Context:
A point is chosen randomly from within a circular region with radius $r$. A related concentric circle with radius $\sqrt{r}$ contains points that are closer to the center than to the boundary. Calculate the probability that a randomly chosen point lies closer to the center than to the boundary.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the line $y=kx-k$ and the parabola $y^{2}=4x$ intersect at points $A$ and $B$, and the distance between $A$ and $B$ is $4$, find the distance from the midpoint of chord $AB$ to the $y$-axis. Express your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If x, y, and z are positive integers and x = 4, y = 7, and z is some value, then the least possible value of x - y - z is 17. What is the value of z?
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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
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which dimensions \(d \geq 1\) does the expectation \(\mathrm{E}\left(\xi^{\top} B \xi\right)^{-1}\) exist, where \(\xi\) is a Gaussian vector in \(\mathbb{R}^{d}\) with a unit covariance matrix, and \(B\) is a symmetric positive definite matrix?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $\sin A + \cos A = \frac{\sqrt{5}}{5}$, find the value of $\sin A - \cos A$. Express your answer as a fraction with a square root if necessary.
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Context:
Suppose that $X_1, X_2, \dots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S = \sum_{i=1}^k X_i/2^i$, where $k$ is the least positive integer such that $X_k < X_{k+1}$, or $k = \infty$ if there is no such integer. Find the expected value of $S$.
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Context:
Fox jeans regularly sell for $15 a pair, and Pony jeans regularly sell for $18 a pair. Additionally, there are jackets available that regularly cost $25 each. During a clearance sale, these regular unit prices are discounted at different rates so that a total of $15 is saved by purchasing 7 items - 3 pairs of Fox jeans, 2 pairs of Pony jeans, and 2 jackets. If the sum of the three discount rates for the jeans and the jackets is 35 percent, given that the jacket's discount rate is 10 percent, what is the discount rate on Pony jeans?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the equation about $x$, $x^{2}e^{x} + t - a = 0$, where $x \in [-1, 1]$, if for any $t \in [1, 3]$, the equation always has a unique real solution, then the range of values for the real number $a$ is __________.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\left( \frac{1}{e} + 3, e + 1 \right]}


Context:
Given the equation of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $a>b>0$, and a point $M$ on the ellipse with a perpendicular line drawn to the $x$-axis, the foot of the perpendicular being $P$, if $\frac{|MP|^2}{|A_1P| \cdot |A_2P|}=\frac{1}{3}$, calculate the eccentricity of the ellipse. Express your answer as a single value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers. Please provide the value of a-b. Here $[x]$ is defined to be the integer such that $[x] \leqq x<[x]+1$ and $\log x$ is the logarithm of $x$ to base $e$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a real number $x$ and the matrix $M_x = \begin{pmatrix} 1 & x+1 & 0 & 0 \\ 0 & x & x-1 & 0 \\ 0 & x-1 & x & 0 \\ 1 & 0 & x+1 & x \end{pmatrix}$, find the value(s) of $x$ for which $M_x$ is diagonalizable. The eigenvalues of $M_x$ are $\lambda_1 = 1$, $\lambda_2 = x$, and $\lambda_3 = 2x-1$. Determine the values of $x$ that make the matrix diagonalizable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Arrange the numbers $1, 2, \ldots, 1962$ in the sequence $a_{1}, a_{2}, \ldots, a_{1962}$ such that the sum $\left|a_{1} - a_{2}\right| + \left|a_{2} - a_{3}\right| + \ldots + \left|a_{1961} - a_{1962}\right| + \left|a_{1962} - a_{1}\right|$ is maximized.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the maximum number of elements in a sigma-algebra generated by a collection of $N$ distinct subsets of a set $X$, given that the intersection of any two subsets in the collection is non-empty.
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Context:
Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly  $ 5\over 4$  times greater than  Vasya's result. What minimum number of portraits can be on the wall?
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Context:
Consider two non-zero square matrices \( A \) and \( B \) such that \( A^2B = BA \). If \( (AB)^{10} = A^kB^{10} \), find the value of \( k \).
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Context:
On New Year's Eve, Santa Claus gave the children the following task: by using all nine digits from 1 to 9 exactly once, insert either "+" or "-" between each pair of adjacent digits so that the result yields all possible two-digit prime numbers. How many such numbers can be obtained?
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Context:
Ninety-nine children are standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If two balls end up with the same child, one of these balls is irrevocably lost. What is the minimum time required for the children to have only one ball left?
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Context:
A random simulation method is used to estimate the probability of an athlete hitting a target in a shooting event. A calculator generates random integers between 0 and 9. The numbers 0, 1, 2, and 3 represent a miss, while 4, 5, 6, 7, 8, and 9 represent a hit. Each group of 4 random numbers corresponds to the results of 4 shots. After random simulation, 20 groups of the following random numbers are generated:
7527, 0293, 7140, 9857, 0347, 4373, 8636, 6947, 1417, 4698,
0371, 6233, 2616, 8045, 6011, 3661, 9597, 7424, 7610, 4281
Estimate the probability that the athlete hits the target at least three times in four shots: $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit for a standard normally distributed random variable \(X \sim \text{Norm}(0,1)\):
\[
\lim_{x\rightarrow\infty}\frac{x}{\overline{F}_{X}(x)}\int_{x}^{\infty}\overline{F}_{X}(s)ds
\]
where \(\overline{F}_{X}(x) = 1 - F_{X}(x)\), and \(F_{X}(x) = \int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}}dx\). Use L'Hopital's rule or other appropriate techniques to solve this limit.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $\Lambda \sim \text{Exp}(1)$ and $X$ has a probability mass function $P_\lambda(X=x)=\frac{\lambda^x e^{-\lambda}}{x!}$, where $\lambda$ is a realization of the random variable $\Lambda$, find the conditional expectation $E(e^{-\Lambda}|X=1)$. Assume $f(\lambda)=e^{-\lambda}$ for $\lambda>0$ and $0$ otherwise.
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Context:
Let there be a point $M(x_0, 1)$. If there exists a point $N$ on the circle $O: x^2 + y^2 = 1$ such that $\angle OMN = 45^\circ$, then the range of values for $x_0$ is ________.
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Context:
A positive integer $n \geq 4$ is called *interesting* if there exists a complex number $z$ such that $|z|=1$ and \[1 + z + z^2 + z^{n-1} + z^n = 0.\] Find how many interesting numbers are smaller than $2022$.
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Context:
Given a graph $G$ with a minimum degree of $\delta(G)=4$, find the connectivity $\kappa(G)$ of the graph.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A certain number less than three times the number of Doberman puppies plus the difference between the number of Doberman puppies and the number of Schnauzers is equal to 90. The number of Doberman puppies is 20, and there are 55 Schnauzers. What is that certain number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In Anchuria, a checkers championship is held over several rounds, with each round taking place in a different city on a different day. There are eight cities and eight days available for the championship. To maximize the probability that no fan correctly guesses the cities and days for all the rounds, how many rounds should be scheduled in the championship?
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Context:
10. (10 points) In the Land of Dwarfs, there are 2011 dwarfs, each of whom wears either a red hat or a blue hat. When a dwarf wears a red hat, they tell the truth; when they wear a blue hat, they can lie; and they can change the color of their hat at any time. One day, every two of them met exactly once, and each said that the other was wearing a blue hat. How many times did they change the color of their hats at least? $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many integers $n$ with $10 \leq n \leq 500$ have the property that the hundreds digit of $17n$ and $17n + 17$ are different?
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Context:
In a theater performance of King Lear, the locations of Acts II-V are drawn by lot before each act. The auditorium is divided into four sections, and the audience moves to another section with their chairs if their current section is chosen as the next location. Assume that all four sections are large enough to accommodate all chairs if selected, and each section is chosen with equal probability. What is the probability that the audience will have to move twice compared to the probability that they will have to move only once?
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many roots does the following equation have:

$$
10^{\lg \cos x}=\frac{1}{7} x-1 ?
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among all triangles formed by selecting any 3 of the 8 vertices of a rectangular prism, how many are acute triangles?
Please reason step by step, and put your final answer within \boxed{}.


Context:
The height of a cone and its slant height are 4 cm and 5 cm, respectively. Find the volume of a hemisphere inscribed in the cone, whose base lies on the base of the cone.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A collection of circles in the upper half-plane, all tangent to the \(x\)-axis, is constructed in layers as follows. Layer \(L_0\) consists of two circles of radii \(70^2\) and \(73^2\) that are externally tangent. For \(k \ge 1\), the circles in \(\bigcup\limits_{j=0}^{k-1}L_j\) are ordered according to their points of tangency with the \(x\)-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer \(L_k\) consists of the \(2^{k-1}\) circles constructed in this way. Let \(S=\bigcup\limits_{j=0}^{6}L_j\), and for every circle \(C\) denote by \(r(C)\) its radius. Find \(\sum\limits_{C\in S} \frac{1}{\sqrt{r(C)}}\). The original answer is in the form of \(\frac{k}{m}\), where the fraction is in its simplest form. Please give the value of k + m.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In rectangle $ABCD,$ $P$ is a point on side $\overline{BC}$ such that $BP = 18$ and $CP = 6.$  If $\tan \angle APD = 2,$ find $AB.$
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Context:
9-8-1. In Midcity, houses stand along one side of a street, each house can have $1,2,3, \ldots, 9$ floors. According to an ancient law of Midcity, if two houses on one side of the street have the same number of floors, then no matter how far apart they are from each other, there must be a house with more floors between them. What is the maximum possible number of houses on one side of the street in Midcity?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two lines pass through point \(N\) and are tangent to a certain circle with center \(O\). Point \(A\) is taken on one of these lines, and point \(B\) is taken on the other line such that \(OA = OB\) and \(OA > ON\). It is known that \(NA = a\), \(NB = b\), and \(OA = c\) where \(a \neq b\). Find \(ON\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a function \( u(x) \) and an operator \( \hat{u} \) defined as \( \hat{u}(f(x))=f(u(x)) \) for any function \( f(x) \) over \( \mathbb{R} \), find \( \hat{u}\left(\dfrac{\partial f(x)}{\partial x}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A shop owner professes to sell his articles at a certain cost price, but he uses false weights with which he cheats by 15% while buying and by 20% while selling. What is his percentage profit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
在一节车厢里，任意$m(m\geq3)$个旅客都有唯一的公共朋友（当甲是乙的朋友时，乙也是甲的朋友.任何人不作为他自己的朋友）.问：这节车厢里有多少人？
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Context:
Consider a triangle with vertices at $(2, 1)$, $(4, 7)$, and $(7, 3)$. This triangle is reflected about the line $y=4$. Find the area of the union of the original and the reflected triangles.
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Context:
Two square napkins with dimensions \(1 \times 1\) and \(2 \times 2\) are placed on a table so that the corner of the larger napkin falls into the center of the smaller napkin. What is the maximum area of the table that the napkins can cover?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the solution interval for the equation $2^{x-2}+x=6$. Express your answer as an interval in the form (a, b).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A hiker walking at a constant rate of 7 miles per hour is passed by a cyclist traveling in the same direction along the same path at a certain speed. The cyclist stops to wait for the hiker 5 minutes after passing her, while the hiker continues to walk at her constant rate. The cyclist must wait 15 minutes until the hiker catches up. What is the cyclist's speed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
James buys 3 dirt bikes for $150 each, 4 off-road vehicles for $300 each, 2 all-terrain vehicles (ATVs) for $450 each, 5 mopeds for $200 each, and 3 scooters for $100 each. It costs him $25 to register each dirt bike and off-road vehicle, $30 to register each ATV, $15 to register each moped, and $20 to register each scooter. Additionally, he purchases maintenance packages for each vehicle: $50 for each dirt bike, $75 for each off-road vehicle, $100 for each ATV, and $60 for each moped. How much did he pay for everything?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a cube. On each edge of the cube, we write a number, either $1$ or $-1$. For each face of the cube, we multiply the four numbers on the edges of this face, and write the product on this face. Finally, we add all the eighteen numbers that we wrote down on the edges and face of the cube.

What is the smallest possible sum that we can get?
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Context:
Sally is asked to multiply 2/3 and 1 1/2. Jane is asked to add them. Calculate the difference between Sally's answer and Jane's answer. Express your answer as a simplified fraction or mixed number.
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Context:
8.3. Solve the inequality $\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}$, and find the sum of its integer solutions that belong to the interval $[-25 ; 25]$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
8. Given a sequence of non-negative integers $\left\{a_{n}\right\}$ satisfying
$$
a_{1}=2016, a_{n+1} \leqslant \sqrt{a_{n}},
$$

and if the number of terms is no less than 2, then any two terms are distinct. Then, the number of such sequences $\left\{a_{n}\right\}$ is $\qquad$
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $\triangle A B C$ 的内角 $\angle A 、 \angle B 、 \angle C$ 的对边分别为 $a 、 b 、 c$, 且 $\angle A-\angle C=\frac{\pi}{2}, a 、 b 、 c$ 成等差数列. 则 $\cos B$ 的值为\_\_\_\_\_\_.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the first half of the 20th century, the population of a particular country increased by 200 percent, with the birth rate at 25 per 1,000, the death rate at 12 per 1,000, and an annual immigration rate of 15,000 individuals. In the second half of the century, the population increased by 300 percent, the birth rate increased to 30 per 1,000, the death rate decreased to 8 per 1,000, and the annual immigration rate doubled to 30,000 individuals. Calculate the overall percent increase in the country's population for the 20th century as a whole, taking into account these demographic factors.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{$a \in \mathbb{R} \setminus \{1\}$}


Context:
Calculate $\sqrt[3]{\frac{x}{2015+2016}}$, where $x$ is the harmonic mean of the numbers

$a=\frac{2016+2015}{2016^{2}+2016 \cdot 2015+2015^{2}}$ and $b=\frac{2016-2015}{2016^{2}-2016 \cdot 2015+2015^{2}}$.

The harmonic mean of two positive numbers $a$ and $b$ is the number $c$ such that $\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The incircle of a right-angled triangle $ABC$ ($\angle ABC =90^o$) touches $AB, BC, AC$ in points $C_1, A_1, B_1$, respectively. One of the excircles touches the side $BC$ in point $A_2$. Point $A_0$ is the circumcenter or triangle $A_1A_2B_1$, point $C_0$ is defined similarly. Find angle $A_0BC_0$.
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Context:
Consider the sequence of numbers defined recursively by \(t_1=1\) and for \(n>1\) by \(t_n=1+t_{(n/2)}\) when \(n\) is even and by \(t_n=\frac{1}{t_{(n-1)}}\) when \(n\) is odd. Given that \(t_n=\frac{19}{87}\), find the sum of the digits of \(n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( a = 3^{100} \) and \( b = 5454 \). Determine how many integers \( z \) exist in the interval \([1, 3^{99})\) such that for every integer \( c \) with \( \gcd(c, 3) = 1 \), the congruences \( x^z \equiv c \pmod{a} \) and \( x^b \equiv c \pmod{a} \) have the same number of solutions.
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Context:
Evaluate the integral \[ I = \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \delta(x_1-x_2+x_3-x_4)\delta(x_2-x_3+x_5-x_6) \, dx_1 \, dx_2 \, dx_3 \, dx_4 \, dx_5 \, dx_6. \]
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Context:
The set \( S \) consists of 9 distinct positive integers. The average of the two smallest integers in \( S \) is 5. The average of the two largest integers in \( S \) is 22. Calculate the greatest possible average of all the integers of \( S \). Express your answer as a single number.
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Context:
Determine the factor by which a random walk on a 2D lattice, with the rule that it cannot go back on itself, is sped up compared to a standard random walk without this rule.
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Context:
Determine the kernel of the matrix \( A = \begin{pmatrix} x_1^0 & x_1^1 & \ldots & x_1^n \\ x_2^0 & x_2^1 & \ldots & x_2^n \\ \vdots & \vdots & \ldots & \vdots \\ x_m^0 & x_m^1 & \ldots & x_m^n \end{pmatrix} \), where \( n < m-1 \) and the \( x_i \) are pairwise distinct.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of \( \lambda > 0 \) such that the function \( f(x) = \begin{cases} 0 & x = 0 \\ |x|^{\lambda} \sin\left(\frac{1}{x}\right) & x \neq 0 \end{cases} \) is differentiable at zero and has a continuous derivative at zero.
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Context:
Compute the standard part of the expression \(\frac{3-\sqrt{c+2}}{c-7}\) given that the standard part of \(c\) is 7 and \(c \neq 7\).
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Context:
What is the name for subgroups $H < G$ that satisfy the condition $C_G(H) \cap H = Z(H)$?
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Context:
\(ABCD\) is a square-based pyramid with base \(ABCD\) and apex \(E\). Point \(E\) is directly above point \(A\), with \(AE = 1024\) units and \(AB = 640\) units. The pyramid is sliced into two parts by a horizontal plane parallel to the base \(ABCD\), at a height \(h\) above the base. The portion of the pyramid above the plane forms a new smaller pyramid. For how many integer values of \(h\) does the volume of this new pyramid become an integer?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Point \(A\) lies on the line \(y = \frac{15}{8}x - 8\), and point \(B\) lies on the parabola \(y = x^2\). What is the minimum length of segment \(AB\)?
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Context:
For each integer $n\ge 3$, let $f(n)$ be the number of 3-element subsets of the vertices of a regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.
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Context:
Determine the minimum number of terms required to approximate Euler's constant \( \gamma \) using the series definition, such that the error is less than \( 2 \times 10^{-3} \). \( \gamma \) is defined as the limit:
\[ \gamma := \lim_{n \to \infty} \left( H_n - \log n \right) \]
where \( H_n \) is the \( n \)-th harmonic number. Use the inequality:
\[ \sum_{n=k}^{\infty} \frac{1}{n} - \int_k^{\infty} \frac{1}{t} \, dt < 2 \times 10^{-3} \]
to find the smallest integer \( k \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The positive integers are arranged in rows and columns as shown below.

| Row 1 | 1 |
| Row 2 | 2 | 3 |
| Row 3 | 4 | 5 | 6 |
| Row 4 | 7 | 8 | 9 | 10 |
| Row 5 | 11 | 12 | 13 | 14 | 15 |
| Row 6 | 16 | 17 | 18 | 19 | 20 | 21 |
|... |

More rows continue to list the positive integers in order, with each new row containing one more integer than the previous row. Calculate the number of integers less than 2000 that are in the column that contains the number 2000. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of real roots of the equation:

$$ \mathcal{T}(e^x,5) = \frac{x^5}{5!} + \frac{x^4}{4!} + \frac{x^3}{3!} + \frac{x^2}{2!} + x + 1 $$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $\{a_n\}$ where $a_3 = 2$ and $a_7 = 1$, if the sequence $\left\{ \frac{1}{a_n+1} \right\}$ forms an arithmetic sequence, find the value of $a_{19}$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the area of the figure bounded by the lines \( y = 0 \), \( y = \{\sin x\} \), \( x = -\frac{\pi}{2} \), and \( x = -\frac{\pi}{4} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X_1, X_2, \ldots$ be a sequence of independent and identically distributed random variables. Define $M$ as the smallest index $n$ such that $X_n > X_{n-1}$. Find the expected value $\mathbb{E}[M]$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_\gamma \frac{dz}{(z^2+1)^2} \), where \( \gamma \) is the semicircle \([-R,R]\cup\{z\in\mathbb{C}:|z|=R\ \text{and}\ \text{Im}{z}>0\}\) traced in the positive direction, and \( R>1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
An ant situated at point \( A \) decides to walk 1 foot east, then \( \frac{1}{2} \) foot northeast, then \( \frac{1}{4} \) foot east, then \( \frac{1}{8} \) foot northeast, then \( \frac{1}{16} \) foot east, and so on (that is, the ant travels alternately between east and northeast, and the distance travelled is decreased by half every time the ant changes its direction). The ant eventually reaches a certain point \( B \). Determine the distance between the ant's initial and final positions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected number of letters needed to be output from a random sequence of the letters \( \{A, B, C, D\} \) until the sequence "ABCD" appears consecutively. Assume each letter is equally likely to appear at each position in the sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$.
Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$.
Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$).
Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$).
Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$.
Find the length of $AP$.
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Context:
Let $A B C$ be a triangle with $A B=2, C A=3, B C=4$. Let $D$ be the point diametrically opposite $A$ on the circumcircle of $A B C$, and let $E$ lie on line $A D$ such that $D$ is the midpoint of $\overline{A E}$. Line $l$ passes through $E$ perpendicular to $\overline{A E}$, and $F$ and $G$ are the intersections of the extensions of $\overline{A B}$ and $\overline{A C}$ with $l$. Compute $F G$.
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Context:
the edge of a cube is 5 a cm . find its surface ?
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Context:
A river has a right-angle bend. Except at the bend, its banks are parallel lines of distance $a$ apart. At the bend the river forms a square with the river flowing in across one side and out across an adjacent side. What is the longest boat of length $c$ and negligible width which can pass through the bend?
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Context:
Let the functions \( f(\alpha, x) \) and \( g(\alpha) \) be defined as
\[ f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} \]
\[ g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2} \]

Then \( g(\alpha) \) is a polynomial in \( \alpha \). Find the leading coefficient of \( g(\alpha) \).
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Context:
Find the value of the following limit:

$$\lim_{q\to 1^-}\frac{\pi K'} {K} \cdot\frac{K} {2\pi^2}\cdot(E-k'^2K)$$

where $K, K', E$ are elliptic integrals and $k, k'$ are moduli corresponding to the nome $q$.
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Context:
Find the greatest common divisor (GCD) of all numbers of the form $p^{12} - 1$, where $p$ is a prime number with a units digit of 1.
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Context:
Find an upper bound for \( \mathbb{E}\left[\frac{1}{1+M}\right] \), where \( M \) follows a Binomial distribution \( B(n,p) \). Express the upper bound in terms of \( n^{-1} \), and consider the case where \( n \) is sufficiently large.
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Context:
The 600 students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. What is the probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group? Express your answer as a fraction.
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Context:
A permutation  $(a_1, a_2, a_3, \dots, a_{2012})$  of  $(1, 2, 3, \dots, 2012)$  is selected at random. If  $S$  is the expected value of
		\[
		\sum_{i = 1}^{2012} | a_i - i |,
		\]
		then compute the sum of the prime factors of  $S$ .

*Proposed by Aaron Lin*
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Context:
In a right-angled triangle $ABC$ with $\angle A = 90^{\circ}$, it is given that $AB = AC = 2AD$. The line $AE$ is perpendicular to $BD$, and the side lengths are denoted by $a$. Let point $O$ be the intersection of $BD$ and $AE$. Given that $AO = \frac{a}{\sqrt 5}$, determine the ratio $BE : CE$. Express your answer as a simplified ratio.
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Context:
Determine the necessary condition for the function $f(x)$ to have an inverse, given the functional relationship \(2(f(x) - 2x) = -\left(f\left(\frac{1}{x}\right) - \frac{2}{x}\right)\).
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Context:
Determine the change of variable that can be used to show the equality \( \int_0^{\infty}\frac{dx}{1+x^2}=2\int_0^1\frac{dx}{1+x^2} \) without performing direct integration.
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Context:
In Rabin's cryptosystem, given the public and private keys with \( p = 127 \), \( q = 131 \), \( N = 16637 \), and \( B = 12345 \), the ciphertext \( c \) is computed as follows for the plaintext \( m = 4410 \):

\[ c = m(m + B) \mod N = 4633 \]

To decrypt, compute:

\[ t = \left(\frac{B^2}{4} + c\right) \mod N \]

Find the value of \( t \).
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Context:
Determine the largest natural number \( m \) such that \[ \int_0^\infty \frac{\ln^m x}{e^x} \, \mathrm{d}x < 2. \]
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Context:
Given that the two sides of a triangle are factors of 72. Then, the number of different such triangles is $\qquad$.
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Context:
In a class, there are \( a_{1} \) students who received at least one failing grade during the year, \( a_{2} \) students who received at least two failing grades, ..., and \( a_{k} \) students who received at least \( k \) failing grades. How many failing grades are there in total in this class?

(It is assumed that no one has more than \( k \) failing grades.)
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Context:
In an arithmetic sequence with a common difference of 3, find the first term if the sum of the squares of the first 1001 terms is equal to the sum of the squares of the next 1000 terms.
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Context:
Evaluate the integral \( \int_{0}^{\infty}\frac{dv}{v\cosh v} \).
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Context:
设$M = \{ 1 , 2 , \cdots , 2 0 \},A _ { 1 }A _ { 2 } , \cdots A _ { n }$是M的互不相同的非空子集，当$i \neq j$时，$A _ { i } \cap A _ { j }$至多有两个元，求n的最大值.
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Context:
Suppose that $X_1, X_2, \ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S=\sum\limits_{i=1}^kX_i/2^i,$ where $k$ is the least positive integer such that $X_k<X_{k+1},$ or $k=\infty$ if there is no such integer. Find the expected value of $S.$
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Context:
A girl, who is an acquaintance of a young man sitting by the window of a moving tram, was walking towards the tram. Eight seconds after she passed by the window, the young man got off the tram and started following her. How much time passed from this moment until he caught up with the girl? The speed of the young man is twice the speed of the girl and one-fifth of the speed of the tram.
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Context:
Find the value of $k$ such that the line $y=k$ intersects the curve $y=2x-3x^3$ in the first quadrant, and the areas of the shaded regions formed are equal.
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Context:
Call a lattice point *visible* if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer $k$, denote by $S_k$ the set of all visible lattice points $(x, y)$ such that $x^2 + y^2 = k^2$. Let $D$ denote the set of all positive divisors of $2021 \cdot 2025$. Compute the sum \[ \sum_{d \in D} |S_d| \]

Here, a lattice point is a point $(x, y)$ on the plane where both $x$ and $y$ are integers, and $|A|$ denotes the number of elements of the set $A$.
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Context:
Find the derivative of an $m \times m$ real matrix $A$ with respect to its vectorized transpose $(\mathrm{vec}(A))^T$, without using tensor notation.
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Context:
Evaluate the limit: $$\lim_{n\to \infty} \int_{0}^{n} \frac{1}{1+n^2\cos^2x}\, dx$$
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Context:
Let \(ABCD\) be a circumscribed quadrilateral such that \(m(\widehat{A})=m(\widehat{B})=120^\circ\), \(m(\widehat{C})=30^\circ\), and \(|BC|=2\). Find the length of \(|AD|\). The original answer is in the form \(\sqrt{k} - m\). Please give the value of k + m.
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Context:
Let \(H\) be a subgroup of a finite group \(G\) with \(|H| = 3\). Determine the value of \(|N_{G}(H) : C_{G}(H)|\).
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Context:
In a soccer league with \( n \) teams (\( n > 3 \)), each pair of teams played at most one match, and there were no draws. Exactly one team remained unbeaten. Additionally, for any two teams \( A \) and \( B \), there exists another team \( C \) such that either \( A \) defeated \( C \) and \( B \) defeated \( C \), or \( C \) defeated \( A \) and \( C \) defeated \( B \). What is the minimum possible number of matches \( k \) that were held?
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Context:
Determine the values of $p$ for which the triangle inequality fails for the function $\lVert (x_1,x_2) \rVert_p = (|x_1|^p+|x_2|^p)^{1/p}$ on $\mathbb{R}^2$. Provide your answer as a range or set of values for $p$.
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Context:
Find the value of \( k \) for which the solution to the differential equation \( y'' + 4y' + ky = 0 \) vanishes at \( t = 2 \) and \( t = 3 \), but not in between.
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Context:
For which real numbers \(\alpha\) does the sequence \(a_n = \sin(\alpha n)\) converge to a limit?
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Context:
Determine the number of partial orders on a set \( S \) where \( |S| = 3 \).
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Context:
Evaluate the limit: $$ \lim _{x\to \infty \:}\left(\frac{\left(\left(2x\right)!\right)}{x^xx!}\right)^{\frac{1}{x}} $$
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Context:
Consider a Markov chain where $X_0 = x$ and each subsequent state $X_i$ is drawn from a discrete uniform distribution on the set $\{0, \dots, X_i-1\}$. Determine the expected absorption time for the process to reach the state 0.
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Context:
Let $L$ be a Lie algebra of dimension $n$ and let $M(n \times n)$ denote the space of square matrices of size $n$. Determine the dimension of $\text{Hom} \big( L, M(n \times n) \big)$. Provide your answer as a single number.
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Context:
Evaluate the integral \( \int_0^{+\infty} \frac{\sin^4{x}}{x^4} \, dx \).
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Context:
Determine the largest integer $k$ for which the equation \[ 25\lfloor x\rfloor - \lfloor x\lfloor x\rfloor \rfloor = 50k - 13\lceil x\rceil \] has a real solution for $x$. Provide your answer as an integer.
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Context:
Let \( \alpha \) be an element in an algebraic closure of \( GF(64) \) such that \( \alpha^4 = \alpha + 1 \). Determine the natural number \( r \) for which \( GF(64)(\alpha) \) is isomorphic to \( GF(2^r) \).
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Context:
Find the stabilizer of the coset $aH$ for the action of the group $G$ on the set $X = G/H$ by left multiplication.
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Context:
Evaluate the integral: $$\int_{0}^{\infty}\frac{\ln x}{(x+1)^{3}}\mathrm{d}x$$
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Context:
Determine the number of steps required to construct the centroid of 10 points using a compass with a fixed opening.
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Context:
Given integers $a$ and $b$, and odd prime numbers $p$ and $q$, suppose there exists a solution to the equation $a^2 + 2b^2 = p^{11}q^{13}$. Determine the number of solutions.
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Context:
Given that $4x^2 + 2xy + 9y^2 = 100$, determine all possible values of the expression $x^2 + 2xy + 3y^2$.
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\boxed{\left[ \dfrac{100}{7}, 40 \right]}


Context:
Suppose there are 3 bank tellers, each currently occupied with a customer, and you are next in line. The service times are independent and identically distributed exponential random variables with parameter \( \lambda \). What is the probability that you will be the last customer to complete service?
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Context:
Find the constant \( c \) such that \( \text{P}(\limsup\limits_{n\to \infty} X_n/\sqrt{\log n}=c)= 1 \), where \( X_n \) are a sequence of independent random variables with \( X_k\sim\mathcal{N}(0,1) \). Use the fact that \( \sqrt{2\pi}\xi\exp\left({\xi^2\over 2}\right)\text{P}(X\geq \xi)\to 1 \) as \( \xi \to \infty \).
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Context:
Given a positive integer N, calculate the number of integers from 1 to N (inclusive) whose binary representation contains exactly K zeroes. For example, if N = 18 and K = 3, determine how many such numbers exist.
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Context:
If three lines \( a, b, c \) in space are pairwise skew lines, determine the number of lines that intersect with all three lines \( a, b, \) and \( c \).
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Context:
Evaluate the integral: $$\int_{-2}^{1} \int_{-2}^{1} \int_{-2}^{1} \frac{x^2}{x^2+y^2+z^2} \,dx \,dy \,dz$$
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Context:
Evaluate the contour integral \( \oint_{|z|=2} z^n(1 - z)^m\ dz \) where \(m\) and \(n\) are negative integers.
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Context:
Find $$\sum^{k=672}_{k=0} \binom{2018}{3k+2} \pmod{3}$$
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Context:
In the sequence $\{a_n\}$, $(a_1=1, a_2=2)$, and $(a_{n+2} - a_n = 1 + (-1)^n (n \in \mathbb{N}^*))$, find the value of $(a_1 + a_2 + \dots + a_{51} = \_\_\_\_\_\_\_)$.
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Context:
There are  $2022$  equally spaced points on a circular track  $\gamma$  of circumference  $2022$ . The points are labeled  $A_1, A_2, \ldots, A_{2022}$  in some order, each label used once. Initially, Bunbun the Bunny begins at  $A_1$ . She hops along  $\gamma$  from  $A_1$  to  $A_2$ , then from  $A_2$  to  $A_3$ , until she reaches  $A_{2022}$ , after which she hops back to  $A_1$ . When hopping from  $P$  to  $Q$ , she always hops along the shorter of the two arcs  $\widehat{PQ}$  of  $\gamma$ ; if  $\overline{PQ}$  is a diameter of  $\gamma$ , she moves along either semicircle.

Determine the maximal possible sum of the lengths of the  $2022$  arcs which Bunbun traveled, over all possible labellings of the  $2022$  points.
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Context:
Given \(\triangle ABC\). On ray \(BA\), take a point \(D\) such that \(|BD| = |BA| + |AC|\). Let \(K\) and \(M\) be two points on rays \(BA\) and \(BC\) respectively, such that the area of \(\triangle BDM\) is equal to the area of \(\triangle BCK\). Find \(\angle BKM\) if \(\angle BAC = \alpha\).
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Context:
Find the total number of non-negative integer solutions to the equation $x_1+2x_2+x_3+x_4+\ldots+x_{10}=3$. Express your answer as a single integer.
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Context:
Determine the value of $v$ that minimizes the integral \[ \int_{-\infty}^{\infty} p(x) \log \frac{p(x)}{g(x,v)} \, dx \] where $p(x)$ and $g(x,v)$ are probability density functions given by \[ p(x) = \frac{1}{2} \left( \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x-2)^2} + \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x+1)^2} \right) \] and \[ g(x,v) = \frac{1}{3} \left( \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x-1)^2} + \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x+3)^2} + \frac{1}{\sqrt{2 \pi}} e^{-\frac{1}{2}(x-v)^2} \right). \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$ \lim_{x\to0}\sum_{n=1}^{\infty}\frac{\sin(x)}{4+n^2x^2} $$
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Context:
Two circles with radii \( R \) and \( r \) intersect at point \( A \). Let \( BC \) be a common tangent to both circles at points \( B \) and \( C \). Find the radius of the circumcircle of triangle \( ABC \).
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Context:
In multivariable calculus, find a constant $c$ such that the tangent planes of the spheres $(x-c)^2 + y^2 + z^2 = 3$ and $x^2 + (y-1)^2 + z^2 = 1$ are perpendicular at any intersection point.
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Context:
Determine the number of homomorphisms from the dihedral group $D_{21}$ to the dihedral group $D_{15}$. Use the fact that any element in $D_{21}$ can be expressed as $s^ir^j$ and that for homomorphisms, $\phi(s^{i + i'}r^{j' - j}) = \phi(s^ir^j)\phi(s^{i'}r^{i'})$. Provide your answer as a single integer.
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Context:
Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers \[ N = a + (a+1) +(a+2) + \cdots + (a+k-1) \] for $k=2017$ but for no other values of $k>1$. Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?
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Context:
Given the group $G = S_7$ and a subgroup $H = \langle (1\ 2\ 3\ 4\ 5\ 6\ 7) \rangle$, determine the number of elements in the intersection of $A_7$ and the normalizer $N_G(H)$. It is given that $|N_G(H)| = 42$ and $|A_7| = 2520$. Use the internal product formula to find $|A_7 \cap N_G(H)|$.
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Context:
On Monday, 5 students in the class received A's in math, on Tuesday 8 students received A's, on Wednesday 6 students, on Thursday 4 students, and on Friday 9 students. None of the students received A's on two consecutive days. What is the minimum number of students that could have been in the class?
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Context:
Let $C$ be the linear code over $\mathbb{F}_5$ with parity check matrix $H$ whose columns are vectors of the form $\begin{bmatrix}1\\ \alpha\\ \alpha^{2} \end{bmatrix}$, where $\alpha\in\mathbb{F}_5$. Determine the number of codewords in $C$ that have a Hamming weight of exactly 3.
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Context:
Given a regular $2 n+1$-sided polygon. How many triangles are there whose vertices are among the vertices of the polygon and the triangle contains the center of the polygon?
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Context:
Determine the number of non-isomorphic random graphs that can be formed with a set of cardinality \( \kappa \), where \( \kappa \) is a regular cardinal.
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Context:
Three customers, A, B, and C, arrive simultaneously at a bank with two tellers on duty. The tellers were idle when the customers arrived. Customer A goes to one teller, B goes to the other, and C waits until either A or B finishes before receiving service. If the service times are exponentially distributed with a mean of 4 minutes, what is the probability that customer A is still in the bank after customers B and C have left?
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Context:
A regular quadrilateral pyramid is intersected by a plane passing through a vertex of the base perpendicular to the opposite lateral edge. The area of the resulting cross-section is half the area of the pyramid's base. Find the ratio of the height of the pyramid to the lateral edge.
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Context:
Evaluate the limit: $$\lim_{x \to 0}\lim_{n \to \infty} \frac{n^{\operatorname{cosec}^{2}x}}{(1^{\tan^{2}x}+2^{\tan^{2}x}+\cdots+n^{\tan^{2}x})^{\cot^{2}x}}$$
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Context:
Find the remainder when the infinite power tower \(7^{7^{7^{7^{.^{.^{.^{\infty}}}}}}}\) is divided by 13.
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Context:
Calculate the sum of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{5}}$ with an accuracy of $10^{-3}$.
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Context:
Determine the number of surjective functions from the set \( \{1, 2, \ldots, 10\} \) to the set \( \{1, 2, \ldots, 9\} \) such that \( f(2i+1) > f(2i-1) \) and \( f(2i) > f(2i+2) \) for \( i = 1, 2, 3, 4 \).
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Context:
Given $F_1$ and $F_2$ are the two foci of the ellipse $E$ on the left and right, respectively, and the parabola $C$ has $F_1$ as its vertex and $F_2$ as its focus. Let $P$ be a point of intersection between the ellipse and the parabola. If the eccentricity $e$ of the ellipse satisfies $|PF_1| = e|PF_2|$, then the value of $e$ is \_\_\_\_\_\_.
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Context:
Determine the values of \(a > 0\) and \(b > 0\) for which the integral \(\int_{1}^{\infty} \frac{1}{x^a(1+x^b)} \,dx\) is convergent.
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Context:
The minimum value of $\frac{a^2+2b^2+\sqrt2}{\sqrt{ab}}$ with $a, b$ positive reals is ....
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Context:
For circles $C_1: x^2+y^2+2ax+a^2-9=0$ and $C_2: x^2+y^2-4by-1+4b^2=0$ having only one common tangent line, given $a\in \mathbb{R}, b\in \mathbb{R}$, and $ab\neq 0$, the minimum value of $\frac{4}{a^2} + \frac{1}{b^2}$ is \_\_\_\_\_\_.
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Context:
Flight number 608 of "Aeroflot" departs from Moscow at 12:00 and arrives in Bishkek at 18:00 (local time).

The return flight number 607 departs at 8:00 and arrives at 10:00. How long does the flight last?
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Context:
Determine the limit of the series \( S = 1 + r + \frac{1}{2}r^2 + \frac{1}{3}r^3 + \cdots \) if it exists, given that the series converges for \( r < 1 \).
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Context:
Reduce the lambda calculus expression \((\lambda x. (\lambda y. x \, y) \, x) \, (\lambda z.p)\) to its simplest form.
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Context:
In the complex plane, the points \( 0, z, \frac{1}{z}, z+\frac{1}{z} \) form a parallelogram with an area of \( \frac{35}{37} \). If the real part of \( z \) is greater than 0, find the minimum value of \( \left| z + \frac{1}{z} \right| \).
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Context:
Determine the number of 8-tuples \((a_1, a_2, \ldots, a_8)\) from the set \([4]^8 = \{1, 2, 3, 4\}^8\) such that each number in \([4]\) appears exactly twice, and for each \(i \in [4]\), the number \(i\) does not appear in the \(i\)-th position.
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Context:
Given that the roots of the polynomial \(x^4 + 2x^3 - 5x^2 + 7x + 10 = 0\) are \(\alpha, \beta, \gamma, \delta\), and the roots of the polynomial \(x^4 + ax^3 + bx^2 + cx + d = 0\) are \(\alpha+\beta+\gamma, \alpha+\beta+\delta, \alpha+\gamma+\delta, \beta+\gamma+\delta\), find the value of \(a + b - c - d\).
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Context:
Calculate the integral \( \int_{\Sigma}(2x \, dy\land dz + 2y \, dz\land dx + (x^2+y^2) \, dx\land dy) \), where \( \Sigma \) is the boundary of the region \( D \) defined by \( 0\leq z\leq 1-x^2-y^2 \). The surface \( \Sigma \) is oriented such that the outward normal vectors point away from the region \( D \).
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Context:
In triangle \(ABC\), \(\angle ABC = 70^\circ\) and \(\angle ACB = 50^\circ\). On \(AB\), a point \(M\) is selected such that \(\angle MCB = 40^\circ\), and on \(AC\), a point \(N\) is chosen such that \(\angle NBC = 50^\circ\). Find \(\angle NMC\).
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Context:
8 is 2% of a, and a certain percentage of b is 2. c equals b / a. What is the value of the percentage of b?
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Context:
A 10000-liter tank, half-full of water, is being filled from a pipe with a flow rate of 1 kiloliter every 2 minutes. At the same time, the tank is losing water from two drains. One drain has a rate of 1 kiloliter every 6 minutes. It takes 60 minutes to fill the tank completely. What is the rate of the other drain?
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Context:
Find a vector field \( \mathbf{F} \) such that the line integral \( \oint_C \mathbf{F} \cdot d\mathbf{r} \) is negative, where \( C \) is a circle of radius \( r \) centered at the origin, traced once in the clockwise direction.
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Context:
In a road traffic survey, cars passing a marker on a road are counted. The streams of cars in the two directions are modeled as independent Poisson processes with intensities of 2 cars per minute and 3 cars per minute, respectively. Counting stops once 400 cars have passed. Let \( T_{400} \) be the time at which this occurs. Compute the time \( \alpha \) such that \( P(T_{400} \leq \alpha) = 0.90 \).
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Context:
A box measuring some inches long by 48 inches wide by 12 inches deep is to be filled entirely with identical cubes. No space is to be left unfilled. The smallest number of cubes that can accomplish this objective is 80. What is the length of the box?
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Context:
A statue is being carved by a sculptor. The original piece of marble weighed 250 kg. In the first week, 30 percent is cut away. In the second week, 20 percent of the remainder is cut away. In the third week, 25 percent of the remaining marble is cut away. In the fourth week, the sculptor cuts away 15 percent. In the fifth week, 10 percent of the remainder is cut away. In the sixth week, the sculptor removes another 5 percent of the remaining marble. The statue is finally completed in the seventh week when 3 percent of the remainder is cut away. What is the weight of the final statue?
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Context:
Evaluate the limit as \( N \to \infty \):
\[
\lim_{{N \to \infty}} \left( \left(\frac{\sin \frac{1}{N}}{\frac{1}{N}}\right)^{N} -1 \right) \cdot \frac{1}{N}.
\]
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Context:
Evaluate the integral \( \int_0^{\infty} e^{-x} \sum_{k=0}^{\infty} \frac{(-1)^k (2015x)^{2k}}{((2k)!)^2} \, dx \).
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Context:
In the Cartesian coordinate system $xOy$, let $A(-1,1)$, and let $B$ and $C$ be two points on the graph of the function $y= \frac {1}{x} (x > 0)$, and $\triangle ABC$ is an equilateral triangle. The height of $\triangle ABC$ is ______.
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Context:
There are births in West Northland every 6 hours, deaths every 2 days, and a net immigration every 3 days. Calculate the approximate annual increase in population.
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Context:
Solve the complex equation $w^3 = 9\overline{w}$, where $\overline{w}$ denotes the conjugate of $w$.
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Context:
Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$.
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Context:
Evaluate the improper integral: $$\int_{0}^{\infty} \frac{\sin x}{x^{3/2}} \, dx.$$
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Context:
Determine the maximum value of the sum
\[S = \sum_{n=1}^\infty \frac{n}{2^n} (a_1 a_2 \cdots a_n)^{1/n}\]
over all sequences $a_1, a_2, a_3, \cdots$ of nonnegative real numbers satisfying
\[\sum_{k=1}^\infty a_k = 1.\]
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Context:
Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). Given that the system of equations \( 2x + y = 2033 \) and \( y = |x-a| + |x-b| + |x-c| \) has exactly one solution, find the minimum possible value of \( c \).
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Context:
Solve for \( n \) in the equation: \[ \frac{365!}{365^{n}(365-n)!} = 0.5 \]
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Context:
In trapezoid \(ABCD\), the base \(AD\) is equal to 16, the sum of the side \(AB\) and the diagonal \(BD\) is equal to 40, and the angle \(CBD\) is \(60^\circ\). The ratio of the areas of triangles \(ABO\) and \(BOC\), where \(O\) is the intersection point of the diagonals, is 2. Find the area of the trapezoid.
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Context:
Let $S$ be a set of intervals defined recursively as follows: Initially, $[1,1000]$ is the only interval in $S$. If $l \neq r$ and $[l, r] \in S$, then both $\left[l,\left\lfloor\frac{l+r}{2}\right\rfloor\right],\left[\left\lfloor\frac{l+r}{2}\right\rfloor+1, r\right] \in S$. An integer $i$ is chosen uniformly at random from the range $[1,1000]$. What is the expected number of intervals in $S$ which contain $i$?
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Context:
In our number system, the base is ten. If the base were changed to four, you would count as follows: $1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, \ldots$ What would be the twentieth number in this system?
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Context:
Let $K/F$ be a finite-dimensional field extension with $[K:F]=n$. Consider the collection of $F$-linear transformations $\mathcal{L}_F(K)$ from $K$ to $K$. This collection forms an algebra over $F$ and is isomorphic to the $n \times n$ matrices over $F$, making it an $n^2$-dimensional vector space over $F$. Determine the dimension of $\mathcal{L}_F(K)$ as a vector space over $K$. \( \text{Express your answer as a single number.} \)
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Context:
Given that \(9^{-1} \equiv 90 \pmod{101}\), find \(81^{-1} \pmod{101}\), as a residue modulo 101. (Give an answer between 0 and 100, inclusive.)
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Context:
Xiao Wu, Xiao Yi, Xiao Jie, Xiao Kuai, and Xiao Le are standing in a row. If Xiao Yi does not appear in the first or last position, and among Xiao Wu, Xiao Jie, and Xiao Le, exactly two are adjacent, how many different arrangements satisfy these conditions?
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Context:
Determine the interval of convergence for the power series representation of \( f(x) = \sqrt{x} \) centered at \( x = 16 \).
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Context:
Let $g: \mathbb{R} \to \mathbb{R}$ be a three-times differentiable function that satisfies $$g(x^3)g''(x)=g'(x)g'(x^3)$$ for all $x \in \mathbb{R}$. If $g(1) = 2$ and $g'''(1) = 12$, determine the value of $g'(1) + g''(1)$.
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Context:
For complex numbers n and q, if m = | | n – 3 | – 2 | q | + | | q - n | + 3 | , for how many values of n is m a purely real number, m = 5 ?
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Context:
What is the limit of the following sum as $n$ approaches infinity?

$$\lim_{n\to\infty} \sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \binom{n}{2k} \left(4^{-k} \binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}$$
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Context:
Let \( R \) be a semicircle with diameter \( XY \). A trapezoid \( ABCD \) in which \( AB \) is parallel to \( CD \) is circumscribed about \( R \) such that \( AB \) contains \( XY \). If \( AD = 4 \), \( CD = 5 \), and \( BC = 6 \), determine \( AB \).
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Context:
By selling some pencils for a rupee, a man loses 35%. In order to gain 30%, he should sell 10 pencils for a rupee. How many pencils was he selling for a rupee initially?
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Context:
1.48 In the expression $x_{1}: x_{2}: \cdots: x_{n}$, use parentheses to indicate the order of operations, and the result can be written in fractional form:
$$\frac{x_{i_{1}} x_{i_{2}} \cdots x_{i_{k}}}{x_{j_{j_{1}}}^{x_{j_{2}}} \cdots x_{j_{n-k}}}$$
(At the same time, each letter in $x_{1}, x_{2}, \cdots, x_{n}$ may appear in the numerator or in the denominator.) How many different fractions can be obtained by adding parentheses in all possible ways?
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Context:
Determine the Galois group of the polynomial \( p(x) = x^4 - 2x^2 - 2 \) over the field \( \mathbb{F}_5 \).
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Context:
In $\triangle ABC$, $\angle A = 60^{\circ}$ and $AB > AC$. Point $O$ is the circumcenter. The two altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, and satisfy $BM = CN$. Find the value of $\frac{MH + NH}{OH}$.
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Context:
Let \(A_{1}, A_{2}, A_{3}, A_{4}, A_{5}\), and \(A_{6}\) be six points on a circle in this order such that \(\overline{A_{1} A_{2}}=\overline{A_{2} A_{3}}\), \(\widehat{A_{3} A_{4}}=\widehat{A_{4} A_{5}}\), and \(\widehat{A_{5} A_{6}}=\widehat{A_{6} A_{1}}\), where \(\widehat{A_{1} A_{2}}\) denotes the arc length of the arc \(A_{1} A_{2}\) etc. It is also known that \(\angle A_{1} A_{3} A_{5}=72^{\circ}\). Find the size of \(\angle A_{4} A_{6} A_{2}\) in degrees.
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Context:
Determine the side length of the smallest square sheet of paper required to wrap a unit cube without cutting or stretching the paper.
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Context:
Determine the coefficient of $z$ in the Laurent series expansion of $\frac{e^z}{z-1}$.
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Context:
In a certain company, there are three departments: IT, HR, and Marketing. Employee attendance at the annual company picnic varies depending on the department, as shown below:

IT Department:
- 25% of the men and 60% of the women attended the picnic.

HR Department:
- 30% of the men and 50% of the women attended the picnic.

Marketing Department:
- 10% of the men and 45% of the women attended the picnic.

The percentage composition of employees in each department is as follows:
- IT Department: 40% men and 25% women
- HR Department: 30% men and 20% women
- Marketing Department: 30% men and 55% women

What percentage of all the employees attended the annual company picnic, considering the departmental break-up and gender proportions?
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Context:
a and b together have $1210. If 1/3 of a's amount is equal to 1/4 of b's amount, how much amount does b have?
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Context:
求最小常数 $a(a>1)$, 使得对正方形 $A B C D$ 内部任一点 $P$, 都存在 $\triangle P A B 、 \triangle P B C$ 、 $\triangle P C D 、 \triangle P D A$ 中的某两个三角形，其面积之比属于区间 $\left[a^{-1}, a\right]$. 原始答案为 $\frac{a+\sqrt{b}}{c}$ 的形式，请给出a+b+c的值。
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Context:
Given that $a$ and $b$ are positive real numbers satisfying $$\frac {1}{a+b}+ \frac {1}{a-b}=1$$, find the minimum value of $3a+2b$.
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Context:
For every non-negative integer \( n \), a polynomial \( p_n(x) \) exists such that \( \int x^n e^x \,dx = p_n(x) e^x + C \). Determine the floor value of \( 1000 \cdot L \), where \( L \) is the limit defined as \( L = \lim_{n \to \infty} \frac{|p_n(2)|}{n!} \).
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Context:
Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n - 1.$
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Context:
In a sphere with a radius of 1, the chord AB has a length of 1 and is positioned at a 60° angle to the diameter CD. Given that AC = \( \sqrt{2} \) and AC < BC, determine the length of the segment BD.
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Context:
Lucca bought 1,500 pieces of bread. On the first day, he ate 3/8 of the pieces of bread. On the second day, he ate 7/10 of the remaining pieces. On the third day, he ate 1/6 of what was left, and on the fourth day, he ate 4/9 of the remaining pieces. On the fifth day, he ate 5/18 of the pieces left. Calculate how many pieces of bread are remaining for Lucca after the fifth day.
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Context:
Evaluate the line integral \( \int_{C} -\frac{y}{x^2 + y^2} \, dx + \frac{x}{x^2 + y^2} \, dy \), where \( C \) is the positively oriented boundary of the region enclosed by the curves \( x^2 + y^2 = 1 \) and \( |x| + |y| = 5 \). Note that the vector field \( F(x,y)=\left(-\frac{y}{x^2 + y^2} , \frac{x}{x^2+y^2}\right) \) is undefined at \((0,0)\).
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Context:
Alicia has 20 gumballs. Pedro has that many gumballs plus an additional number of gumballs equal to a certain multiple of the number Alicia has. They put their gumballs in a bowl, and later Pedro takes out 40% of the gumballs to eat. There are 60 gumballs remaining in the bowl. What is the ratio of the additional gumballs Pedro has to the number of gumballs Alicia has?
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Context:
Given $0 \leq a-b \leq 1$ and $1 \leq a+b \leq 4$, find the value of $8a+2002b$ when $a-2b$ reaches its maximum value.
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Context:
How many ways are there to list the numbers 1 to 10 in some order such that every number is either greater or smaller than all the numbers before it?
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Context:
Let $O$ be the origin, and $F$ be the right focus of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b > 0$. The line $l$ passing through $F$ intersects the ellipse $C$ at points $A$ and $B$. Two points $P$ and $Q$ on the ellipse satisfy
$$
\overrightarrow{O P}+\overrightarrow{O A}+\overrightarrow{O B}=\overrightarrow{O P}+\overrightarrow{O Q}=0,
$$
and the points $P, A, Q,$ and $B$ are concyclic. Find the eccentricity of the ellipse $C$.
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Context:
Find the residue of the function \( f(z) = \frac{\cot(z)\coth(z)}{z^3} \) at \( z = 0 \).
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Context:
Inside the circle $\omega$ are two intersecting circles $\omega_{1}$ and $\omega_{2}$, which intersect at points $K$ and $L$, and are tangent to the circle $\omega$ at points $M$ and $N$. It turns out that the points $K$, $M$, and $N$ are collinear. Find the radius of the circle $\omega$, given that the radii of the circles $\omega_{1}$ and $\omega_{2}$ are 3 and 5, respectively.
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Context:
Determine if there is a known bound for the VC-dimension of the set of binary classifiers \(\mathcal{H} = \{\boldsymbol{x}\mapsto\boldsymbol{1}\hspace{-0.9ex}\mathrm{1}(\boldsymbol{h}^\mathsf{T}\boldsymbol{x}\ge 0)\mid  \boldsymbol{h} \in \mathcal{C} \}\), where \(\mathcal{C}\) is a pointed, salient, full-dimensional cone in \(\mathbb{R}^d\), that is significantly better than the VC-dimension of the set of all linear classifiers in \(\mathbb{R}^d\) (i.e., \(d\)).
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Context:
Two weeks ago, Vartan spent 25 percent of his wages on recreation. Last week, his wages were 5 percent less than two weeks ago, and he spent 35 percent of his wages on recreation. This week, his wages are 15 percent less than last week's wages, and he spent 45 percent of his wages on recreation. The amount he spends on recreation this week is what percent of the amount he spent on recreation two weeks ago?
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Context:
A tetrahedron has five edges of length 3 and a circumradius of 2. What is the length of the sixth edge?
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Context:
In the triangle \(ABC\), angle \(C\) is obtuse, \(D\) is the intersection point of the line \(DB\) perpendicular to \(AB\) and the line \(DC\) perpendicular to \(AC\). The altitude of triangle \(ADC\) drawn from vertex \(C\) intersects side \(AB\) at point \(M\). Given that \(AM = a\) and \(MB = b\), find \(AC\).
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Context:
Determine the minimum value of \( n \) such that it is possible to create at least 100 ordered lists of length 5 using numbers from 1 to \( n \), with the condition that any two lists share at most one common element at the same index.
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Context:
Find the supremum of the fraction \( \frac{|c_1^\top z + b_1|}{c_2^\top z+b_2} \) with respect to \( z \in S = \{x \in \mathbb{R}^n: c_2^\top x + b_2 > 0 \} \), where \( c_1, c_2 \in \mathbb{R}^n \).
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Context:
Let \(I\) be the center of the inscribed circle of triangle \(ABC\), and let \(A^{\prime}, B^{\prime}\), and \(C^{\prime}\) be the symmetric points of \(I\) with respect to lines \(BC\), \(CA\), and \(AB\) respectively. The circumcircle of \(A^{\prime} B^{\prime} C^{\prime}\) passes through \(B\). Find \(\widehat{ABC}\).
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Context:
There are five concentric circles \(\Gamma_{0}, \Gamma_{1}, \Gamma_{2}, \Gamma_{3}, \Gamma_{4}\) whose radii form a geometric sequence with a common ratio \(q\). Find the maximum value of \(q\) such that a closed polyline \(A_{0} A_{1} A_{2} A_{3} A_{4}\) can be drawn, where each segment has equal length and the point \(A_{i} (i=0,1, \ldots, 4)\) is on the circle \(\Gamma_{i}\).
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Context:
有八个盒子，每个盒子中有 6 个球，每个球均被染成 n 种颜色中的一种. 已知同一个盒子中的球均不同色，且任意两个盒子中的球至多有一种相同颜色，求 n 的最小可能值.
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Context:
Calculate the maximum value of the summation \( \sum_{k=1}^{n} \left(\frac{(-1)^k}{k!} \int_{0}^{k} 3x^2 \, dx\right) \) for a given positive integer \( n \).
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Context:
求使得s$\sin x = \frac { x } { 1 0 0 }$成立的实数x的个数.
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Context:
If $x, y, z, p, q, r$ are real numbers such that:
\[ \frac{1}{x+p} + \frac{1}{y+p} + \frac{1}{z+p} = \frac{1}{p} \]
\[ \frac{1}{x+q} + \frac{1}{y+q} + \frac{1}{z+q} = \frac{1}{q} \]
\[ \frac{1}{x+r} + \frac{1}{y+r} + \frac{1}{z+r} = \frac{1}{r} \]
Find the numerical value of $\frac{1}{p} + \frac{1}{q} + \frac{1}{r}$.
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Context:
Find the result of the inverse Fourier transform of the complex conjugate of a function in the Fourier domain.
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Context:
In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle?
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Context:
Suppose an article has a cost price that is 60% of the marked price for the first half of the quantity and 70% of the marked price for the second half of the quantity. The store offers four successive discounts of 10%, 8%, 5%, and 3% during a sale. Calculate the net gain percentage after accounting for all these factors and applying all the discounts.
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Context:
Given the set $U={1,2,3,4,5}$, select 4 numbers from set $U$ to form a four-digit number with no repeating digits. This four-digit number should be greater than $2345$ and less than $4351$. The number of four-digit numbers that meet these conditions is \_\_\_\_\_\_.
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Context:
Ten points in the plane are given, with no three collinear . Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m$ and $n$ are relatively prime positive integers . Find $m+n.$
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Context:
What operation should be applied to 49 and then squared to get the result of 49?
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Context:
What is the largest power of 2 by which the number \(10^{10} - 2^{10}\) is divisible?
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Context:
What is the probability that in a group of 13 people, at least two were born in September? (For simplicity, assume that each month is equally likely.)
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Context:
Altitudes \( BE \) and \( CF \) of acute triangle \( ABC \) intersect at \( H \). Suppose that the altitudes of triangle \( EHF \) concur on line \( BC \). If \( AB=3 \) and \( AC=4 \), then \( BC^2=\frac{a}{b} \), where \(a\) and \(b\) are relatively prime positive integers. Compute \(100a+b\).
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Context:
Two mathematics teachers administer a geometry test, assessing the ability to solve problems and knowledge of theory for each 10th-grade student. The first teacher spends 5 and 7 minutes per student, and the second teacher spends 3 and 4 minutes per student. What is the minimum time needed to assess 25 students?
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Context:
有四所学校的学生参加一项数学竞赛，每所学校派出 3 名选手.组委会要抽选其中若干名选手做一项调研，要求任意两所学校被抽中的选手数之和至少为 1 、至多为 3 ，则不同的抽选方式数为 $\qquad$ （结果用数值表示）.
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Context:
Given a sequence of independent and identically distributed (iid) random variables $X_k$, for $k \geq 0$, with a probability mass function (pmf) defined as follows:
$$p_j = P(X_1 = j), \quad 1 \leq j \leq m, \quad \sum_{j=1}^{m}p_j = 1$$
Let $T$ denote the first time the initial value is observed again, i.e., $T = \min{n > 0 : X_n = X_0}$. Determine the expected value of $T$, $E[T]$, given that $X_0$ can take any value from $\{1, 2, \ldots, m\}$.
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Context:
在三棱雉 $P-A B C$ 中, 若 $P A \perp$ 底面 $A B C$, 且棱 $A B, B P, B C, C P$ 的长分别为 $1,2,3,4$ ，则该三棱雉的体积为 $\qquad$.  原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
设整数 n\geq 4，A=\{a_1,a_2,\cdots,a_n\} 是整数集，满足 2023\in A，且 A 中任意两个元素的差的绝对值是素数. 则这样的集合 A 共有_________个.
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Context:
At a joint conference of the Parties of Liars and Truth-lovers, 32 people were elected to the presidium and seated in four rows of 8 people each. During the break, each member of the presidium claimed that among their neighbors there are representatives of both parties. It is known that liars always lie, and truth-lovers always tell the truth. What is the minimum number of liars in the presidium for the described situation to be possible? (Two members of the presidium are neighbors if one of them is seated to the left, right, in front, or behind the other).
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Context:
How many integer triples $(x,y,z)$ are there satisfying the equation $x^3+y^3=x^2yz+xy^2z+2$?
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Context:
Sasha chose five numbers from 1, 2, 3, 4, 5, 6, and 7 and informed Anya of their product. Based on this information, Anya realized that she could not uniquely determine the parity of the sum of the numbers chosen by Sasha. What number did Sasha inform Anya of?
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Context:
Given a circle $C: (x-1)^{2} + (y-2)^{2} = 25$ and a line $l: mx-y-3m+1=0$ intersect at points $A$ and $B$. Find the minimum value of $|AB|$.
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Context:
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, Neymar chooses a divisor $d > 1$ of $n$, erases $n$, and writes $n + d$. If the initial number on the board is $2022$, what is the largest composite number that Neymar will never be able to write on the blackboard?
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Context:
For a sequence defined by $a_1 = 1$ and $a_n = n(1 + a_{n-1})$ for $n \geq 2$, find the value of the following limit:

$$\lim_{n\to \infty} \frac{\prod_{k=1}^{n}\left(1+\frac{1}{a_k}\right)}{\sum_{k=0}^{n-1}\frac{1}{k!}}$$
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Context:
Let $f(x) = \frac{x^6 - 2}{4}$. Find $f^{-1}\left(-\frac{1}{8}\right)$.
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Context:
Max has drawn a pair of concentric circles. He begins to draw chords $\overline{PQ}$, $\overline{QR}$, and so on, of the larger circle, each chord being tangent to the smaller circle. Given that $\angle PQR = 60^\circ$, how many sections will Max draw before returning to his starting point at $P$?
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Context:
Find a non-zero element in the set \( S = \mathbb{C} \times \mathbb{C} \) that is not invertible, where the multiplication on \( S \) is defined as \((a,b) \cdot (c,d) = (ac - db, ad + bc)\).
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Context:
a bookseller has two display windows . she plans to display 3 new fiction books in the left window , and 3 new non - fiction books in the right window . assuming she can put the 4 fiction books in any order , and separately , the 3 non - fiction books in any order , how many total configurations will there be for the two display windows ?
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Context:
Let \( r=\sqrt{\frac{\sqrt{53}}{2}+\frac{3}{2}} \). There is a unique triple of positive integers \((a, b, c)\) such that
\[
r^{100}=2 r^{98}+14 r^{96}+11 r^{94}-r^{50}+a r^{46}+b r^{44}+c r^{40}
\]
Calculate the value of \( a^{2}+b^{2}+c^{2} \). Express your answer as a single integer.
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Context:
In a puzzle, 111 = 09, 444 = 12, and 777 = 15. What number equals 15 in this puzzle?
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Context:
8.5. In the tetrahedron $K L M N$, the lengths of the edges are known: $K L = M N = 9$, $K M = L N = 15$, $K N = L M = 16$. Points $P, Q, R, S$ are the centers of the circles inscribed in triangles $K L M$, $K L N$, $K M N$, and $L M N$. Find the volume of the tetrahedron $P Q R S$. Round the answer to the nearest hundredth if necessary.
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Context:
What is the expected number of coin flips needed to obtain more heads than tails?
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Context:
Let \(f:\mathbb{R} \to \mathbb{R}\) and \(g:\mathbb{R} \to \mathbb{R}\) be two non-constant differentiable functions such that \(f'(x) - g'(x) = g'(x)(f(x))^2 - f'(x)(g(x))^2\) for all \(x \in \mathbb{R}\). Given that \(f(1) = 1\), \(g(1) = \frac{1}{3}\), and there is no \(x\) for which \(f(x) = -2\) or \(g(x) = 2\), find the minimum positive value that \((f(x) - g(x) + 3)\) may attain.
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Context:
Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. Calculate the area, in square yards, of the region outside of the doghouse that Spot can reach. Express your answer in terms of pi.
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Context:
Given the functions $y= \frac {1}{x^{2}}$, $y=2x$, $y=x^{2}+x$, and $y=\sqrt[3]{x^{5}}$, how many of these functions are power functions, i.e., can be expressed in the form $y=x^{\alpha}$ where $\alpha$ is a real number? Provide your answer as a whole number.
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Context:
A triangle is inscribed in a circle. A second circle, concentric with the first, touches one side of the triangle and divides each of the other two sides into three equal parts. Find the ratio of the radii of these circles.
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Context:
Calculate the integral \( \int_\gamma \frac{dz}{z} \) where \( \gamma = a \cos(t) + i b \sin(t) \) for \( t \in [0, 2\pi] \), using the complex logarithm function.
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Context:
For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1} = f_1 \circ f_n$. Evaluate $f_{2007}(2^{2006})$.
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Context:
A set of six edges of a regular octahedron is called Hamiltonian cycle if the edges in some order constitute a single continuous loop that visits each vertex exactly once. How many ways are there to partition the twelve edges into two Hamiltonian cycles?
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Context:
Each frog needs to eat 30 flies per day to live. Each fish needs to eat 8 frogs per day to live. Each gharial needs to eat 15 fish a day to live. Additionally, each heron needs to eat 5 frogs and 3 fish per day to live, and each caiman needs to eat 2 gharials and 2 herons per day to live.

How many flies get eaten every day in this swamp if it has 9 gharials, 12 herons, and 7 caimans?
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Context:
Evaluate the limit: $$\lim_{z \to \infty}\int_{\mathbb{C}}|w|e^{-|z-w|^2}dA(w)$$ where $dA = rdrd\theta$ is the area measure in polar coordinates.
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Context:
Let $\vartriangle ABC$ be an equilateral triangle. Points $D,E, F$ are drawn on sides $AB$,$BC$, and $CA$ respectively such that $[ADF] = [BED] + [CEF]$ and $\vartriangle ADF \sim  \vartriangle BED \sim \vartriangle CEF$. The ratio $\frac{[ABC]}{[DEF]}$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$.
(Here $[P]$ denotes the area of polygon $P$.)
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Context:
The championship finals of the Chinese Volleyball Super League adopt a seven-game, four-win system, meaning that if one team wins four games first, that team will be the overall champion and the competition will end. Now, Team A and Team B are competing in the finals. Since the two teams have similar strengths, the probability of each team winning a game is $\frac{1}{2}$. According to past data, the ticket revenue for the first game is 5 million yuan, and the ticket revenue for each subsequent game increases by 100,000 yuan compared to the previous game. The probability that the total ticket revenue for the finals is exactly 45 million yuan is ____.
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Context:
Find the sum of the real roots of $x^4 - 4x - 1 = 0.$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
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Context:
Find the Taylor series expansion of the operator \( D = e^{bx \frac{d}{dy}} \) when it operates on an infinitely differentiable function \( g(y) \).
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Context:
Two parallel lines are intersected by a third line. Find the angle between the bisectors of the interior angles on the same side.
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Context:
Evaluate the surface integral \( \int_Y xyz \, dS \), where \( Y \) is the part of the sphere \( x^2 + y^2 + z^2 = 1 \) that lies above the region \( \{ y \leq x, x \leq 0, 0 \leq x^2 + y^2 \leq 1 \} \).
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Context:
A side of beef undergoes three stages of processing. In the first stage, it loses 20 percent of its weight. In the second stage, it loses 15 percent of its remaining weight. In the third stage, it loses 25 percent of its remaining weight. If the side of beef weighed 350 pounds after the third stage of processing, how many pounds did it weigh before processing?
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Context:
Find the largest integer \( k \) that divides the product \( abcdef \) for all integer solutions \((a, b, c, d, e, f)\) to the equation \( a^2 + b^2 + c^2 + d^2 + e^2 = f^2 \).
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Context:
For a two-by-two matrix $A$, determine the flux of the linear vector field ${\bf F}(x,y)=A\begin{bmatrix}x\\ y\end{bmatrix}$ through a unit circle centered at the origin in $\mathbb{R}^2$, expressed in terms of the entries of $A$. Assume the unit circle is parameterized by $x = \cos t$ and $y = \sin t$ for $0 \leq t < 2\pi$. Provide your answer in terms of the matrix entries.
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Context:
Evaluate the limit: $$\lim_{x\to\infty} \left(\int_0^1 t^{-tx} \, dt\right)^{\frac{1}{x}}$$
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Context:
Point $P$ is taken inside the square $ABCD$ such that $BP + DP = 25$, $CP - AP = 15$, and $\angle ABP = \angle ADP$. What is the radius of the circumcircle of $ABCD$?
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Context:
In the diagram, \(C\) lies on \(AE\) and \(AB=BC=CD\). If \(\angle CDE=t^{\circ}, \angle DEC=(2t)^{\circ}\), and \(\angle BCA=\angle BCD=x^{\circ}\), determine the measure of \(\angle ABC\).
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Context:
Folklore

On a chessboard, $n$ white and $n$ black rooks are placed such that rooks of different colors do not attack each other. Find the maximum possible value of $n$.

#
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Context:
Given the cubic equation
\[
x^3 + Ax^2 + Bx + C = 0 \quad (A, B, C \in \mathbb{R})
\]
with roots \(\alpha, \beta, \gamma\), find the minimum value of \(\frac{1 + |A| + |B| + |C|}{|\alpha| + |\beta| + |\gamma|}\).
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Context:
The base of a prism is an isosceles trapezoid inscribed around a circle with radius $r$, and its acute angle is $\alpha$. A plane passed through one side of the base and the acute angle endpoint of the opposite side of the top plane also forms an angle of $\alpha$ with the base plane. What is the height of the prism?
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Context:
Determine the numbers of all squares in which a radar beam, directed along the line \( y = x + 2013 \), will detect a mysterious object moving along the trajectory \( y = \left(\left(\left(x^{5}-2013\right)^{5}-2013\right)^{5}-2013\right)^{5} \).
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Context:
11 个黑球， 9 个红球，依次取出，剩下全是一种颜色就结束，求最后只剩下红球的概率？  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
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Context:
Mr. and Mrs. A are opposite to each other. The distance between Mr. A and Mrs. A is 120 km. Mr. A's speed is 30 kmph, and Mrs. A has a certain speed. One bee is running between Mr. A's nose to Mrs. A's nose and returning back to Mr. A's nose. The speed of the bee is 60 kmph. The bee traveled 180 km. What is the speed of Mrs. A?
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Context:
Evaluate the integral: $\int \frac{1}{t\left(t^{2}-4\right)} d t $
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Context:
Given the transition matrix of a Markov chain:
\[
\begin{bmatrix}
\frac{1}{5} & \frac{4}{5} & 0 & 0 & 0 \\
\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\
\frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} & \frac{1}{5} \\
0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\
0 & 0 & 0 & 0 & 1
\end{bmatrix}
\]
find the expected long-run proportion of time the chain spends at state $a$, given that it starts at state $b$. Use the stationary distribution to determine this proportion.
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Context:
12.3.8 ** Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1. A sphere is constructed with vertex $A$ as the center and $\frac{2 \sqrt{3}}{3}$ as the radius. The length of the curve formed by the intersection of the sphere's surface with the cube's surface is equal to
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Context:
Determine the minimum number of blue squares required on a $7 \times 7$ board such that each square has at least two neighboring blue squares. Two squares are considered neighbors if they share a side.
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Context:
Consider the system of ordinary differential equations: \[\begin{array}{ll}\ddot y + y + \ddot x + x = 0 \\ y+\dot x - x = 0 \end{array}\] How many initial conditions are required to determine a unique solution for this system?
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Context:
Given point $P\left(a+1,2a-3\right)$, if the point is symmetric about the $x$-axis in the first quadrant, then the range of $a$ is ____.
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Context:
Given an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ such that $AB = 0$ and $\operatorname{dim}(\operatorname{ker}B) = 0$, find the minimum possible value of $\operatorname{dim}(\operatorname{ker}A)$. \( \text{Provide your answer as a non-negative integer.} \)
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Context:
What is the sum of the value(s) of $n$ for which $|2n - 7| = 3$?
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Context:
The inclination angle of the line $\sqrt {3}x-y+1=0$ is \_\_\_\_\_\_.
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Context:
We place a hemispherical cap on a sphere of radius $R$, where the radius of the cap is $\frac{R \sqrt{3}}{2}$. By what percentage is the surface area of the resulting shape larger than the surface area of the sphere?
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Context:
Suppose  $a_{1} < a_{2}< \cdots < a_{2024}$  is an arithmetic sequence of positive integers, and   $b_{1} <b_{2} < \cdots <b_{2024}$  is a geometric sequence of positive integers. Find the maximum possible number of integers that could appear in both sequences, over all possible choices of the two sequences.

*Ray Li*
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Context:
A cylinder with radius 15 and height 16 is inscribed in a sphere. Three congruent smaller spheres of radius \( x \) are externally tangent to the base of the cylinder, externally tangent to each other, and internally tangent to the large sphere. What is the value of \( x \)?
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Context:
Define the derivative of the $(n-1)$th derivative as the $n$th derivative $(n \in N^{*}, n \geqslant 2)$, that is, $f^{(n)}(x)=[f^{(n-1)}(x)]'$. They are denoted as $f''(x)$, $f'''(x)$, $f^{(4)}(x)$, ..., $f^{(n)}(x)$. If $f(x) = xe^{x}$, then the $2023$rd derivative of the function $f(x)$ at the point $(0, f^{(2023)}(0))$ has a $y$-intercept on the $x$-axis of ______.
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Context:
Determine the number of real roots of the equation \( x^{3} - x - \frac{2}{3 \sqrt{3}} = 0 \).
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Context:
In a school, there are 10 rowing athletes, 3 of whom can only row on the right side, 2 can only row on the left side, and the remaining 5 can row on both sides. Now, 6 people need to be selected to row, evenly distributed on both sides. There are $\boxed{\text{different methods}}$ to do this (without considering the order of teammates on the same side).
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Context:
On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn from two rectangles $ABCD$ and $EFGH$, with sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$.
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Context:
What is the minimum number of non-adjacent swaps required to sort a list of distinct integers in ascending order?
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Context:
Two boys start moving from the same point A on a circular track but in opposite directions. Their speeds are 5 ft. per second and 9 ft. per second. If they start at the same time and finish when they first meet at the point A again, then the number of times they meet, excluding the start and finish, is
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Context:
Determine the minimum distance between the curves $x^2 = y + 2$ and $y^2 = x + 2$ using the method of common normals.
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Context:
Example 8. Calculate the flux of the vector field $\mathbf{a}=x \mathbf{i}+y \mathbf{j}+\sqrt{x^{2}+y^{2}-1} \mathbf{k}$ through the outer side of the one-sheet hyperboloid $z=\sqrt{x^{2}+y^{2}-1}$, bounded by the planes $z=0, z=\sqrt{3}$
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Context:
Determine the number of solutions to the equation $t \sin(2t) = 2$ for $0 < t < 3\pi$. Provide your answer as a single integer.
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Context:
Determine the value of the real number \(a\) for which the integral \(\int_0^{\infty} \frac{1-\cos x}{x^a} \, dx\) converges.
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Context:
Given that the amount of wine in a bottle is normally distributed as \( N(1.05l, 0.004l^2) \) and the volume of a cask is normally distributed as \( N(22l, 0.16l^2) \), calculate the probability that the contents of 20 randomly chosen bottles will fit inside a randomly chosen cask.
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Context:
Determine the value of \( \alpha \) such that the integral \( \int_{0}^{\infty} x^{\alpha}\sin\left(\frac{1}{x}\right)dx \) converges.
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Context:
Determine the VC-dimension of the class of parity functions defined on the boolean hypercube $\{0,1\}^N$. The class is given by $H_{N-parity} = \{h_I:I\subseteq \{1,2,..., N\}\}$, where for a binary vector $x = (x_1, x_2, ...,x_N) \in \{0,1\}^N$, the parity function $h_I(x) = \bigg(\sum_{i\in I}x_i\bigg) \mod 2$. Provide your answer as a single integer.
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Context:
In a game of Paper, Scissor, Rock with $n$ players $(n \geq 6)$, determine the number of strategic equilibria that exist when each player must choose at least two pure strategies.
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Context:
Let $a=\sqrt{2}$, $b=\log_{0.5}e$, and $c=\ln{2}$. Arrange the values in increasing order.
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Context:
Given the sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n=3^{n}(\lambda-n)-6$, if the sequence $\{a_n\}$ is a decreasing sequence, then the range of $\lambda$ is ______.
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Context:
$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$. What is the area of $R$ divided by the area of $ABCDEF$?
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Context:
The volume of a cylinder circumscribed around a sphere with radius $r$ is $V_{1}$, and the volume of a cone circumscribed around the same sphere is $V_{2}$. What is the minimum value of the ratio $V_{2} / V_{1}$?
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Context:
Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying $f(1)=1$ and for $n \in \mathbb{N}, f(2 n)=2 f(n)$ and $f(2 n+1)=2 f(n)-1$. Determine the sum of all positive integer solutions to $f(x)=19$ that do not exceed 2019.
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Context:
A fair six-sided die has faces numbered 1, 2, 3, 4, 5, 6. The die is rolled four times, and the results are a, b, c, d. What is the probability that one of the numbers in the set {a, a+b, a+b+c, a+b+c+d} is equal to 4?
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