Context:
Define
\[A = \frac{1}{1^2} + \frac{1}{5^2} - \frac{1}{7^2} - \frac{1}{11^2} + \frac{1}{13^2} + \frac{1}{17^2} - \dotsb,\]which omits all terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3, and
\[B = \frac{1}{3^2} - \frac{1}{9^2} + \frac{1}{15^2} - \frac{1}{21^2} + \frac{1}{27^2} - \frac{1}{33^2} + \dotsb,\]which includes only terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3.

Determine $\frac{A}{B}.$
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Context:
A and B play a number-changing game on a $5 \times 5$ grid: A starts and both take turns filling empty spaces, with A filling each space with the number 1 and B filling each space with the number 0. After the grid is completely filled, the sum of the numbers in each $3 \times 3$ square is calculated, and the maximum sum among these squares is denoted as $A$. A tries to maximize $A$, while B tries to minimize $A$. What is the maximum value of $A$ that A can achieve?

(The problem is from the 35th IMO selection test)
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Context:
Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a strictly increasing function such that $f(1)=1$ and $f(2n)f(2n+1)=9f(n)^{2}+3f(n)$ for all $n \in \mathbb{N}$. Compute $f(137)$.
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Context:
Jack and Jill run \(10\) kilometers. They start at the same point, run \(5\) kilometers up a hill, and return to the starting point by the same route. Jack has a \(10\) minute head start and runs at the rate of \(15\) km/hr uphill and \(20\) km/hr downhill. Jill runs \(16\) km/hr uphill and \(22\) km/hr downhill. How far from the top of the hill are they when they pass going in opposite directions? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
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Context:
Evaluate the limit: $$L=\lim_{x \to -0.5^{-}} \left\lfloor\frac{1}{x} \left\lfloor \frac{-1}{x} \right\rfloor\right\rfloor$$
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Context:
Find all solutions for the equation \( \tan(\alpha) = \cos(\alpha + 33.44^\circ) \) within the interval \( 0 < \alpha < 90^\circ \).
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Context:
Find the value of \(\theta_0\) such that the distance from the origin to the point \(P(x(\theta_0), y(\theta_0), z(\theta_0))\), where \(x(\theta) = 5\cos \theta\), \(y(\theta) = 5\sin \theta\), and \(z(\theta) = \theta\), is equal to the arclength from the origin to that point.
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Context:
Marge planted 23 seeds in her garden. Five of the seeds never grew into plants. A third of the remaining seeds grew, but the plants were eaten by squirrels and rabbits. A third of the number of uneaten plants were strangled by weeds. Marge pulled some weeds, but liked the flowers on one weed and let the plant grow as part of her garden. Marge ended up with 9 plants. How many weeds did Marge pull?
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Context:
Square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. It is given that $\angle BAF = 18^\circ$. The lines $EF$ and $GH$ intersect at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angle FAH$ in degrees is $x$, find the nearest integer to $x$.
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Context:
In a modern art museum, visitors who watched a certain Picasso painting were asked to fill in a short questionnaire. According to the results of the survey, all 110 visitors who did not enjoy the painting also did not feel they had understood the painting, and the number of visitors who enjoyed the painting was equal to the number of visitors who felt they had understood the painting. If 440 visitors answered the questionnaire, what fraction of the visitors both enjoyed the painting and felt they had understood the painting?
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Context:
At a large round table, 90 people are feasting: 40 barons, 30 counts, and 20 marquises, all facing the center of the table. At a signal, exactly those who have both neighbors (left and right) with the same title should stand up. What is the maximum number of people who can stand up?

For example, in order for a count to stand up, both of their neighbors must have the same title, though not necessarily the title of count.
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Context:
The average of some numbers is 5 more than the average of 10, 60, and 35. If the other numbers are 20 and 60, what is the second number?
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Context:
A burger at Ricky C's weighs $120$ grams, of which $30$ grams are filler. What percent of the burger is not filler?
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Context:
In a 5x6 seating arrangement for 30 students, two students (A and B) from the same school are assigned to sit in the same examination room. However, they cannot sit immediately next to each other in front, back, left, or right directions. Calculate the total number of different seating arrangements for students A and B.
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Context:
Two circles with radius $R$ and centers at points $O_1$ and $O_2$ touch each other. A straight line intersects them at points $A$, $B$, $C$, and $D$ such that $AB = BC = CD$. Find the area of the quadrilateral $O_1AD O_2$.
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Context:
Given natural numbers $a$, $b$, and $c$ such that $a, b, c > 100$ and the greatest common divisor $(a, b, c) = 1$, find the minimal value of $b$ under the conditions:
1. $c$ divides $a + b$ (i.e., $c \mid a + b$).
2. $a$ divides $b + c$ (i.e., $a \mid b + c$).
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Context:
An ant walks an average of 1000 meters in 30 minutes on smooth terrain. However, on forest terrain, which makes up 40% of its walking path, it can only achieve 75% of its smooth terrain speed due to the difficulties of the terrain. A beetle, although smaller, is better equipped for tough terrain and walks 10% less distance on smooth terrain at the same time as the ant but maintains 90% of its speed on forest terrain.

Assuming both the ant and the beetle split their walk equally between the two terrains, calculate the beetle's average speed in km/h. Consider that both creatures are not taking a rest and are walking continuously.
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Context:
A retail appliance store priced three different video recorders (V1, V2, and V3) at 20 percent, 25 percent, and 30 percent above their respective wholesale costs of $200, $250, and $300. In addition, the store has a fluctuating wholesale discount policy for its employees, where they get 20 percent off on V1, 15 percent off on V2, and 10 percent off on V3 in the first week, and the discount rates change to 25 percent, 20 percent, and 15 percent respectively in the second week. If a store employee intends to purchase all three video recorders, one per week for three weeks, what would be the minimum amount they would have to spend in total?
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Context:
Determine the number of solutions the equation ||$|x-a|-1|-1|=|b|$ can have. Express your answer as a single integer.
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Context:
A pentagon has vertices labeled $A, B, C, D, E$ in that order counterclockwise, such that $AB$ and $ED$ are parallel, and \( \angle EAB = \angle ABD = \angle ACD = \angle CDA \). Furthermore, suppose that \( AB = 8 \), \( AC = 12 \), \( AE = 10 \). If the area of triangle $CDE$ can be expressed as \( \frac{a \sqrt{b}}{c} \), where $a$, $b$, $c$ are integers such that $b$ is square-free, and $a$, $c$ are relatively prime, find $a + b + c$.
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Context:
The equation

$$
(x-1) \times \ldots \times(x-2016) = (x-1) \times \ldots \times(x-2016)
$$

is written on the board. We want to erase certain linear factors so that the remaining equation has no real solutions. Determine the smallest number of linear factors that need to be erased to achieve this objective.
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Context:
A car traveled the first quarter of a certain distance at 4 times the speed it traveled the second quarter, and 6 times the speed it traveled the third quarter. During the last quarter of the distance, the car traveled at half the speed of the second quarter. What proportion of the total time traveled, was the time taken to travel the first quarter of the distance?
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Context:
In triangle $ ABC$ with $ \left|BC\right|>\left|BA\right|$, $ D$ is a point inside the triangle such that $ \angle ABD=\angle DBC$, $ \angle BDC=150{}^\circ$ and $ \angle DAC=60{}^\circ$. What is the measure of $ \angle BAD$?
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Context:
Determine the value of \( n \in \mathbb{N} \) for which the following limit is a non-zero finite number:

\[ \lim_{x \to 0} \frac{\cos^2x - \cos x - e^x\cos x + e^x - \frac{x^3}{2}}{x^n} \]
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Context:
143. Find the general solution of the equation $y^{\prime \prime}-6 y^{\prime}+9 y=0$.
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Context:
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_1$, $R_2$, $R_3$, $R_4$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$, find $R_4$.
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Context:
Three thin, straight sticks of equal length are bound together in a shape resembling the letter "Π" and tossed into the air. The adjacent sticks are connected by a short string, allowing them to rotate freely relative to each other. When the sticks fall to the ground, they form a random figure. Calculate the probability that the first and third sticks do not intersect when they land.
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Context:
Given that \(a\) is a positive integer and the equation \( \cos^{2} \pi(a-x) - 2 \cos \pi(a-x) + \cos \frac{3 \pi x}{2a} \cos \left(\frac{\pi x}{2a} + \frac{\pi}{3}\right) + 2 = 0 \) in terms of \(x\) has real solutions, find the smallest value of \(a\).
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Context:
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\ldots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t\geq t_0$ by the
following properties:
(a) $f(t)$ is continuous for $t\geq t_0$,and is twice differentiable for all $t>t_0$ other than $t_1,\ldots,t_n;$
(b)$f( t_0) = 1/ 2;$
(c) $\lim _{r\to t_{k}^{+ }}f^{\prime }( t) = 0$ for $0\leq k\leq n;$
(d) For $0\leq k\leq n-1$, we have $f^{\prime\prime}(t)=k+1$ when
$t_k<t<t_{k+1}$, and $f^{\prime\prime}(t)=n+1$ when $t>t_n.$

Considering all choices of $n$ and $t_0,t_1,\ldots,t_n$ such that $t_k\geq t_{k-1}+1$ for l$\leq k\leq n$,what is the least possible value of $T$ for which $f(t_0+T)=2023?$
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Context:
The sequence $\left\{a_{n}\right\}$ is defined by the following conditions: $a_{1}=1, a_{n+1}=a_{n}+\frac{1}{a_{n}}$ for $n \geq 1$. Find the integer part of $a_{100}$.
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Context:
Find the number of integers $n$ such that $1 \le n \le 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$.
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Context:
Determine the number of linear functions from \( \{0, 1\}^n \) to \( \{0, 1\} \). A Boolean function \( F: \{0, 1\}^n \rightarrow \{0, 1\} \) is said to be linear if \( F(x \oplus y) = F(x) \oplus F(y) \) for all \( x, y \in \{0, 1\}^n \), where \( x \oplus y \) denotes the component-wise exclusive-or of \( x \) and \( y \).
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Context:
A frog is positioned at the origin in the coordinate plane. From the point $(x,y)$, the frog can jump to any of the points $(x+1, y), (x+2, y), (x, y+1),$ or $(x, y+2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0,0)$ and ends at $(4,4)$.
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Context:
A class has 54 students, and there are 4 tickets for the Shanghai World Expo to be distributed among the students using a systematic sampling method. If it is known that students with numbers 3, 29, and 42 have already been selected, then the student number of the fourth selected student is ▲.
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Context:
Real numbers between \(0\) and \(1\), inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is \(0\) if the second flip is heads, and \(1\) if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval \([0,1]\). Two random numbers \(x\) and \(y\) are chosen independently in this manner. Find the probability that \(|x-y| > \tfrac{1}{2}\). The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
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Context:
Your friend sitting to your left (or right?) is unable to solve any of the eight problems on his or her Combinatorics  $B$  test, and decides to guess random answers to each of them. To your astonishment, your friend manages to get two of the answers correct. Assuming your friend has equal probability of guessing each of the questions correctly, what is the average possible value of your friend’s score? Recall that each question is worth the point value shown at the beginning of each question.
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Context:
Simplify the expression \(\frac{(a-b)^{2}+ab}{(a+b)^{2}-ab} \div \frac{a^{5}+b^{5}+a^{2}b^{3}+a^{3}b^{2}}{\left(a^{3}+b^{3}+a^{2}b+ab^{2}\right) \left(a^{3}-b^{3}\right)}\).
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Context:
Ten years ago, the claim size under homeowner insurance policies at a particular insurance company had an exponential distribution, with 25% of claims being less than \$1000. Due to inflation, today's claims are twice the size of equivalent claims from a decade ago. What is the probability that a claim made today is less than \$1000?
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Context:
9 kg of toffees cost less than 10 rubles, and 10 kg of the same toffees cost more than 11 rubles. How much does 1 kg of these toffees cost?
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Context:
A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a $7 \times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).
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Context:
Given the sets \( M = \left\{(x, y) \mid \tan \pi y + \sin^2 \pi x = 0\right\} \) and \( N = \left\{(x, y) \mid x^2 + y^2 \leqslant 2\right\} \), find the number of elements in \( M \cap N \). Express your answer as a single integer.
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Context:
A pyramid with a square base is cut by a plane that is parallel to its base and $2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. The original answer is in the form k + m\sqrt{2}. Please provide the value of k + m, where k and m are integers.
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Context:
Given a triangle with internal angles A, B, C and their respective opposite sides a, b, c, if a = $\sqrt {2}$, $b^2 - c^2 = 6$, find the maximum area of triangle ABC when angle A is at its maximum.
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Context:
Calculate the covariance between the first and third order statistics of three independent random variables with uniform distribution in $[0,1]$, given that the variance of the $k$th order statistic is $\frac{k(4-k)}{80}$.
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Context:
A book is bought for $600 and has an additional government tax of 5%, a shipping fee of $20, and a 3% seller's discount. If the book is sold for $624, what is the profit in percentage?
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Context:
The area enclosed by the curve $y=\sin x (0\leqslant x\leqslant \pi)$ and the line $y=\frac{1}{2}$ is $\_\_\_\_\_\_\_.$
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Context:
Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at 4¢ per glob and $J$ blobs of jam at 5¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is \$ 2.53. Assume that $B$, $J$, and $N$ are positive integers with $N>1$. What is the cost of the jam Elmo uses to make the sandwiches?
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Context:
Four students participate in a knowledge contest, each student must choose one of the two questions, A or B, to answer. Correctly answering question A earns 60 points, while an incorrect answer results in -60 points. Correctly answering question B earns 180 points, while an incorrect answer results in -180 points. The total score of these four students is 0 points. How many different scoring situations are there in total?
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Context:
If the function $f(x) = |\frac{e^{x}}{2} - \frac{a}{e^{x}}| (a \in \mathbb{R})$ is monotonically increasing in the interval $[1, 2]$, the range of the real number $a$ is _______.
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Context:
从正整数列 $1,2,3,4, \cdots$ 中划去 3 及 4 的倍数, 但其中凡是 5 的倍数均保留 (例如 $15,20,60,120$ 均不划去), 划完后, 剩下的数依原有次序组成一个数列: $a_1=1, a_2=$ $2, a_3=5, a_4=7, \cdots$. 试求 $a_{2003}$.
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Context:
Determine the number of subsets $S$ of $\{1,2, \ldots, 1000\}$ that satisfy the following conditions: - $S$ has 19 elements, and - the sum of the elements in any non-empty subset of $S$ is not divisible by 20 .
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Context:
The coordinates of the point symmetric to the point (-1, 1) about the line x - y - 1 = 0 are _______.
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Context:
A standard six-sided die is rolled twice, and the uppermost numbers are recorded as m and n. When m is either 2 or 4, find the probability that m + n > 5. Express your answer as a simplified fraction.
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Context:
If angle $α$ is in the second quadrant, then point $P(\tan α, \sec α)$ is located in the \_\_\_\_\_\_ quadrant.
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Context:
Sisyphus has three boxes of stones. Each hour, he moves a stone from one box to another. For each transfer, he earns coins equal to the number of stones in the box from which the stone is taken minus the number of stones in the recipient box, excluding the stone just moved. If this number is negative, he pays that amount. After 1000 years, all stones are back in their initial boxes. What is the greatest possible earning of Sisyphus at that moment?
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Context:
In the cells of a $3 \times 3$ square, the numbers $0,1,2, \ldots, 8$ are placed. It is known that any two consecutive numbers are located in neighboring (side-adjacent) cells. What number can be in the central cell if the sum of the numbers in the corner cells is 18?
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Context:
Juku has the first $100$ volumes of the Harrie Totter book series at his home. For every $i$ and $j$, where $1 \le i < j \le 100$, call the pair $(i, j)$ reversed if volume No. $j$ is before volume No. $i$ on Juku’s shelf. Juku wants to arrange all volumes of the series in one row on his shelf in such a way that there do not exist numbers $i, j, k$, where $1 \le i < j < k \le 100$, such that pairs $(i, j)$ and $(j, k)$ are both reversed. Find the largest number of reversed pairs that can occur under this condition.
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Context:
给定正整数 $6$ 。求最大的实数 $M$ ，使得对于任意的实数 $x_{i}(1 \leqslant i \leqslant 2 n+1)$ ，若 $x_{1} \leqslant x_{2} \leqslant \cdots \leqslant x_{2 n+1}$, 则 $\sum_{i=1}^{2 n+1} x_{i}^{2} \geqslant M x_{n+1}^{2}$.
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Context:
Let  $ABCD$  be a cyclic quadrilateral with circumradius  $100\sqrt{3}$  and  $AC=300$ . If  $\angle DBC = 15^{\circ}$ , then find  $AD^2$ .

*Proposed by Anand Iyer*
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Context:
In a complex division problem, you are given that the quotient is 256, the divisor is 3892, and the remainder is 354. Furthermore, you must multiply the quotient by 3, add the product to the remainder, and then subtract 1500 from this result. What is the dividend of this operation?
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Context:
Let's call a number greater than 25 semi-prime if it is the sum of two distinct prime numbers. What is the maximum number of consecutive natural numbers that can be semi-prime?
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Context:
Determine the number of group homomorphisms from \( S_3 \) to \( S_3 \).
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Context:
In the triangle \(ABC\), \(AB = AC\), \(\angle A = 90^\circ\). \(D\) is the midpoint of \(BC\), \(E\) is the midpoint of \(AC\), and \(F\) is a point on \(AB\) such that \(BE\) intersects \(CF\) at \(P\). Points \(B\), \(D\), \(P\), and \(F\) lie on a circle. Let \(AD\) intersect \(CP\) at \(H\). Given that \(AP = \sqrt{5} + 2\), find the length of \(PH\).
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Context:
Determine the cardinality of the set of all subsets \( A \subset \mathbb{R} \) such that \( \mathbb{R} \setminus A \) is countably infinite.
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Context:
Calculate the integral \( \frac{1}{2\pi i}\oint_{|z|=1}\frac{2(1-\cos(z))e^z}{z^4}dz \).
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Context:
If every $k-$element subset of $S=\{1,2,\dots , 32\}$ contains three different elements $a,b,c$ such that $a$ divides $b$, and $b$ divides $c$, $k$ must be at least ?
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Context:
Given the hyperbola $C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ (where $a > 0$ and $b > 0$), with its left and right foci being $F_1$ and $F_2$, respectively. Suppose that there exists a point $P$ on the hyperbola $C$ such that the triangle $PF_1F_2$ is isosceles with $\cos\angle PF_1F_2 = \frac{1}{8}$. Find the eccentricity of the hyperbola.
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Context:
Let $S$ be the set of all polynomials $Q(x,y,z)$ with coefficients in \{0,1\} such that there exists a homogeneous polynomial $P(x,y,z)$ of degree 2016 with integer coefficients and a polynomial $R(x,y,z)$ with integer coefficients so that
\[P(x,y,z) Q(x,y,z) = P(yz,zx,xy) + 2R(x,y,z)\]
and $P(1,1,1)$ is odd. Determine the size of $S$.

Note: A homogeneous polynomial of degree $d$ consists solely of terms of degree $d$.
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Context:
Evaluate the integral \( \frac{1}{2\pi}\int\limits_{-\infty}^{0}\int\limits_{x}^{\infty}e^{-\frac{(x^2+y^2)}{2}}dydx \) without using polar coordinates.
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Context:
Two logs of length 10 are laying on the ground touching each other. Their radii are 3 and 1, and the smaller log is fastened to the ground. The bigger log rolls over the smaller log without slipping, and stops as soon as it touches the ground again. The volume of the set of points swept out by the larger log as it rolls over the smaller one can be expressed as $n \pi$, where $n$ is an integer. Find $n$.
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Context:
Example 1 A city has $n$ high schools, the $i$-th high school sends $c_{i}$ students $\left(1 \leqslant c_{i} \leqslant 39\right)$ to watch a ball game at the gymnasium, where $\sum_{i=1}^{n} c_{i}=1990$. Each row in the stand has 199 seats, and it is required that students from the same school sit in the same row. Question: What is the minimum number of rows that need to be arranged so that all students can definitely be seated?
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Context:
Given sets A = {x ∈ ℕ | -1 < x < 3} and B = {x | -2 ≤ x < 2}, find the intersection of sets A and B. Express your answer as a set, including all elements of the intersection.
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Context:
Given $x^3y = k$ for a positive constant $k$, find the percentage decrease in $y$ when $x$ increases by $20\%$.
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Context:
Given points $P_{1}(2,-1)$, $P_{2}(0,5)$, and point $P$ is on the extension line of $P_{1}P_{2}$, with $| \overrightarrow{P_{1}P}|=2| \overrightarrow{PP_{2}}|$, find the coordinates of point $P$.
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Context:
Determine the number of ways to select a positive number of squares on an \(8 \times 8\) chessboard such that no two lie in the same row or the same column and no chosen square lies to the left of and below another chosen square.
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Context:
Find the mass percentage of H in Ammonia
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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. Find the volume of the original cone.
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Context:
Determine the number of solutions to the equation \[{(\sqrt{3}\sin x+\cos x)}^{\sqrt{\sqrt{3}\sin{2x}-\cos{2x}+2}}=4.\]
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Context:
Consider the family of holomorphic functions $\mathcal F = \{f$ holomorphic in $\Bbb D$ with $f(\Bbb D)\subset\Bbb D,$ $f\left(\frac12\right)=f'\left(\frac12\right)=0\}$, where $\Bbb D$ is the unit disc. Compute $\sup_{f\in\mathcal F}|f(0)|$.
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Context:
Johnny TV makes 25 percent more movies than L&J Productions each year, with a 5 percent annual growth rate in movie production. If L&J Productions produces 220 movies in the first year with a 3 percent annual growth rate, how many movies do the two production companies produce combined in five years?
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Context:
Determine the number of natural numbers \( n \) such that \( 0 \leq n \leq 10,000,000 \) and the sum of the digits of \( n \) is at most 34.
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Context:
Find the counting formula for the number of n-covers of an n-element set, denoted as \( C_{n, n} \).
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Context:
In $\triangle ABC$, $3\sin A + 4\cos B = 6$, $4\sin B + 3\cos A = 1$, find the size of angle $C$.
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Context:
Find the smallest three-digit divisor of the number \(1\underbrace{00\ldots 0}_{100\text{ zeros}}1\underbrace{00\ldots 0}_{100\text{ zeros}}1.\)
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Context:
Given $4^{a}=9^{b}=k$ and $\frac{1}{a} + \frac{1}{b} = 2$, find the value of $k$.
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Context:
In the "Joy Tonight" program, a TV station took out two mailboxes, which contained letters from audiences who performed well in two consecutive quizzes. There are 30 letters in mailbox A and 20 letters in mailbox B. Now, the host draws lots to determine the lucky audience. If one lucky star is determined first, and then one lucky partner is determined from each mailbox, how many different outcomes are there?
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Context:
已知实数 $x_{1}, x_{2}, \cdots, x_{2018}$ 满足：对任意的 $1 \leqslant i<j \leqslant 2018$, 均有 $x_{i}+x_{j} \geqslant(-1)^{i+j}$. 求 $\sum_{i=1}^{2018} i x_{i}$ 的最小值。原始答案为$\frac{m}{n}$的格式，请给出m + n的值。
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Context:
The polynomial $$P(x)=(1+x+x^2+\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\left[\cos(2\pi\alpha_k)
+i\sin(2\pi\alpha_k)\right]$, $k=1,2,3,\ldots,34$, with $0<\alpha_1\le\alpha_2\le\alpha_3\le\dots\le\alpha_{34}<1$ and $r_k>0$. Find $\alpha_1+\alpha_2+\alpha_3+\alpha_4+\alpha_5.$
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Context:
Given a regular pentagon with side length 1, draw the two diagonals from one vertex. This creates a net for a tetrahedron with a regular triangular base. What is the volume of this tetrahedron?
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Context:
How many triples of integers $(a, b, c)$ with $-10 \leq a, b, c \leq 10$ satisfy $a^2 + b^2 + c^2 = (a + b + c)^2$?
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Context:
Consider the Hopf bundle \( \mathbb{S}^3 \to \mathbb{S}^2 \) and the usual irreducible representations \( \rho_n: U(1) \to \operatorname{Gl}(\mathbb{C}) \) for \( n \in \mathbb{Z} \). Given that the first Chern class of the Hopf bundle is \( c_1(\operatorname{Hopf}) = -1 \), determine the first Chern class \( c_1\left(\operatorname{Hopf} \times_{\rho_n} \mathbb{C}\right) \).
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Context:
初始时黑板上有六个向量(1,0,0),\ (-1,0,0),\ (0,1,0),\ (0,-1,0),\ (0,0,1),\ (0,0,-1).一次操作可以将黑板上的两个向量 v,w 换为 \df{1}{\sqrt{2}}(v+w), \df{1}{\sqrt{2}}(v-w). 求若干次操作后，黑板上所有向量之和的模的最大可能值.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
Given the polynomial

\[ P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_{1} x + a_{0} \]

where each coefficient \( a_i \) belongs to the interval \([100, 101]\). For what minimum \( n \) can such a polynomial have a real root?
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Context:
There is convex $2009$-gon on the plane.

b) Find the greatest number of vertices of $2009$-gon such that among any three of them there is one that is not connected with other two by side.
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Context:
A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows:

- It has  $3$  digits.
- It is a palindrome.
- The middle digit is a multiple of  $4$ .
- It is divisible by  $11$ .

What is the sum of all possible numbers of cows that the herder has?
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Context:
Given the function $f(x)=2x-\cos x$, the sequence $\{a_n\}$ is an arithmetic sequence with a common difference of $\frac{\pi}{8}$. If $f(a_1)+f(a_2)+f(a_3)+f(a_4)+f(a_5)=5\pi$, find the value of $[f(a_3)]^2-a_1a_5$.
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Context:
Determine the leftmost three digits of the number \[1^1 + 2^2 + 3^3 + \ldots + 999^{999} + 1000^{1000}.\]
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Context:
In $\triangle ABC$, $AB = BC = 2$, $\angle ABC = 120^\circ$. A point $P$ is outside the plane of $\triangle ABC$, and a point $D$ is on the line segment $AC$, such that $PD = DA$ and $PB = BA$. Find the maximum volume of the tetrahedron $PBCD$.
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Context:
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and the area is $S$. Given that $a^2 + b^2 - c^2 = 4\sqrt{3}S$, find the value of $C$.
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Context:
Evaluate the limit: $$\lim_{x \to 0} \frac{(1 + \sin x + \sin^2 x)^{1/x} - (1 + \sin x)^{1/x}}{x}$$
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Context:
Right triangle \( XYZ \) has a right angle at \( Y \) with \( XY = 228 \) and \( YZ = 2004 \). Angle \( Y \) is trisected, and the angle trisectors intersect \( XZ \) at \( P \) and \( Q \) such that \( X, P, Q, Z \) lie on \( XZ \) in that order. Find the value of \((PY + YZ)(QY + XY) \).
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Context:
Two transformations are said to commute if applying the first followed by the second
 gives the same result as applying the second followed by the first. Consider these four
 transformations of the coordinate plane:
 - A translation 2 units to the right
 - A \( {90}^{ \circ  } \) - rotation counterclockwise about the origin.
 - A reflection across the \( x \) -axis, and
 - A dilation centered at the origin with scale factor 2.
 Of the 6 pairs of distinct transformations from this list, how many commute?
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Context:
Let \( p, q, r \) be the roots of the polynomial \( x^3 - 8x^2 + 14x - 2 = 0 \). Define \( t = \sqrt{p} + \sqrt{q} + \sqrt{r} \). Find \( t^4 - 16t^2 - 12t \).
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Context:
A line passes through the intersection of the medians of triangle \( ABC \) and intersects the sides \( AB \) and \( AC \).

The distances from vertices \( B \) and \( C \) to this line are \( b \) and \( c \) respectively. Find the distance from vertex \( A \) to this line.
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Context:
A robot starts at the origin of the number line and moves along the direction of the number line, following a pattern where it advances 4 steps then retreats 3 steps. The robot moves one step per second, each step being one unit length. Let $x_n$ denote the position of the robot on the number line at second $n$ (for example, $x_4=4$, $x_5=3$, $x_7=1$). What is the value of $x_{2007} - x_{2011}$?
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Context:
The edge of a cube is 7 cm. If the edge length is increased by 15%, find the new surface area of the cube.
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Context:
For a positive integer $p$, define the positive integer $n$ to be $p$<i>-safe</i> if $n$ differs in absolute value by more than $2$ from all multiples of $p$. For example, the set of $10$-safe numbers is $\{ 3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23, \ldots\}$. Find the number of positive integers less than or equal to $10,000$ which are simultaneously $7$-safe, $11$-safe, and $13$-safe.
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Context:
Determine the set of real numbers \( p \) for which the integral \( \int_{\mathbb{R}^n}|\log|x||^p \, dx \) converges in \( \mathbb{R}^n \).
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Context:
设$a$、$b$、$c$、$a＋b-c$、$a＋c-b$、$b＋c-a$、$a＋b＋c$是7个两两不同的质数，且$a$、$b$、$c$中有两数之和是800．设$d$是这7质数中最大数与最小数之差．求$d$的最大可能值．
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Context:
Given that the solution set of the inequality $ax^{2}+bx+c < 0$ is $((-∞,1)∪(3,+∞))$, determine the solution set of the inequality $cx^{2}+bx+a > 0$.
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Context:
All positive integers whose digits add up to 14 are listed in increasing order: $59, 68, 77, ...$. What is the fifteenth number in that list?
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Context:
Let $K$ be a number field where the rational prime $2$ splits as $(2)\mathcal{O}_K = \mathfrak{P}_1\mathfrak{P}_2$. Determine the intersection $K_{\mathfrak{P}_1} \cap K_{\mathfrak{P}_2}$, where $K_{\mathfrak{P}_i}$ denotes the completion of $K$ at the prime $\mathfrak{P}_i$.
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Context:
$n$ trains circulate in one direction at equal distances along a circular track. Stations $A$, $B$, and $C$ are located at the vertices of an equilateral triangle along the track, labeled in the direction of movement. Ira enters at station $A$, and Lexa enters at station $B$ at the same time to board the nearest trains. It is known that if they enter the stations when the driver Roma is passing through the forest, Ira will board the train before Lexa. In all other cases, Lexa will board before Ira or simultaneously with her. Determine the portion of the track that passes through the forest.
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Context:
On the standard $12$ hour clock, each number from $1$ to $12$ is replaced by the sum of its divisors. On this new clock, what is the number of degrees in the measure of the non-reflex angle between the hands of the clock at the time when the hour hand is between $7$ and $6$ while the minute hand is pointing at $15$?
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Context:
Find the minimum value of the function \( f = \sin\theta_1 + \sin\theta_2 \), given that \( \theta_1 + \theta_2 + \phi = \pi \), where \( \phi \) is a variable in the range \((0, \pi)\). Here, \( \theta_1, \theta_2, \phi \) represent the interior angles of a triangle formed by connecting a point on an ellipse to its focus, with \( \phi \) being the angle that sees the major axis of the ellipse.
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Context:
Let \( S \) be the surface defined by \( \{(x,y,z) : x^{2} + y^{2} + 2z = 2, z \geq 0 \} \). Given the vector field \( \mathbf{F} = (y, xz, x^{2} + y^{2}) \), find the net flux of \( \mathbf{F} \) through the surface \( S \). Assume \( \mathbf{n} \) is the outward normal vector.
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Context:
Calculate the Laplace Transform of the function \(-e^{-at}u(-t)\).
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Context:
How many mutually non-isomorphic graphs are there with 30 vertices and 3 edges?
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Context:
Determine the number of distinct integers that can be obtained from the function \( f(x) = [3x] + [4x] + [5x] + [6x] \) for \( 0 \leq x \leq 100 \), where \([x]\) denotes the greatest integer less than or equal to \( x \).
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Context:
Determine the Euler characteristic of a sphere with three non-intersecting holes cut out.
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Context:
Determine the dimension of the field of rational functions \( \mathbb{Q}(t) \) as a vector space over the field \( \mathbb{Q}(t^2) \).
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Context:
Determine the smallest integer $n$ such that for all $x \in \left(-\frac{1}{2}, \frac{1}{2}\right)$, the inequality \[ \left|e^x - \sum_{k=0}^n \frac{x^k}{k!}\right| \leq \frac{|e^x|}{10^{16}} \] holds.
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Context:
Two players take turns drawing one number randomly from the set \( \{1, 2, 3, 4\} \), with each number having a probability of \( \frac{1}{4} \) of being drawn. The winner is the player who draws a number that makes the sum of all numbers drawn by both players divisible by 3. What is the probability that the first player will be the winner?
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Context:
Let $A$ be a square matrix of order $n$ with real entries such that $A^3 + I = 0$. Determine the value of $\operatorname{trace}(A)$. \(\text{Note: } I \text{ is the identity matrix of the same order as } A.\)
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Context:
Determine the asymptotic probability that the random matrix $A(p, n)$ is non-singular as $n$ approaches infinity, given that $p = 1/2$. Provide your answer as a probability expression or value.
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Context:
Calculate the condition number of the matrix \(A = \begin{bmatrix} 0 & 0 & -10^4 & 0 \\ 0 & 0 & 0 & -10 \\ 0 & 10^{-3} & 0 & 0 \\ 10^{-2} & 0 & 0 & 0 \\ \end{bmatrix}\) by determining \(\|A\|\), \(\|A^{-1}\|\), and their product.
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Context:
Pentagon \(ABCDE\) is inscribed in a circle. The distances from point \(A\) to the lines \(BC\), \(CD\), and \(DE\) are \(a\), \(b\), and \(c\) respectively. Calculate the distance from vertex \(A\) to the line \(BE\).
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Context:
Determine the values of \( x > 0 \) for which the series \( \sum_{n=1}^\infty \frac{\ln(1+nx)}{nx^n} \) converges.
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Context:
Calculate the expected length of runs in a sequence of 52 binary digits (0s and 1s) that contains an equal number of 0s and 1s.
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Context:
Calculate the volume of the solid enclosed by the surface \((x^2 + y^2 + z^2)^2 = x\).
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Context:
Find the values of \(\alpha\) for which the series \(\sum_{k=0}^{\infty}(-1)^{k}\frac{x^{2k}}{(2k)!^{\alpha}}\) converges for all real numbers \(x\).
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Context:
Evaluate the double integral \( \iint \frac{x^2}{y^3} \, dA \) over the region enclosed by the curves \( y = 2x \), \( y = x \), \( y = x^2 \), and \( y = 2x^2 \) using a change of variables.
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Context:
In a circular arrangement, particles are initially placed in the sequence 0, 1, -0, and -1. Each second, a new particle is randomly added to the sequence, chosen equally among the four types: 0, 1, -0, and -1. If a 0 particle neighbors a -0 or a 1 particle neighbors a -1, both particles disappear. What is the probability that eventually the entire sequence will vanish?
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Context:
Determine the intervals where the Fourier series of the function \(f(x) = \sin(x)\) for \(-2 < x \leq 2\) converges uniformly.
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Context:
Determine the points in the complex plane where the function \( f(z) = e^z(z-\bar{z})^2 \) is differentiable.
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Context:
Evaluate the integral \( \int_a^\infty \frac{1}{x} e^{-x} \,dx \).
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Context:
Find the maximum value of \( u_x(0) \) for a harmonic function \( u: D \rightarrow [0,1] \), where \( D \subset \mathbb{R}^2 \) is the open unit disk centered at 0.
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Context:
Find the highest degree $ k$ of $ 1991$ for which $ 1991^k$ divides the number\[ 1990^{1991^{1992}} + 1992^{1991^{1990}}.\]
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Context:
Determine the integer \( D \) such that the quadratic subfield of \( \mathbb{Q}(\zeta_7) \), where \( \zeta_7 = e^{i2\pi/7} \) is a 7th root of unity, can be expressed as \( \mathbb{Q}(\sqrt{D}) \).
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Context:
Consider all labeled trees with nodes \(\{1, 2, 3, \ldots, n\}\). How many of these trees contain the edge \(\{1, 2\}\)?
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Context:
Given a random sample \(X_1, X_2, \dots, X_n\) from a distribution with probability density function (p.d.f.) \[f(x, \theta) = e^{-(x - \theta)}, \quad \theta < x < \infty, \quad -\infty < \theta < \infty,\] find the sufficient statistic for the parameter \(\theta\).
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Context:
Determine the minimum order \( n \) of an \( n \)-th order homogeneous differential equation with constant coefficients that has the particular solutions \( t^3 \) and \( t(e^t)(\sin(t)) \). What is the minimum value of \( n \)?
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Context:
Given the vectors $x_1=(1,-1,1,0)^T$, $x_2=(1,1,0,1)^T$, $x_3=(1,3,1,0)^T$, and $x_4=(1,1,1,0)^T$, determine the rank of the matrix $A = \sum_{i=1}^4 x_ix_i^T$ without explicitly computing $A$. Note that $x_1 = 2x_4 - x_3$. What is the rank of $A$?
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Context:
Integrate the vector field \( F(x,y,z) = \frac{1}{\sqrt{x^2+y^2}}(y,-y,1) \) over the surface of the paraboloid defined by \( z = 1 - x^2 - y^2 \) for \( 0 \leq z \leq 1 \), which corresponds to the region \( x^2 + y^2 \leq 1 \).
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Context:
Determine the value of $p$ for which the integral \( \int \frac{dx}{\sqrt{x+2}(x-3)} \) equals \( \frac{2}{\sqrt{3}}\arctan\left(\frac{\sqrt{x+2}}{\sqrt{3}} \right) \).
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Context:
From the consecutive natural numbers \(1, 2, 3, \ldots, 2014\), select \(n\) numbers such that no two of the selected numbers include one number being seven times another. What is the maximum value of \(n\)?
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Context:
Calculate the limit: \[ \lim_{n\to\infty}\int_{0}^{\infty}\frac{e^{-x}\cos(x)}{\frac{1}{n}+nx^2}dx \]
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Context:
Determine the number of roots of the polynomial \( f(z) = 1 + 6z^3 + 3z^{10} + z^{11} \) that lie within the annulus defined by \( 1 < |z| < 2 \).
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Context:
Find the residue of the function \( f(z) = \frac{1}{z^2+1} \) at the point \( z = i \).
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Context:
Determine the number of roots of the equation \(z^4 + z^3 + 1 = 0\) that lie in the first quadrant of the complex plane using Rouche's Theorem.
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Context:
Find the domain of definition for the function \( g(x) = \int_{0}^{\infty}\frac{t^x}{1+t^2 \cos^2 t}dt \).
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Context:
Find a tight upper bound, expressed as an arithmetic function \( f(n, k) \), for the length of the shortest word in the language generated by a context-free grammar \( G = (\Sigma, N, R, S) \), where \(|N| = n\) is the number of non-terminals, \(k\) is the maximum length of a production rule, and \(L(G)\) is non-empty.
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Context:
Find a closed-form expression for the recursive formula \( f(n) = 1+\frac{1}{n}\sum_{i = 0}^{n - 1} f(i) \) with the base case \( f(0) = 0 \).
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Context:
Evaluate the line integral \( \int_{C_2} \frac{x\vec{e_1}+y\vec{e_2}}{x^2+y^2} d\vec{r} \) where \( C_2 \) is a square with vertices \((1,1), (-1,1), (-1,-1), (1,-1)\).
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Context:
Determine the greatest integer $a$ such that the inequality \(x^2 - ax - \ln x + e^{\sin x} - 1 > 0\) is satisfied for all \(x > 0\).
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Context:
Calculate the residue of \( \frac{1}{(e^z-e)^3} \) at \( z = 1 \).
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Context:
A tank with a diameter of $40\,\rm m$ and a height of $15\,\rm m$ is initially $50\%$ full with $92$-Octane fuel. The fuel is being drained at a rate of $500\,\rm m^3$ per hour, while $87$-Octane fuel is being added at a rate of $3000\,\rm m^3$ per hour. Assuming the fuel mixes evenly, determine how many hours it will take for the fuel in the tank to become $91$-Octane.
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Context:
Let \(X_{1}, X_{2}, \ldots\) be a sequence of independent random variables, where \(X_{2k-1}\) follows a Rademacher distribution with \(P(X_{2k-1}=1) = P(X_{2k-1}=-1) = \frac{1}{2}\), and \(X_{2k}\) follows a Cauchy distribution with probability density function \(f(x) = \frac{1}{\pi(1+x^2)}\). Determine the limit:

\[ \lim_{n \to \infty} P\left(\frac{X_{1} + \cdots + X_{2n}}{\sqrt{n}} \ge 2 \right) \]
Please reason step by step, and put your final answer within \boxed{}.


Context:
What is the minimum number of orthogonal matrices required to express the matrix $M_{i,j}$, which is a standard basis matrix of size $n > 1$, as a linear combination of those orthogonal matrices?
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Context:
Evaluate the double integral \( \iint_{D} xy \, dA \) where \( D \) is the region bounded by the lines \( y=x \), \( y=x+4 \), \( x=0 \), and \( x=2 \). Use the change of variables \( x=2u \) and \( y=2u+v \). Express your answer as a decimal.
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Context:
Find the minimum value of the sum \(r_1 + r_2\) for two distinct non-negative integers \(r_1\) and \(r_2\) such that the binomial coefficient \(\binom{1999}{r}\) is even for both \(r_1\) and \(r_2\).
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Context:
Let $A$ be an $n \times n$ matrix with integer entries, each with an absolute value bounded by a constant $C$. Determine whether the entries of the inverse matrix $A^{-1}$ can grow more rapidly than exponentially with respect to $n$, or if they are bounded by $(C + O(1))^n$.
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Context:
An Abelian group G is generated by elements x and y, where the order of x is 16 and the order of y is 24. Given that x^2 = y^3, determine the order of the group G.
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Context:
Given a connected graph with an independence number $k > 1$, determine the maximum possible value for the path number of the graph in terms of $k$. The independence number of a graph $\mathbf{G}$ is the maximum number of pairwise non-adjacent vertices in $\mathbf{G}$. Express your answer in terms of $k$. \( \text{path number} = f(k) \)
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let \((a_1, a_2, \ldots, a_{2014})\) be a permutation of \((1, 2, 3, \ldots, 2014)\). Define \(S_k = a_1 + a_2 + \cdots + a_k\) for \(k = 1, 2, \ldots, 2014\). Find the maximum possible size of the set \(A = \{S_i \mid S_i \equiv 1 \pmod{2}\}\).
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Context:
Find the abelianization of the group \( G = \langle a, b \mid ab^2 = b^2a,\ a^4 = b^3 \rangle.\)
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Context:
Given that $689\Box\Box\Box20312 \approx 69$ billion (rounded), find the number of ways to fill in the three-digit number.
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Context:
Five diamonds are to be embedded at five equally spaced positions around a circle. If there are three different colors of diamonds available, how many distinct ways can the diamonds be embedded, considering that the positions are indistinguishable?
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Context:
Determine the number of solutions to the inequality \( \frac{p_{n+1} - p_n}{2} \ge \pi(p_n) \), where \( p_n \) and \( p_{n+1} \) are consecutive primes and \( \pi(n) \) is the prime counting function.
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Context:
A park contains two adjacent equilateral triangles formed by five straight paths, each of length \(a\). Additionally, six shorter paths, each of length \(b\), connect the centers of these triangles to their vertices. Determine the shortest route a watering machine must take to travel along all the paths and return to the starting point \(A\).
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Context:
Find the smallest prime number \( p \) such that \( p \), \( p+1 \), and \( p-1 \) each have at least three distinct prime factors.
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Context:
For a group with prime order $p$, determine the conditions under which the function $f(k) = k^n$ is invertible.
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Context:
Calculate \( \frac{dF(x)}{dx} \bigg|_{x=0} \) for the function \( F(x) = \int_0^2 \sin(x+l)^2\ dl \).
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Context:
Let $x,y,z$ be positive real numbers that satisfy at least one of the inequalities, $2xy>1$, $yz>1$. Find the least possible value of
$$xy^3z^2+\frac{4z}{x}-8yz-\frac{4}{yz}$$
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Context:
Solve the inequality \( \frac{\left(\left|x^{2}-2\right|-7\right)(|x+3|-5)}{|x-3|-|x-1|} > 0 \). Find and write the greatest integer that is a solution to the inequality.
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Context:
Find the abelianization of the free product of the groups $\mathbb{Z}_4$ and $\mathbb{Z}_5$. Express your answer in terms of known groups.
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Context:
Let $x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$, $x_7$ be natural numbers, and $x_1 < x_2 < x_3 < \ldots < x_6 < x_7$, also $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is.
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Context:
On the side $AC$ of triangle $ABC$, a circle is constructed such that it intersects sides $AB$ and $BC$ at points $D$ and $E$ respectively, and is constructed with $AC$ as its diameter. The angle $\angle EDC$ is equal to $30^\circ$, $AE = \sqrt{3}$, and the area of triangle $DBE$ is related to the area of triangle $ABC$ as \(1:2\). Find the length of segment $BO$, where $O$ is the point of intersection of segments $AE$ and $CD$.
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Context:
We will call a natural number interesting if all its digits are different, and the sum of any two adjacent digits is the square of a natural number. Find the largest interesting number.
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Context:
Ship \( P \) spots ship \( Q \), which is moving in a direction perpendicular to \( PQ \), maintaining its course. Ship \( P \) chases \( Q \), always heading directly towards \( Q \); the speed of both ships at any moment is the same (but can change over time). It is clear without calculations that \( P \) travels along a curved path; if the chase lasts long enough, the trajectory of the pursuing ship and the trajectory of the fleeing ship eventually become nearly identical. What will the distance \( PQ \) be then, if initially it was 10 nautical miles?
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Context:
How many positive three-digit integers with a $7$ in the units place are divisible by $21$?
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Context:
In a right triangle $ABC$ (right angle at $C$), the bisector $BK$ is drawn. Point $L$ is on side $BC$ such that $\angle C K L = \angle A B C / 2$. Find $KB$ if $AB = 18$ and $BL = 8$.
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Context:
$在某次投篮测试中，有两种投篮方案.方案甲：先在A点投篮一次，以后都在B点投篮；方案乙：始终在B点投篮.每次投篮之间相互独立.某选手在A点命中的概率为 \frac{3}{4} ，命中一次得3分，没有命中得0分；在B点命中的概率为 \frac{4}{5} ，命中一次得2分，没有命中得0分.用随机变量\xi 表示该选手一次投篮测试的累计得分，如果\xi 的值不低于3分，则认为通过测试并停止投篮，否则继续投篮，但一次测试最多投篮3次.$
$若该选手选择方案甲，求测试结束后所得分\xi的数学期望；$
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Context:
Given $f(x) = \frac{x^2}{2x+1}$, $f_1(x) = f(x)$, $f_n(x) = \overset{\overset{f(\cdots f(x)\cdots)}{}}{n\text{个}f}$, then $f_{10}\left(\frac{1}{2}\right) = \_\_\_\_\_\_$.
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Context:
A plane is drawn through the vertex of the base and the midpoints of two lateral edges of a regular triangular pyramid. Find the ratio of the lateral surface area of the pyramid to the area of its base, given that the cutting plane is perpendicular to one of the lateral faces.
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Context:
Consider the set $S$ of the eight points $(x,y)$ in the Cartesian plane satisfying $x,y \in \{-1, 0, 1\}$ and $(x,y) \neq (0,0)$. How many ways are there to draw four segments whose endpoints lie in $S$ such that no two segments intersect, even at endpoints?
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Context:
Given the hyperbola \( P : \frac{x^{2}}{9}-\frac{y^{2}}{16}=1 \) with left and right foci \( B \) and \( C \), point \( A \) lies on \( P \). \( I \) is the incenter of triangle \( ABC \) and the line \( AI \) passes through the point \( (1,0) \). If \( \overrightarrow{A I}=x \overrightarrow{A B}+y \overrightarrow{A C} \), then \( x+y \) equals ____.
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Context:
A trapezoid $ABCD$, where side $AB$ is parallel to side $CD$, is divided by its diagonals, which intersect at point $M$, into four parts. Determine its area, given that the triangle $AMD$ has an area of $8 \ \mathrm{cm}^{2}$ and the triangle $DCM$ has an area of $4 \ \mathrm{cm}^{2}$.
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Context:
Find the number of ordered quadruples of positive integers $(a, b, c, d)$ such that $a, b, c$, and $d$ are all (not necessarily distinct) factors of 30 and $abcd>900$.
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Context:
Beatrix is going to place six rooks on a $6 \times 6$ chessboard where both the rows and columns are labeled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The $value$ of a square is the sum of its row number and column number. The $score$ of an arrangement of rooks is the least value of any occupied square. The average score over all valid configurations is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
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Context:
A natural number $n$ is called a "good number" if the column addition of $n$, $n+1$, and $n+2$ does not produce any carry-over. For example, 32 is a "good number" because $32+33+34$ does not result in a carry-over; however, 23 is not a "good number" because $23+24+25$ does result in a carry-over. The number of "good numbers" less than 1000 is \_\_\_\_\_\_.
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Context:
Given a tetrahedron with four faces having areas \( S_{1}, S_{2}, S_{3}, S_{4} \), let \( S \) be the largest of these areas. Define \( \lambda = \frac{\sum_{i=1}^{4} S_{i}}{S} \). Find the range of values that \( \lambda \) must satisfy. Express your answer as an inequality.
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Context:
Calculate the flux of the vector field \(\mathbf{a}=x \mathbf{i}+y \mathbf{j}+\sqrt{x^{2}+y^{2}-1} \mathbf{k}\) through the external side of the one-sheet hyperboloid \(z=\sqrt{x^{2}+y^{2}-1}\), bounded by the planes \(z=0\) and \(z=\sqrt{3}\).
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Context:
A sequence \( a_1, a_2, \cdots, a_n, \cdots \) is defined such that \( S(a_i) \) denotes the sum of all digits of \( a_i \). For example, \( S(22) = 2 + 2 = 4 \). Given \( a_1 = 2017 \), \( a_2 = 22 \), and \( a_n = S(a_{n-1}) + S(a_{n-2}) \), find the value of \( a_{2017} \).
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Context:
The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 55 and 31 respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).
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Context:
方程$\sin x = \frac { x } { 100 }$的实根个数有 个.
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Context:
Compute $\sin 12^\circ \sin 36^\circ \sin 72^\circ \sin 84^\circ.$
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Context:
The diagonals of a quadrilateral \(ABCD\) inscribed in a circle of radius \(R\) intersect at point \(M\). It is known that \(AB = BC = a\) and \(BD = m\).

Find the radius of the circumcircle of triangle \(BCM\).
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Context:
How many roots does the equation \(8x(1 - 2x^2)(8x^4 - 8x^2 + 1) = 1\) have on the interval \([0, 1]\)?
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Context:
What is the minimum number of convex pentagons needed to form a convex 2011-gon?
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Context:
Simplify the following fraction:

$$
\frac{x y^{2}+2 y z^{2}+y z u+2 x y z+2 x z u+y^{2} z+2 z^{2} u+x y u}{x u^{2}+y z^{2}+y z u+x u z+x y u+u z^{2}+z u^{2}+x y z}
$$
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Context:
There is a reservoir that receives a certain amount of water per unit of time, and at the same time, water is released from it. With the current inflow and outflow rates, the water in the reservoir can last for 40 days. Due to recent rainfall at the source, the inflow of water to the reservoir has increased by 20%. If the outflow rate is increased by 10%, the water can still last for 40 days. If the water is released at the original rate, how many days can the water last?
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Context:
Right triangles $T_1$ and $T_2$, have areas of 1 and 2, respectively. A side of $T_1$ is congruent to a side of $T_2$, and a different side of $T_1$ is congruent to a different side of $T_2$. What is the square of the product of the lengths of the other (third) sides of $T_1$ and $T_2$?
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Context:
Let $f$ be the function defined on the open unit disk:

$$f(x)=\sum_{n=0}^{+ \infty} x^{n^2}$$

Find the limit of $f(x)$ as $x$ approaches $-1$.
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Context:
A positive integer is called primer if it has a prime number of distinct prime factors. A positive integer is called primest if it has a primer number of distinct primer factors. Find the smallest primest number.
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Context:
A rectangle of size $1000 \times 1979$ is divided into cells. Into how many parts will it be divided if one diagonal is drawn in it?
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Context:
The photographer wants to arrange three boys and three girls in a row such that a boy or a girl could be at each end, and the rest alternate in the middle, calculate the total number of possible arrangements.
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Context:
What is \( \frac{1}{4} \) more than 32.5?
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Context:
Find the largest integer $x$ such that the number

$$
4^{27} + 4^{1000} + 4^{x}
$$

is a perfect square.
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Context:
There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns.
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Context:
Let \( m=30030=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \) and let \( M \) be the set of its positive divisors which have exactly two prime factors. Determine the minimal integer \( n \) with the following property: for any choice of \( n \) numbers from \( M \), there exist three numbers \( a, b, c \) among them satisfying \( a \cdot b \cdot c = m \).
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Context:
Given an arithmetic sequence $\{a_n\}$ with a common difference of $\dfrac{\pi}{9}$, the sum of the first 8 terms is $6\pi$. Let $\tan\dfrac{\pi}{9}=k$. Find the sum of the first 7 terms of the sequence $\{\tan a_n\tan a_{n+1}\}$.
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Context:
Compute the number of subsets $S$ of $\{0,1,\dots,14\}$ with the property that for each $n=0,1,\dots,
6$, either $n$ is in $S$ or both of $2n+1$ and $2n+2$ are in $S$.

[i]Proposed by Evan Chen[/i]
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Context:
Given the parabola $x^{2}=4y$ with focus $F$ and point $A(-1,8)$, let point $P$ be a point on the parabola, then the minimum value of $|PA|+|PF|$ is \_\_\_\_\_\_.
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Context:
Given the variance of a sample is $$s^{2}= \frac {1}{20}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+\ldots+(x_{n}-3)^{2}]$$, then the sum of this set of data equals \_\_\_\_\_\_.
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Context:
As shown in the figure, the large circle is a 400-meter track, with the track length from \(A\) to \(B\) being 200 meters, and the straight-line distance being 50 meters. A father and his son start running simultaneously from point \(A\) in a counterclockwise direction around the track for long-distance jogging exercise. The son runs around the large circle, while the father runs along the straight line whenever he reaches point \(B\). The father takes 20 seconds to run every 100 meters, and the son takes 19 seconds to run every 100 meters. With this pace, during which lap will the son first meet the father again?
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Context:
Given that the quadratic inequality $2kx^2+kx+\frac{1}{2}\geq0$ holds for all real numbers $x$, the range of the real number $k$ is \_\_\_\_\_\_.
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Context:
Betty Lou and Peggy Sue take turns flipping switches on a $100 \times 100$ grid. Initially, all switches are "off". Betty Lou always flips a horizontal row of switches on her turn; Peggy Sue always flips a vertical column of switches. When they finish, there is an odd number of switches turned "on'' in each row and column. Find the maximum number of switches that can be on, in total, when they finish.
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Context:
9. Let $A_{1} B_{1} C_{1}, A_{2} B_{2} C_{2}$, and $A_{3} B_{3} C_{3}$ be three triangles in the plane. For $1 \leq i \leq 3$, let $D_{i}, E_{i}$, and $F_{i}$ be the midpoints of $B_{i} C_{i}, A_{i} C_{i}$, and $A_{i} B_{i}$, respectively. Furthermore, for $1 \leq i \leq 3$ let $G_{i}$ be the centroid of $A_{i} B_{i} C_{i}$.
Suppose that the areas of the triangles $A_{1} A_{2} A_{3}, B_{1} B_{2} B_{3}, C_{1} C_{2} C_{3}, D_{1} D_{2} D_{3}, E_{1} E_{2} E_{3}$, and $F_{1} F_{2} F_{3}$ are $2,3,4,20,21$, and 2020 , respectively. Compute the largest possible area of $G_{1} G_{2} G_{3}$.
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Context:
Evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}(x^2-2xy)e^{-(3x^2+2y^2+z^2+2xy)}dxdydz$.
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Context:
Determine the smallest natural number which can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.
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Context:
Of the 500 soccer balls that Nova went to inflate, 65 percent had holes in them and could not inflate, 25% of the remaining balls were overinflated and exploded, and 10% had manufacturing defects preventing proper inflation. How many balls were inflated successfully and could be used?
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Context:
A graph has 8 points and no squares (a square is 4 distinct points A, B, C, D with A and C both joined to B and D). What is the largest number of edges it can have?
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Context:
Find the value of $f(n) = \sum\limits_{i > j \ge 0} {{}^{n + 1}{C_i}{}^n{C_j}} $
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Context:
Find the total length of the intervals on the number line where the inequalities $|x|<1$ and $\sin \log _{3}|x|>0$ hold.
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Context:
Given that all the edges of a regular tetrahedron are $\sqrt{2}$, and all four vertices lie on the same sphere, the surface area of this sphere is ______.
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Context:
In a movie theater, five friends took seats numbered 1 to 5 (with seat number 1 being the leftmost seat). During the movie, Anya left to buy popcorn. When she returned, she found that Varya had moved two seats to the right, Galya had moved one seat to the left, and Diana and Ella had switched their seats, leaving the end seat for Anya. In which seat was Anya sitting before she left?
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Context:
Calculate the covariance between the following processes: $X_t=\int_0^t(B_s+s)ds$ and $Y_t=\int_0^t(s+1)dB_s$, where $B$ is a one-dimensional Brownian Motion.
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Context:
Let $\theta$ be a constant number such that $0\leq \theta \leq \pi.$
Evaluate

\[ \int_0^{2\pi} \sin 8x|\sin (x-\theta)|\ dx. \]
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Context:
In Vila Par, all the truth coins weigh an even quantity of grams and the false coins weigh an odd quantity of grams. The eletronic device only gives the parity of the weight of a set of coins. If there are $2020$ truth coins and $2$ false coins, determine the least $k$, such that, there exists a strategy that allows to identify the two false coins using the eletronic device, at most, $k$ times.
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Context:
Find the sum placed on simple interest (SI) if the SI on that sum for 3 years at different increasing interest rates for each year (8% for the first year, 12% for the second year, and a 5% increase on the previous year's rate for the third year) is half the compound interest (CI) on Rs. 4000 for 2 years at 10% per annum, given that the interest rate increases by 2% every 6 months during the 2-year period and compounded semi-annually.
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Context:
The function \( f(x) \) has a domain of \( \mathbf{R} \). For any \( x \in \mathbf{R} \) and \( y \neq 0 \), \( f(x+y)=f\left(x y-\frac{x}{y}\right) \), and \( f(x) \) is a periodic function. Find one of its positive periods.
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Context:
What is the result of evaluating the following integral using the orthogonality of Hermite polynomials: $$\int_{-\infty}^\infty H_{n+1}\left(x+\frac{x_0}{2}\right)H_n\left(x+\frac{x_0}{2}\right)e^{^{-\frac{x^2}{2}}}dx$$
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Context:
The math teacher of the ninth grade teaches an average of 6 tutoring classes per month. If the classes are taught by female teachers, each female teacher should teach 15 classes per month. Let $a$ be the number of classes taught by each male teacher if only male teachers are teaching. Using the information given, calculate the value of $a$. Express your answer as a whole number.
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Context:
There are $n$ schools in a city. The $i$th school dispatches $C_i$ students to watch a football match, where $1 \leq C_i \leq 39$ and $1 \leq i \leq n$. The total number of students is $\sum_{i=1}^{n} C_{i} = 1990$. Each line contains 199 seats, but students from the same school must sit in the same line. How many lines of seats are needed to ensure all students have a seat?
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Context:
Let C be the boundary of the quarter-circle x^2 + y^2 = r^2, x >= 0, y >= 0, traversed in the clockwise direction. Calculate the line integral ∫ (x^2 - y^2) dx + (x^2 + y^2) dy.
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Context:
12. What is the sum of all possible $\binom{i}{j}$ subject to the restrictions that $i \geq 10$, $j \geq 0$, and $i+j \leq 20$? Count different $i, j$ that yield the same value separately - for example, count both $\binom{10}{1}$ and $\binom{10}{9}$.
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Context:
Let \( ABC \) be an isosceles triangle with \( AB = AC \). Let \( D \) and \( E \) be the midpoints of segments \( AB \) and \( AC \), respectively. Suppose that there exists a point \( F \) on ray \( \overrightarrow{DE} \) outside of \( ABC \) such that triangle \( BFA \) is similar to triangle \( ABC \). Compute \( \frac{AB}{BC} \).
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Context:
Nylah has her living room lights on a timer. Each evening, the timer switches the lights on at exactly 7:00 p.m., 7:30 p.m., 8:00 p.m., 8:30 p.m., or 9:00 p.m. Later in the evening, the timer switches the lights off at any random time between 11 p.m. and 1 a.m. On a given night, Nylah's lights are on for \( t \) hours. Find the probability that \( 4 < t < 5 \). Express your answer as a fraction.
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Context:
Two sheets of 11 cm x 8 cm paper are placed on top of each other, forming an overlapping 8 cm x 8 cm square in the center. Calculate the area of rectangle WXYZ formed by the overlapping sheets. Express your answer in square centimeters (cm^2) as a whole number.
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Context:
Given \( -1 < a < 0 \) and \( \theta = \arcsin a \), what is the solution set for the inequality \( \sin x < a \)?
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Context:
Given an arithmetic-geometric sequence ${a_n}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $\frac{S_6}{S_3} = -\frac{19}{8}$ and $a_4 - a_2 = -\frac{15}{8}$. Find the value of $a_3$.
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Context:
The monotonic increasing interval of the function $y=a^{x^{2}-3x+2}$ ($a>1$) is __________.
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Context:
Given $a_{1}$ in the sequence $\{a_{n}\}$, and the function $f(x)=\frac{1}{3}{x^3}-{a_{n+1}}sinx+({{a_n}+2})x+1$ has a unique zero for its derivative, and the function $g\left(x\right)=8x+\sin \left(\pi x\right)-\cos \left(\pi x\right)$ with $g(a_{1})+g(a_{2})+\ldots +g(a_{9})=18$. Find $a_{5}$.
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Context:
Let $x_1,$ $x_2,$ $x_3,$ $\dots,$ $x_{100}$ be positive real numbers such that $x_1^2 + x_2^2 + x_3^2 + \dots + x_{100}^2 = 1.$  Find the minimum value of
\[\frac{x_1}{1 - x_1^2} + \frac{x_2}{1 - x_2^2} + \frac{x_3}{1 - x_3^2} + \dots + \frac{x_{100}}{1 - x_{100}^2}.\]
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Context:
Given the sequence \( a_{1}, a_{2}, \cdots, a_{n}, \cdots \) that satisfies \( a_{1}=a_{2}=1, a_{3}=2 \), and for any natural number \( n \), \( a_{n} a_{n+1} a_{n+2} \neq 1 \). Furthermore, it is given that \( a_{n} a_{n+1} a_{n+2} a_{n+3} = a_{1} + a_{n+1} + a_{n+2} + a_{n+3} \). Find the value of \( a_{1} + a_{2} + \cdots + a_{100} \).
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Context:
在 $\triangle A B C$ 中, $\sin A=\frac{\sqrt{2}}{2}$, 求 $\cos B+\sqrt{2} \cos C$ 的取值范围，请提供该取值范围端点之和的整数部分。
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Context:
Three people are playing table tennis, with the player who loses a match giving up their place to the player who did not participate in that match. As a result, it turned out that the first player played 10 matches, and the second player played 21 matches. How many matches did the third player play?
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Context:
In the number $2016 * * * * 02 *$, you need to replace each of the 5 asterisks with any of the digits $0, 2, 4, 6, 7, 8$ (digits may repeat) so that the resulting 11-digit number is divisible by 6. How many ways can this be done?
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Context:
Lee can make 24 cookies with four cups of flour. If the ratio of flour to sugar needed is 2:1 and he has 3 cups of sugar available, how many cookies can he make?
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Context:
Calculate (without calculators, tables, etc.) with accuracy to $0.00001$ the product \(\begin{pmatrix}1-\frac{1}{10}\end{pmatrix}\begin{pmatrix}1-\frac{1}{10^2}\end{pmatrix}\cdots\begin{pmatrix}1-\frac{1}{10^{99}}\end{pmatrix}.\) The original answer is a decimal number with five decimal places, please provide the value as an integer by removing the decimal point.
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Context:
The total age of a and b is 15 years more than the total age of b and c. Additionally, the total age of a and d is 12 years more than the total age of c and d. If a is 3 years older than d, how many years younger is c than a?
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Context:
A frog is located at the origin of the Cartesian coordinate system. Every second, the frog jumps either horizontally or vertically to one of the four adjacent points with integer coordinates. Determine the number of different points where the frog can be located after 2015 seconds.
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Context:
Find the constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$
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Context:
Let \( M \) be a set consisting of a finite number of positive integers, and suppose \( M = A_{1} \cup A_{2} \cup \cdots \cup A_{20} = B_{1} \cup B_{2} \cup \cdots \cup B_{20} \), where \( A_{i} \neq \varnothing, B_{i} \neq \varnothing \) for \( i = 1, 2, \ldots, 20 \). For any \( 1 \leq i < j \leq 20 \), we have \( A_{i} \cap A_{j} = \varnothing \) and \( B_{i} \cap B_{j} = \varnothing \).

Given that for any \( 1 \leq i \leq 20 \) and \( 1 \leq j \leq 20 \), if \( A_{i} \cap B_{j} = \varnothing \), then \( \left|A_{i} \cup B_{j}\right| \geq 18 \). Find the minimum number of elements in the set \( M \) (where \( |X| \) denotes the number of elements in set \( X \)).
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Context:
How many positive integers $N$ are there such that for every multiple of $N$ denoted as $\overline{\mathrm{abc}}$, the numbers $\overline{\mathrm{bca}}$ and $\overline{\mathrm{cab}}$ are also multiples of $N$? Here, $a$, $b$, and $c$ are digits from 0 to 9, and numbers like 123 represent 123, 028 represents 28, and 007 represents 7.
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Context:
What is the maximum number of \(4 \times 1\) rectangles that can be placed inside a \(6 \times 6\) square (without breaking cell boundaries)?
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Context:
Four copper weights together weigh 40 kilograms; with their help, every whole number weight from 1 to 40 kilograms can be measured; what are the weights of these copper weights?
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Context:
Given the function $f(x)=x^{5}-3x^{3}-6x^{2}+x-1$, calculate the value of $f(-2)$ using the Horner's method. Express your answer as a single integer.
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Context:
Find the limit
\[\lim_{n\to\infty} \frac{1}{n\ln n}\int_{\pi}^{(n+1)\pi} (\sin ^ 2 t)(\ln t)\ dt.\]
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Context:
设实数 $a$ 使得不等式 $|2 x-a|+|3 x-2 a| \geqslant a^{2}$ 对任意实数 $x$ 恒成立，求满足条件的 $a$ 所组成的集合的区间端点。如果区间为 $[m, n] = \left[\frac{k}{l}, \frac{p}{q}\right]$，请给出k+l+p+q的值。
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Context:
[ Ratio of areas of similar triangles ]

In a right triangle, the sine of the smaller angle is $\frac{1}{3}$. A line perpendicular to the hypotenuse divides the triangle into two equal areas. In what ratio does this line divide the hypotenuse?
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Context:
A person travels from P to Q at a certain speed and returns by increasing his speed by 30%. His average speed for both the trips is 34.5 km/hr. What is his initial speed from P to Q?
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Context:
Machiavelli added together $1+3i$ and $2-4i$. What number did he get?
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Context:
Evaluate the limit using L'Hopital's rule: \[ \lim_{x \to 0} \log (\cot x)^{\tan x} \]
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Context:
How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 8?
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Context:
Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in \(\{(x, y) \mid 1 \leq x, y \leq 5\}\) blue such that the distance between any two blue points is not an integer.
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Context:
How many ordered triples $(a, b, c)$ of integers with $-15 \le a, b, c \le 15$ are there such that the three equations $ax + by = c$, $bx + cy = a$, and $cx + ay = b$ correspond to lines that are distinct and concurrent?
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Context:
Determine the prime \( p \) such that the number of groups of order \( 31p^2 \) is maximized. Consider the cases where \( p = 31 \) and \( p \neq 31 \).
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Context:
Let $n$ be a positive integer. How many polynomials $P$ with coefficients in the set $\{0,1,2,3\}$ are there such that $P(2)=n$?
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Context:
In the prism \(A B C A_1 B_1 C_1\), the medians of the bases \(A B C\) and \(A_1 B_1 C_1\) intersect at points \(O\) and \(O_1\) respectively. A line is drawn through the midpoint of the segment \(O O_1\) parallel to the line \(C A_1\). Given that \(C A_1 = a\), find the length of the segment of this line that lies inside the prism.
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Context:
Compute the number of ordered triples of integers $(a, b, c)$ between $1$ and $12$, inclusive, such that, if \[ q = a + \frac{1}{b} - \frac{1}{b + \frac{1}{c}}, \] then $q$ is a positive rational number and, when $q$ is written in lowest terms, the numerator is divisible by $13$.
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Context:
In a certain sports competition, the scoring rules are: winning a match gives 3 points, drawing a match gives 1 point, and losing a match gives 0 points. If a particular athlete plays 15 matches and scores a total of 22 points, calculate the number of possible scenarios. Express your answer as a single integer.
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Context:
Determine the number of real values of $x$ that satisfy the equation:

$$\left| 3 -x \right|^{ \log_7(x^2) - 7\log_x (49)} = (3-x)^3$$
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Context:
An arrangement of 12 real numbers in a row is called *good* if for any four consecutive numbers, the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. Determine how many good arrangements exist in which the first and last numbers are 1, and the second number is the same as the third.
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Context:
Two irrational numbers in the interval $(0,1)$ are chosen at random, and their decimal parts are considered up to 2014 places. The probability that the smaller number can be subtracted from the larger one without borrowing is expressed as $p^k$, where $p$ is a rational number and $k > 1000$ is a positive integer. Given that the first decimal digit of the larger number is greater than that of the smaller number, find the last three digits of $\lfloor{100(p+k+1)}\rfloor$. Provide your answer as a three-digit number.
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Context:
对于给定的正整数 $k$, 定义 $f_1(k)$ 为 $k$ 的数字和的平方, 并令

$$
f_{n+1}(k)=f_1\left(f_n(k)\right)
$$

求 $f_{1991}\left(2^{1990}\right)$ 的值。
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Context:
Find the smallest positive distance $d$ such that for any line in the plane not parallel to the $x$- or $y$-axis, there exists a lattice point whose distance to the line does not exceed $d$. Express your answer in simplest form.
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Context:
Find a closed form expression for the sum $\sum^{n}_{k=0}{n\choose k} (-1)^k {n+m-k\choose r-k}$ using generating functions, where $n,m,r$ are natural numbers and $n \leq r \leq n+m$.
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Context:
Evaluate the integral \( \oint_{|z|=2} \frac{f^{\prime}(z)}{f(z)} \, dz \) for the function \( f(z)=\frac{2 z^{4}-7 z^{3}-2 z-i}{(z-i)^{2}\left(3 z^{2}-z+9\right)^{2}} \).
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Context:
Calculate the expectation of a random variable $Z$ with the cumulative distribution function given by: $$F_Z(z) = 1 - 0.2e^{-0.4z} + 0.7e^{-0.1z}.$$
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Context:
Consider a cuboid with base edge lengths $a$ and $b$ and height $h$. Given that the base area $L = ab$ and the lateral area $S = 2h(a + b)$ are constant, determine the minimum value of the perimeter $P = 4(a + b + h)$. Provide your answer as a single number.
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Context:
Let \( \phi \) be a continuous function on the interval \([0,b]\) such that \( \phi(x) > 0 \) for all \( x \in [0,b] \). Assume \( \phi(x) \sim A x^r \) for some constant \( A > 0 \) and \( r \geq 0 \). Define the function \( F(\varepsilon) = \int_0^b \frac{dx}{\varepsilon + \phi(x)} \). Find \( \lim_{\varepsilon \to 0} F(\varepsilon) \) for \( r = 1 \).
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Context:
Consider the system of equations:

\[ a + 2b + 3c + \ldots + 26z = 2020 \]
\[ b + 2c + 3d + \ldots + 26a = 2019 \]
\[ \vdots \]
\[ y + 2z + 3a + \ldots + 26x = 1996 \]
\[ z + 2a + 3b + \ldots + 26y = 1995 \]

where each equation is a rearrangement of the first equation with the variables cycling and the coefficients staying in place. Find the value of:

\[ z + 2y + 3x + \ldots + 26a \]
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Context:
It is known that $\int_1^2x^{-1}\arctan (1+x)\ dx = q\pi\ln(2)$ for some rational number $q.$ Determine $q.$ Here, $0\leq\arctan(x)<\frac{\pi}{2}$ for $0\leq x <\infty.$
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Context:
At some time between 9:30 and 10 o'clock, the triangle determined by the minute hand and the hour hand is an isosceles triangle. If the two equal angles in this triangle are each twice as large as the third angle, what is the time?
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Context:
Given the function \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that \( f(f(x+y)) = f(x^2) + f(y^2) \), and \( f(f(2020)) = 1010 \), determine the value of \( f(2025) \).
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Context:
We say that a subset \( T \) of \(\{1, 2, \dots, 2024\}\) is **kawaii** if \( T \) has the following properties:

1. \( T \) has at least two distinct elements;
2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \).

For example, the subset \( T = \{31, 71, 2024\}\) is **kawaii**, but \( T = \{5, 15, 75\}\) is not **kawaii** because \( 15 - 5 = 10 \) divides \( 15 + 5 = 20 \).

What is the largest possible number of elements that a **kawaii** subset can have?
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Context:
On Mary's birthday, her brother surprised her with $200. She decided to spend some of it on a shopping trip. She bought a new video game which originally cost $60, but she had a 15% discount coupon. Additionally, she spent 20% of the remaining amount on swimming goggles, which had an 8% sales tax added to the final price.

Mary also decided to buy a sports jacket that had a 25% discount, and the jacket's original price was $80. She then bought a book with a 5% sales tax, spending 10% of her remaining money. Finally, she used a $20 gift card to buy a pair of stylish socks.

How much money did Mary have left after her shopping trip?
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Context:
Determine the smallest number \( n \) such that, when 1000 balls of 40 different colors (25 balls of each color) are placed in a circle in any arrangement, there are always \( n \) adjacent balls that include at least 20 different colors.
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Context:
In a larger geoboard grid of $7 \times 7$, points are evenly spaced vertically and horizontally. Points $A$ and $B$ are at $(3,3)$ and $(5,3)$ respectively. How many of the remaining points in the grid will allow for triangle $ABC$ to be isosceles?
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Context:
Evaluate the expression \(A + B + C + AB + BC + AC + ABC\) where:
\[ A = \frac{2\sqrt{3}}{3}i \cdot \cos\left(\frac{\pi}{6} + \frac{i}{3}\text{arcsinh}\left(\frac{3\sqrt{3}}{2}\right)\right) \]
\[ B = \frac{2\sqrt{3}}{3}i \cdot \cos\left(\frac{5\pi}{6} + \frac{i}{3}\text{arcsinh}\left(\frac{3\sqrt{3}}{2}\right)\right) \]
\[ C = \frac{2\sqrt{3}}{3}i \cdot \cos\left(\frac{3\pi}{2} + \frac{i}{3}\text{arcsinh}\left(\frac{3\sqrt{3}}{2}\right)\right) \]
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Context:
A regular triangular pyramid is circumscribed around a sphere of volume $V$. What is the smallest possible volume of this pyramid?
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Context:
Determine the number of ordered pairs of positive integers $(m,n)$ with $1 \leq m \leq 100$ and $1 \leq n \leq 100$ such that \[ \gcd(m+1,n+1) = 10\gcd(m,n). \]
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Context:
方程 $\sin x=\cos 2 x$ 的最小的 $20$ 个正实数解之和为$m\pi$, 请给出$m$的值。
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Context:
In a jar, there are 3409 pieces of candy, which includes 1462 chocolate candies and 1947 gummy candies. There are also 145 secret eggs, where 98 eggs have 1 prize each, 38 eggs have 2 prizes each, and 9 eggs have 3 prizes each. Calculate the total number of items in the jar, including candies and prizes in the eggs.
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Context:
实数 $a_{1}, a_{2}, \cdots, a_{n}(n \geqslant 3)$ 满足
$a_{1}+a_{2}+\cdots+a_{n}=0$,
且 $2 a_{k} \leqslant a_{k-1}+a_{k+1}(k=2,3, \cdots, n-1)$.
求最小的 $\lambda(n)$, 使得对所有的 $k \in\{1$, $2, \cdots, n\}$ ，都有
$\left|a_{k}\right| \leqslant \lambda(n) \cdot \max \left\{\left|a_{1}\right|,\left|a_{n}\right|\right\}$。若$n=5$，答案格式为\frac{p}{q}，其中$p,q$互质，求$p+q$。
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Context:
Teena is driving at 55 miles per hour and is currently 7.5 miles behind Loe, who is driving at a certain speed in the same direction. In 90 minutes, Teena will be 15 miles ahead of Loe. What is Loe's speed in miles per hour?
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Context:
Determine the number of non-isomorphic semidirect products of the group \( \mathbb{Z}/p^2\mathbb{Z} \) by the group \( \mathbb{Z}/p\mathbb{Z} \), where \( p \) is an odd prime.
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Context:
In a non-homogeneous Poisson process $(N_t)$ with rate function $\lambda(t)$ given by:

$$\lambda(t) = \begin{cases} 1 & t<1 \\ 2 &1\leq t < 2 \\ 3 & t\geq 2 \\ \end{cases} $$

it is known that $N_3 = 10$ events occur over the interval $[0, 3]$. Calculate the conditional probability $P(N_1 = 2 \mid N_3 = 10)$, which is the probability that exactly 2 of these events occurred in the interval $[0, 1]$. Provide your answer as a decimal rounded to four decimal places.
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Context:
Determine the number of subgroups \( G < \mathbb{Z}^3 \) such that \( \mathbb{Z}^3/G \cong \mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z} \).
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Context:
Given $((x-\sqrt{3})^{2017}=a_{0}x^{2017}+a_{1}x^{2016}+…+a_{2016}x+a_{2017})$, find the value of $((a_{0}+a_{2}+…+a_{2016})^{2}-(a_{1}+a_{3}+…+a_{2017})^{2})$.
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Context:
若三位数$n=\overline{abc}$是一个平方数，并且其数字和$a+b+c$也是一个平方数，称$n$为超级平方数，设超级平方数的集合为$A$，$A$中元素的个数为$\left| A \right|$，$A$中所有元素之和为$S\left( A \right)$，则与$\frac{S\left( A \right)}{\left| A \right|}$最接近的整数是_______.
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Context:
There are 10 numbers written in a circle, and their sum is 100. It is known that the sum of any three consecutive numbers is not less than 29.

Determine the smallest number \( A \) such that in any such set of numbers, each number does not exceed \( A \).
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Context:
Each day a man meets his wife at the train station after work, and then she drives him home. She always arrives exactly on time to pick him up. One day he catches an earlier train and arrives at the station some time early. He immediately begins walking home along the same route the wife drives. Eventually, his wife sees him on her way to the station and drives him the rest of the way home. When they arrive home, the man notices that they arrived 30 minutes earlier than usual. How early did the man arrive at the station?
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Context:
Given that point A has coordinates (-3, 2), and F is the focus of the parabola $y^2 = -4x$, and point P is a moving point on the parabola, when $|PA| + |PF|$ takes the minimum value, the coordinates of P are.
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Context:
Express in the form of an irreducible fraction:

$$
6 \frac{16}{2015} \times 9 \frac{17}{2016}-2 \frac{1999}{2015} \times 17 \frac{1999}{2016}-27 \times \frac{16}{2015}
$$
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