
Please reason step by step, and put your final answer within \boxed{}.


Context:
King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The numbers from 1 to 8 are placed at the vertices of a cube so that the sum of the numbers at any three vertices, which lie on the same face, is at least 10. What is the minimum possible sum of the numbers on one face?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which integers \( n \) do there exist integers \( a \) and \( b \) such that \( n + a^{2} = b^{2} \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are nine reduced quadratic polynomials written on a board: \(x^{2}+a_{1} x+b_{1}, x^{2}+a_{2} x+b_{2}, \ldots, x^{2}+a_{9} x+b_{9}\). It is known that the sequences \(a_{1}, a_{2}, \ldots, a_{9}\) and \(b_{1}, b_{2}, \ldots, b_{9}\) are arithmetic progressions. It turns out that the sum of all nine polynomials has at least one root. What is the maximum number of the original polynomials that can have no roots?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two identical rectangular crates are packed with cylindrical pipes, using different methods. Each pipe has a diameter of 8 cm. In Crate A, the pipes are packed directly on top of each other in 25 rows of 8 pipes each across the width of the crate. In Crate B, pipes are packed in a staggered (hexagonal) pattern that results in 24 rows, with the rows alternating between 7 and 8 pipes.

After the crates have been packed with an equal number of 200 pipes each, what is the positive difference in the total heights (in cm) of the two packings?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider a sequence $x_{n}$ such that $x_{1}=x_{2}=1, x_{3}=\frac{2}{3}$. Suppose that $x_{n}=\frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2}-x_{n-1} x_{n-3}}$ for all $n \geq 4$. Find the least $n$ such that $x_{n} \leq \frac{1}{10^{6}}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many people are needed to be certain that at least two of them have the same birthday?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A convex hexagon has interior angles with measures $2x+2$, $3x$, $4x+3$, $5x$, $6x-1$, and $7x$. What is the measure of the largest angle?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Arun, Tarun, and Chun can do a work in 10 days. Arun can do the work alone in 60 days, and Tarun can do the work alone in 40 days. After working together for 4 days, Tarun leaves for his village, and Chun, who can finish the work alone in 20 days, joins. How many days are required for Arun and Chun to complete the remaining work together?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a, x, y \) be three positive integers greater than 100, satisfying the equation \( y^{2} - 1 = a^{2}(x^{2} - 1) \). Find the minimum value of \( \frac{a}{x} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the probability that \( G(x,y,z) > 100 \), where \( G(x,y,z) = 20000 - 800x - 100000y + 200000z \). The variables \( x, y, z \) are independent and normally distributed with the following parameters: \( x \sim N(150, 20^2) \), \( y \sim N(2, 0.8^2) \), and \( z \sim N(0.2, 0.02^2) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the Ackermann function at the point (4, 0) to find the value of \( \operatorname{ack}(4, 0) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two persons start running simultaneously around a circular track of a certain length from the same point at speeds of 20 kmph and 40 kmph. They meet for the first time anywhere on the track after 14.998800095992321 seconds when they are moving in the opposite direction. What is the length of the track?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute
\[\frac{\lfloor \sqrt{1} \rfloor \cdot \lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{3} \rfloor \cdot \lfloor \sqrt{5} \rfloor \dotsm \lfloor \sqrt{15} \rfloor}{\lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{4} \rfloor \cdot \lfloor \sqrt{6} \rfloor \dotsm \lfloor \sqrt{16} \rfloor}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five super-intelligent and vindictive pirates, ranked by height, must distribute 500 gold coins. The shortest pirate proposes a distribution, and if at least half of the pirates, including the proposer, agree, the distribution is accepted. Otherwise, the proposer is eliminated, and the next shortest proposes. What is the optimal number of coins the shortest pirate can receive in a Nash Equilibrium?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A room is illuminated by two light bulbs. The bulbs are replaced only when both have burned out. The lifetimes of the bulbs are independent and identically distributed, following an exponential distribution with parameter \( \lambda = 1 \). What fraction of the time is the room only half-lit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_k$, $k \geq 0$, be iid random variables with pmf $p_j = P(X_1 = j)$ for $1 \leq j \leq m$, where $\sum_{j = 1}^{m}p_j = 1$. Define $T = \min\{n > 0 : X_n = X_0\}$ as the first time we observe the initial value again. Compute $E[T]$. \(E[T]\) is the expected value of the first time the random variable $X_n$ equals $X_0$ again.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( k \) such that \[ \int_{0}^{\infty} \left(\dfrac{\ln x}{1-x}\right)^{2} \mathrm{d}x + k \times \int_{0}^{1} \dfrac{\ln (1-x)}{x} \mathrm{d}x = 0. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
John ends up serving on jury duty for a case involving three defendants with different roles in the crime. Jury selection takes 2 days. The trial itself lasts 4 times as long as jury selection. However, on each day of the trial, 2 hours are spent on reviewing evidence and 1 hour for lunch, apart from the regular 6 hours daily trial session. It is a complicated trial.

The jury must deliberate separately for each defendant. The number of hours spent in jury deliberation for the first defendant was the equivalent of 6 full days. For the second defendant, the jury took 4 full days of deliberation. The third defendant required 5 full days of deliberation. They spend 14 hours a day in deliberation, considering a 2-hour break for lunch and rest in between.

How many days does John spend on jury duty in total, taking into account the extra time spent on evidence review, lunch, and breaks during the trial and deliberation periods for all three defendants?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: $$\int _{0}^{1} \frac{7x^4+5x^2+4x}{\sqrt {x^3+x+1}} \, dx$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit \( \lim_{n\rightarrow\infty} \sum^{n}_{k=0}\left|\frac{2\pi\cos(k\pi(3-\sqrt{5}))}{n}\right| \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the asymptotic behavior of the integral \((n+1)\int_{0}^{1}x^n\ln(x+1)dx\) as \(n\) approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the flux of the vector field \( v = (3y_1 - 2y_2^2 + y_3^2, 3y_1^2 - 2y_2 + y_3^2, 3y_1^2 - 2y_2^2 + y_3) \) through the surface \( \tilde{Y} \), where \( \tilde{Y} \) is the image of the parameterization \( \varphi: K \to \mathbb{R}^3 \) with \( K = [0, 2\pi] \times [0, 1] \) and \( \varphi(x) = ((\cos x_1)(x_2 - x_2^3), (\sin x_1)(x_2 - x_2^3), x_2^2 + \frac{1}{3} x_2^6) \).
Please reason step by step, and put your final answer within \boxed{}.