Context:
How many positive integers $N$ less than $10^{1000}$ are such that $N$ has $x$ digits when written in base ten and $\frac{1}{N}$ has $x$ digits after the decimal point when written in base ten? For example, 20 has two digits and $\frac{1}{20}$= 0.05 has two digits after the decimal point, so $20$ is a valid N.
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Context:
For which $a>0$ does the series $$\sum_{n=1}^\infty \left(2-2 \cos\frac{1}{n} -\frac{1}{n}\cdot \sin\left( \sin\frac{1}{n} \right) \right)^a $$ converge?
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Context:
In triangle $\triangle ABC$, the sides opposite to angles A, B, and C are of lengths $a$, $b$, and $c$ respectively. Given the vectors $\overrightarrow {m} = (a+b, \sin C)$ and $\overrightarrow {n} = (\sqrt {3} a+c, \sin B-\sin A)$. If $\overrightarrow {m}$ is parallel to $\overrightarrow {n}$, find the magnitude of angle B.
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Context:
Father is aged 3 times more than his son Ronit. After some years, he would be two and a half times Ronit's age. After further 8 years, he would be 2 times Ronit's age. How many years later will the father be two and a half times Ronit's age?
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Context:
Two players, A and B, take turns rolling a standard six-sided die. Player A rolls first. If a player rolls a 1 or 2, they win the game. If a player rolls a 3, 4, or 5, the other player gets a turn. If two consecutive 6's are rolled, the game ends in a tie. What is the probability that player A wins the game?
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Context:
Evaluate the limit: \[ \lim_{x\to0^+} \frac{(1-x)^{x}-\cos x e^{-\frac{x^2}{2}}}{\arctan(x^2)\sin x +x^3 \ln x} \]
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Context:
Determine the cardinality of the set \( \{ f \in \mathbb{N}^{\mathbb{N}} \mid f \le h \} \), where \( h: \mathbb{N} \to \mathbb{N} \) is defined by \( h(n) = n + 1 \) and the partial order relation \( f \le g \) is defined as \( \forall n \in \mathbb{N}, f(n) \le g(n) \).
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Context:
Determine the number of rotation matrices that can rotate line A to line B in 3D space, given that both lines are of the same length, centered at the origin, and are not identical.
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Context:
Determine the Galois group of the polynomial \( g(x) = x^4 + x^3 + x^2 + x + 1 \) over the finite field \( \mathbb{F}_2 \).
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Context:
2. A circle with center $O$ is inscribed in triangle $A B C$. Point $L$ lies on the extension of side $A B$ beyond vertex $A$. A tangent line drawn from $L$ to the circle intersects side $A C$ at point $K$. Find $\angle K O L$, if $\angle B A C=50^{\circ}$.
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Context:
Two sectors of a circle of radius $15$ overlap in the same manner as the original problem, with $P$ and $R$ as the centers of the respective circles. The angle at the centers for both sectors is now $45^\circ$. Determine the area of the shaded region.
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Context:
Consider the ellipse given by the equation x^2/12 + y^2/3 = 1. Let F1 denote the left focus of the ellipse, and let P be a point on the ellipse. If the midpoint M of the line segment PF1 lies on the y-axis, then find the y-coordinate of point P. Express your answer as a numerical value, including the ± symbol to indicate both positive and negative values.
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Context:
All the faces of a cube are painted with red color. It is then cut into some equal small cubes. There are 8 small cubes whose three faces are colored. Into how many equal small cubes is the large cube cut?
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Context:
Consider a series of random variables defined by $a_1=1$, and for $n \geq 2$, $a_n$ is either $a_{n-1}$ or $\frac{1}{2} a_{n-1}$, each with equal probability. What is the probability that the series $\sum_{i=1}^{\infty} a_i$ converges?
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Context:
What is the maximum number of points that can be placed on a segment of length 1 such that on any subsegment of length \( d \) within this segment, there are no more than \( 1 + 1000d^{2} \) points?
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Context:
Calculate the value of \( f'(\sqrt{\pi}) + g'(\sqrt{\pi}) \) for the functions \( f(x) = \left(\int_{0}^{x} e^{-t^2}dt\right)^2 \) and \( g(x) = \int_{0}^{1} \frac{e^{-x^2(1+t^2)}}{1+t^2} dt \).
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Context:
Determine the number of coordinate patches required to cover the boundary of the unit ball in \(\mathbb{R}^n\), which is the sphere \(S^{n-1}\).
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Context:
Given that $f(x)$ is a differentiable function satisfying the equation $f(x) = \left(1 + \frac{x^3}{3}\right) + \int_0^x e^{-t} f(x-t) \, dt$, find the value of $\int_0^1 f(x) \, dx$. \(\text{Express your answer as a single number.}\)
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Context:
Solve the equation \((\cos x + 2 \cos 6x)^{2} = 9 + \sin^{2} 3x\).
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Context:
Given 4 distinct positive integers, how many ways are there to sum them, considering the order of addition?
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Context:
给定 $n$ 个不同实数，其所有全排列组成的集合为 $A_{n}$ ，对于 $\left(a_{1}, a_{2}, \cdots, a_{n}\right) \in A_{n}$ ，若恰有两个不同的整数 $i, j \in\{1,2, \cdots, n-1\}$ 使得 $a_{i}>a_{i+1}, a_{j}>a_{j+1}$ 成立，则称该排列为 "好排列"。求 $A_{n}$ 中 "好排列"的个数。当 $n = 5$ 时，请给出 "好排列" 的个数。
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Context:
Solve the equation: $x^x = 10$.
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Context:
30. On the Magic Island, the divine monkeys feed on immortal peaches. Every noon, each divine monkey eats one immortal peach; by evening, both the monkeys and the peaches on the island can clone themselves. If there are 2000 divine monkeys and 50000 immortal peaches on the first morning, then by the afternoon of the $\qquad$th day, all the immortal peaches will be eaten up.
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Context:
Three cones are placed on a table, standing on their bases and touching each other. The radii of their bases are \(2r\), \(3r\), and \(10r\). A truncated cone (frustum) is placed on the table with its smaller base downward, sharing a slant height with each of the other cones. Find \(r\), if the radius of the smaller base of the truncated cone is 15.
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Context:
A company sells a brand of cars in two locations, A and B, with profits (in terms of ten thousand yuan) given by $L_1 = -x^2 + 21x$ and $L_2 = 2x$, respectively, where $x$ is the number of cars sold. If the company sells a total of 15 cars between the two locations, what is the maximum profit they can achieve?
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Context:
When a certain number is divided by 179, the quotient is 89 and the remainder is 37. What is the number being divided?
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Context:
If x is less than y by some percentage, then y exceeds x by 25%. What is the percentage by which x is less than y?
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Context:
A cylinder with a volume of 9 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 63. What is the volume of the original cone?
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Context:
Evaluate the limit: \[ L = \lim_{x \rightarrow \infty} \left(\sqrt[11]{\prod_{n=1}^{11}\left(2x+\frac{n(2n+1)}{2}\right)}-2x\right) \]
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Context:
Given sets $A=\{x|x^2+x-6=0\}$ and $B=\{x|x^2+x+a=0\}$, and $B \subseteq A$, find the range of the real number $a$.
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\boxed{\{-6\} \cup \left(\frac{1}{4}, +\infty\right)}


Context:
In triangle ABC, points D, E, and F are located on sides BC, AC, and AB, respectively. Given that \( \frac{AE}{EB} = \frac{3}{2} \) and \( \frac{AF}{FC} = \frac{3}{1} \), and the intersection of lines AD and EF is point O, determine the ratio \( \frac{EO}{OF} \).
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Context:
【Question 15】
Zhao, Qian, Sun, and Li, four children, all like playing chess. Among them, someone likes playing Chinese chess, someone likes playing Go, and someone likes both types of chess. Therefore, the different ways these four people can like these two types of chess are $\qquad$ kinds.
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Context:
There are 15 students, with 3 students each from 5 different high schools. Five admission officers, each from a different college, will each select 3 students to attend their college. In how many ways can the officers select the students such that no officer selects all 3 students from the same high school?
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Context:
$已知O为坐标原点，点E(\frac{3}{4},0)，过动点W作直线x=-\frac{1}{4}的垂线，垂足为F，\overrightarrow{OW}\cdot \overrightarrow{EF}=0，记W的轨迹为曲线C。$
$若A_1，B_1，A_2，B_2均在C上，直线A_1B_1，A_2B_2的交点为P(\frac{1}{4},0)，A_1B_1\perp A_2B_2，求四边形A_1A_2B_1B_2面积的最小值.$
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Context:
A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at a point $(x,y)$, she takes a step based on the following rules:
(a) if $x+y$ is even she moves to either $(x+1,y)$ or $(x+1,y+1)$;
(b) if $x+y$ is odd she moves to either $(x,y+1)$ or $(x+1,y+1)$.
How many distinct paths can she take to go from $(0,0)$ to $(8,8)$ given that she took exactly three steps to the right $((x,y)$ to $(x+1,y))$?
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Context:
Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,\ldots,\dfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
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Context:
James has some barrels that he uses to store water. They store 3 gallons more than twice as much as a large cask. He has a certain number of barrels and a cask stores 20 gallons. He can store 172 gallons of water. How many barrels does he have?
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Context:
Calculate the number of ways to divide 13 people into 5 indistinguishable groups with sizes 2, 2, 3, 3, and 3.
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Context:
Let $M= \{ 1, 2, \cdots, 19 \}$ and $A = \{ a_{1}, a_{2}, \cdots, a_{k}\}\subseteq M$. Find the least $k$ so that for any $b \in M$, there exist $a_{i}, a_{j}\in A$, satisfying $b=a_{i}$ or $b=a_{i}\pm a_{i}$ ($a_{i}$ and $a_{j}$ do not have to be different) .
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Context:
Solve the inequality in integers:

$$
\frac{1}{\sqrt{x-2y+z+1}} + \frac{2}{\sqrt{2x-y+3z-1}} + \frac{3}{\sqrt{3y-3x-4z+3}} > x^2 - 4x + 3
$$
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Context:
Given the equation \( \int_0^1 (1+\cos^8 x)(ax^2+bx+c) \, dx = \int_0^2 (1+\cos^8 x)(ax^2+bx+c) \, dx \), determine the interval in which the root of the quadratic equation \( ax^2 + bx + c = 0 \) lies, where \( a, b, \) and \( c \) are non-zero numbers.
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Context:
Evaluate the limit: \[ \lim\limits_{n\to \infty}\lim\limits_{x\searrow 0}f^{(n)}(x) \] where the function \( f:[0,\infty)\to \mathbb{R} \) is given by \( f(x)=\exp(\sqrt{x})+\exp(-\sqrt{x}) \). Assume that \( f \) and its derivatives satisfy the differential equation \( 2f'(x)+4xf''(x)=f(x) \).
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Context:
Find whole numbers $\heartsuit$ and $\clubsuit$ such that $\heartsuit \cdot \clubsuit = 48$ and $\heartsuit$ is even, then determine the largest possible value of $\heartsuit + \clubsuit$.
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Context:
For a standard Brownian motion $(B_t)_{t\geq0}$, let $T$ be the first time the process hits $1$ or $-1$. Compute the expected value \( \mathbb{E}\left(\int_{0}^{T} |B_{t}|^{2} \, dt\right) \).
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Context:
Using Qin Jiushao's algorithm, calculate the polynomial expression for the given input values: n = 5, v = 1, and x = 2. Express your answer as a single numerical value.
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Context:
A caretaker is giving candy to his two babies. Every minute, he gives a candy to one of his two babies at random. The five possible moods for the babies to be in, from saddest to happiest, are "upset," "sad," "okay," "happy," and "delighted." A baby gets happier by one mood when they get a candy and gets sadder by one mood when the other baby gets one. Both babies start at the "okay" state, and a baby will start crying if they don't get a candy when they're already "upset".
    The probability that 10 minutes pass without either baby crying can be expressed as $\frac{p}{q}$. Compute $p+q$.

[i]2022 CCA Math Bonanza Team Round #7[/i]
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Context:
Evaluate the limit: $$\lim\limits_{x \to -\infty} \left(\frac{\left((x+3)^2+x\cos(x)\right)(x)!}{(x-1)!+2^x}\right)$$
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Context:
Define the sequence  $ (a_p)_{p\ge0}$  as follows:  $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$ .
Find  $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$ .
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Context:
\[
\left(\frac{4-2x+x^{2}}{4-2x}+\frac{6x^{2}+8+12x}{4-x^{2}}-\frac{x^{2}+2x+4}{2x+4}\right)^{-1/3} \cdot (x+2)
\]
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Context:
In a cube labeled with vertices \(A, B, C, D, E, F, G, H\), where \(ABCD\) is the top face and \(H, G, F, E\) are directly below \(A, B, C, D\) respectively, a real number is assigned to each vertex. At each vertex, the average of the numbers at the three adjacent vertices is calculated. The averages at vertices \(A, B, C, D, E, F, G, H\) are 1, 2, 3, 4, 5, 6, 7, and 8 respectively. Determine the number assigned to vertex \(F\).
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Context:
Inside a right angle with vertex \(O\), there is a triangle \(OAB\) with a right angle at \(A\). The height of the triangle \(OAB\), dropped to the hypotenuse, is extended past point \(A\) to intersect with the side of the angle at point \(M\). The distances from points \(M\) and \(B\) to the other side of the angle are \(2\) and \(1\) respectively. Find \(OA\).
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Context:
Lotta is a consulting mathematician with 100 clients ranked 1 through 100 in order of importance. Each day, she visits the most important client who feels mistreated. A client feels mistreated if Lotta has never visited them, or if she has visited someone less important since the last time she visited them. On the first day, she visits client 1; on the second day, she visits client 2; on the third day, she goes back to client 1, and so on. Determine the number of days it will take for Lotta to retire when none of her clients feel mistreated.
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Context:
Determine if there exists an integer $n > 3$ such that the average digit of $2^n$ is greater than or equal to twice the average digit of $5^n$. If such an $n$ exists, provide an example.
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Context:
设 $n(n \geqslant 2)$ 为给定的整数。求最小的实数 $\lambda$ ，使得对于任意实数 $x_{1}, x_{2}, \cdots, x_{n} \in$ $[0,1]$ ，存在 $\varepsilon_{1}, \varepsilon_{2}, \cdots, \varepsilon_{n} \in\{0,1\}$ ，满足：对
于任意的 $1 \leqslant i \leqslant j \leqslant n$ ，均有
$$
\left|\sum_{k=i}^{j}\left(\varepsilon_{k}-x_{k}\right)\right| \leqslant \lambda.
$$
若$n=3$，答案写为$\frac{m}{n}$，其中$m,n$互质，求$m+n$.
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Context:
The monotonic increasing interval of the function $y=2\cos \left(2x- \frac{\pi}{4}\right)\left(x\in\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]\right)$ is ______.
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Context:
The residents of an accommodation need to pay the rent for the accommodation. If each of them contributes $10 \mathrm{Ft}$, the amount collected falls $88 \mathrm{Ft}$ short of the rent. However, if each of them contributes $10.80 \mathrm{Ft}$, then the total amount collected exceeds the rent by $2.5 \%$. How much should each resident contribute to collect exactly the required rent?
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Context:
A car traveled the first quarter of a 300-mile journey at 75 mph, the second quarter at 45 km/h, the third quarter at 50 km/h, and the last quarter at 90 mph. What was the average speed of the car in miles per hour for the entire trip?
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Context:
Cyclic quadrilateral $ABCD$ has $AC \perp BD$, $AB + CD = 12$, and $BC + AD = 13$. Find the greatest possible area of $ABCD$.
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Context:
Compute the second mixed partial derivative \( \frac{\partial^2 f}{\partial y \partial x}(0,0) \) for the function \( f(x,y)=\frac{xy(x^2-y^2)}{x^2+y^2} \) when \((x,y) \neq (0,0)\), and \( f=0 \) otherwise.
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Context:
In the following two equations, the same Chinese character represents the same digit, and different Chinese characters represent different digits:
数字花园 + 探秘 = 2015,
探秘 + 1 + 2 + 3 + ... + 10 = 花园
So the four-digit 数字花园 = ______
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Context:
Calculate the probability of observing $x$ particles given that $y$ particles were created, where $x > y$. Provide your answer as a probability value.
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Context:
In a 6 by 5 grid, how many 10-step paths are there from $W$ to $X$ that must pass through a point $H$? Assume $W$ is located at the top-left corner, $X$ at the bottom-right corner, and $H$ is three squares to the right and two squares down from $W$.
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Context:
 Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$?
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Context:
Two balls, A and B, are dropped simultaneously from the same height. Ball A bounces back to 90% of its previous height after each bounce, while Ball B bounces back to 80% of its previous height. How many times will the two balls cross each other during their bounces?
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Context:
A diagnostic test for a disease has a 95% chance of returning a true positive and a 10% chance of a false positive. The true proportion of diseased individuals in the population is \(d\). Everyone is screened, and those who test positive are tested again. Of these, 25% test positive on the second test, assuming the tests are independent. What is the value of \(d\)?
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Context:
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day’s job.At what time will the doughnut machine complete the job?
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Context:
A certain debt will be paid in 104 installments from January 1 to December 31 of a certain year. Each of the first 24 payments is to be $520; each of the remaining payments is to be $95 more than each of the first 24 payments. What is the average (arithmetic mean) payment that will be made on the debt for the year?
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Context:
Compute the sum of all positive integers $N$ for which there exists a unique ordered triple of non-negative integers $(a, b, c)$ such that $2a + 3b + 5c = 200$ and $a + b + c = N$.
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Context:
In a strip of size \(1 \times 2021\), some cells contain one chip each. In each empty cell, a number is recorded equal to the absolute value of the difference in the number of chips to the left and right of that cell. All recorded numbers are distinct and non-zero. Determine the minimum number of chips that can be placed in the cells.
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Context:
Using Clarke's Theorem, determine the number of labeled trees with 12 vertices where one specified vertex has a degree of 5.
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Context:
Let $k$ be the largest integer such that $2^k$ divides $$\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}\right)^2\right)\left(\prod_{n=1}^{25}\left(\sum_{i=0}^n\binom{n}{i}^2\right)\right).$$ Find $k$.
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Context:
Let \( D \) be the closed unit disc in \( \mathbb{R}^2 \) and \( S^1 = \partial D \). Suppose \( f: S^1 \rightarrow \mathbb{R}^3 \) is a smooth embedding such that \( f(S^1) \) intersects the plane \( z = 0 \) at exactly two points. Define \( \lambda: S^1 \rightarrow S^2 \) by

\[\lambda(x) = \frac{f(x)}{\|f(x)\|}.\]

Compute the degree of \( \lambda \).
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Context:
一个盒中有 9 个正品和 3 个废品，每次取一个产品，取出后不再放回. 则在取出 2 个正品前已取出 3 个废品的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
On Monday, a school library was attended by 5 students; on Tuesday, by 6; on Wednesday, by 4; on Thursday, by 8; and on Friday, by 7. None of them attended the library two days in a row. What is the least possible number of students who visited the library during the week?
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Context:
Let $(X_1,X_2)$ be a bivariate normal random variable with mean vector $\mu = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and covariance matrix $\Sigma = \begin{pmatrix} 3 & -1\\ -1 & 3 \end{pmatrix}$. Calculate the probability that $X_1 + X_2 < 2$ and $X_1 - X_2 > 0$. \(\text{Express your answer as a probability.}\)
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Context:
Compute the number of values of $x$ in the interval $[-11\pi,-2\pi]$ that satisfy $\tfrac{5\cos(x)+4}{5\sin(x)+3}=0$.
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Context:
Find the range of values of \(n\) for which the equation \(x^n - x - n = 0\) has more than one solution in the interval \((1,2]\).
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Context:
17th Chinese 2002 Problem A3 18 people play in a tournament of 17 rounds. There are 9 games in each round and each person plays in one game in each round. Each person plays every other person just once in the tournament. What is the largest n such that however the tournament is arranged we can find 4 people amongst whom only one game has been played by the end of round n?
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Context:
What is the value of n if the sum of the consecutive prime numbers from the smallest prime number until n equals 2500?
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Context:
设 $n \equiv 1(\bmod 4)$ 且 $n>1, P=\left\{a_1, a_2, \cdots, a_n\right\}$ 是 $\{1,2,3, \cdots, n\}$ 的任意排列, $k_p$表示使下列不等式成立的最大下标 $k$,

$$
a_1+a_2+\cdots+a_k<a_{k+1}+a_{k+2}+\cdots+a_n .
$$

试对一切可能的不同排列 $P$, 求对应的 $k_p$ 的总和.
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Context:
Let \( G \) be a cubic graph with \( q \) edges and girth \( \ge 4 \). Determine the number of paths of length 3 in \( G \).
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Context:
Let \( f(x) = \sin{x} + x^2 + 1 \). Find the value of \( \frac{d^4}{dy^4}(f^{-1}(y)) \bigg|_{y=1} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the line y = 3x - 5 and the square PQRS with R at (2009, 2009), find the number of lattice points on the line that are inside the square. A lattice point is a point with integer coordinates. Express your answer as a single integer.
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Context:
Xiao Wang places some equilateral triangle paper pieces on the table. The first time he places 1 piece; the second time he places three more pieces around the first triangle; the third time he places more pieces around the shape formed in the second placement, and so on. The requirement is: each piece placed in each subsequent placement must share at least one edge with a piece placed in the previous placement, and apart from sharing edges, there should be no other overlaps. Calculate the total number of equilateral triangle pieces used after the 20th placement.
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Context:
The number 770 is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either 40 or 41 from the number on the board. She performs moves until the number is not positive, and then she stops. Let \( N \) be the number of sequences of moves that Melody could perform. Suppose \( N = a \cdot 2^b \) where \( a \) is an odd positive integer and \( b \) is a nonnegative integer. Compute \( 100a + b \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute \( \mathbb{E} \left( \int_0^1 \left| B_s \right| dB_s \right)^2 \) using Ito's formula, where \(B_s\) is a Brownian motion.
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Context:
Determine the domain of definition of the function $$f(x)=\int_{0}^{\infty}\frac{t^x}{1+t^4 \sin ^2 t}dt$$
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Context:
Determine the number of group homomorphisms from the dihedral group $D_4 = \langle r_4, f_4 : r_4^4 = f_4^2 = (r_4f_4)^2 = e \rangle$ to the symmetric group $S_3$. Consider the possible images of the generators $r_4$ and $f_4$ under a homomorphism $\phi: D_4 \rightarrow S_3$ that satisfy the relation $\phi((r_4f_4)^2) = \phi(e) = \varepsilon$. What is the total number of such homomorphisms?
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Context:
Determine the minimum number of infinite subsets that must be added to a cofinite topology on a countably infinite set to obtain the discrete topology.
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Context:
A candidate mistakenly gave his height as a certain percentage more than his actual height in a certificate. In the interview panel, he clarified that his height was 5 feet 8 inches. The candidate made a 17.012448132780083% correction from his stated height to his actual height. What was the percentage by which he overstated his height?
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Context:
Inside a square with side length \(a\), a semicircle is constructed on each side, using the side as the diameter. Find the area of the rosette enclosed by the arcs of the semicircles.
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Context:
Let $\kappa$ be an infinite cardinal and let $X$ be a compact Hausdorff space of size $2^\kappa$. Let $U$ be a dense open subset of $X$. Determine a lower bound for the cardinality of $U$.
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Context:
Given a right triangle \(ABC\) with a right angle at \(C\), a circle is drawn with diameter \(BC\) of length 26. A tangent \(AP\) from point \(A\) to this circle (distinct from \(AC\)) is drawn. The perpendicular \(PH\) dropped onto segment \(BC\) intersects segment \(AB\) at point \(Q\). Find the area of triangle \(BPQ\) given \(BH:CH = 4:9\).
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Context:
Suppose $f(x)$ and $g(x)$ are nonconstant smooth functions satisfying the equations $$\frac{f'(x)}{f(x)} + \frac{g'(x)}{g(x)} = 2$$ $$ \frac{f''(x)}{f(x)} - \frac{f'^2(x)}{f^2(x)} + \frac{g''(x)}{g(x)} - \frac{g'^2(x)}{g^2(x)} = 0$$ (assume $f$ and $g$ are always nonzero). Compute the value of $$ \frac{f''(2023)}{f'(2023)} + \frac{g''(2023)}{g'(2023)}.$$
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Context:
Given a regular triangular prism \( S-ABC \) with height \( S O = 3 \) and base side length of 6, a perpendicular is drawn from point \( A \) to the opposite face \( SBC \), with the foot of the perpendicular denoted as \( O' \). A point \( P \) is chosen on \( AO' \) such that \( \frac{AP}{PQ} = 8 \). Find the area of the cross-section passing through point \( P \) and parallel to the base.
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Context:
Starting at $12:00:00$ AM on January $1, 2022$, after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365$. Find $y + d$.
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Context:
Given the cubic equation \( x^3 - 3x + 1 = 0 \), let \( a, b, c \) be the roots such that \( a < b < c \). Find the value of the expression \( \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \) without solving the original equation.
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Context:
a and b can do a work in 4 hours and 12 hours respectively. a starts the work at 6 am and they work alternately for one hour each. When will the work be completed?
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Context:
Given that $X$ and $Y$ are independent and identically distributed standard normal random variables, calculate the covariance $Cov(XY|X+Y>0)$. Provide your answer as a single value.
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Context:
Given a group homomorphism \( \phi: \mathbb{Z}_{50} \rightarrow \mathbb{Z}_{15} \) with \( \phi(7) = 6 \), determine the expression for \( \phi(x) \).
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Context:
A grocer starts with 400 pounds of coffee in stock, 20 percent of which is decaffeinated. They then buy another 100 pounds of coffee, of which 60 percent is decaffeinated, and 150 pounds of coffee, of which 45 percent is decaffeinated. Next, the grocer sells 50 pounds of decaffeinated coffee and buys 40 pounds of coffee that is 10 percent decaffeinated, followed by selling 70 pounds of caffeinated coffee. Afterward, the grocer purchases 90 pounds of coffee, of which the decaffeinated percentage fluctuates varying from 25 percent at the beginning to 50 percent at the end. Finally, the grocer sells an additional 30 pounds of caffeinated and 20 pounds of decaffeinated coffee, then buys another 80 pounds of coffee, of which 30 percent is decaffeinated. What percent, by weight, of the grocer's stock of coffee is decaffeinated after all these transactions?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the coefficient of the term \((a+d)^{11} b^4 c^2\) in the expansion of \((a - 2b + 3c + d)^{17}\) using the multinomial theorem.
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Context:
We have four registers, $R_1, R_2, R_3, R_4$, such that $R_i$ initially contains the number $i$ for $1 \leq i \leq 4$. We are allowed two operations:

1. Simultaneously swap the contents of $R_1$ and $R_3$ as well as $R_2$ and $R_4$.

2. Simultaneously transfer the contents of $R_2$ to $R_3$, the contents of $R_3$ to $R_4$, and the contents of $R_4$ to $R_2$. For example, if we do this once, then $(R_1, R_2, R_3, R_4) = (1, 4, 2, 3)$.

Using these two operations as many times as desired and in whatever order, what is the total number of possible outcomes?
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Context:
Determine the leftmost three digits of the number\[1^1+2^2+3^3+...+999^{999}+1000^{1000}.\]
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Context:
Evaluate the following limit: $$\lim_{x\to 0} \frac{\sin(\sin(\sin(\sin x))) - x}{x^3}$$
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Context:
Determine the number of solutions in natural numbers for the equation \((a+1)(b+1)(c+1)=2abc\).
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Context:
Given that $P$ is a point on the circle $x^2+y^2=4$ and is not on the coordinate axes, with $A(2,0)$, $B(0,2)$. The line $PA$ intersects the $y$-axis at point $M$, and the line $PB$ intersects the $x$-axis at point $N$. The minimum value of $|AN|+2|BM|$ is \_\_\_\_\_\_.
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Context:
Given the function $f(x)=2\sin (\omega x+\frac{π}{4})$ where $\omega\ \ \gt 0$, and the range of $f(x)$ on the interval $\left[-1,1\right]$ is $\left[m,n\right]$ with $n-m=3$, find the value of $\omega$.
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Context:
Let $\phi: \mathbb{Z} \rightarrow \mathbb{Z}_8$ be a homomorphism such that $\phi(1) = 6$. Determine all values of $n$ for which $\phi(n) = 0$.
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Context:
Let $X = Y = \mathbb{N}$, where $\mathbb{N}$ is the set of natural numbers. Define functions $f: X \to Y$ and $g: Y \to X$ by $f(x) = 2x$ and $g(y) = 3y$. Given $D = f(X)$ and $A = Y \setminus D$, find an element $x \in X$ that is not in the set $W = \bigcup_{n=0}^\infty g(fg)^n(A)$. Find such an element $x$. \( \boxed{} \)
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Context:
Determine all real numbers $a$ such that the sequence \( \{S_n(a)\}_{n\geq 2} \) converges to a positive real number, where \( S_n(a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}} \).
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Context:
What is the area of a quadrilateral with vertices at $(0,0)$, $(4,3)$, $(7,0)$, and $(4,4)$?
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Context:
Given that $\sqrt{x+1}$ is the simplest quadratic root and can be combined with $\sqrt{\frac{5}{2}}$, find the value of $x$.
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Context:
Tom went to 4 hockey games this year, but missed some games. He went to 9 games last year. Tom went to 13 hockey games in all. How many hockey games did Tom miss this year?
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Context:
Evaluate the limit: \[ \lim_{n\to\infty}\left(\frac{3}{2}\cdot\frac{5}{3}\cdot\dots\frac{2n+1}{n+1}\right)^\frac{1}{n} \]
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Context:
A 24-ton mixture of crushed rock is needed in a construction job; it will cost $800. The mixture is composed of rock costing $30 per ton and another type of rock costing a certain amount per ton. 8 tons of each should be added. How much does the more expensive rock cost per ton?
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Context:
Given the function $f(x)$ and its derivative $f''(x)$ on $\mathbb{R}$, and for any real number $x$, it satisfies $f(x)+f(-x)=2x^{2}$, and for $x < 0$, $f''(x)+1 < 2x$, find the minimum value of the real number $a$ such that $f(a+1) \leqslant f(-a)+2a+1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the definite integral: \[ \int_0^{\frac{\pi}{2}}{\sqrt{\sin(2\theta)} \cdot \sin(\theta)\,d\theta} \]
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Context:
Evaluate the integral: $$\int_1^3 \sqrt{x^4+1} \, dx$$
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Context:
In the arithmetic sequence $\{a_n\}$, $a_3 - a_2 = -2$, and $a_7 = -2$, calculate the value of $a_9$.
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Context:
Evaluate the double integral \( \iint_{D} x \, dx \, dy \) where the region \( D \) is bounded by the lines \( y^{2} - 4y + x^{2} = 0 \), \( y^{2} - 8y + x^{2} = 0 \), \( y = \frac{x}{\sqrt{3}} \), and \( x = 0 \).
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Context:
Find an explicit value of $\varepsilon >0$ such that for every $x\in \left [ 0,1 \right ] $, there always is $$\left | \sqrt[]{x} -\sqrt[]{x+\varepsilon} \right | < \frac{1}{200}$$
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Context:
A player plays 9 games each day with his computer. In each game, he will win, draw, or lose with probabilities \(\frac{1}{2}\), \(\frac{1}{3}\), and \(\frac{1}{6}\) respectively. A day is considered “great” if he wins all 9 games. Approximate the probability that within 128 days, he experiences exactly three “great days.”
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Context:
Let \( d(n) \) denote the number of positive divisors of the positive integer \( n \). For instance, we have \( d(1) = 1 \), \( d(2) = 2 \), and \( d(12) = 6 \). We further define the function \( f(n) = \frac{d(n)}{\sqrt[3]{n}} \).

Given that there exists a unique positive integer \( N \) such that for any positive integer \( n \neq N \), the inequality \( f(N) > f(n) \) holds, find the sum of the digits of \( N \). Express your answer as a single integer.
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Context:
9.1. (12 points) Two circles touch each other externally at point $K$. On their common internal tangent, point $P$ is marked such that $K P=14$. Through point $P$, two secants are drawn to the circles, with one of them intercepting a chord $A B=45$ on the first circle, and the other intercepting a chord $C D=21$ on the second circle, with point $A$ lying between points $B$ and $P$, and point $C$ lying between points $D$ and $P$. Find the ratio $B C: A D$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 $x_{k} \in[-2,2](k=1,2, \cdots, 2013)$ ，
且 $x_{1}+x_{2}+\cdots+x_{2013}=0$. 试求
$$M=x_{1}^{3}+x_{2}^{3}+\cdots+x_{2013}^{3}$$
的最大值.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6, and the distance between their centers is 13. If a plane is made tangent to these two spheres and intersects with the cylinder to form an ellipse, then the sum of the lengths of the major and minor axes of this ellipse is $\qquad$
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Context:
7.5. 16 travelers, each of whom is either a liar or a knight (liars always lie, knights always tell the truth), settled into 3 rooms of a hotel. When everyone gathered in their rooms, Basil, who was staying in the first room, said: "There are more liars than knights in this room right now. Although no - there are more knights than liars in this room right now." After that, Basil went to the second room and said the same two statements there. And then he went to the third room and said the same two statements there as well. How many knights could there have been among these 16 travelers?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit of the sequence: \[ \lim_{n \to +\infty} \frac{1}{\sqrt{n}}\sum_{k=1}^n \frac{k}{\sum_{j=1}^k j^{-1/2}} \]
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Context:
Approximate the value of \( \int_{0}^{0.5}{\frac{\sin(x)}{x}}dx \) using the Maclaurin series to within an error of 0.0001, where \(x\) is in radians.
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Context:
Let $n$ be a positive integer. There is a collection of cards that meets the following properties:
$\bullet$Each card has a number written in the form $m!$, where $m$ is a positive integer.
$\bullet$For every positive integer $t\le n!$, it is possible to choose one or more cards from the collection in such a way
$\text{   }$that the sum of the numbers of those cards is $t$.
Determine, based on $n$, the smallest number of cards that this collection can have.
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Context:
Inside a circle, 16 radii of this circle and 10 circles with the same center as the circle are drawn. Into how many regions do the radii and circles divide the circle?
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Context:
记集合 $T=\{0,1,2,3,4,5,6\}, M=\left\{\left.\frac{a_{1}}{7}+\frac{a_{2}}{7^{2}}+\frac{a_{3}}{7^{3}}+\frac{a_{4}}{7^{4}} \right\rvert\, a_{i} \in T, i=1,2,3,4\right\}$ ，将 $M$ 中的元素按从大到小排列，求第 2005 个数。原始答案的形式为 \frac{b_1}{7}+\frac{b_2}{7^{2}}+\frac{b_3}{7^{3}}+\frac{b_4}{7^{4}}，请给出b_1 + b_2 + b_3 + b_4的值。
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Context:
Calculate the surface area of a Lambertian hemisphere, given that the intensity of radiation is proportional to the absolute value of the cosine of the angle between the surface normal and the direction of radiation.
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Context:
Find the volume of the solid generated by revolving the region bounded by the parabola $y=2x^2$ and the line $y=6x-2$ about the line $y=12$ using the shell method.
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Context:
Compute \( \mathbb{E}\left(\int_{0}^{T} |B_{t}|^{2} dt\right) \) where \((B_t)_{t\geq0}\) is a standard Brownian motion and \(T\) is the first time the Brownian motion hits \(1\) or \(-1\).
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Context:
We randomly select 5 distinct positive integers from the set \( \{1, 2, \ldots, 90\} \). Calculate \( \lfloor 10 \times E(X_4) \rfloor \), where \( E(X_4) \) is the expected value of the fourth largest number in the selection.
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Context:
Outside the temple, there is a shop which charges 11 dollars for each object. Please note that one shoe is counted as an object. Same is true for socks, mobiles, and umbrellas. Nathan went to the temple with both parents, his grandmother, and his younger sibling. All of them kept their shoes, socks, mobiles, and umbrellas in the shop. Nathan and his dad have a pair of gloves each which they also kept in the shop. How much do they have to pay?
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Context:
The true discount on a bill of Rs. 2660 is Rs. 360. What is the banker's discount?
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Context:
设 $E 、 F 、 G$ 分别是正四面体 $A B C D$ 的棱 $A B 、 B C 、 C D$ 的中点。求二面角 $C-F G-E$ 的大小，并用表达式 $\pi-\operatorname{arccot} \frac{\sqrt{m}}{n}$ 表示，其中$m$和$n$是整数，且$\frac{\sqrt{m}}{n}$是最简分数。请给出$m + n$的值。
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Context:
Find the result of the division $0111/1111$ in the finite field $GF(16)$, where $GF(16)$ is generated by the irreducible polynomial $\Pi(\alpha)=1+\alpha+\alpha^4$. Express $0111$ as $\alpha+\alpha^2+\alpha^3$ and $1111$ as $1+\alpha+\alpha^2+\alpha^3$. Compute the result of this division as an element of $GF(16)$. What is the result?
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Context:
Given that there are $24$ primes between $3$ and $100$, inclusive, determine the number of ordered pairs $(p, a)$ where $p$ is a prime number such that $3 \le p < 100$, and $1 \le a < p$. The condition is that the sum:
\[
a + a^2 + a^3 + \cdots + a^{(p-2)!}
\]
is not divisible by $p$.
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Context:
Evaluate the limit: \[ \lim_{n\to\infty} \int_0^\infty \frac{\sin(x/n)}{x+x^2} \, dx. \]
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Context:
Let $0 < \theta < 2\pi$ be a real number for which $\cos (\theta) + \cos (2\theta) + \cos (3\theta) + ...+ \cos (2020\theta) = 0$ and $\theta =\frac{\pi}{n}$ for some positive integer $n$. Compute the sum of the possible values of $n \le  2020$.
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Context:
In the circle shown, diameter \(AB\) has points \(C\) and \(D\) on the circle such that \(BD\) bisects \(\angle CBA\). The chords \(AC\) and \(BD\) intersect at point \(E\). Given that \(AE = 169\) cm and \(EC = 119\) cm, find the length of \(ED\) in centimeters.
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Context:
A company has a total of nine factories, each with the same amount of inventory and the ability to produce the same amount of new products daily. There are two groups of inspectors, A and B, to accept the products, and each inspector works at the same speed. Group A's 8 inspectors took two days to inspect all the products of two factories and another three days to inspect all the products of another two factories. Meanwhile, Group B's inspectors just finished inspecting all the products of the remaining five factories within these five days. How many inspectors are there in Group B?
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Context:
A particle starts at the origin in a Cartesian coordinate plane and moves every second to one of the eight nearest lattice points with equal probability (\(\frac{1}{8}\)) independently of previous moves. A square is drawn with vertices at \((2,2)\), \((-2,2)\), \((-2,-2)\), and \((2,-2)\). The particle will eventually hit the square for the first time, either at a corner or at an interior point on a side. The probability of hitting a corner rather than an interior point is in the form \(\frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers. Find \(m + n\).
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Context:
Let $\Gamma$ be an ellipse with two foci $F_{1}$ and $F_{2}$. A line passing through $F_{1}$ intersects $\Gamma$ at points $P$ and $Q$. Given that $|P F_{2}| = |F_{1} F_{2}|$, and $3|P F_{1}| = 4|Q F_{1}|$, find the ratio of the minor axis to the major axis of the ellipse $\Gamma$.
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Context:
Evaluate the integral: $$\int_{0}^{\infty}{\sin(x^n)\over x[\cos(x^n)+\cosh(x^n)]}dx$$
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Context:
Acute triangle $ABC$ has altitudes $AD$, $BE$, and $CF$.  Point $D$ is projected onto $AB$ and $AC$ to points $D_c$ and $D_b$ respectively.  Likewise, $E$ is projected to $E_a$ on $BC$ and $E_c$ on $AB$, and $F$ is projected to $F_a$ on $BC$ and $F_b$ on $AC$.  Lines $D_bD_c$, $E_cE_a$, $F_aF_b$ bound a triangle of area $T_1$, and lines $E_cF_b$, $D_bE_a$, $F_aD_c$ bound a triangle of area $T_2$.  What is the smallest possible value of the ratio $T_2/T_1$?
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Context:
Let $ABCD$ be a quadrilateral with side lengths $AB=2, BC=3, CD=5$, and $DA=4$. What is the maximum possible radius of a circle inscribed in quadrilateral $ABCD$?
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Context:
The average of a set of distinct primes is 27. What is the largest prime that can be in this set?
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Context:
Determine the number of ways to color the edges of a regular hexagon using three colors such that each color is used on exactly two edges, and the coloring is invariant under reflection across a diagonal.
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Context:
Given a natural number $n = 2016^{155}$, repeatedly subtract the largest divisor of $n$ (other than $n$ itself) until you reach 1. How many steps are required to reach 1?
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Context:
Evaluate the double integral \( \iint |\cos(x+y)| \, dx \, dy \) over the region \( 0 \leq x \leq \pi \) and \( 0 \leq y \leq \pi \) using the transformation \( x = u-v \) and \( y = v \).
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Context:
Let \( f(x) \) be a function such that \( f : \mathbb{R} \to \mathbb{R} \) and \( f(x) + f\left(1-\frac{1}{x}\right) = \tan^{-1}(x) \) for all real \( x \) except \( 0 \). Find the value of \( \int_0^1 f(x) \, \mathrm{d}x \).
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Context:
Let's call a natural number "curious" if, after subtracting the sum of its digits from it, the resulting number consists of identical digits. How many three-digit curious numbers exist?
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Context:
If a trader sold three cars, with the first car being sold at Rs. 404415 with a 15% gain, the second car being sold at Rs. 404415 with a 15% loss, and the third car being sold at Rs. 550000 with a 10% gain, then what is his overall profit or loss percentage on the entire transaction?
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Context:
Let \( g:[-1,1]\to \mathbb{R} \) be a monotonic function such that \( g(-1)=-1 \) and \( g(1)=1 \). Determine the number of solutions \( M \) to the equation \( g(g(x)) = \sin(\pi x) \).
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Context:
Given the stochastic process \(X_t = B_t - tB_1\) for \(0\leq t\leq 1\), where \(B_t\) is a standard Brownian motion, find the conditional expectation \(\mathbb{E}(X_s | X_t = 1)\) for \(0\leq s<t\leq 1\).
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Context:
In $\triangle ABC$, if $a=1$, $B=45^{\circ}$, and the area of $\triangle ABC$ is $2$, then the diameter of the circumcircle of $\triangle ABC$ is ______.
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Context:
What is the maximum number of intersection points that the outlines of two quadrilaterals can have?
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Context:
When written out in full, the number \((10^{2020}+2020)^{2}\) has 4041 digits. Calculate the sum of the digits of this 4041-digit number. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{\infty} \frac{\sin^2(\tan x)}{x^2} \, \mathrm{d}x \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABC$ be a triangle with $\angle C=90^\circ$. A line joining the midpoint of its altitude $CH$ and the vertex $A$ meets $CB$ at point $K$. Let $L$ be the midpoint of $BC$ ,and $T$ be a point of segment $AB$ such that $\angle ATK=\angle LTB$. It is known that $BC=1$. Find the perimeter of triangle $KTL$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Two real numbers x and y are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x / y$ is even? The original answer is in the form $(m-\pi)/n$, please provide the value of m+n.
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Context:
In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most  $100$  flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest  $d$  for which one can always claim that any city can be reached from any other by a sequence of an even number of flights not exceeding  $d$ ?
Please reason step by step, and put your final answer within \boxed{}.


Context:
已知集合 $A, B, C \subseteq\{1,2,3, \cdots, 2020\}$ ，且 $A \subseteq B \subseteq C$ ，求有序集合组 $(A, B, C)$的个数。答案的形式为x^y，请给出x+y的值。
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Context:
Given the function $f(x)=-x^{3}+ax^{2}-4$ has an extremum at $x=2$, and if $m, n \in [-1,1]$, then the minimum value of $f(m)+f'(n)$ is ______.
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Context:
Doug and Ryan are competing in the 2005 Wiffle Ball Home Run Derby. In each round, each player takes a series of swings. Each swing results in either a home run or an out, and an out ends the series. When Doug swings, the probability that he will hit a home run is \( \frac{1}{3} \). When Ryan swings, the probability that he will hit a home run is \( \frac{1}{2} \). In one round, what is the probability that Doug will hit more home runs than Ryan?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a non-empty subset family \( U \) of \( S = \{a_1, a_2, \ldots, a_n\} \) that satisfies the property: if \( A \in U \) and \( A \subseteq B \), then \( B \in U \); and a non-empty subset family \( V \) of \( S \) that satisfies the property: if \( A \in V \) and \( A \supseteq B \), then \( B \in V \). Determine the maximum possible value of \( \frac{|U \cap V|}{|U| \cdot |V|} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the sum of the series: \((2^2-1)(6^2-1) + (4^2-1)(8^2-1) + \cdots + (100^2-1)(104^2-1)\).
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Context:
Two adjacent faces of a tetrahedron, which are isosceles right triangles with a hypotenuse of length 2, form a dihedral angle of 60 degrees. The tetrahedron rotates about the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing the given edge.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^{\pi/2} (\cos^{10} x)(\sin 12x) \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of 3-Sylow subgroups in the semidirect product \( G = \mathbb{Z}_7 \rtimes_{\rho} \mathbb{Z}_6 \) where \( |\ker\rho| = 2 \).
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Context:
Given the interval \([1, 2014]\), determine the number of integers \(a\) for which the system of equations \[\begin{align*} x + y &= a, \\ \frac{x^2}{x-1} + \frac{y^2}{y-1} &= 4 \end{align*}\] has a finite number of integer solutions \((x, y)\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the maximum number of 2 by 2 squares that can be placed on a 7 by 7 checkerboard so that any two placed squares share at most one common cell? The 2 by 2 squares are placed along the grid lines so that each covers exactly 4 cells. The squares do not go beyond the boundary of the board.
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Context:
Find the number of polynomials $f(x)=ax^3+bx$ satisfying both following conditions:
(i) $a,b\in\{1,2,\ldots,2013\}$;
(ii) the difference between any two of $f(1),f(2),\ldots,f(2013)$ is not a multiple of $2013$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \( \int_{\mathcal{C}} \log(z) \, \mathrm{d}z \), where \( \log(z) \) is the branch of the complex logarithm with a branch cut along the negative real axis, and \( \mathcal{C}(t) = e^{it} \) for \( t \) from \(-\pi\) to \(\pi\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
The positive integers $v, w, x, y$, and $z$ satisfy the equation \[v + \frac{1}{w + \frac{1}{x + \frac{1}{y + \frac{1}{z}}}} = \frac{222}{155}.\] Compute $10^4 v + 10^3 w + 10^2 x + 10 y + z$.
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Context:
Determine the number of poor numbers less than or equal to 2018, where a number is defined as "poor" if there is no positive integer $m$ such that $m!$ has exactly that number of trailing zeros.
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Context:
An urn contains 18 differently-colored balls. You draw one ball at a time, with replacement. How many draws are required to have a 90% probability of obtaining at least one ball of each color?
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Context:
Calculate the total time taken to perform binary addition from 0 to \(2^m - 1\), given that the time for each addition is proportional to the number of bits that change during the operation.
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Context:
Find the largest integer value of \( k \) such that the following holds as \( x \to 0 \): \[ e^{\tan x} = 1 + x + \frac{x^2}{2!} + o(x^k), \] where \( e^{\tan x} - \left(1 + x + \frac{x^2}{2!}\right) = o(x^k). \]
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Context:
Find all real solutions to the equation\n$$(x+1)^{2001}+(x+1)^{2000}(x-2)+(x+1)^{1999}(x-2)^2+...+(x+1)^2(x-2)^{1999}+(x+1)^{2000}(x-2)+(x+1)^{2001}=0$$
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Context:
Somewhere in the universe, $n$ students are taking a $10$-question math competition.  Their collective performance is called [i]laughable[/i] if, for some pair of questions, there exist $57$ students such that either all of them answered both questions correctly or none of them answered both questions correctly.  Compute the smallest $n$ such that the performance is necessarily laughable.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In the village of Matitika, five friends live along a straight road in the following order: Alya, Bella, Valya, Galya, and Dilya. Each of them calculated the sum of distances (in meters) from her house to the houses of the others. Bella reported the number 700, Valya reported 600, Galya reported 650. How many meters are there between Bella's and Galya's houses?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given proposition $p$: $x^{2}+2x-3 > 0$, and proposition $q$: $x > a$, if $\neg p$ is a sufficient but not necessary condition for $\neg q$, then the range of values for the real number $a$ is \_\_\_\_\_\_.
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Context:
Find the limit of \( P\left(Y_n \geq \frac{n}{2} - n^{2/3} \right) \) as \( n \to \infty \), where \( Y_n \) is the sum of \( n \) i.i.d. random variables each uniformly distributed over \( (0, 1) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many distinct ways can the numbers 1 to 20 be assigned to the faces of a regular icosahedron, considering all faces are indistinguishable?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the parabola $C: x^{2}=8y$ and its focus $F$, the line $PQ$ and $MN$ intersect the parabola $C$ at points $P$, $Q$, and $M$, $N$, respectively. If the slopes of the lines $PQ$ and $MN$ are $k_{1}$ and $k_{2}$, and satisfy $\frac{1}{{k_1^2}}+\frac{4}{{k_2^2}}=1$, then the minimum value of $|PQ|+|MN|$ is ____.
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Context:
A propane tank is formed by revolving the region enclosed by the right half of the ellipse \(x^2 + 16y^2 = 144\) and the y-axis about the y-axis. Given that the dimensions are in meters, calculate the depth of the propane in the tank when it is filled to one-quarter of the tank's total volume. Provide your answer rounded to three decimal places.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Matias rides his bicycle around four blocks near his house, arranged as shown in the figure. His ride consists of making the longest possible route, but he has devised a rule for more fun: he can pass through street crossings several times, but he cannot pass through the same street more than once, and he must end his bike ride when he can no longer respect this condition. The four blocks are squares, each with 100 meters on each side, and the width of the streets is not considered relevant. Starting from the point indicated by $P$ in the figure, what is the longest bike ride that Matias can take?
Please reason step by step, and put your final answer within \boxed{}.


Context:
For a proposition $p$: "If two lines $l_1: a_1x + b_1y + c_1 = 0$ and $l_2: a_2x + b_2y + c_2 = 0$ are parallel, then $a_1b_2 - a_2b_1 = 0$." Let $f(p)$ denote the number of true propositions among the four forms of proposition $p$ (the original proposition, the converse, the negation, and the contrapositive). Find $f(p)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
From the three-digit numbers \(100, 101, 102, \ldots, 699, 700\), choose \(n\) different numbers such that you can always find three of them with the same sum of digits. What is the minimum value of \(n\)?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the 2-form $dxdy + 3dxdz$ on the oriented triangle with vertices $(0,0,0)$, $(1,2,3)$, and $(1,4,0)$ in that order.
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Context:
In a report on a ski race, it is stated that 96% of its participants met the standard. It is known that this figure is accurate to within 0.5%. What is the minimum number of participants in this race?
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Context:
Evaluate the integral \( \int_{0}^{\infty} e^{-2ix} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Toward the end of a game of Fish, the 2 through 7 of spades, inclusive, remain in the hands of three distinguishable players: DBR, RB, and DB, such that each player has at least one card. If it is known that DBR either has more than one card or has an even-numbered spade, or both, in how many ways can the players' hands be distributed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine whether the function \( f(x,y) = \frac{x |y|^\alpha}{x^2 + y^2} \) is continuous at the origin, where \( \alpha > 0 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the order of $E[||\sigma||_p]$ with respect to $m$, where $\sigma=(\sigma_1,...,\sigma_m)$ are i.i.d. uniform binary 0-1 valued variables. Given that Jensen's inequality provides an upper bound of $(m/2)^{1/p}$, find a lower bound of the same order.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of subgroups of order \(p^2\) in the group \(\mathbb{Z}_{p^3} \oplus \mathbb{Z}_{p^2}\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted by $a$, $b$, and $c$ respectively. Given that angle $A = \frac{\pi}{4}$, $\sin A + \sin (B - C) = 2\sqrt{2}\sin 2C$, and the area of triangle $ABC$ is $1$. Find the length of side $BC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the volume enclosed by the paraboloid \(4z = x^2 + y^2\) and the cone \((z+1)^2 = x^2 + y^2\) above the xy-plane using a triple integral.
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Context:
Let $A = \{1, 2, \ldots, 100\}$ and $f(k)$, where $k \in \mathbb{N}$, be the size of the largest subset of $A$ such that no two elements differ by $k$. How many solutions are there to $f(k) = 50$?
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Context:
Businessmen Ivanov, Petrov, and Sidorov decided to create a car company. Ivanov bought 70 identical cars for the company, Petrov bought 40 identical cars, and Sidorov contributed 44 million rubles to the company. It is known that Ivanov and Petrov can share the money among themselves in such a way that each of the three businessmen's contributions to the business is equal. How much money is Ivanov entitled to receive? Provide the answer in million rubles.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Quadrilateral $EFGH$ has right angles at $F$ and $H$, and $EG=5$. If $EFGH$ has three sides with distinct integer lengths and $FG = 1$, then what is the area of $EFGH$? Express your answer in simplest radical form.
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Context:
Find the limit of the sequence \((x_n^n, y_n^n)\) defined by the system \(\dot x = -y\), \(\dot y = x\), with initial point \((x_0, y_0) = (0, 1)\) and time-step \(h = 1/n\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest natural number \( n \) such that different natural numbers can be placed in several circles connected by line segments, satisfying the following conditions:

- If numbers \( a \) and \( b \) are not connected by a line segment, then \( a - b \) is relatively prime to \( n \).
- If numbers \( a \) and \( b \) are connected by a line segment, then \( a - b \) and \( n \) share a common divisor greater than 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
现安排 7 名同学去参加 5 个运动项目，要求甲、乙两同学不能参加同一个项目，每个项目都有人参加，每人只参加一个项目，则满足上述要求的不同安排方案数为 $\qquad$. (用数字作答)
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Context:
Calculate the area of the region defined by the inequality \( \sin^2(\pi x) + \sin^2(\pi y) \le 1 \) for \( x, y \in [-1, 1] \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given sequences of positive numbers $a_n$ and $b_n$ such that \( \lim_{n\to\infty}a_n^n = a \) and \( \lim_{n\to\infty}b_n^n = b \), where \( a, b \in (0, \infty) \), determine the limit \( \lim_{n\to\infty}(pa_n + qb_n)^n \) for nonnegative numbers \( p \) and \( q \) with \( p + q = 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the value of the summation:
\[1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{2013}+\frac{1}{1\cdot2}+\frac{1}{1\cdot3}+\cdots+\frac{1}{2012\cdot 2013}+\frac{1}{1\cdot2\cdot3\cdots2013}\]
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Context:
内接于椭圆 $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ 的菱形周长的最大值和最小值之和是以\frac{m \sqrt{k}}{n}的形式，请给出m + n+k的值。
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Context:
A circle with radius 2 is centered at $(2,0)$. Another circle with radius 1 is centered at $(5,0)$. A line is tangent to both circles at points located in the first quadrant. The original answer is in the form k\sqrt{m}. Please find the value of k + m, where k and m are integers.
Please reason step by step, and put your final answer within \boxed{}.


Context:
On a horizontal plane, touching each other, there lie 4 spheres with a radius of \( 24-12\sqrt{2} \), and their centers form a square. On top above the center of the square, a fifth sphere of the same radius is placed over the four spheres. Find the distance from its highest point to the plane.
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Context:
Evaluate the determinant \( \begin{vmatrix} \lambda +1 &\alpha &\beta\\ \alpha &\lambda + \beta &1\\ \beta &1 &\lambda + \alpha \end{vmatrix} \), where \( \alpha \) and \( \beta \) are non-real roots of the equation \( x^3 - 1 = 0 \).
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Context:
Among all the five-digit numbers formed by digits 1, 2, 3, 4, 5 without repetition, count how many numbers satisfy the "wave number" pattern, where the digits follow the alternating pattern "less-greater-less-greater...less-greater". Express your answer as a single integer representing the total count of such "wave numbers".
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Context:
In a 100x60 grid, you can place only one chocolate in each square. Chocolates are considered neighbors if they are in the same row or column with no other chocolate between them. Determine the maximum number of chocolates that can be placed in the grid such that no chocolate has more than two neighbors.
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Context:
7. The center of a unit square coincides with the center of a circle, and the square is inside the circle. If a point is randomly selected on the circle, the probability that this point can see two complete sides of the square is $\frac{1}{2}$, then the radius of the circle is . $\qquad$
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Context:
A group of 6 students decided to form "study groups" and "service activity groups" based on the following principle:

- Each group must have exactly 3 members.
- For any pair of students, there are the same number of study groups and service activity groups that both students are members of.

Given that there is at least one group and no three students belong to both the same study group and service activity group, find the minimum number of groups.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \(a = \sum\limits_{r = 1}^{11} {{{\tan }^2}\left( {\frac{{r\pi }}{{24}}} \right)} \) and \(b = \sum\limits_{r = 1}^{11} {{{\left( { - 1} \right)}^{r - 1}}{{\tan }^2}\left( {\frac{{r\pi }}{{24}}} \right)} \). Find the value of \(\log_{(2b - a)}(2a - b)\).
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Context:
It is known that 20% of people own at least 80% of all the money in the world. What is the smallest percentage of all people that can be guaranteed to own 90% of all the money?
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Context:
Dan had 97 baseball cards, and 8 were torn. Sam bought 15 of Dan's baseball cards. How many baseball cards does Dan have now?
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Context:
Let \( A \) be a \( 7 \times 7 \) matrix such that \( 2A^2 - A^4 = I \), where \( I \) is the identity matrix. If \( A \) has two distinct eigenvalues, each with geometric multiplicity 3, how many non-zero entries are there in the Jordan canonical form of \( A \)?
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Context:
Determine the number of distinct values that the difference between the principal branch of the logarithm and any other branch of the logarithm defined on a simply connected region \( \Omega \) that does not contain 0 can take.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the Galois group of the polynomial \( g(X) = (X^2 - 5)(X^3 + 2) \) over \( \mathbb{Q} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the limit: \[ \lim_{x \rightarrow +\infty} \frac{\sqrt x + x^3 -\sqrt{x^6-2x^4} -x}{\log(1+e^{3\sqrt x})} \]
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Context:
Evaluate the integral \( \int_{0}^{1} x f(x) \, dx \) given that \( f \in \mathcal{C}^2[0,1] \), \( f(1) = 2 \), \( \int_{0}^{1} f(x) \, dx = \frac{1}{3} \), and \( \int_{0}^{1} f'(x) \, dx = \frac{25}{3} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the limit \( \lim_{n \to \infty} P\left( \sigma^2 \in (A_n, B_n) \right) \), where \( A_n = \frac{(n-1)S_{n}^{2}}{\chi^{2}_{n-1,\alpha/2}} \) and \( B_n = \frac{(n-1)S_{n}^{2}}{\chi^{2}_{n-1,1-\alpha/2}} \). Assume \( X_1, X_2, \ldots, X_n, \ldots \) are i.i.d. random variables with \( \mathbb{E}(X_1^4) < \infty \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a radioactive decay process that follows a Poisson distribution with a rate of 10,000 decays per second, calculate the approximate probability that the 1,000,000th decay occurs within 100.2 seconds.
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Context:
Find the fourth derivative of the inverse function \(f^{-1}(y)\) evaluated at \(y=1\), where \(f(x) = \sin{x} + x^2 + 1\).
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Context:
Calculate the line integral \( K=\oint_{l} e^{x} \, dx + z\left(x^{2}+y^{2}\right)^{\frac{3}{2}} \, dy + yz^{3} \, dz \) using Stokes' theorem, where \( l \) is the closed curve \( OCBAO \) formed by the intersection of the surfaces \( z=\sqrt{x^{2}+y^{2}} \), \( x=0 \), \( x=2 \), \( y=0 \), and \( y=1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the expected value \( \mathbb{E}\left(\frac{X}{X+Y}\right) \) where \( X \) and \( Y \) are independent random variables with exponential distributions such that \( \mathbb{E}X=1 \) and \( \mathbb{E}Y=\frac{1}{2} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the fourth-last digit of the number \(N=3^{3^{3^{3^{3^3}}}}\) (six threes) without directly computing the entire exponentiation.
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Context:
Given the matrix \(A=\begin{pmatrix} 2 & 0 & -1 \\ 0 & 4 & 0 \\ -1 & 0 & 2 \end{pmatrix}\) and the vector sequence defined by \(y_1=\begin{pmatrix} 2 \\ 0 \\ 0 \end{pmatrix}\) and \(y_{n+1}=Ay_n\), find the limit \(\lim_{n \to \infty} \frac{1}{n}\log\|y_n\|.\)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the surface integral of the unit normal vector over the surface defined by $z = x^2 + y^2$, where $z$ ranges from $1$ to $4$ and $\theta$ ranges from $0$ to $2\pi$. What is the value of this integral?
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Context:
Consider a uniform random tournament with \(n\) vertices. For any two vertices \(x\) and \(y\), an edge is drawn from \(x\) to \(y\) with probability 0.5, and from \(y\) to \(x\) otherwise. Let \(S\) be the set of all out-degrees, with \(s_1\) as the largest element and \(s_2\) as the second largest (or equal to \(s_1\) if \(S\) is a singleton). Given a constant \(c\in (0,1)\), determine \(\lim_{n\rightarrow\infty}\text{Pr}[s_1-s_2<cn]\].
Please reason step by step, and put your final answer within \boxed{}.


Context:
A plane intersects a triangular pyramid and divides the medians of the faces emanating from one vertex in the ratios \(2:1, 1:2, 4:1\) respectively, measured from the vertex. Determine the ratio in which this plane divides the volume of the pyramid.
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Context:
Determine the smallest integer $k$ such that the product space $S^2 \times S^2$ can be embedded in $\mathbb{R}^k$. What is this optimal dimension $k$?
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Context:
An encoder maps source codewords $x \in \{0,1\}^K$ to channel codewords $y \in \{0,1\}^{N'}$ with $K/N' < 1$. The corresponding decoder maps a codeword $r \in \{0,1\}^{N'}$ to a source codeword $\hat{x} \in \{0,1\}^K$. Assume this coding scheme can correctly decode codewords at the output of a BSC with transition probability $q$, satisfying $K/N' \leq 1-H(q)$. Now, consider the reverse operation: an arbitrary codeword $r \in \{0,1\}^{N'}$ is input to the decoder, outputting $\hat{x} \in \{0,1\}^K$. The codeword $\hat{x}$ is then input to the encoder, outputting $y \in \{0,1\}^{N'}$. What is the probability that a certain bit of $y$ differs from the corresponding bit of $r$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the flow rate of water across a net submerged in a river. The velocity vector field of the river is given by \( \vec{v}=(x-y)\vec{i}+(z+y)\vec{j}+z^2\vec{k} \). The net is described by the surface \( y=\sqrt{1-x^2-z^2}, y\geq 0 \), and is oriented in the positive \( y \)-direction. Calculate the flow rate by evaluating the surface integral \( \iint_{S}\vec{v}\,d\vec{S} \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find a closed-form formula for \( \frac{d\text{vec}(X^T)}{d\text{vec}(X)} \), where \( X \) is an \( n \times n \) matrix with its entry indices increasing along every column. For example, when \( n = 2 \), \( X = \begin{pmatrix} x_1 & x_3 \\ x_2 & x_4 \end{pmatrix} \). The vectorization \( \text{vec}(X) \) is formed by stacking the columns of \( X \) into a single vector. Provide the formula for any \( n \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A system consists of three components. Component 1 is in series with two parallel components, 2 and 3. Each component operates independently and has a lifetime that follows an exponential distribution with rate parameter \( \lambda = 0.002 \). Calculate the probability that the system will last more than 500 hours.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the sequences \( a_m = \sum_{k=0}^{m} (\sqrt{k} + \sqrt{k+1})^{-1} \) and \( b_m = \sum_{k=0}^{m} (\sqrt{k} + \sqrt{k+2})^{-1} \), define \( c_n = \prod_{m=0}^{n} \frac{a_m}{a_m - 2(b_m - b_{m+1})} \). Find the smallest natural number \( n \) for which \( c_n < 0.01 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A 1992-digit number is written such that each two-digit number formed by adjacent digits is divisible by either 17 or 23. If the last digit of the number is 1, what is the first digit?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of connected components in the graph \( H_n \), where \( H_n \) is defined for each positive integer \( n \) as a graph whose vertices are \( n \)-tuples with binary entries (0 or 1), and two vertices are adjacent if they differ in precisely two coordinates. Find the number of connected components for any \( n \geq 1 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: $$\lim_{x\to\infty} \sum ^\infty_{n=3} \frac{n\cdot x^{n-2}\cdot(-1)^n\cdot(n-1)}{(2n)!}$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the number of possible arrangements of 5 tigers, 5 lions, and 5 cheetahs in 15 consecutive cages, given the following restrictions: tigers cannot be placed in the 5 leftmost cages, lions cannot be placed in the 5 middle cages, and cheetahs cannot be placed in the 5 rightmost cages.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A nonzero polynomial with rational coefficients has all of the numbers \(1+\sqrt{2}, \; 2+\sqrt{3}, \;3+\sqrt{4},\; \dots, \;1000+\sqrt{1001}\) as roots. What is the smallest possible degree of such a polynomial?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a sequence $a_n$ defined as $a_n = 2^{x_n}$, where $x_n = p_1^{p_2^{\cdots ^{p_n}}}$, express the sum of the differences between consecutive terms, $\sum\limits_{i=1}^n (a_i - a_{i-1})$, in terms of $a_n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a city, there are 2019 metro stations, and some pairs of stations are connected by tunnels, allowing travel between any two stations. The mayor wants to organize several metro lines such that each line includes multiple distinct stations connected sequentially by tunnels, and each station is part of at least one line. The number of lines should not exceed $k$. However, the mayor's plan is impracticable. Determine the largest value of $k$ for which this situation could occur.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the conditions on the parameter $p$ such that the curve $y = x^p\cos\left(\frac{\pi}{x}\right)$, for $0 < x \leqslant 1$, has a finite length.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the sum \( \sum_{b=1}^r |S(r,b,\psi)|^2 \), where \( S(r,b,\psi) = \sum_{y=1}^r \psi(y) e(by/r) \) and \( \psi \) is an arbitrary character modulo \( r \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the conditions on $m$ such that the sequence $\{b_n\}_{n=1}^\infty$ defined by \( b_n = \frac{n^m + 1}{n^{m+1} + 1} \) is monotonically decreasing.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that $a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$ for distinct real numbers $a$, $b$, and $c$, find the value of $abc$. \( \text{Express your answer as a single number.} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( \lim_{x\to2}\frac{\ xf(x)-4}{x-2}=3, \) find the value of \( \lim _{ x\rightarrow 2 }{ \frac { f\left( x \right) -2 }{ \sqrt { x-1 } -1 }  }. \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
For any finite set $X$, let $|X|$ denote the number of elements in $X.$ Define $$S_n = \\sum |A \\cap B|,$$ where the sum is taken over all ordered pairs $(A, B)$ such that $A$ and $B$ are subsets of $\\{1, 2, 3, \u2026, n\\}$ with $|A| = |B|.$ For example, $S_2 = 4$ because the sum is taken over the pairs of subsets $$(A, B) \\in \\{ (\\emptyset, \\emptyset), (\\{1\\}, \\{1\\}), (\\{1\\}, \\{2\\}), (\\{2\\}, \\{1\\}), (\\{2\\}, \\{2\\}), (\\{1, 2\\}, \\{1, 2\\})\\},$$ giving $S_2 = 0 + 1 + 0 + 0 + 1 + 2 = 4.$ Let $\\frac{S_{2022}}{S_{2021}} = \\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find the remainder when $p + q$ is divided by $1000.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the line integral \( \oint_C F \cdot dr \) using Green's Theorem, where \( C \) is a closed \( C^1 \) path oriented counterclockwise, consisting of a piece of the curve \( y^2 = 2(x+2) \) and the vertical segment \( x = 2 \). The vector field \( F \) is given by \( F(x,y) = \left(\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2}\right) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest grid size such that, when colored with 5 different colors, there must exist a square with cells of the same color directly above, below, to the left, and to the right. What is this minimal grid size?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area of the region bounded by the parametric equations \( x = 32 \cos^{3} t \) and \( y = \sin^{3} t \), and the vertical line \( x = 4 \) for \( x \geq 4 \).
Please reason step by step, and put your final answer within \boxed{}.


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


Context:
A circle with radius 1 is situated between two parallel lines, touching both lines. An isosceles triangle is positioned such that its base lies on one of the lines, and its vertex is on the other line. It is known that the triangle and the circle have exactly one common point, and this point lies on the inscribed circle of the triangle. Find the radius of the inscribed circle of the triangle.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider permutations of the set \(\{1, 2, 3, 4, 5, 6\}\) where the third element is 4. Calculate the sum of the inversion numbers for all such permutations.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the exact length of the curve given by \(x= \frac{1}{3}(y^2+2)^\frac{3}{2}\) for \(1 \leq y \leq 2\) using the arc length formula \(\int_{a}^{b}\sqrt{1+\left(\frac{\text{d}y}{\text{d}x}\right)^2}\space\space\text{d}x\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A tradesman has three types of items which he deals with. For Item A, he uses a false balance that defrauds to the extent of 30% in buying and selling. For Item B, he defrauds to the extent of 20% in buying and 10% in selling. For Item C, he defrauds 10% in buying and 20% in selling. Considering he spends equal amounts on purchasing all three items, calculate his overall percentage gain on his total outlay.

Note: In each case, the defrauding percentage is applied during both buying and selling of the respective item.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $ABCD$ be a trapezoid such that $AD\parallel BC$. The interior angle bisectors of the corners $A$ and $B$ meet on $[DC]$. If $|BC|=9$ and $|AD|=4$, find $|AB|$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
There are seven students taking a graduation photo in a row. Among them, student A cannot stand next to student B, and students B and C must stand together. Calculate the total number of different arrangements possible under these conditions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a class of 25 students arranged in five rows of five, each student must sit next to every other student at least once, either side by side, one behind the other, or diagonally. What is the minimum number of weeks required for this seating arrangement to occur?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider two polynomials, $P_1(x)$ and $P_2(x)$, with integer coefficients, and integers $n$ and $a$. Determine the number of distinct ordered pairs $(P_1(x), P_2(x))$ such that $(P_1(x))^2 + (P_2(x))^2 = (x^n - a)^2$, excluding the trivial solution $(x^n - a, 0)$. Assume that $x^n - a$ is irreducible over the Gaussian integers $\mathbb{Z}[i][x]$. Provide your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the function \( g \) defined on the set of integers with the following conditions:

1) \( g(1) - 1 > 0 \)
2) \( g(x) g(y) + x + y + xy = g(x+y) + x g(y) + y g(x) \) for any \( x, y \in \mathbb{Z} \)
3) \( 3 g(x+1) = g(x) + 2x + 3 \) for any \( x \in \mathbb{Z} \).

Find the value of \( g(-6) \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
A soccer coach named  $C$  does a header drill with two players  $A$  and  $B$ , but they all forgot to put sunscreen on their foreheads.  They solve this issue by dunking the ball into a vat of sunscreen before starting the drill.  Coach  $C$  heads the ball to  $A$ , who heads the ball back to  $C$ , who then heads the ball to  $B$ , who heads the ball back to  $C$ ;
this pattern  $CACBCACB\ldots\,$  repeats ad infinitum.  Each time a person heads the ball,  $1/10$  of the sunscreen left on the ball ends up on the person's forehead.  In the limit, what fraction of the sunscreen originally on the ball will end up on the coach's forehead?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Compute the number of tuples \(\left(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)\) of (not necessarily positive) integers such that \(a_{i} \leq i\) for all \(0 \leq i \leq 5\) and
\[ a_{0} + a_{1} + \cdots + a_{5} = 6. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a school, there are 1200 students. Each student must join exactly $k$ clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of $k$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the polynomial \( p(x) = x^3 + ax^2 + bx + c \) with roots \( \alpha, \beta, \gamma \), where \( \alpha \leq \beta \leq \gamma \) are all positive integers. Given that \( 2^2a + 2^1b + 2^0c = -2^{11} \), find the sum of all possible values of \( \gamma \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
凸四边形具有面积 $30$ 和边长 $5, 6, 9,$  $7,$ 并按此顺序排列。由 $\theta$ 四边形对角线形成的锐角的量度表示。那么 $\tan \theta$ 可以写成  $\tfrac{m}{n}$ 形式 ， 其中 $m$ 和 $n$ 是相对素数的正整数。查找  $m + n$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f(n)=n^2+100.$ Compute the remainder when $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots ))$ is divided by $10^4.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
a, b, c, and d are integers. a + b + c + d < 25, a > 8, b < 5, c is an odd number, and d is an even number. What is the smallest possible value of a - b + c - d?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the limit: \[ \lim_{x\to 0}\frac{x^{(\sin x)^{x}}-(\sin x)^{x^{\sin x}}}{x^3} \] without using Taylor's formula.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A 3D cake is cut by $n$ planes, resulting in the maximum number of pieces. Given that the maximum number of pieces is defined by the Cake Number $N=\frac{1}{6}(n^3+5n+6)$, determine the number of pieces that have the cake crust as one of their boundaries. Assume the entire surface of the cake is crust.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of models of the real numbers up to isomorphism.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the following algorithm:

Step 0. Set \( n = m \).

Step 1. If \( n \) is even, divide \( n \) by two. If \( n \) is odd, increase \( n \) by one.

Step 2. If \( n > 1 \), go to Step 1. If \( n = 1 \), end the algorithm.

How many natural numbers \( m \) exist such that Step 1 will be performed exactly 15 times when this algorithm is executed?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Every edge of a square-based pyramid is 2 units. Another pyramid with edges of equal length is attached to one of the pyramid's faces as a base. The total length of the edges of the resulting solid is 18 units. What is the volume of this solid?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Equilateral triangle $ABC$ has circumcircle $\Omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\Omega$ respectively such that $BC=DE$. Given that triangle $ABE$ has area 3 and triangle $ACD$ has area 4, find the area of triangle $ABC$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Five people are seated around a circular table. Let \( f \) represent the number of people sitting next to at least one female and \( m \) represent the number of people sitting next to at least one male. How many possible ordered pairs \( (f, m) \) are there? Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{{th}}$ pirate receive?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Calculate the area of the common part of two rhombuses, where the lengths of the diagonals of the first one are 4 cm and 6 cm, and the second one is obtained by rotating the first one by 90 degrees around its center.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Each of the 2009 distinct points in the plane is colored either blue or red. On every unit circle centered at a blue point, there are exactly two red points. Find the greatest possible number of blue points.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The tax on a commodity is diminished by 22 % but its consumption is increased by 9 %. What is the decrease percent in the revenue derived from it?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of non-empty strings of length less than or equal to \( N \) that can be formed using \( k \) different characters, such that no string is a prefix of another. Consider the constraints \( 1 \le N \le 10^9 \) and \( 1 \le k \le 50 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral: \[ \int_{0}^{1}\frac{x^4+1}{x^6+1} \, dx. \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $f$ be a function on nonnegative integers such that $f(0)=0$ and $$f(3n+2)=f(3n+1)=f(3n)+1=3f(n)+1$$ for all integers $n$ $\ge$ $0.$ Compute the sum of all nonnegative integers $m$ such that $f(m)=13.$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the closure of the subspace \(D=\{f\in C([0,1]):f(1)=0\}\) in the space \(C([0,1])\) with the norm \[||f||=\left(\int_0^1 |f(x)|^2\right)^{1/2}.\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the double integral \( \iint\limits_D (x^2+y^2) \,dA \) where \( D \) is the region enclosed by the curves \( xy=1 \), \( xy=2 \), \( x^2-y^2 =1 \), and \( x^2-y^2 =2 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the contour integral \(\oint_C (1 + z + z^2)(e^\frac{1}{z}+e^\frac{1}{z-1}+e^\frac{1}{z-2}) \, dz\), where \(C\) is a circle of radius 3 centered at the origin. Use the residue theorem to find the value of the integral.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_{0}^{3} (x^2+1) \, d[x] \), where \([x]\) is the greatest integer function.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Evaluate the integral \( \int_0^1 \left((1-x^{2023})^{\frac{1}{2022}} - (1-x^{2022})^{\frac{1}{2023}}\right)\;dx \) using special functions.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the ellipse given by the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) where \(a > b > 0\). The ellipse intersects the positive direction of the \(y\)-axis at point \(B\). Determine the number of right-angled isosceles triangles that can be inscribed in the ellipse with one vertex at point \(B\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the splitting field of the polynomial \( f(x) = x^3 - x + 1 \) over the field \( \mathbb{Z}_5 \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that \( \int_{0}^{\infty} \left(\frac{\sin x}{x}\right)^3 \, dx = A \), find the value of \( k \) such that \( \int_{0}^{\infty} \frac{x - \sin x}{x^3} \, dx = kA \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the growth rate of the longest worm-like path in a random configuration on the lattice $[1,3n]^2$ in $\mathbb{Z}^2$, where each lattice point with coordinates $\equiv 2 \pmod{3}$ is randomly assigned one of six patterns. Express the growth rate with respect to $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the minimum growth rate of a sequence $w_n$ such that $P(\overline{X}_n > w_n) \to 1$ as $n \to \infty$, where $X_i$ are independent and identically distributed random variables with $X_1 \geq 0$ and $E[X_1] = \infty$. Provide your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let three non-identical complex numbers \( z_1, z_2, z_3 \) satisfy the equation \( 4z_1^2 + 5z_2^2 + 5z_3^2 = 4z_1z_2 + 6z_2z_3 + 4z_3z_1 \). Denote the lengths of the sides of the triangle in the complex plane, with vertices at \( z_1, z_2, z_3 \), from smallest to largest as \( a, b, c \). Find the ratio \( a : b : c \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
How many different lines pass through at least two points in this 4-by-4 grid of lattice points?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given N trucks, each assigned a number from 1 to N, assume the prior distribution of N is proportional to \( \frac{1}{x} \) for \( x = 1, \ldots, 500 \). Find the posterior mean of N when observing a truck numbered 50, assuming the likelihood is \( \frac{1}{N} \) for \( N \geq 50 \) and 0 otherwise.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In triangle $\triangle ABC$, $AB = AC$ and $\angle A > 90^\circ$. Point $D$ lies on segment $AB$ such that $AD = BC$ and $\angle ADC = \frac{3}{4} \cdot \angle ABC$. Determine the measure of $\angle A$. \( \boxed{\text{Answer}} \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Identify the topological space that is homeomorphic to the partition of \(\mathbb{R}^2 - \{0\}\) into lines of the form \(y = mx\) for all real \(m\), including vertical lines.
Please reason step by step, and put your final answer within \boxed{}.