Context:
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad \(N\), \(0<N<10\), it will jump to pad \(N-1\) with probability \(\frac{N}{10}\) and to pad \(N+1\) with probability \(1-\frac{N}{10}\). Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake? Provide your answer as a fraction in simplest form, \(\frac{m}{n}\), and give the value of \(m + n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $X = \{1, 2, \ldots, n\}$. Determine the number of ways to choose subsets $A$, $B$, and $C$ such that $A \cap B = \{1\}$, $B \cap C = \{2\}$, $A \cap C = \{3\}$, and $A$, $B$, and $C$ form a partition of $X$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For which sets $X$ is the composition operation $f \circ g$ on the set of all bijections $f: X \rightarrow X$ commutative?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the genus of the subset \( S = \{[x_0, x_1, x_2, x_3] \in \mathbb{RP}^3 : x_0^2 + x_1^2 - x_2^2 - x_3^2 = 0\} \), which is an embedded submanifold of \( \mathbb{RP}^3 \) with codimension 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the function \( f : [-\pi, \pi] \to [-\pi,\pi] \) defined by \( f(x)= \frac{\sin(x)}{2} \). This function is a contraction. Using the fixed point iteration \( x_{k+1}= f(x_{k}) \) starting from \( x_{0}=\frac{\pi}{2} \), determine the minimum number of iterations \( k \) required such that the error satisfies \(|x_{k}-x^{*}| \leq \frac{1}{1024}\), where \( x^{*} \) is the fixed point.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find all functions \( f: \mathbb{Z} \to \mathbb{Z} \) such that for all integers \( k \), the equation \( f(f(k+1)+3) = k \) holds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the square root of a positive number $m$ is $x+1$ and $5+2x$, then the value of $m$ is ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 3 valves in a pool system. With all 3 valves open, the pool is filled with water in 36 minutes. The first valve alone would fill the pool in 3 hours, while the second valve would fill the pool in 4 hours when functioning separately. If the third valve emits 30 cubic meters more water per minute than the first valve and 10 cubic meters more water per minute than the second valve, and there's a constant leakage of 20 cubic meters of water per minute from the pool, what is the capacity W of the pool?
Please reason step by step, and put your final answer within \boxed{}.


Context:
In rectangle $PQRS$, $PQ = 150$. Let $T$ be the midpoint of $\overline{PS}$. Given that line $PT$ and line $QT$ are perpendicular, find the greatest integer less than $PS$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose we calculate the Taylor polynomial of the $n^{\text{th}}$ degree centered at $a=0$ for the function $\sin(x)$. Determine the smallest integer $n$ such that the residual error $R_{n}f(x)$ is less than or equal to $0.01$ for all $x \in [0, \frac{1}{3}]$, and provide the value of this $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the geometric mean of 1 and 9. Express your answer in the form ±n, where n is a whole number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
4*. Find all such positive integers $k$ for which the number

$$
\underbrace{1 \ldots 1 \overbrace{2 \ldots 2}^{k}}_{2000}-\underbrace{2 \ldots 2}_{1001}
$$

is a perfect square.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设有理数 $r=\frac{p}{q} \in(0,1)$, 其中 $p, q$ 为互素的正整数, 且 $p q$ 整除 3600. 这样的有理数 $r$ 的个数为 $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( y = \sqrt{a x^2 + b x + c} \) (where \(a, b, c \in \mathbb{R}\) and \(a < 0\)), the domain is \( D \). If the points \( (s, f(t)) \) (where \( s, t \in D \)) form a square, then the real number \( a \) equals ______.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a parallelepiped $A B C D A_1 B_1 C_1 D_1$, points $M$, $L$, and $K$ are located on edges $A D$, $A_1 D_1$, and $B_1 C_1$, respectively. Given that $B_1 K = \frac{1}{3} A_1 L$, $A M = \frac{1}{2} A_1 L$, and $K L = 2$, determine the length of the segment by which the plane $K L M$ intersects the parallelogram $A B C D$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the side \( BC \) of triangle \( ABC \), points \( A_1 \) and \( A_2 \) are marked such that \( BA_1 = 6 \), \( A_1A_2 = 8 \), and \( CA_2 = 4 \). On the side \( AC \), points \( B_1 \) and \( B_2 \) are marked such that \( AB_1 = 9 \) and \( CB_2 = 6 \). Segments \( AA_1 \) and \( BB_1 \) intersect at point \( K \), and segments \( AA_2 \) and \( BB_2 \) intersect at point \( L \). Points \( K \), \( L \), and \( C \) lie on the same line. Find \( B_1B_2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The pentagon $ABCDE$ is inscribed in a circle. Points $M, Q, N,$ and $P$ are the bases of the perpendiculars dropped from vertex $E$ to the sides $AB, BC, CD$ (or their extensions) and diagonal $AD$ respectively. It is known that $|EP|=d$, and the ratio of the area of triangle $MQE$ to the area of triangle $PNE$ is $k$. Find $|EM|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The least possible value of the expression e = (x - 1)(x - 3)(x - 4)(x - 6) + some constant for real values of x is 1. What is the missing constant?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If the quadratic inequality $2kx^{2}+kx- \frac {3}{8} < 0$ holds for all real numbers $x$, find the range of $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve the system in positive numbers:

$$
\begin{cases}x^{y} & =z \\ y^{z} & =x \\ z^{x} & =y\end{cases}
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of elements \( \alpha \in \mathbb{F}_{3^5} \) such that \( \mathbb{F}_{3}(\alpha) = \mathbb{F}_{3^5} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the general form of the nth derivative of the function $$g(x)=\frac{x^7}{(1-x^2)^{2}}$$ at x = 0.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the circles $S_1: x^2 + y^2 + 4y - 1 = 0$, $S_2: x^2 + y^2 + 6x + y + 8 = 0$, and $S_3: x^2 + y^2 - 4x - 4y - 37 = 0$, which are mutually tangent, let $P_1, P_2, P_3$ be the points of contact and $C_1, C_2, C_3$ be their centers. Determine the ratio of the area of triangle $P_1P_2P_3$ to the area of triangle $C_1C_2C_3$. Express your answer as a simplified fraction.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The number of edges of a convex polyhedron is 99. What is the greatest number of edges a plane, not passing through its vertices, can intersect?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of homomorphisms from the dihedral group $D_{21}$ to the dihedral group $D_{15}$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Using Rouché's theorem, determine the number of zeros of the polynomial \( f(z) = z^{47} - z^{23} + 2z^{11} - z^5 + 4z^2 + 1 \) within the disk \( D(0, \frac{2}{3}) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a moving large circle $\odot O$ tangent externally to a fixed small circle $\odot O_{1}$ with radius 3 at point $P$, $AB$ is the common external tangent of the two circles with $A$ and $B$ as the points of tangency. A line $l$ parallel to $AB$ is tangent to $\odot O_{1}$ at point $C$ and intersects $\odot O$ at points $D$ and $E$. Find $C D \cdot C E = \quad$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Francis and Kiera had breakfast at a cafe. Muffins cost $2 each, fruit cups cost $3 each, and coffee costs $1.50 per cup. There is also a 10% discount for customers who order a combination of 2 muffins and 1 fruit cup. Francis had 2 muffins, 2 fruit cups, and 1 coffee, while Kiera had 2 muffins, 1 fruit cup, and 2 cups of coffee. How much did their breakfast cost, taking into account any applicable discounts?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A production line is checked daily by inspecting a sample of 100 units. Historically, the process has a yield of 80%, with a proportion defective of 20% and a standard deviation of 0.04 for the proportion defective. On a particular day, the sample contains 38 defective units. Assuming the prior distribution of the proportion defective, θ, is a Beta distribution, calculate the mean of the posterior distribution of θ for that day.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of solutions to the equation
\[\sin x = \left( \frac{1}{3} \right)^x\]
on the interval \( (0, 150 \pi) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $O$ is the coordinate origin, and the line $l$ passing through point $P(1,1)$ intersects the graph of the function $f(x)=\frac{{2}^{x+1}}{{2}^{x}-2}$ at points $A$ and $B$, then $(\overrightarrow{OA}+\overrightarrow{OB})\cdot \overrightarrow{OP}=\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知函数 $y=f(x)$ 的图像既关于点 $(1,1)$ 中心对称, 又关于直线 $x+y=0$轴对称。若 $x \in(0,1)$ 时, $f(x)=\log _{2}(x+1)$, 则 $f\left(\log _{2} 10\right)$ 的值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum of the sequence $81 + 891 + 8991 + 89991 + \dots + 8\underbrace{99\dots99}_{n-1\text{ nines}} = \_\_\_\_\_\_\_\_\_\_.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence of events $\{A_n\}$ such that the covariance between any pair of distinct events satisfies $\operatorname{Cov}(I_{A_i}, I_{A_j}) \leq 0$ for all $i \neq j$, where $I_{A_k}$ denotes the indicator function for $A_k$. If $\sum P(A_n) = \infty$, determine $P[\limsup A_n]$. Provide your answer as a probability value or statement.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the sign of the determinant of the matrix \(\begin{bmatrix} A & B \\ C & D \end{bmatrix}\) in terms of the determinant of \(AD - BC\), where \(A, B, C, D\) are commuting \(n \times n\) matrices.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the differentiable functions $z$ and $w$ of $x$ and $y$ that satisfy the equations:

$$xw^3 + yz^2 + z^3 = -1$$
$$zw^3 - xz^3 + y^2w = 1$$

Find the partial derivative of $z$ with respect to $x$, $\frac{\partial z}{\partial x}$, at the point $(x, y, z, w) = (1, -1, -1, 1)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
An isosceles trapezoid is circumscribed around a circle with radius $r$. The legs of the trapezoid form an angle $\alpha$ with the shorter base. Find the radius of the circle circumscribed around the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\frac{r \sqrt{1 + \sin^2 \alpha}}{\sin^2 \alpha}}


Context:
On a tough week, Haji's mother sells goods worth $800, which is half the amount she sells on a good week. In addition to the revenue from selling goods, she also offers a 10% discount on purchases during tough weeks. She incurs a 5% tax on her total sales and has to spend $50 per week on transportation costs to get the goods. What's the total amount of money she makes (net profit) if she has 5 good weeks and 3 tough weeks?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mary has a sequence $m_{2}, m_{3}, m_{4}, \ldots$, such that for each $b \geq 2, m_{b}$ is the least positive integer $m$ for which none of the base-$b$ logarithms $\log _{b}(m), \log _{b}(m+1), \ldots, \log _{b}(m+2017)$ are integers. Find the largest number in her sequence.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a real number \( x \), find the maximum value of
\[
\frac{x^6}{x^{12} + 3x^8 - 6x^6 + 12x^4 + 36}
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let  $ABCD$  be a trapezoid of bases  $AB$  and  $CD$  . Let  $O$  be the intersection point of the diagonals  $AC$  and  $BD$ . If the area of the triangle  $ABC$  is  $150$  and the area of the triangle  $ACD$  is  $120$ , calculate the area of the triangle  $BCO$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Xavier starts from point P towards point Q at a speed of 40 kmph and after a certain time interval increases his speed by 20 kmph. The distance between point P and point Q is 56 km. He takes 48 minutes to cover the distance. After how many minutes does he increase his speed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
A monkey is typing randomly on a typewriter with three keys: A, B, and C. Each key is pressed with equal probability, independently of previous presses. Let \( T \) be the random time at which the sequence "ABC" first appears in the typed sequence. What is the expected value of \( T \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Tanya and Vera are playing a game. Tanya has cards with numbers from 1 to 30. She arranges them in some order in a circle. For every two neighboring numbers, Vera calculates their difference by subtracting the smaller number from the larger one and writes down the resulting 30 numbers in her notebook. After that, Vera gives Tanya the number of candies equal to the smallest number written in the notebook. Tanya wants to arrange the cards to get as many candies as possible. What is the maximum number of candies Tanya can receive?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the work required to fill a hemispherical tank with water through a hole in its base. The tank has a radius of 2 feet and is positioned with its base circular. The water source is at the base, and the weight-density of water is 62.4 pounds per cubic foot. Provide the work done in foot-pounds.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \( P \) be a regular 2006-sided polygon. A diagonal of \( P \) is called a "good edge" if its endpoints divide the polygon into two parts, each containing an odd number of the polygon's sides. Each side of \( P \) is also considered a "good edge".

Given that 2003 non-intersecting diagonals divide \( P \) into several triangles, determine the maximum number of isosceles triangles, under this division, that have two "good edges".
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sets $S=\{ x\mid | x-2 | > 3 \}$ and $T=\{ x\mid a < x < a+8 \}$, and their union $S\cup T=R$, find the range of values for $a$.
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Context:
In an acute-angled triangle \( \mathrm{ABC} \), angle \( \mathrm{B} = 75^\circ \). A point \( K \) is chosen on side \( \mathrm{AC} \). Circles are circumscribed around triangles \( \mathrm{ABK} \) and \( \mathrm{CBK} \), with centers \( \mathrm{O}_1 \) and \( \mathrm{O}_2 \) respectively. Find the radius of the circumcircle of triangle \( \mathrm{ABC} \) if the smallest possible length of segment \( \mathrm{O}_1 \mathrm{O}_2 \) is 2 cm.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{2 \sqrt{6} - 2 \sqrt{2} \text{ cm}}


Context:
Determine the number of non-isomorphic groups of order 2013.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Diane wants to buy 3 packages of cookies for her friends. Each package of cookies costs 75 cents. There is also a 10% sales tax on the purchase. Diane currently has 55 cents. What's the difference between the total cost of the cookies (including sales tax) and the money Diane has?
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $(1+x+x^{2})^{6}=a\_{0}+a\_{1}x+a\_{2}x^{2}+...+a\_{12}x^{12}$, then $a\_{2}+a\_{4}+...+a\_{12}=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$ and $f(x+ \frac{5}{2})=-\frac{1}{f(x)}$, and when $x \in [-\frac{5}{2}, 0]$, $f(x)=x(x+ \frac{5}{2})$, find $f(2016)=$ \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ways to arrange 6 red roses and 4 white roses in a garland such that no two white roses are adjacent.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the real values of $n$ for which the matrix \( M = \begin{bmatrix} n & 1 & 1 \\ 1 & n & 1 \\ 1 & 1 & -n \end{bmatrix} \) is negative definite.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a circle $(x^2 + y^2 - 4x - 4y = 0)$, there are at least three distinct points on the circle such that the distance from each point to the line $l: y = kx$ is $\sqrt{2}$. Determine the range of the slope $k$ of the line $l$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral $$J=\int_0^{\infty} \frac{\operatorname{artanh}\left(\frac{1}{\sqrt{1+x^2}}\right)}{\sqrt{1+x^2}} d x$$
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Context:
Let $V$ be a finite-dimensional complex vector space and $G \subset GL(V)$ be a finite group. Consider the quotient variety $V/G$. Determine the codimension of the singular locus in $V/G$.
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Context:
a and b are two multiples of a certain number, and q is the set of consecutive integers between a and b, inclusive. q contains 9 multiples of that number and there are 17 multiples of 7 in q. What is that certain number?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Mark does a gig every other day for 2 weeks. For each gig, he plays a certain number of songs. 2 of the songs are 5 minutes long and the last song is twice that long. He played for 280 minutes. How many songs does he play at each gig?
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are 100 chairs arranged in a circle. If \( n \) people are sitting on these chairs, such that any new person sitting down will always sit on a chair adjacent to one of the \( n \) people, what is the minimum value of \( n \)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知数列 $\left\{a_{n}\right\}$ 满足
$$
a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right]\left(n \in \mathbf{Z}_{+}\right),
$$
其中， $[x]$ 表示不超过实数 $x$ 的最大整数。设 $C$ 为实数，且对于任意的正整数 $n$ ，均有
$$
\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C
$$
则 $C$ 的最小值为 $\qquad$原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.


Context:
11. There are 11 students who have signed up for the volunteer tour guide activity at the museum. The activity runs from 9 AM to 5 PM, with a public mini-lecture every hour. Each session requires 1 student to provide tour guide services to visitors. To avoid overworking the students, the museum will not schedule the same person for two consecutive sessions, and the first and last sessions must be handled by two different students. How many scheduling arrangements are possible for the museum?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Zoey read $15$ books, one at a time. The first book took her $1$ day to read, the second book took her $2$ days to read, the third book took her $3$ days to read, and so on, with each book taking her $1$ more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day the week did she finish her $15$th book?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two circles touch each other externally at point \( C \). A line is tangent to the first circle at point \( A \) and to the second circle at point \( B \). The line \( AC \) intersects the second circle at point \( D \), distinct from \( C \). Find \( BC \), given that \( AC = 9 \) and \( CD = 4 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
x and y are two towns. Ganesh covers the distance from x to y at an average speed of 43 km/hr. However, he covers the distance from y to x at an average speed of 34 km/hr. What is his average speed during the whole journey?
Please reason step by step, and put your final answer within \boxed{}.


Context:
On the leg \(BC\) of the right triangle \(ABC\), a circle is constructed with \(BC\) as its diameter. This circle intersects the hypotenuse \(AB\) at point \(P\). Chord \(PQ\) is parallel to leg \(BC\). Line \(BQ\) intersects leg \(AC\) at point \(D\). Given that \(AC = b\) and \(DC = d\), find \(BC\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Medians $\overline{AD}$ and $\overline{BE}$ of $\triangle ABC$ intersect at an angle of $45^\circ$. If $AD = 12$ and $BE = 16$, then calculate the area of $\triangle ABC$.
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Context:
We fill a graduated cylinder with a diameter of $2.9 \mathrm{~cm}$ with water up to a height of $4 \mathrm{~cm}$. Then, we place solid copper cubes with an edge length of $2 \mathrm{~cm}$ into the cylinder. What is the maximum number of cubes that can be submerged in the water in the cylinder?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Among the four-digit numbers formed by the digits 6, 7, 8, and 9, where each digit is different, there are $\boxed{\text{several}}$ numbers that are divisible by 11.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the curve $y=\sin x$ ($0 \leq x \leq \pi$) and the line $y= \frac {1}{2}$, calculate the area of the enclosed shape formed by these two functions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a geometry problem, Isabella has drawn a pair of concentric circles. She starts drawing chords $\overline{CD}$, $\overline{DE}$, and so on, of the larger circle, each tangent to the smaller one. If $m\angle CDE=60^\circ$, how many chords will Isabella draw before she returns to her starting point at $C$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Under the new AMC $10, 12$ scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, the only way to obtain a score of $146.5$ is to have $24$ correct answers and one unanswered question. Some scores can be obtained in exactly two ways, for example, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect, and also with $12$ correct answers and $13$ unanswered questions. There are (three) scores that can be obtained in exactly three ways. If the sum of these scores is $x$, what is the value of $10x$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the sum of the digits in base ten of the expression (10^{4n^2 + 8} + 1)^2, where n is a positive integer. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
\( p(x) \) is a polynomial of degree less than 1992 such that \( p(0) \), \( p(1) \), \( p(-1) \) are all non-zero. The 1992nd derivative of \( \frac{p(x)}{x^3 - x} \) is \( \frac{f(x)}{g(x)} \) for polynomials \( f(x) \) and \( g(x) \). Find the smallest possible degree of \( f(x) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The negation of the proposition "For all $x > 0$, $\sin x \geq -1$":
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\exists x > 0, \text{ such that } \sin x < -1}


Context:
Let f(x) be a monotonically decreasing odd function defined on R. Given x1+x2>0, x2+x3>0, and x3+x1>0, determine the sign of f(x1)+f(x2)+f(x3). Express your answer as a single inequality in the format f(x1)+f(x2)+f(x3) [inequality] 0.
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, \ldots, k_n$ such that
$k^2_1 + k^2_2 + \ldots + k^2_n = 2002?$
Please reason step by step, and put your final answer within \boxed{}.


Context:
# 2.1. Condition:

In the campus, rooms are numbered consecutively $1,2,3,4 \ldots, 10,11, \ldots$ For room numbering, stickers with digits were purchased, with the digits 1, 2, and 3 being purchased in equal quantities, and the digit 5 being purchased three more than the digit 6. How many rooms are there in the campus, given that there are fewer than 100?
Please reason step by step, and put your final answer within \boxed{}.


Context:
As shown in the diagram, in the plane Cartesian coordinate system, the ellipse $\Gamma$: $\frac{x^{2}}{2} + y^{2} = 1$ has left and right foci $F_{1}$ and $F_{2}$, respectively. Let $P$ be a point on $\Gamma$ in the first quadrant, and let the extended lines $P F_{1}$ and $P F_{2}$ intersect $\Gamma$ at points $Q_{1}\left(x_{1}, y_{1}\right)$ and $Q_{2}\left(x_{2}, y_{2}\right)$, respectively. Find the maximum value of $y_{1} - y_{2}$.
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Context:
Yann writes down the first $n$ consecutive positive integers, $1,2,3,4, \ldots, n-1, n$. He removes four different integers $p, q, r, s$ from the list. At least three of $p, q, r, s$ are consecutive and $100<p<q<r<s$. The average of the integers remaining in the list is 89.5625. What is the number of possible values of $s$?
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Context:
In triangle $XYZ$, which is equilateral with a side length $s$, lines $\overline{LM}$, $\overline{NO}$, and $\overline{PQ}$ are parallel to $\overline{YZ}$, and $XL = LN = NP = QY$. Determine the ratio of the area of trapezoid $PQYZ$ to the area of triangle $XYZ$.
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Context:
In the vertices of a regular 100-gon, 100 chips numbered $1, 2, \ldots, 100$ are placed in exactly that order in a clockwise direction. During each move, it is allowed to swap two chips placed at adjacent vertices if the numbers on these chips differ by no more than $k$. What is the smallest $k$ such that, in a series of such moves, every chip is shifted one position clockwise relative to its initial position?
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Context:
The number 175 can be written as the sum of the squares of 4 different positive integers. What is the sum of these 4 integers?
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Context:
Let $\mathcal{S}$ be a sphere with radius $2.$ There are $8$ congruent spheres whose centers are at the vertices of a cube, each has radius $x,$ each is externally tangent to $3$ of the other $7$ spheres with radius $x,$ and each is internally tangent to $\mathcal{S}.$ There is a sphere with radius $y$ that is the smallest sphere internally tangent to $\mathcal{S}$ and externally tangent to $4$ spheres with radius $x.$ There is a sphere with radius $z$ centered at the center of $\mathcal{S}$ that is externally tangent to all $8$ of the spheres with radius $x.$ Find $18x + 5y + 4z.$
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Context:
Given that the sequence $\{a\_n\}$ satisfies $a\_1$, $a\_2-a\_1$, $a\_3-a\_2$, $…$, $a\_n-a_{n-1}$, $…$, is an arithmetic sequence with the first term being $1$ and the common ratio being $2$, find the value of $a\_n$ equal to _____.
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Context:
Through the vertex \(B\) of triangle \(ABC\), a line \(\ell\) is drawn parallel to side \(AC\). The angle bisector of angle \(A\) intersects \(\ell\) at point \(A'\), and the angle bisector of angle \(C\) intersects \(\ell\) at point \(C'\). It is known that the area of triangle \(ABA'\) is \(50\%\) greater and the area of triangle \(CBC'\) is \(25\%\) greater than the area of triangle \(ABC\). The bisectors intersect at point \(O\). What fraction of the area of triangle \(AOC\) is the area of triangle \(ABC\)?
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Context:
There are $n$ integers $a$ such that $0 \leq a < 91$ and $a$ is a solution to the congruence $x^3 + 8x^2 - x + 83 \equiv 0 \pmod{91}$. What is $n$?
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Context:
If $ABCDE$ is a regular pentagon and $X$ is a point in its interior such that $\triangle CDX$ is equilateral, compute $\angle AXE$ in degrees.
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Context:
If the positive sequence $\{a_n\}$ satisfies $\log a_{n+1} - \log a_n = 1$, and $a_{2001} + a_{2002} + a_{2003} + \ldots + a_{2010} = 2015$, then the value of $a_{2011} + a_{2012} + a_{2013} + \ldots + a_{2020}$ is \_\_\_\_\_\_.
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Context:
Given $A=\{y|y=|x+1|, x\in[-2,4]\}$, $B=[2,5)$, then $\complement_{A}B=$ ?
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Context:
Compute the integer $m > 3$ for which
\[\log_{10} (m - 3)! + \log_{10} (m - 1)! + 3 = 2 \log_{10} m!.\]
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Context:
Al told Bob that he was thinking of $2011$ distinct positive integers. He also told Bob the sum of those $2011$ distinct positive integers. From this information, Bob was able to determine all $2011$ integers. How many possible sums could Al have told Bob?

[i]Author: Ray Li[/i]
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Context:
Given a sequence $\{a_n\}$ that satisfies $a_n-(-1)^n a_{n-1}=n$ ($n\geqslant 2, n\in \mathbb{N}^*$), and $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, then $S_{40}=$_______.
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Context:
If there exists a real number $x$ such that the inequality $\left(e^{x}-a\right)^{2}+x^{2}-2ax+a^{2}\leqslant \dfrac{1}{2}$ holds with respect to $x$, determine the range of real number $a$.
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Context:
设整数 $a_{1}, a_{2}, a_{3}$ 满足 $1 \leqslant a_{k} \leqslant 24(k=1,2,3)$ ，且对任意整数 $x, 2 a_{1} x^{2}+3 a_{2} x+4 a_{3}$ 是 24 的倍数。求满足条件的有序数组 $\left(a_{1}, a_{2}, a_{3}\right)$ 的个数。
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Context:
Given the equation of the ellipse is $\frac{x^2}{4}+\frac{y^2}{3}=1$, $F_1$ and $F_2$ are the left and right foci of the ellipse, respectively. The chord $PQ$ passes through the right focus $F_2$ of the ellipse. The maximum area of $\Delta PF_1Q$ is \_\_\_\_
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Context:
Dad bought a cylindrical birthday cake for his daughter Yuanyuan. Yuanyuan wants to cut the cake into several pieces (not less than 10 pieces) of different sizes to share with 10 friends. If the cake is cut vertically, at least how many cuts are needed?
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Context:
Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.
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Context:
From Sunday to Thursday, Prudence sleeps 6 hours a night. On Friday and Saturday, she sleeps for a different number of hours a night. She also takes a 1-hour nap on Saturday and Sunday. Prudence gets 200 hours of sleep in 4 weeks. How many hours does she sleep per night on Friday and Saturday?
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Context:
Let $ABC$ be a triangle such that $AB=6,BC=5,AC=7.$ Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X.$ Let $Z$ be a point on the circumcircle of $ABC.$ Let $Y$ be the foot of the perpendicular from $X$ to $CZ.$ Let $K$ be the intersection of the circumcircle of $BCY$ with line $AB.$ Given that $Y$ is on the interior of segment $CZ$ and $YZ=3CY,$ compute $AK.$
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Context:
If the function $f(x)=a\sin^2x+b\tan x+1$, and $f(2)=5$, then calculate $f(\pi-2)+f(\pi)$.
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Context:
Eight celebrities meet at a party. Each celebrity shakes hands with exactly two others. A fan makes a list of all unordered pairs of celebrities who shook hands with each other. If the order does not matter, how many different lists are possible?
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Context:
Let \( n \in \mathbf{N}^{*} \). Consider the set \( S = \{1, 2, \ldots, 2n\} \) and its \( k \) subsets \( A_{1}, A_{2}, \ldots, A_{k} \) that satisfy the following conditions:

1. For any \( i \neq j \) (with \( i, j \in \{1, 2, \ldots, k\} \)), the intersection \( A_{i} \cap A_{j} \) has exactly an odd number of elements.
2. For any \( i \) (where \( i = 1, 2, \ldots, k \)), \( i \notin A_{i} \).
3. If \( i \in A_{j} \), then \( j \in A_{i} \).

Determine the maximum value of \( k \).
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Context:
## Task A-1.3.

Determine how many six-digit natural numbers exist such that by removing the first two, or the last two digits, we obtain two four-digit numbers that give the same remainder when divided by 99.
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Context:
Points \( K, L, M, N \) are the centers of the circles inscribed in the faces \( SAB, SAC \), \( SBC \), and \( ABC \) of the tetrahedron \( SABC \). It is known that \( AB = SC = 5 \), \( AC = SB = 7 \), and \( BC = SA = 8 \). Find the volume of the tetrahedron \( KLMN \). If necessary, round the answer to two decimal places.
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Context:
In triangle $ABC$, $BC = 40$ and $\angle C = 45^\circ$. Let the perpendicular bisector of $BC$ intersect $BC$ at $D$ and extend to meet an extension of $AB$ at $E$. Find the length of $DE$.
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Context:
实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数，其中两个根的和为 $2+\mathrm{i}$, 另两个根的积为 $5+6 \mathrm{i}$。求 $b$ 的值。
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Context:
If the negation of the proposition "There exists a real number $x_0$ such that $x + ax_0 + 1 < 0$" is true, then the range of the real number $a$ is ______.
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Context:
Given that the complex number $z$ and $(z+2)^2 - 8i$ are both purely imaginary, find $z$.
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Context:
In a $2 \times 8$ squared board, you want to color each square red or blue such that on each $2 \times 2$ sub-board there are at least $3$ squares painted blue. In how many ways can this coloring be done?

**Note:** A $2 \times 2$ board is a square made up of $4$ squares that have a common vertex.
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Context:
Given the complex number \( z \) satisfies \( |z|=1 \) and \( \operatorname{Re} \frac{z+1}{\bar{z}+1}=\frac{1}{3} \), find the value of \( \operatorname{Re} \frac{z}{\bar{z}} \).
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Context:
Density is defined as the ratio of mass to volume. There are two cubes. The second cube is made from a material with twice the density of the first, and the side length of the second cube is 100% greater than the side length of the first. By what percentage is the mass of the second cube greater than the mass of the first?
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Context:
If \( k \) is an integer greater than 1 and \( a \) is a root of \( x^2 - kx + 1 = 0 \), for any natural number \( n \) greater than 10, the units digit of \( a^{2n} + a^{-2n} \) is always 7. What is the units digit of \( k \)?
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Context:
Two right triangles, $ABC$ and $ACD$, are joined at side $AC$. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. Determine the number of square units in the area of the fourth square.
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Context:
Given rectangle ABCD, AB=4, BC=8, points K and L are midpoints of BC and AD, respectively, and point M is the midpoint of KL. What is the area of the quadrilateral formed by the rectangle diagonals and segments KM and LM.
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Context:
For each integer $n>1$, let $F(n)$ be the number of solutions to the equation $\sin{x}=\sin{(nx)}$ on the interval $[0,\pi]$. What is $\sum_{n=2}^{2007} F(n)$?
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Context:
In what ratio does the line $TH$ divide the side $BC$?
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Context:
These points form a new triangle with angles \(45^{\circ}, 60^{\circ},\) and \(75^{\circ}\). Find the ratio of the areas of the original and the new triangles.
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Context:
Let
$$p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3.$$Suppose that
\begin{align*}
p(0,0) &=p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1)= p(1,1) = p(1, - 1) = p(2,2) = 0.
\end{align*}There is a point $(r,s)$ for which $p(r,s) = 0$ for all such polynomials, where $r$ and $s$ are not integers.  Find the point $(r,s).$
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Context:
Evaluate the line integral \( \int_c \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy \) along the curve \( c(t) = (1+2\cos t, \sin t) \) for \( 0 \leq t \leq 2\pi \).
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Context:
Compute the integral \( \int_\Omega  \left|\,x\,\right|\,dx\,dy\,dz \) over the volume of a torus. In \(\mathbb{R}^3\), let \(C\) be the circle in the \(xy\)-plane with radius 2 and the origin as the center, i.e., \(C= \Big\{ \big(x,y,z\big) \in \mathbb{R}^3 \mid x^2+y^2=4, \ z=0\Big\}.\) Let \(\Omega\) consist of all points \((x,y,z) \in \mathbb{R}^3\) whose distance to \(C\) is at most 1. Use appropriate coordinate transformations and symmetry considerations to simplify the computation.
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Context:
What is the maximum number of idempotent and linearly independent matrices in $M_n(F)$ (considered as a vector space over the field $F$)?
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Context:
Find the distance from the ellipsoid \(x^2 + y^2 + 4z^2 = 4\) to the plane \(x + y + z = 6\). Use the method of Lagrange multipliers to solve the problem.
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Context:
A square is divided into 2022 rectangles, with no two rectangles sharing an interior point. What is the maximum number of distinct lines that can be formed by the sides of these rectangles?
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Context:
There are 1000 toy bears in a row. The bears come in three colors: white, brown, and black. Among any three consecutive bears, there is at least one bear of each color. Iskander tries to guess the colors of the bears. He made five guesses:

- The 2nd bear from the left is white;
- The 20th bear from the left is brown;
- The 400th bear from the left is black;
- The 600th bear from the left is brown;
- The 800th bear from the left is white.

It turns out that exactly one of his guesses is wrong. What could be the number of the bear whose color Iskander did NOT guess correctly?
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Context:
The probability that a set of three distinct vertices chosen at random from among the vertices of a regular n-gon determine an obtuse triangle is $\frac{93}{125}$ . Find the sum of all possible values of $n$.
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Context:
Determine the infimum of the set \( \{\lvert\lvert f\rvert\rvert_{\infty} : f \in F\} \), where \( F = \{ f \in C[0,1] : f(0)=1, \int_0^1 (1-t)f(t)dt =2 \} \).
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Context:
Determine the number of homomorphisms from the ring of Gaussian integers \( \mathbb{Z}[i] \) onto the ring \( \mathbb{Z}_{13}(i) \).
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Context:
Calculate the covariance between the first and third order statistics of three independent random variables with a uniform distribution on the interval $[0,1]$. The variance of the $k$th order statistic is given by $\frac{k(4-k)}{80}$.
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Context:
Evaluate the limit:
\[ \lim_{n \to \infty} \left( \sum_{k=10}^{n+9} \frac{2^{11(k-9)/n}}{\log_2 e^{n/11}} - \sum_{k=0}^{n-1} \frac{58}{\pi\sqrt{(n-k)(n+k)}} \right) \]
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Context:
Find the largest natural number in which all digits are different and each pair of adjacent digits differs by 6 or 7.
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Context:
An empty fuel tank is filled with brand Z gasoline. When the tank is 1/4 empty, it is filled with brand Y gasoline. When the tank is half empty again, it is filled with brand Z gasoline. When the tank is half empty again, it is filled with brand Y gasoline. At this time, what percent of the gasoline in the tank is brand Z?
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Context:
In a village with $N$ people, each person owns one dog. A certain number of dogs, $K$, have contracted rabies. Every morning, villagers take their dogs for a walk and can identify which dogs, except their own, have rabies. They agree to shoot their own dog at night if they can conclude it has rabies. The disease is not contagious, and $K$ is constant with $K \geq 1$.

On the first night, no dogs are shot. On the second night, no dogs are shot. On the third night, no dogs are shot. On the fourth night, several gunshots are heard.

How many dogs were killed on the fourth night?
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Context:
Given that the graph of the power function $y=(m^{2}-2m-2)x^{m^{2}+4m}$ is symmetric with respect to the origin and does not intersect the $x$-axis or $y$-axis, determine the value of the integer $m$.
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Context:
Determine the operator norm of $R_t$ as an element of $M_2(A)$, where $A$ is a normed real vector space and $A^2$ is equipped with a norm derived from an inner product satisfying the parallelogram identity.
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Context:
A quadrilateral $ABCD$ is inscribed in a circle of radius $R.$ It is known that $\measuredangle ADB = 90^\circ$ and the distance between the centers of the circles inscribed in the triangles $ADB$ and $BCD$ is $6.$ Calculate $R.$
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Context:
Given a random sample $X_1$ and $X_2$ of size $n=2$ from a uniform distribution on the interval $[0, \theta]$, determine the efficiency of the maximum likelihood estimator of $\theta$ relative to the estimator $\frac{3}{2} \bar{X}$, where $\bar{X}$ is the sample mean.
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Context:
An isosceles triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of one of the equal sides of the triangle to the width of the rectangle? Express your answer as a common fraction.
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Context:
In the city of liars and knights, there are 366 inhabitants, each born on a different day of a leap year. All inhabitants of the city answered two questions. To the question "Were you born in February?" 100 people answered affirmatively, and to the question "Were you born on the 30th day?" 60 people answered affirmatively. How many knights were born in February?
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Context:
Given that the cross-section of a cylinder is a square with side length $2$, what is the surface area of the cylinder?
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Context:
In a finite graph represented by the adjacency matrix \( B \) and transition matrix \( P \), calculate the probability of reaching state D before reaching state E, starting from state A. The matrices are given as follows:

\[ B = \begin{bmatrix}
0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 \\
\frac{1}{3} & 0 & \frac{1}{3} & 0 & \frac{1}{3} \\
\frac{1}{2} & \frac{1}{2} & 0 & 0 & 0 \\
\frac{1}{2} & 0 & 0 & 0 & \frac{1}{2} \\
0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 \\
\end{bmatrix} \]

\[ P = \begin{pmatrix}
0 & 1 & 1 & 1 & 0 \\
1 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 & 0 \\
\end{pmatrix} \]

Determine the probability \( p_A \) of reaching state D before state E from state A.
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Context:
A right circular cylindrical tank lies on its side with a length of 12 feet and a radius of 4 feet. The water level in the tank is 18 inches above the lowest part. Calculate the volume of water that must be drained from the tank. Round your answer to the nearest tenth of a cubic foot.
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Context:
Rosencrantz and Guildenstern are playing a game where they repeatedly flip coins. Rosencrantz wins if 1 heads followed by 2009 tails appears. Guildenstern wins if 2010 heads come in a row. They will flip coins until someone wins. What is the probability that Rosencrantz wins?
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Context:
Among the 2019 natural numbers from 1 to 2019, how many of them, when added to the four-digit number 8866, result in at least one carry?
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Context:
For which values of \( n \) does the cyclotomic extension \( \mathbb{Q}(\zeta_n) \) over \( \mathbb{Q} \) contain the imaginary unit \( i \)? Determine the condition on \( n \) such that \( i \in \mathbb{Q}(\zeta_n) \).
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Context:
Given a twice differentiable function $k: \Bbb R \to \Bbb R$ such that $k(a) = k(e) = 0$, $k(b) = k(d) = 2$, and $k(c) = 1$, with $a < b < c < d < e$, determine the minimum number of solutions for the equation $(k'(x))^2 + k(x)k''(x) = 0$ within the interval $[a, e]$. Find the minimum number of solutions.
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Context:
Compute the determinant of the matrix $A = I + yx^H$, where $x, y \in \mathbb{C}^n$.
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Context:
Find the general form of all $4 \times 4$ matrices that commute with every other $4 \times 4$ matrix.
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Context:
Determine whether the set \( X = \{ f \in \mathbb{L}_1(\mathbb{R}^n) \mid m(f^{-1}((0,\infty))) = 0 \} \) is open, closed, or neither in \( \mathbb{L}_1(\mathbb{R}^n) \).
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Context:
For a 3x3 matrix \( A \) with the characteristic polynomial \( c^3 - 3c^2 + 4 = 0 \), find the trace of the adjoint of \( A \).
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Context:
Determine the number of 8-tuples of nonnegative integers $\left(a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}\right)$ satisfying $0 \leq a_{k} \leq k$, for each $k=1,2,3,4$, and $a_{1}+a_{2}+a_{3}+a_{4}+2 b_{1}+3 b_{2}+4 b_{3}+5 b_{4}=19$.
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Context:
Given natural numbers \( m \) and \( n \). There are two piles on the table: the first pile contains \( n \) stones, and the second pile contains \( m \) stones. Petya and Vasya play the following game. Petya starts. In one move, a player can break one of the available piles on the table into several smaller piles. The player who cannot make a move loses. For which values of \( m \) and \( n \) can Petya ensure victory regardless of Vasya's play?
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Context:
Find the surface area of the part of the sphere \(x^2+y^2+z^2=a^2\) that lies inside the cylinder \(x^2+y^2=ax\).
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Context:
Find a non-zero solution to the equation \(3\sin(\sqrt{x})=x\).
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Context:
Given that Xiao Ming's elder brother was born in a year that is a multiple of 19, and the year 2013 is the first time in his life he would experience a year with no repeated digits, how old is he in 2013?
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Context:
$\begin{array} {l}\text{In the convex quadrilateral}ABCD,\mathrm{point}X\text{is selected on side}AD,\text{and the diagonals intersect at point}E.\text{It is known that} \\ AC=BD,\angle ABX=\angle AXB=50^o,\angle CAD=51^o,\angle AED=80^o.\text{Find the value of angle}\angle AXC. \end{array}$
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Context:
Eleven points are marked from left to right on a straight line. The sum of all the distances between the first point and the other points is $2018 \mathrm{~cm}$. The sum of all the distances between the second point and the other points, including the first one, is $2000 \mathrm{~cm}$. What is the distance between the first and second points? Express your answer in centimeters (cm) as a whole number.
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Context:
Evaluate the iterated integral \( \int_{0}^{2}\left (\int_{0}^{\sqrt{2x-x^2}} \sqrt{x^2+y^2} \, dy\right ) \, dx \).
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Context:
What is the expected value of a random variable $X$ that follows the negative binomial distribution with parameters $r$ and $p$?
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Context:
Given a twice differentiable function $f(x)$ with $f(a) = 0, f(b) = 2, f(c) = -1, f(d) = 2, f(e) = 0$, where $a < b < c < d < e$, determine the minimum number of zeros of the function $g(x) = (f'(x))^2 + f(x)f''(x)$ in the interval $[a, e]$. What is this minimum number?
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Context:
You and a friend are playing a game with a fair coin. You go first, and you each take turns flipping the coin, recording the result each time. A player wins the game when they get two consecutive heads. What is the probability that you win?
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Context:
In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
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Context:
Let $S$ be the set of the first $2018$ positive integers, and let $T$ be the set of all distinct numbers of the form $ab$, where $a$ and $b$ are distinct members of $S$. What is the $2018$th smallest member of $T$?
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Context:
Friends Vasya, Petya, and Kolya live in the same house. One day, Vasya and Petya set off to go fishing at the lake on foot. Kolya stayed home, promising his friends to meet them on the way back with a bicycle. Vasya was the first to head back home, and at the same time, Kolya set off to meet him on the bicycle. Petya, traveling at the same speed as Vasya, left the lake for home at the moment Vasya and Kolya met. Kolya, after meeting Vasya, immediately turned around and brought Vasya home on the bicycle, then immediately set off again on the bicycle towards the lake. Upon meeting Petya, Kolya once again turned around and brought Petya home. As a result, the time Petya spent traveling from the lake to home was $5 / 4$ of the time Vasya took for the same journey. How much slower would Vasya have been if he walked the entire way home by himself? (8 points)
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Context:
Calculate the area of the region where the polar curves \( r = 2 - \cos(\theta) \) and \( r = 3 \cos(\theta) \) intersect, given that the intersection angles are \( \theta = \pm \frac{\pi}{3} \).
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Context:
Find the value of $x$ such that \( \sum_{k=1}^{2014} k^k \equiv x \pmod{10} \).
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Context:
Let \( A \) be the sum of the decimal digits of \( 4444 \, 4444 \), and \( B \) be the sum of the decimal digits of \( A \). Find the sum of the decimal digits of \( B \).
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Context:
Compute the Lebesgue measure of the set \( S = \{(x, y, z) \in \mathbb{R}^3 : x \geq 1, \ y^2 + z^2 \leq \frac{1}{x^2} \} \).
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Context:
Given a right triangular prism ABC-A1B1C1 with volume V, points P and Q lie on the edges AA1 and OC1 respectively, such that AP = C1Q. Calculate the volume of the quadrangular pyramid B-APQC as a fraction of the original volume V. Express your answer in the form of V/n, where n is the denominator.
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Context:
Evaluate the limit \( L = \lim\limits_{n\to\infty} \frac{2^{n^{1.1}}}{n!} \).
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Context:
A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.
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Context:
A positive integer \( b \) is called good if there exist positive integers \( 1 = {a}_{1},{a}_{2},\ldots ,{a}_{2023} = b \) such that \( \left| {{a}_{i + 1} - {a}_{i}}\right|  = {2}^{i} \) . Find
 the number of the good integers.
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Context:
There are 6 movie tickets numbered 1, 2, 3, 4, 5, and 6. They need to be distributed among four people: A, B, C, and D. Each person should receive at least one ticket and a maximum of two tickets. If a person gets two tickets, they must be consecutive. Calculate the number of different ways the tickets can be distributed. Express your answer as a whole number.
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Context:
Find the value of \( \eta = \int_0^1 f(x) \, \mathrm{d}x \) given the integral equation \( f(x) = x + \int_0^1 t(x+t)f(t) \, \mathrm{d}t \).
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Context:
Let \(S\) be the set of ordered triples \((x,y,z)\) of real numbers for which

\(\log_{10}(x+y) = z\) and \(\log_{10}(x^{2}+y^{2}) = z+1.\)
There are real numbers \(a\) and \(b\) such that for all ordered triples \((x,y,z)\) in \(S\) we have \(x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.\) The original answer is in \(\frac{m}{n}\) format, please find the value of \(m+n\).
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Context:
When Minhyuk asked Kihyun's math test score, Kihyun answered it would be 90 if he rounded it off to 10 digits. What is the biggest number possible with Kihyun's math score?
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Context:
The greatest common divisor of natural numbers \( m \) and \( n \) is 1. What is the greatest possible value of \(\text{GCD}(m + 2000n, n + 2000m) ?\)
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Context:
For every integers $ a,b,c$ whose greatest common divisor is $n$, if
\[ \begin{array}{l} {x + 2y + 3z = a} \\
{2x + y - 2z = b} \\
{3x + y + 5z = c} \end{array}
\]
has a solution in integers, what is the smallest possible value of positive number $ n$?
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Context:
三角形的三边长分别是整数l,m,n，且l>m>n. 已知$\{ \frac { 3 ^ { l } } { 1 0 ^ { 4 } } \} = \{ \frac { 3 ^ { m } } { 1 0 ^ { 4 } } \} = \{ \frac { 3 ^ { n } } { 1 0 ^ { 4 } } \}$，其中{x}=x-[x]，而［[x]表示不超过x的最大整数.求这种三角形周长的最小值.
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Context:
Calculate the variance of a random variable X that follows the hyperbolic secant distribution, given that the expected value E[X] = 0.
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Context:
Sonya the frog chooses a point uniformly at random lying within the square \([0, 6] \times [0, 6]\) in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from \([0, 1]\) and a direction uniformly at random from \{north, south, east, west\}. All of her choices are independent. Find the probability that she lands outside the square. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Let \( a \) and \( b \) be real numbers, and consider the function \( f(x) = x^3 + ax^2 + bx \). Suppose there exist three real numbers \( x_1, x_2, x_3 \) such that \( x_1 + 1 \leq x_2 \leq x_3 - 1 \) and \( f(x_1) = f(x_2) = f(x_3) \). Find the minimum value of \( |a| + 2|b| \).
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Context:
The shopkeeper increased the price of a product by 25%, making it harder for customers to purchase as much as they needed. However, a particular customer, with a budget of $150, managed to buy only 64% of the quantity they originally intended to purchase. Assuming there was also a maximum limit of 40 units of the product available, what is the net difference in the expenditure on that product?
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Context:
Determine the set of all cluster points of subsequences of the sequence \( \frac{n}{e} - \left[\frac{n}{e}\right] \), where \( e \) is Euler's number.
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Context:
16. Check Digit (8th grade. 3 points). The Simple-Dairy Telephone Company “Simple-Telecom” uses three-digit telephone numbers. The equipment is old, so during connection, errors in individual digits of the transmitted subscriber number are possible - each digit, independently of the others, has a probability of $p=0.02$ of being replaced by some other random digit. To reduce the probability of incorrect connection, the company uses the following rule: the first digit of the number is always equal to the remainder of the sum of the two other digits divided by 10. For example, the numbers 000 and 156 are possible, while the number 234 is not. If the check digit is incorrect during connection, an error message is displayed. Find the probability that, despite the measures taken, an incorrect connection will occur.
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Context:
Evaluate the improper integral: $$\int_{-\infty}^{\infty} \frac{\sin^{4}(x)}{x^2}dx.$$
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Context:
The sum of the house numbers on one side of a street from corner to corner is 117. What is the house number of the fifth house from the beginning of this section?
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Context:
A fighting game club has $2024$ members. One day, a game of Smash is played between some pairs of members so that every member has played against exactly $3$ other members. Each match has a winner and a loser. A member will be [i]happy[\/i] if they won in at least $2$ of the matches. What is the maximum number of happy members over all possible match-ups and all possible outcomes?
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Context:
Find the angle between the tangents drawn to the curve \(3x^2+4xy+5y^2-4=0\) at the points of intersection with the lines \(y=-\frac{3}{2}x\) and \(y=-\frac{2}{5}x\).
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Context:
The area of a rectangular grassy park is $4000 \mathrm{~m}^{2}$. Two concrete paths parallel to the sides intersect in the park. The area of one path is $400 \mathrm{~m}^{2}$, and the area of the other path is $250 \mathrm{~m}^{2}$. What percentage of the park's area is covered with grass?
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Context:
Given the joint density function \( f_{XY}(x, y) = kxy^2 \) for \( 0 \leq x, y \leq 1 \) and 0 otherwise, where \( k = 6 \), compute the probability \( \Pr(|X - Y| < 0.5) \).
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Context:
Evaluate the limit of the integral as \( n \to \infty \):
\[
\lim_{n\to\infty} n\int_{-\infty}^{\infty} e^{-2n\vert x\vert}\frac{x+3}{x^4+3x^2+1} \mathrm{d}x
\]
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Context:
A dragon has 40 piles of gold coins, with the number of coins in any two piles differing. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the minimum number of different piles of coins that could result?
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Context:
How many roots of the equation $f(z)=z^4+8z^3+3z^2+8z+3=0$ lie in the right half plane?
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Context:
Determine the limit of the function $\frac{(\lg n)!}{(\lg(2/5))^{\lg n}(2/5)^n}$ as $n$ approaches infinity.
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Context:
Find the limit of \( \frac{(x+3)^x}{\sum_{n=3}^{x+2}n^x} \) as \( x \) approaches infinity.
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Context:
Determine a lower bound for the cardinality of the set \( A \), where \( A = \{ R \subset \mathbb{N} \times \mathbb{N} : R \text{ is transitive} \} \).
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Context:
Given the homomorphism \(\varphi: \mathbb{Z}^2 \to S_{15}\) defined by:
\[\varphi(1,0)=(2,7,3)(3,11,5)(12,13)\]
\[\varphi(0,1)=(14,15)(1,4,6,8,9,10)\]
Calculate the number of elements in the group \(\mathbb{Z}^2/\!\ker\varphi\).
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Context:
Alex, a film critique with a passion for old Christmas movies, is planning a holiday movie marathon. He has a collection of 12 classic Christmas films, each with a soundtrack composed by different artists. However, 5 of these soundtracks were composed by his favorite composer, Harry Gregson-Williams.

Alex decides to watch 3 movies per day leading up to Christmas. On the first day, he randomly selects and watches 2 movies with soundtracks not composed by Harry Gregson-Williams and 1 movie with a soundtrack by him.

If Alex continues to watch the same pattern of movies (2 non-Gregson-Williams and 1 Gregson-Williams) each day, how many days will it take him to watch all 12 movies from his collection?
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Context:
Let $Y = \operatorname{Pois}(n)$. Using the Normal approximation, also known as the Central Limit Theorem (CLT), estimate the probability \( p\Big[|Y-n| \geq 2\sqrt{n}\Big] \).
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Context:
Bill is stocking the kitchenware section of the Walmart. He needs to stack 60 pots, but they come in different shapes and sizes. On each shelf, he can stack the following combinations of pots:

- 3 large pots vertically and two sets of vertically stacked large pots side-by-side
- 5 medium pots vertically and three sets of vertically stacked medium pots side-by-side
- 7 small pots vertically and four sets of vertically stacked small pots side-by-side

Bill has 25 large pots, 20 medium pots, and 15 small pots to stack. How many shelves does he need to stock all the pots?
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Context:
Consider a sequence of figures starting with an equilateral triangle $ABC$ with side length 1. The leading edge of the triangle, going clockwise around $A$, is labeled $\overline{AB}$ and is darkened. The second figure in the sequence is formed by attaching a square with side length 1 to the leading clockwise edge of the triangle. The third figure is formed by attaching a regular pentagon with side length 1 to the leading clockwise edge of the square. This process continues, with each subsequent figure formed by attaching a regular polygon with an increasing number of sides (hexagon, heptagon, etc.) to the leading clockwise edge of the previous polygon. By the eighth figure, a regular decagon is attached. After attaching the decagon, its leading clockwise edge forms an angle of less than $180^\circ$ with the side $\overline{AC}$ of the equilateral triangle. Express this angle in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, and find $m + n$. \( \text{[asy diagram omitted]} \)
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Context:
Determine the number of distinct $3 \times 3$ matrices with entries in \(\{0,1\}\) such that each row is non-zero, distinct, and the matrix has a real rank of 2 or 3, excluding matrices that are permutations of each other.
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Context:
Let \( S = \{1, 2, 3, 4\} \); a sequence \( a_{1}, a_{2}, \ldots, a_{n} \) of \( n \) terms has the following property: for any non-empty subset \( B \) of \( S \) (denoted by \( |B| \) as the number of elements in set \( B \)), there exists a sequence of \( |B| \) consecutive terms in the sequence that exactly forms the set \( B \). Find the minimum value of \( n \).
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Context:
Evaluate the surface integral \( \iint_S z \, dx \, dy \) over the complete surface \( S \) of a circular cylinder with radius \( a \) and axis along the \( OZ \) direction, where the ends of the cylinder are located at \( z = 0 \) and \( z = 2a \).
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Context:
设$a$为整数，若对任意的$x\in (0,+\infty )$，不等式$\frac{{{e}^{x}}+3}{x}\ge {{ea}}$恒成立，则$a$的最大值是__________．
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Context:
Find a quadratic polynomial that is orthogonal to both \( p(x) = 1 \) and \( q(x) = x \).
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Context:
Given a geometric sequence $\{a_n\}$ with a common ratio $q$, and the product of its first $n$ terms is $T_n$, where the first term $a_1 > 1$, and $a_{2014}a_{2015} - 1 > 0$, $\frac{a_{2014} - 1}{a_{2015} - 1} < 0$, find the largest natural number $n$ such that $T_n > 1$.
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Context:
Evaluate the limit: \[ \lim_{x \to 0} \left( 1 + \sin \left( \frac{3}{x} \right) \right)^x \]
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Context:
A rectangle has dimensions 12 by 15, and a circle centered at one of its vertices has a radius of 15. What is the area of the union of the regions enclosed by the rectangle and the circle? Express your answer in terms of \( \pi \).
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Context:
Given the vector field $\mathbf{F}: \mathbb{R}^2 \setminus\{(0,0)\} \to \mathbb{R}^2$ defined by $\mathbf{F}(x,y) = \frac{y}{x^2+y^2}\mathbf{i} - \frac{x}{x^2+y^2}\mathbf{j}$, consider the curves $\gamma, \alpha : [0,1] \to \mathbb{R}^2$ defined by $\gamma(t) = (8\cos(2\pi t), 17\sin(2\pi t))$ and $\alpha(t) = (26\cos(2\pi t), -10\sin(2\pi t))$. If $3\int_{\alpha} \mathbf{F \cdot dr} - 4\int_{\gamma} \mathbf{F \cdot dr} = 2m\pi$, find the value of $m$. \[ \text{(Hint: Consider the properties of the vector field and the curves.)} \]
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Context:
Patty ordered a pizza to be delivered. The bill was $\$$12.00. Patty pays $\$$15 and tells the delivery person to keep the change as a tip. What percent tip did she give?
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Context:
Find another value of $x$ that satisfies both equations $a=x^{x^{x^{.^{.^{.}}}}}$ and $x=a^{1/a}$, besides $x=3$.
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Context:
Given the parabola $y=x^{2}+2x-3$ intersects the line $l_{1}$: $y=-x+m$ at points $A$ and $C$, and the line $l_{2}$ is symmetric with respect to the axis of symmetry of the parabola. Line $l_{2}$ intersects the parabola at points $B$ and $D$, where points $A$ and $D$ are above the $x$-axis, and points $B$ and $C$ are below the $x$-axis. If $AC \cdot BD = 26$, then the value of $m$ is ______.
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Context:
On side $AC$ of triangle $ABC$, points $M$ and $N$ are marked such that $M$ lies on segment $AN$. It is known that $AB = AN$ and $BC = MC$. The circumcircles of triangles $ABM$ and $CBN$ intersect at points $B$ and $K$. What is the angle $\angle AKC$ if $\angle ABC = 68^\circ$?
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Context:
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$, the symmetric point $Q$ of the right focus $F(c, 0)$ with respect to the line $y = \dfrac{b}{c}x$ is on the ellipse. Find the eccentricity of the ellipse.
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Context:
The least number which, when increased by 7, is divisible by some numbers including 24, 36, and 54 is 857. What is the other number that the least number should be divisible by when increased by 7?
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Context:
The time on a cell phone is 3:52. Calculate the number of minutes that will pass before the phone next shows a time using each of the digits 2, 3, and 5 exactly once. Express your answer in minutes.
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Context:
Petra had natural numbers from 1 to 9 written down. She added two of these numbers, erased them, and wrote the resulting sum in place of the erased addends. She then had eight numbers left, which she was able to divide into two groups with the same product.

Determine the largest possible product of these groups.
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Context:
A ray passes through points $P(-3,1)$ and $Q(a,0)$ and reflects off the $x$-axis, then it is tangent to the circle $x^{2}+y^{2}=1$. Find the value of $a$.
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Context:
Does the equation

$$
x^{2}+y^{2}+z^{2}=2 x y z
$$

have a solution in non-negative integers?
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Context:
There are 2000 nodes, and every pair of nodes is connected by a wire. Varia and Petya take turns cutting these wires, with Varia starting first. Varia can only cut one wire at a time, while Petya can cut 2 or 3 wires at a time. The rule is that the player who cuts the last wire loses. Who will be the winner?
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Context:
设函数$f\left( x \right)=\frac{3}{2}{{x}^{2}}-2ax(a>0)$与$g\left( x \right)={{a2}}\ln x+b$有公共点，且在公共点处的切线方程相同，则实数$b$的最大值为__________．
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Context:
In the plane, there are $10$ lines in general position, meaning that no two are parallel and no three are concurrent. At each intersection of two lines, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?
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Context:
A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths $13, 19, 20, 25$ and $31$, although this is not necessarily their order around the pentagon. The area of the pentagon is
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Context:
Calculate the expectation of the product of the minimum values of two independent uniform random variables from the distribution $\mathrm{Unif}(0, 1)$. Specifically, find $E[\min\{U_{1}, U_{2}\}\times \min\{U_{1}, U_{3}\}]$ where $U_{1}, U_{2}, U_{3}$ are independent and identically distributed.
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Context:
Determine the number of lattice points \((v, w, x, y, z)\) that lie on or inside a 5-sphere centered at the origin with radius 3.
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Context:
Let $S$ be the set of points on the rays forming the sides of a $120^{\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. Count the number of distinct equilateral triangles that can be formed. Express your answer as a single integer.
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Context:
If the equation $1-k(x-2e)\cdot\ln x=0$ has two different solutions on $(1,+\infty)$, where $e$ is the base of the natural logarithm, then the range of the real number $k$ is ______.
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Context:
Determine the value of \(x\) for which the matrices \(A = \begin{pmatrix} 3&0&-1 \\ -2&1&1 \\ 2&0&0 \end{pmatrix}\) and \(B = \begin{pmatrix} 1&x&0 \\ 0&1&0 \\ -1&x&2 \end{pmatrix}\) are similar.
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Context:
There are $15$ (not necessarily distinct) integers chosen uniformly at random from the range from $0$ to $999$, inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $100m + n$.
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Context:
已知椭圆 $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ 的左右焦点分别为 $F_{1}$ 与 $F_{2}$, 点 $P$ 在直线 $l: x-\sqrt{3} y+8+2 \sqrt{3}=0$ 上. 当 $\angle F_{1} P F_{2}$ 取最大值时, 比 $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}$ 的值为\sqrt{m}-n的形式, 请给出m - n的值.
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Context:
For positive integers \( n \), let \( L(n) \) be the largest factor of \( n \) other than \( n \) itself. Determine the number of ordered pairs of composite positive integers \( (m, n) \) for which \( L(m) L(n) = 80 \).
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Context:
Katie writes a different positive integer on the top face of each of the fourteen cubes in the pyramid shown. The sum of the nine integers written on the cubes in the bottom layer is 50. The integer written on each of the cubes in the middle and top layers of the pyramid is equal to the sum of the integers on the four cubes underneath it. What is the greatest possible integer that she can write on the top cube?
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Context:
## Task B-1.6.

All letters in the equality $(a+b)(c+d)(e+f)(g+h)=5005$ should be replaced with different numbers from 1 to 8 so that the equality is correct. In how many ways can this be done?
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Context:
Define a polynomial $f_{n}(x)$, where $n \geqslant 0$ is $f_{0}(x)=1$, $n \geqslant 1$ is $f_{n}(0)=0$, and $n \geqslant 0$ is
$$
\frac{d}{d x}\left(f_{n+1}(x)\right)=(n+1) f_{n}(x+1)
$$
. Clearly decompose $f_{100}(1)$ into powers of different prime numbers, $a^{b}$, and find the value of $a+b$.
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Context:
Each Kinder Surprise contains exactly 3 different Smurfs, and there are a total of 11 types of Smurfs. The box contains a sufficient number of Kinder Surprises, where no two of them have identical sets of three Smurfs. What is the minimum number of Kinder Surprises that need to be purchased so that after opening them, there is at least one of each of the 11 types of Smurfs?
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Context:
Consider a "Modulo $m$ graph paper," with $m = 17$, forming a grid of $17^2$ points. Each point $(x, y)$ represents integer pairs where $0 \leq x, y < 17$. We graph the congruence $$5x \equiv 3y + 2 \pmod{17}.$$ The graph has single $x$-intercept $(x_0, 0)$ and single $y$-intercept $(0, y_0)$, where $0 \leq x_0, y_0 < 17$. Determine $x_0 + y_0$.
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Context:
The function \( f(x) \) satisfies the equation \( f(x+y) = f(x \cdot y) \) for any real numbers \( x \) and \( y \), and given \( f\left(-\frac{1}{2}\right) = -\frac{1}{2} \), find \( f(2000) \).
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Context:
If \(60^a=360^b=5\), then what is the value of \(12^{\frac{1-a-b}{2(1-b)}}\)?
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Context:
In the first week, New York recorded 5,000 new coronavirus cases. In the second week, there was a 30% increase in new cases compared to the first week. In the third week, there was a 20% decrease in new cases compared to the second week. In the fourth week, the daily number of vaccinations increased by 10% compared to the third week, which resulted in a 25% decrease in new cases. During this time, the population of New York grew by 0.5%. Taking into account the population growth, what is the total number of recorded new coronavirus cases in the state after the four weeks?
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Context:
A swing is initially stationary with its footboard one foot off the ground. When pushed forward by 10 feet (two steps of 5 feet each), the footboard becomes level with a person who is 5 feet tall. If the rope of the swing is pulled very straight at this point, what is the length of the rope? Express your answer in feet, rounded to one decimal place.
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Context:
In a \(7 \times 7\) table, some cells are black while the remaining ones are white. In each white cell, the total number of black cells located with it in the same row or column is written; nothing is written in the black cells. What is the maximum possible sum of the numbers in the entire table?
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Context:
For real  $\theta_i$ ,  $i = 1, 2, \dots, 2011$ , where  $\theta_1 = \theta_{2012}$ , find the maximum value of the expression

\[
\sum_{i=1}^{2011} \sin^{2012} \theta_i \cos^{2012} \theta_{i+1}.
\]
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Context:
Given the sequence $\{a_n\}$ where $a_n = (\sqrt{2}+1)^n - (\sqrt{2}-1)^n \quad (n \in \mathbf{N})$, and let $[x]$ denote the greatest integer not exceeding the real number $x$, what is the units digit of $\left[ a_{2017} \right]$?
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Context:
Suppose the function $\psi$ satisfies $\psi(1) = \sqrt{2 + \sqrt{2 + \sqrt{2}}}$ and
\[\psi(3x) + 3 \psi(x) = \psi^3(x)\]for all real $x.$  Determine $\prod_{n = 1}^{100} \psi(3^n).$The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
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Context:
Given points E and D are on sides AB and BC of triangle ABC, where AE:EB=1:3 and CD:DB=1:2, find the value of EF/FC + AF/FD.
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Context:
Given a subgroup \( H \) of \( \mathbb{Z}^2 \) generated by the vectors \((1, 6), (3, 5), (7, 11)\), how many distinct cosets are there in the quotient group \( \mathbb{Z}^2/H \)?
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Context:
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6 and with their centers 13 units apart. If a plane \(\alpha\) is tangent to both spheres and intersects the cylindrical surface forming an ellipse, what is the length of the major axis of this ellipse?
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Context:
求所有的正整数 n，使得存在正整数 a,b，满足 a^2+b+3 不能被任何质数的立方整除，且n=\df{ab+3b+8}{a^2+b+3}.
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Context:
If you draw a card with a value of five or less from a standard deck of cards, I will pay you \$49. If not, you pay me \$25. (Aces are considered the highest card in the deck.)

Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.
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Context:
Mary is baking a cake. The recipe calls for 5 cups of flour and a certain amount of sugar. She already put in 10 cups of flour and 2 cups of sugar. She needs to add 12 more cups of sugar. How many cups of sugar does the recipe call for?
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Context:
Call a positive integer 'mild' if its base-3 representation never contains the digit 2. How many values of $n(1 \leq n \leq 1000)$ have the property that $n$ and $n^{2}$ are both mild?
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Context:
The coefficients of the fifth and sixth terms in the binomial expansion of $(x- \dfrac {2}{ \sqrt {x}})^{n}$ are the largest. Find the sum of the coefficients of all the terms.
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Context:
The base side length of the regular triangular prism \( ABC-A_1B_1C_1 \) is \( a \), and the side edge length is \( \sqrt{2}a \). Find the angle between \( AC_1 \) and the lateral face \( ABB_1A_1 \).
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Context:
Find the formula for the summation \( S = \sum_{i=1}^{n} \frac{n}{i} \).
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Context:
The base of a quadrilateral pyramid is a square, and all the lateral faces are right triangles with the vertices of the right angles lying on the base of the pyramid. Find the volume of the pyramid if its height is 1 and one of the dihedral angles at the apex is $120^{\circ}$.
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Context:
Given the functions $f(x)=x+ \frac {4}{x}$ and $g(x)=2^{x}+a$, if there exists $x_{1} \in \left[ \frac {1}{2},3\right]$ and for all $x_{2} \in \left[2,3\right]$, such that $f(x_{1}) \geqslant g(x_{2})$, then the range of the real number $a$ is \_\_\_\_\_\_.
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Context:
Heath spent his weekend helping at his uncle's farm planting carrots and potatoes. He planted 400 rows in total; 250 rows were carrots and 150 rows were potatoes. The number of plants in each row and the time taken to plant them varied.

For the carrot rows, the first 100 rows had 275 plants each and took Heath 10 hours to plant, while the remaining 150 rows had 325 plants each and took 20 hours to complete. Meanwhile, Heath's cousin, Jake, planted the 150 rows of potatoes. In the first 50 rows, Jake planted 300 plants per row, taking 12 hours to complete. The remaining 100 rows had 400 plants per row and took 18 hours to finish.

Calculate the weighted average of the number of carrot and potato plants planted per hour by both Heath and Jake during the entire weekend.
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Context:
Given an ordered triplet \((x, y, z)\) of real numbers satisfying \(x < 1\), \(y < 2\), \(z < 3\), and \(x + \frac{y}{2} + \frac{z}{3} > 0\), determine the values of \(a\), \(b\), and \(c\) that maximize the expression \((1 - x)(2 - y)(3 - z)(x + \frac{y}{2} + \frac{z}{3})\). Find the sum \(a + b + c\).
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Context:
Joe is given a permutation $p = (a_1, a_2, a_3, a_4, a_{5})$ of $(1, 2, 3, 4, 5)$. A swap is an ordered pair $(i, j)$ with $1 \le i < j \le 5$, and this allows Joe to swap the positions $i$ and $j$ in the permutation. For example, if Joe starts with the permutation $(1, 2, 3, 4, 5)$, and uses the swaps $(1, 2)$ and $(1, 3)$, the permutation becomes
\[(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5). \]Out of all $\tbinom{5}{2} = 10$ swaps, Joe chooses $4$ of them to be in a set of swaps $S$. Joe notices that from $p$ he could reach any permutation of $(1, 2, 3, 4, 5)$ using only the swaps in $S$. How many different sets are possible?
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Context:
Given six thin wooden sticks, the two longer ones are $\sqrt{3}a$ and $\sqrt{2}a$, while the remaining four are of length $a$. If they are used to form a triangular prism, find the cosine of the angle between the lines containing the two longer edges.
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Context:
In the first pile, there were 7 nuts, and in the rest, there were 12 nuts each. How many piles were there in total?
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Context:
4. On the table, there are 100 weights of different masses. A weight is called successful if its mass is equal to the sum of the masses of some two other weights on the table. What is the smallest number of successful weights that would allow us to assert with certainty that the masses of some two weights differ by at least a factor of three?
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Context:
In a variant of SHORT BINGO, a $5\times5$ card is modified such that the middle square is marked as FREE, and the remaining 24 squares are populated with distinct numbers. The modification includes:
- First column contains 5 distinct numbers from the set $1-15$.
- Second column contains 5 distinct numbers from the set $16-30$.
- Third column (excluding the FREE middle square) contains 4 distinct numbers from $31-45$.
- Fourth column contains 5 distinct numbers from the set $46-60$.
- Fifth column contains 5 distinct numbers from the set $61-75$.

How many distinct possibilities are there for the values in the first column of this modified SHORT BINGO card?
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Context:
Given the differential equation \( \frac{dx}{dt} = -x + x^7 \) with the initial condition \( x(0) = \lambda \), where \( x = x(t, \lambda) \), find \( \frac{ \partial x(t, \lambda)}{\partial \lambda} \bigg| _{\lambda = 0} \).
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Context:
For every integers  $ a,b,c$  whose greatest common divisor is  $n$ , if
\[ \begin{array}{l} {x \plus{} 2y \plus{} 3z \equal{} a}
{2x \plus{} y \minus{} 2z \equal{} b}
{3x \plus{} y \plus{} 5z \equal{} c} \end{array}
\]
has a solution in integers, what is the smallest possible value of positive number  $ n$ ?
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Context:
In triangle \( \triangle ABC \), \( AB = AC \), \( AD \) and \( BE \) are the angle bisectors of \( \angle A \) and \( \angle B \) respectively, and \( BE = 2 AD \). What is the measure of \( \angle BAC \)?
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Context:
Two players play alternately on a $ 5 \times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \times 3$ squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player?
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Context:
Calculate 469111111 x 99999999 = ?
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Context:
Jon's laundry machine can do 8 pounds of laundry at a time. The weights of different types of clothes are as follows:

- 4 shirts weigh 1 pound
- 2 pairs of pants weigh 1 pound
- 6 pairs of socks weigh 1 pound
- 1 jacket weighs 2 pounds

Jon needs to wash 20 shirts, 20 pairs of pants, 18 pairs of socks, and 6 jackets. If the machine works at 100% efficiency with shirts and pants but loses 10% efficiency for each jacket in the load, how many loads of laundry does Jon have to do to complete the washing?
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Context:
Find the smallest possible value of the moment generating function $m_X(2)$ for a random variable $X$ with an expected value of $0$ and a moment generating function $m_X(1) = z \geq 1$. Provide your answer in terms of $z$.
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Context:
5. Given 9 points in space, where no four points are coplanar, a line segment is drawn between every pair of points. These line segments can be colored blue or red, or left uncolored. Find the minimum value of $n$ such that if any $n$ of these line segments are arbitrarily colored red or blue, the set of these $n$ line segments will necessarily contain a triangle with all sides of the same color.
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Context:
Given conditions $p$: $|x+1| > 2$, and $q$: $x > a$, where $\neg p$ is a sufficient but not necessary condition for $\neg q$, determine the range of values for $a$.
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Context:
9. A positive integer that can be written in the form $20a + 8b + 27c$ $(a, b, c \in \mathbf{N})$ is called a "good number." The number of good numbers in the set $\{1, 2, \cdots, 200\}$ is $\qquad$.
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Context:
2. We will call a set of distinct natural numbers from 1 to 9 good if the sum of all numbers in it is even. How many good sets are there?
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Context:
Consider the following sequence of sets of natural numbers. The first set \( I_{0} \) consists of two ones, 1,1. Then, between these numbers, we insert their sum \( 1+1=2 \); we obtain the set \( I_{1}: 1,2,1 \). Next, between each pair of numbers in \( I_{1} \) we insert their sum; we obtain the set \( I_{2}: 1,3,2,3,1 \). Proceeding in the same way with the set \( I_{2} \), we obtain the set \( I_{3}: 1,4,3,5,2,5,3,4,1 \), and so on. How many times will the number 1973 appear in the set \( I_{1000000} \)?
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Context:
Given the joint density function \( f(y_1,y_2) = \frac{1}{8}y_1 e^{-(y_1 + y_2)/2} \) for \( y_1 > 0 \) and \( y_2 > 0 \), and \( f(y_1,y_2) = 0 \) elsewhere, find the expected value of \( D = 50 + 2Y_1 + 4Y_2 \).
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Context:
Determine the radius of convergence of the function \( f(z) = \frac{\sin(z)}{z} \) for any point \( z_0 \in \mathbb{C} \setminus \{0\} \).
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Context:
Determine the kernel of the homomorphism \( \phi: R/I \oplus R/J \rightarrow R/(I+J) \) defined by \( (a+I, b+J) \mapsto (a-b) + I + J \), where \( I \) and \( J \) are ideals in a Noetherian ring \( R \).
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Context:
Determine the highest power of two that divides the expression \(32! + 33! + 34! + \dots + 90!\).
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Context:
Calculate the probability $\mathbb{P}\{X_{(1)}+X_{(2)} > X_{(3)}\}$, where $X_1, X_2, X_3$ are independent uniform $(0, 1)$ random variables, and $(X_{(1)}, X_{(2)}, X_{(3)})$ denotes their corresponding order statistics.
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Context:
A permutation $a_1, a_2, \ldots, a_{13}$ of the numbers from $1$ to $13$ is given such that $a_i > 5$ for $i=1,2,3,4,5$. Determine the maximum possible value of
\[ a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}. \]
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Context:
Evaluate the limit: $$\lim_{n \to \infty} \int_{0}^{1} \frac{n^4u^2e^{-nu}}{1 + n^2u}du$$
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Context:
How do you solve the following limit?

$$\lim_{n\to \infty }\left[\lim_{x\to \:0}\left(1+\sin^2x+\sin^22x+\ldots+\sin^2nx\right)^{1/(n^3x^2)}\right]$$
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Context:
Find the approximate interval in which the root of the equation $\ln \left( x+1
ight)-\frac{2}{x}=0$ ($x > 0$) exists. Express your answer as an interval.
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Context:
The area of trapezoid \(ABCD\) is 30. Point \(P\) is the midpoint of the lateral side \(AB\). Point \(R\) on side \(CD\) is chosen such that \(2CD = 3RD\). Lines \(AR\) and \(PD\) intersect at point \(Q\). Find the area of triangle \(APQ\) if \(AD = 2BC\).
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Context:
Determine all values of the parameter \(a\) such that the integral \(\int_0^{+\infty} \frac{dx}{1+x^a \sin^2x}\) converges.
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Context:
Let $\triangle ABC$ be a right triangle with $B$ as the right angle. A circle with diameter $AC$ intersects side $BC$ at point $D$. If $AB = 18$ and $AC = 30$, find the length of $BD$.
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Context:
Kelvin the Frog is hopping on a number line (extending to infinity in both directions). Kelvin starts at $0$. Every minute, he has a $\frac{1}{3}$ chance of moving $1$ unit left, a $\frac{1}{3}$ chance of moving $1$ unit right, and $\frac{1}{3}$ chance of getting eaten. Find the expected number of times Kelvin returns to $0$ (not including the start) before he gets eaten.
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Context:
Troy's and Emily's homes are 75 meters and 98 meters away from school, respectively. However, there is a park, a river, a construction site, and a closed road along their routes, which force them to take different detours each day.

On Monday, Troy takes a detour adding 15 meters to his route due to the park, while Emily takes a detour adding 10 meters because of the river. On Tuesday, Troy takes another detour adding 20 meters due to the construction site, while Emily takes one adding 25 meters because of the closed road. On Wednesday, both take a detour around the park, adding 10 meters each. On Thursday, Troy has no detours while Emily takes one adding 15 meters to avoid the construction site. On Friday, Troy takes a detour adding 5 meters due to the closed road, and Emily takes one adding 10 meters because of the river.

Additionally, on Thursday, both Troy and Emily encounter an unexpected road closure that adds another 10 meters to each of their routes.

Given the varying detours and distances added on each day due to the obstacles and sudden changes, find out how much farther Emily walks to school and back in five days.
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Context:
Mr. Harmon is a teacher known for his calm and patient nature. Every day, he takes a peaceful 25-minute walk in the morning to start his day. His partner, Alex, on the other hand, loves to run and completes the same distance in just 15 minutes. Over the course of a week, Mr. Harmon walks every morning and Alex runs every other day. How many minutes do they spend exercising together in one week?
Please reason step by step, and put your final answer within \boxed{}.