Context:
Calculate the flux of the vector field \( \bar{V} = z\bar{i} + x\bar{j} + y\bar{k} \) through the surface defined by \( x^2 + y^2 + z^2 = 1 \), \( x = 0 \), \( y = 0 \), \( z = 0 \) with \( x > 0 \), \( y > 0 \), \( z > 0 \) using Gauss's theorem.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given that the derivative of the even function f(x) (x ≠ 0) is f'(x), and it satisfies f(1) = 0, when x > 0, xf'(x) < 2f(x). Find the range of x that makes f(x) > 0 true. Express your answer as an interval or union of intervals.
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Context:
Find the least number of balanced quadruples \((a, b, c, d)\) in a set \(\mathcal{S}\) where \(1 \leq a < b < d < c \leq 20\) and the cardinality of \(\mathcal{S}\) is 4411. A quadruple is balanced if \(a + c = b + d\).
Please reason step by step, and put your final answer within \boxed{}.


Context:
设 P_1,P_2,\cdots,P_{12} 是平面上的 12 个点（允许重合），其中任意两点之间的距离不超过 1. 求 \ds{1\leq i<j\leq 12}{}|P_iP_j|^2 的最大可能值.
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Context:
In the polar coordinate system, the maximum distance from a point on the circle $ρ=8\sin θ$ to the line $θ=\frac{π}{3}\ (ρ∈R)$ is $\_\_\_\_\_\_$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the moment generating function of a random variable $X$:
\[ M_{X}(t) = \frac{6}{\pi^{2}}\sum_{n\ge1}\frac{e^{\frac{t^2}{2n}}}{n^2}\,,\;t\in \mathbb{R} \]
Determine the probability that $X$ is a rational number.
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Context:
Evaluate :
 \( \sqrt[8]{{2207} - \frac{1}{{2207} - \frac{1}{{2207} - \cdots }}} \)
 Express your expression in the form \( \frac{a + b\sqrt{c}}{d} \) where \( a,b,c,d \in  \mathbb{Z} \) .
Please reason step by step, and put your final answer within \boxed{}.


Context:
If $f$ is a quadratic function such that $f(0)=1$ and $\int\frac{f(x)}{x^2(x+1)^3}dx $ is a rational function, find the value of $f'(0)$.
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Context:
Given the linear programming problem: Minimize \(x+y+z-\alpha t\) subject to \(2x - y + z = 6\), \(-x + y + t = 3\), and \(x, y, z, t \geq 0\). If \((3, 0, 0, \beta)\) is an optimal solution, determine the maximum value of \(\beta - \alpha\).
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Context:
The average weight of some pigs is 15 kg. Weight less than 16 kg is considered as underweight. The maximum number of pigs that can be underweight is 4. How many pigs are there in total?
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Context:
Find the total length of the intervals on the number line where the inequalities \( x < 1 \) and \( \operatorname{tg} \log _{4} x > 0 \) hold.
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Context:
For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1)=1,c(2n)=c(n),$ and $c(2n+1)=(-1)^nc(n)$. Find the value of $$S = \sum_{n=1}^{2013}c(n)c(n+2).$$
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Context:
Evaluate the line integral \( \oint_C \mathbf{G} \cdot d\mathbf{r} \), where \( C \) is the boundary of the surface \( T \) defined by \( z = 10x - 20y + 60 \) over the rectangle \( R = \{ (x,y) \mid -2 < x < 3, 0 < y < 2 \} \) and \( \mathbf{G}(x,y,z) = -y^2 \mathbf{i} + xy \mathbf{j} + zx \mathbf{k} \).
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Context:
Three integers form a geometric progression. If 8 is added to the second number, an arithmetic progression is formed. If 64 is added to the third term of this arithmetic progression, a geometric progression is formed again. What are the original numbers?
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Context:
$\underline{\text { Folklore }}$

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

#
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Context:
Find the smallest prime divisor of the sum \(1^{60} + 2^{60} + \cdots + 33^{60}\).
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Context:
9.1. Find the number of roots of the equation

$$
\operatorname{arctg}\left(\operatorname{tg}\left(\sqrt{13 \pi^{2}+12 \pi x-12 x^{2}}\right)\right)=\arcsin \left(\sin \sqrt{\frac{13 \pi^{2}}{4}+3 \pi x-3 x^{2}}\right)
$$
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Context:
Find the coefficient of  $x^2$  after expansion and collecting the terms of the following expression (there are  $k$  pairs of parentheses):  $((... (((x - 2)^2 - 2)^2 -2)^2 -... -2)^2 - 2)^2$ .
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Context:
A train 220 m long is running with a certain speed. It passes a man who is running at 6 km/hr in the direction opposite to that in which the train is going in 11.999040076793857 seconds. What is the speed of the train?
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Context:
Determine the minimum number of elements that a sample space must have to ensure the existence of $n$ independent events, each with a probability strictly between 0 and 1.
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Context:
Chinese shooter Huang Yuting, 17 years old, won three gold medals for the Chinese delegation at the 2023 Hangzhou Asian Games with her tenacious style and superb skills, demonstrating a spirit of striving for progress and climbing to new heights. Below are the scores (in rings) she achieved in the final elimination stage of the women's 10-meter air rifle individual event:

| 1   | 2   | 3   | 4   | 5   | 6   | 7   | 8   | 9   | 10  | 11  | 12  | 13  | 14  |
|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| 10.3| 10.3| 10.4| 10.4| 10.8| 10.8| 10.5| 10.4| 10.7| 10.5| 10.7| 10.7| 10.3| 10.6|

Find the variance of this data set. (Round to $0.001$)
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Context:
Find all common points of the graphs

$$
y=8 \cos \pi x \cdot \cos ^{2} 2 \pi x \cdot \cos 4 \pi x \quad \text{and} \quad y=\cos 9 \pi x
$$

with abscissae belonging to the segment \( x \in [0, 1] \). In the answer, indicate the sum of the abscissae of the found points.
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Context:
Two runners started simultaneously in the same direction from the same point on a circular track. The first runner, moving ahead, caught up with the second runner at the moment when the second runner had only run half a lap. From that moment, the second runner doubled their speed. Will the first runner catch up with the second runner again? If so, how many laps will the second runner complete by that time?
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Context:
In the geometric sequence {a_n}, given that a_2a_5 = -3/4 and a_2 + a_3 + a_4 + a_5 = 5/4, find the sum of the reciprocals 1/a_2 + 1/a_3 + 1/a_4 + 1/a_5. Express your answer as a single fraction or decimal.
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Context:
Evaluate the integral \( \frac{1}{2\pi} \int_{0}^{2\pi} e^{\cos\theta} \cos(n\theta) \,d\theta \) using the residue theorem.
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Context:
Evaluate the integral \( \int_{0}^{\infty} f(x) g(x) \, dx \), where \( f(x) = x - \frac{x^3}{2} + \frac{x^5}{2 \cdot 4} - \frac{x^7}{2 \cdot 4 \cdot 6} + \cdots \) and \( g(x) = 1 + \frac{x^2}{2^2} + \frac{x^4}{2^2 \cdot 4^2} + \frac{x^6}{2^2 \cdot 4^2 \cdot 6^2} + \cdots \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
At 15:00 there were 20 students in the computer lab. At 15:03 and every three minutes after that, some students entered the lab. If at 15:10 and every ten minutes after that 8 students left the lab, there were 27 students in the computer lab at 15:44. How many students entered the lab every three minutes?
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Context:
Consider the set  $$ \mathcal{S}=\{(a, b, c, d, e): 0<a<b<c<d<e<100\} $$  where  $a, b, c, d, e$  are integers. If  $D$  is the average value of the fourth element of such a tuple in the set, taken over all the elements of  $\mathcal{S}$ , find the largest integer less than or equal to  $D$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Convert the polar equation \(\rho^{2} \sin \theta - \rho \cos^{2} \theta - \sin \theta = 0\) for \(\theta \in (0, \pi)\) into its Cartesian coordinate system representation.
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Context:
Find the number of elements that a set $B$ can have, contained in $(1, 2, ... , n)$, according to the following property: For any elements $a$ and $b$ on $B$ ($a \ne b$), $(a-b) \not| (a+b)$.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{ \left\lceil \frac{n}{3} \right\rceil }


Context:
One of the receipts for a math tournament showed that  $72$  identical trophies were purchased for  $\$  $-$ 99.9$-, where the first and last digits were illegible.  How much did each trophy cost?
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Context:
The area of a right triangle is $2 r^{2} / 3$, where $r$ is the radius of the circle that touches one leg, the extension of the other leg, and the hypotenuse. Find the sides of the triangle.
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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.

Find the distance from vertex \(A\) to the line \(BE\).
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Context:
A school buys 60% of its light bulbs from supplier A. The light bulbs from supplier A have exponentially distributed lifetimes with a mean of 600 hours, and the lifetimes of all bulbs are independent of each other. Using a normal approximation, find the probability that 100 light bulbs from supplier A last more than 50,000 hours in total.
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Context:
Given the function $y = \log_a(x + 3) - 1$ where $a > 0$ and $a \neq 1$, its graph always passes through a fixed point $A$. If point $A$ lies on the line $mx + ny + 1 = 0$, where $mn > 0$, then the minimum value of $\frac{1}{m} + \frac{2}{n}$ is $\_\_\_\_\_\_\_\_\_\_.$
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Context:
A positive integer $n$ is picante if $n$ ! ends in the same number of zeroes whether written in base 7 or in base 8 . How many of the numbers $1,2, \ldots, 2004$ are picante?
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Context:
For complex number $u = a+bi$ and $v = c+di$ (where $i=\sqrt{-1}$), define the binary operation $u \otimes v = ac + bdi$ Suppose $z$ is a complex number such that $z\otimes z = z^{2}+40$. What is $|z|$?
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Context:
Evaluate the limit: \[ \lim_{n \to +\infty}\sum_{k=n}^{3n} \binom{k-1}{n-1} \left(\frac{1}{3}\right)^n \left(\frac{2}{3}\right)^{k-n} \]
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Context:
In triangle \(\triangle ABC\), vertices \(A(0,a)\) and \(B(0,b)\) are fixed, where \(a, b > 0\), and vertex \(C(x,0)\) is variable. Determine the value of \(x\) that maximizes the angle \(ACB\).
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Context:
Determine the mass percentage of Oxygen in a mixture of Bromic acid (HBrO3) and Sulfurous acid (H2SO3), given that the mixture contains 1 mol of HBrO3 and 2 mol of H2SO3.
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Context:
The school principal, the janitor, and the parent committee, unable to agree with each other, each bought a rug for the school assembly hall sized $10 \times 10$. After thinking about what to do, they decided to place all three rugs as shown in the picture: the first rug $6 \times 8$ in one corner, the second rug $6 \times 6$ in the opposite corner, and the third rug $5 \times 7$ in one of the remaining corners (all dimensions are given in meters).

Find the area of the part of the hall covered by rugs in three layers (give your answer in square meters).
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Context:
Let $\triangle ABC$ be a triangle with side lengths $AB = 9$, $BC = 10$, $CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC$, $E = BO \cap CA$, $F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{CF}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.
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Context:
Aarti can do a piece of work in 5 days. In how many days will she complete a certain multiple of work of the same type, if it takes her 15 days to complete that amount of work?
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Context:
For what integers  $ n\ge 3$  is it possible to accommodate, in some order, the numbers  $ 1,2,\cdots, n$  in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?
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Context:
Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c) = P(b, c, a)$ and $P(a, a, b) = 0$ for all real $a$, $b$, and $c$. If $P(1, 2, 3) = 1$, compute $P(2, 4, 8)$.

Note: $P(x, y, z)$ is a homogeneous degree $4$ polynomial if it satisfies $P(ka, kb, kc) = k^4P(a, b, c)$ for all real $k, a, b, c$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
In a $4 \times 4$ grid, there should be 30 squares. As shown in the figure, 4 points have already been removed. How many more points need to be removed so that there is exactly one square left in the grid?
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Context:
Let $M$ and $m$ be the largest and the smallest values of $x$, respectively, which satisfy $4x(x - 5) \le 375$. Find $M - m$.
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Context:
Find the smallest positive number $\lambda$ such that for any triangle with side lengths $a, b, c$, if $a \geqslant \frac{b+c}{3}$, then the following inequality holds:
$$
ac + bc - c^2 \leqslant \lambda \left( a^2 + b^2 + 3c^2 + 2ab - 4bc \right).
$$
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given an ellipse $M$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a > b > 0\right)$ with foci $F_{1}$ and $F_{2}$, and intersecting points with the coordinate axes at $A$, $B$, $C$, and $D$, find the possible values for the eccentricity of the ellipse $M$ that allow three of the points $F_{1}$, $F_{2}$, $A$, $B$, $C$, and $D$ to form an equilateral triangle. Express your answer as a list of possible values, separated by commas, and enclosed in a boxed notation.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the smallest positive integer $n \ge 3$ for which
\[ A \equiv 2^{10n} \pmod{2^{170}} \]
where $A$ denotes the result when the numbers $2^{10}$, $2^{20}$, $\dots$, $2^{10n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A = 10241048576$).
Please reason step by step, and put your final answer within \boxed{}.


Context:
4. $1991^{2000}$ divided by $10^{6}$, the remainder is
Please reason step by step, and put your final answer within \boxed{}.


Context:
Adam is playing Minesweeper on a  $9\times9$  grid of squares, where exactly  $\frac13$  (or 27) of the squares are mines (generated uniformly at random over all such boards). Every time he clicks on a square, it is either a mine, in which case he loses, or it shows a number saying how many of the (up to eight) adjacent squares are mines.

First, he clicks the square directly above the center square, which shows the number  $4$ . Next, he clicks the square directly below the center square, which shows the number  $1$ . What is the probability that the center square is a mine?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Solve
\[\arcsin (\sin x) = \frac{x}{2}.\]Enter all the solutions, separated by commas.
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Context:
A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. The number of degrees of angle $ADB$ is:
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Context:
Let \( f(x) = x^3 - 3x + 1 \). Find the number of distinct real roots of the equation \( f(f(f(x))) = 3 \).
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Context:
Given the following equations: $$\sqrt {2}=2\cos \frac {π}{4}$$, $$\sqrt {2+ \sqrt {2}}=2\cos \frac {π}{8}$$, $$\sqrt {2+ \sqrt {2+ \sqrt {2}}}=2\cos \frac {π}{16}$$, ... please deduce the nth equation: $$\overset{}{n\text{ of }2}\_\_\_\_\_\_$$
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Context:
5. In triangle $A B C, \angle A=45^{\circ}$ and $M$ is the midpoint of $\overline{B C}$. $\overline{A M}$ intersects the circumcircle of $A B C$ for the second time at $D$, and $A M=2 M D$. Find $\cos \angle A O D$, where $O$ is the circumcenter of $A B C$.
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Context:
Inside a circle with radius $6$ lie four smaller circles with centers $A$, $B$, $C$, and $D$. These circles touch each other as shown. The point where the circles with centers $A$ and $C$ touch each other is the center of the big circle. Calculate the area of quadrilateral $ABCD$.
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Context:
Given the following program, determine the condition for the UNTIL statement that yields an output of s = 132.
i = 12
s = 1
DO
s = s * i
i = i - 1
LOOP UNTIL "condition"
PRINT s
END.
Express your answer as a conditional statement (e.g., i < 11).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a cyclic quadrilateral \(ABCD\) where \(AB = BC = AD + CD\) and \(\angle BAD = \alpha\), and \(AC = d\). Find the area of \(\triangle ABC\).
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Context:
Evaluate the constant \( \alpha \) such that \( \iiint_{g(S)}(2x+y-2z)\,dxdydz = \alpha \iiint_S z\,dxdydz \), where the transformation \( g: \mathbb{R}^3 \to \mathbb{R}^3 \) is defined by \( g(x,y,z) = (3x+4z, 2x-3z, x+3y) \) and \( S = \{(x,y,z) \in \mathbb{R}^3 : 0 \leq x \leq 1, 0 \leq y \leq 1, 0 \leq z \leq 1\} \).
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Context:
The diagram shows a rectangle $AEFJ$ inside a regular decagon $ABCDEFGHIJ$. What is the ratio of the area of the rectangle to the area of the decagon?
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Context:
A child lines up  $2020^2$  pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below  $250$ . How many bricks remain in the end?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Jurgen is traveling to Waterloo by bus. He packs for 25 minutes. He then walks to the bus station, which takes 35 minutes. He arrives 60 minutes before his bus leaves. His bus leaves at 6:45 p.m. At what time did he start packing? Express your answer in p.m. time format (e.g., 4:45 p.m.).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the value of the integral:

$$\int_0^1 \int_1^2 \frac{\log(x) - \log(y)}{x - y} \, dy \, dx$$
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Context:
2. Find all eight-digit numbers with this property: if we strike out the first two and the last two digits of the number, we get a four-digit number that is 2019 times smaller than the original number.

(Pavel Calábek)
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Context:
For some complex number  $\omega$  with  $|\omega| = 2016$ , there is some real  $\lambda>1$  such that  $\omega, \omega^{2},$  and  $\lambda \omega$  form an equilateral triangle in the complex plane. Then,  $\lambda$  can be written in the form  $\tfrac{a + \sqrt{b}}{c}$ , where  $a,b,$  and  $c$  are positive integers and  $b$  is squarefree. Compute  $\sqrt{a+b+c}$ .
Please reason step by step, and put your final answer within \boxed{}.


Context:
Determine the number of ideals of norm 100 in the ring of integers of the field $K=\mathbb{Q}(\sqrt{-7})$. Provide your answer as a single integer.
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Context:
The cells of a $5 \times 5$ grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through 9 cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly 3 red cells, exactly 3 white cells, and exactly 3 blue cells no matter which route he takes.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Three cards are dealt at random from a standard deck of 52 cards.  What is the probability that the first card is a 4, the second card is a $\clubsuit$, and the third card is a 2?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
Please reason step by step, and put your final answer within \boxed{}.


Context:
6. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system

$$
\left\{\begin{array}{l}
y=-b-x^{2}, \\
x^{2}+y^{2}+8 a^{2}=4+4 a(x+y)
\end{array}\right.
$$

has at least one solution $(x ; y)$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The function $f(x) = x^3 - 3ax + 2$ is a decreasing function on the interval $(-2, 2)$. Determine the range of values for $a$.
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Context:
What are the numbers whose triples, when added to 1, result in a prime number between 70 and 110?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Let $n\ge 1$  be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$  , where $p$  runs over polynomials of degree less than $n$  with real coefficients and $f$  runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$   defined on the closed interval $[0,1]$ , where $c_k \ge 0$  and $\sum_{k=n}^{\infty} c_k=1$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Consider the sum $$S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|.$$
The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$, satisfying $2c < d$. Find the value of $c + d$.
Please reason step by step, and put your final answer within \boxed{}.


Context:
For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$?
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given the ellipse $\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$ with foci $F_1$ and $F_2$ at $(-\frac{\sqrt{2}}{2}, 0)$ and $(\frac{\sqrt{2}}{2}, 0)$ respectively, a tangent line $l$ is drawn at any point $P$ on the ellipse. Let $d_1$ and $d_2$ denote the distances from $F_1$ and $F_2$ to the line $l$, respectively. Calculate the product $d_1 \cdot d_2$. Express your answer as a single numerical value.
Please reason step by step, and put your final answer within \boxed{}.


Context:
Suppose $W(t)$ is a standard Brownian motion. Calculate the probability $P(-1 < W(3) - W(1) < 1)$. Provide your answer as a percentage.
Please reason step by step, and put your final answer within \boxed{}.


Context:
The polynomial \( x^3 - 3x^2 + 1 \) has three real roots \( r_1, r_2, \) and \( r_3 \). Compute:
\[
\sqrt[3]{3r_1 - 2} + \sqrt[3]{3r_2 - 2} + \sqrt[3]{3r_3 - 2}.
\]
Please reason step by step, and put your final answer within \boxed{}.


Context:
Find the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1.
Please reason step by step, and put your final answer within \boxed{}.


Context:
A soccer store typically sells replica jerseys at various discounts ranging from 20 percent to 80 percent off the list price, depending on the popularity of the team and the age of the jersey design. During the annual summer sale, everything in the store has an additional discount policy. Items that were already discounted 40 percent or less receive an additional 25 percent off, while items that were discounted between 41 percent and 80 percent receive an additional 10 percent off the original list price. If a replica jersey's list price is $120 and comes with varying discounts and additional summer sale offers, approximately what percent of the list price is the lowest possible total sale price for the jersey?
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Context:
Rs. 116000 is divided among 3 persons A, B, and C such that the ratio of the shares of A and B is 3 : 4 and that of B : C is 5 : 6. What is the share of A?
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Context:
Find \( \mathop{\lim }\limits_{{n \rightarrow  \infty }}\frac{\pi }{n}\left\{  {\frac{1}{\sin \frac{\pi \left( {n + 1}\right) }{4n}} + \frac{1}{\sin \frac{\pi \left( {n + 2}\right) }{4n}} + \cdots  + \frac{1}{\sin \frac{\pi \left( {n + n}\right) }{4n}}}\right\} \)
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Context:
In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims:
(i) Less than  $12$  employees have a more difficult work;
(ii) At least  $30$  employees take a higher salary.
Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise.
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Context:
Consider  $7$ -gons inscribed in a circle such that all sides of the  $7$ -gon are of different length. Determine the maximal number of  $120^\circ$  angles in this kind of a  $7$ -gon.
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Context:
For any positive real numbers \(a_1, a_2, \cdots, a_5\), if \(\sum_{i=1}^{5} \frac{a_i}{\sqrt{a_i^2 + 2^{i-1} a_{i+1} a_{i+2}}} \geq \lambda\), find the maximum value of \(\lambda\).
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Context:
A merry mouse moves in a maze. If it is at time $n$ in a room with $k$ horizontal or vertical adjacent rooms, it will be at time $n+1$ in one of the $k$ adjacent rooms, choosing one at random, each with probability $1 / k$. A fat lazy cat remains all the time in room $3,$ and a piece of cheese waits for the mouse in room $5$. The mouse starts in room $1$. The cat is not completely lazy: If the mouse enters the room inhabited by the cat, the cat will eat it. Also, if the mouse eats the cheese, it rests forever. Let $X_{n}$ be the position of the mouse at time $n$. What is the probability that the mouse ever gets to eat the cheese?

The transition matrix is as follows:
$$P=\begin{pmatrix}0 & 1/2 & 0 & 1/2 & 0 \\
1/2 & 0 & 1/2 & 0 & 0 \\
0 & 1/2 & 0 & 1/2 & 0 \\
1/3 & 0 & 1/3 & 0 & 1/3 \\
0 & 0 & 0 & 1 & 0 \\
\end{pmatrix}$$
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Context:
Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$, tens digit $y$, and units digit $z$, and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$.  How many three-digit numbers $\underline{abc}$, none of whose digits are 0, are there such that $\underline{ab}>\underline{bc}>\underline{ca}$?
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Context:
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Over all integers $1 \leq n \leq 100$, find the maximum value of $\phi(n^2+2n) - \phi(n^2)$.
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Context:
Evaluate the contour integral \( \int_C \frac{z+2}{(z^2-2z+1)^2}dz \) using the residue theorem, where \( C \) is the semi-circle in the right half-plane with radius \( R > 1 \) and its diameter lies on the imaginary axis. The contour is traced counterclockwise.
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Context:
A sequence of positive integers is given by $a_{1}=1$ and $a_{n}=\operatorname{gcd}\left(a_{n-1}, n\right)+1$ for $n>1$. Calculate $a_{2002}$.
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Context:
Compute the integral \( \int_0^{\infty} xe^{-x^2/2}y \, dx \), where \( y = \sum\limits_{n \ge 0} \frac{(x/2)^{2n}}{(n!)^2} \) is the unique solution to the initial value problem \( xy''+y'-xy=0, \ y(0)=1, \ y'(0) = 0. \)
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Context:
Given a circle with center \( O \) and radius 1. From point \( A \), tangents \( AB \) and \( AC \) are drawn. Point \( M \) lies on the circle such that the quadrilaterals \( OBMC \) and \( ABMC \) have equal areas. Find \( MA \).
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Context:
Over all real numbers $x$ and $y$, find the minimum possible value of $$ (x y)^{2}+(x+7)^{2}+(2 y+7)^{2} $$
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Context:
Given $T' = \{(0,0),(5,0),(0,4)\}$ be a triangle in the coordinate plane, determine the number of sequences of three transformations—comprising rotations by $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, and reflections across the lines $y = x$ and $y = -x$—that will map $T'$ onto itself.
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Context:
Let \(\left\{a_{n}\right\}\) be a sequence of positive integers such that \(a_{1}=1\), \(a_{2}=2009\) and for \(n \geq 1\), \(a_{n+2} a_{n} - a_{n+1}^{2} - a_{n+1} a_{n} = 0\). Determine the value of \(\frac{a_{993}}{100 a_{991}}\).
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Context:
Determine the mass percentage of Al in a mixture containing 2 moles of Aluminium sulfide (Al2S3), 3 moles of Calcium carbonate (CaCO3), and 5 moles of Potassium chloride (KCl).
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Context:
In triangle \( PQR \), points \( N \) and \( L \) are located on sides \( PQ \) and \( PR \) respectively, such that \( NQ = LR \). The intersection point \( A \) of segments \( QL \) and \( NR \) divides \( QL \) in the ratio \( m:n \) from point \( Q \). Determine the ratio \( \frac{PN}{PR} \).
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Context:
Given a function $y=f(x)$ ($x \in \mathbb{R}$) that satisfies $f(x+2)=f(x)$, and when $x \in (-1,1]$, $f(x)=|x|$, find the number of real roots of $\log_{3}|x|-f(x)=0$. Express your answer as a single integer.
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Context:
Evaluate the line integral \(\int_{C} \left(\frac{\sin(3x)}{x^2+1}-6x^2y\right) dx + \left(6xy^2+\arctan\left(\frac{y}{7}\right)\right) dy\) using Green's Theorem, where \(C\) is the circle \(x^2+y^2=8\) oriented counterclockwise.
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Context:
设三位数 $n=abc$，若以 $a, b, c$ 为三条边长可以构成一个等腰（含等边）三角形，则这样的三位数 $n$ 有多少个？
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Context:
Let  $ABC$  be a triangle with  $\angle  A = 90^o, \angle B = 60^o$  and  $BC = 1$  cm. Draw outside of  $\vartriangle ABC$  three equilateral triangles  $ABD,ACE$  and  $BCF$ . Determine the area of  $\vartriangle DEF$ .
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Context:
Solve the equation: \(2^x + x = 4\).
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Context:
There are 99 identical square tiles, each with a quarter-circle drawn on it. When the tiles are randomly arranged in a 9 by 11 rectangle, what is the expected value of the number of full circles formed?
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Context:
Let  $\Omega$  be a unit circle and  $A$  be a point on  $\Omega$ . An angle  $0 < \theta < 180^\circ$  is chosen uniformly at random, and  $\Omega$  is rotated  $\theta$  degrees clockwise about  $A$ . What is the expected area swept by this rotation?
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Context:
Consider a triangle \( ABC \) with angle \( A = \theta \) where \( \theta \) is in the range \( (0, \pi) \), and side \( BC = 1 \). Determine the number of incongruent triangles for which the points \( A \), the centroid of the triangle, the midpoint of \( AB \), and the midpoint of \( AC \) are concyclic.
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Context:
For two lines $ax+2y+1=0$ and $3x+(a-1)y+1=0$ to be parallel, determine the value of $a$ that satisfies this condition.
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Context:
On a circle of radius 12 with center at point \( O \), there are points \( A \) and \( B \). Lines \( AC \) and \( BC \) are tangent to this circle. Another circle with center at point \( M \) is inscribed in triangle \( ABC \) and touches side \( AC \) at point \( K \) and side \( BC \) at point \( H \). The distance from point \( M \) to line \( KH \) is 3. Find \( \angle AOB \).
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Context:
$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150.$ Find the $\angle MT D.$
[i](K. Ivanov )[/i]
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Context:
Evaluate the integral \(\int_{S^2} \frac{dS}{\sqrt{(x-a)^2 + y^2 + z^2}}\) over the unit sphere \(S^2\), given that \(a > 1\), using only multivariable calculus techniques.
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Context:
Given that $f(x)$ is an odd function defined on $(−∞,0)∪(0,+∞)$, and $f(x+ \frac{5}{2})f(x)=1$, if $f(-1) > 1$ and $f(2016)= \frac{a+3}{a-3}$, then the range of $a$ is _______.
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Context:
Find the residue at infinity for the function \( \frac{e^{z^2}}{z^{2n+1}} \).
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Context:
Find the large number from the below question. The difference of two numbers is 24672. On dividing the larger number by the smaller, we get 13 as quotient and 257 as remainder.
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Context:
A bus stop has two independent Poisson arrival processes: buses arrive according to a Poisson process with rate $\mu$, and passengers arrive according to a Poisson process with rate $\lambda$. Assuming that all passengers waiting at the stop board the bus when it arrives, find the probability that there are exactly $k$ passengers on the third bus.
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Context:
In a math competition, there are 6 questions, each worth 7 points, with scores ranging from 0 to 7. The total score of a participant is the product of their scores on the 6 questions. If two participants have the same product score, their ranking is determined by the sum of their scores. If these are also the same, the participants are tied. There are 262,144 participants with no ties. What is the total score of the participant ranked 117,649th?
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Context:
A hay farmer harvested 560 bales of hay from 5 acres of grass per month last year. This year, he planted an additional 7 acres of grass. If the farmer also owns 9 horses and each horse consumes 3 bales of hay a day, how many bales of hay would the farmer have left by the end of December if he starts feeding them this year's hay beginning the first day of September?
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Context:
Determine the minimum number of tokens that must be placed in the cells of a $99 \times 99$ grid such that every $4 \times 4$ subgrid contains at least eight tokens.
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Context:
Consider all the permutations of the 26 English alphabets that start with the letter 'z'. In how many of these permutations is the number of letters between 'z' and 'y' less than the number of letters between 'y' and 'x'?
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Context:
Determine the values of \( \alpha \) for which the generalized Gaussian probability density \( p_{\alpha}(t) \propto \exp (- |t|^\alpha) \) is infinitely divisible, given that \( 0<\alpha<\infty \).
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Context:
Julia is learning how to write the letter C. She has 6 differently-colored crayons, and wants to write Cc Cc Cc Cc Cc. In how many ways can she write the ten Cs, in such a way that each upper case C is a different color, each lower case C is a different color, and in each pair the upper case C and lower case C are different colors?
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Context:
Given that \(2 \cdot 50N\) is an integer and its representation in base \(b\) is 777, find the smallest positive integer \(b\) such that \(N\) is a fourth power of an integer.
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Context:
Among all the line segments connecting a point outside a line to every point on the line, the shortest is ___.
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Context:
Smith, Brown, and Robinson each bought shares with their wives. Mary bought 50 more shares than Mr. Brown, and Mr. Robinson bought 120 more than Jane. Each man paid as many shillings per share as he bought shares, and each wife as many pence per share as she bought shares. Each man spent one guinea more than his wife. Determine Eliza's surname given these conditions.
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Context:
Given that the terminal side of angle $\alpha$ passes through the fixed point $P$ on the function $y=\log _{a}(x-3)+2$, find the value of $\sin 2\alpha+\cos 2\alpha$.
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Context:
In Figure 1, \( P, B \) and \( C \) are points on a circle with center \( O \) and diameter \( BC \). If \( A, B, C, D \) are collinear such that \( AB = BC = CD \), \( \alpha = \angle APB \) and \( \beta = \angle CPD \), find the value of \( (\tan \alpha)(\tan \beta) \).
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Context:
Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability $\frac{9}{10}$ . Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability $\frac{3}{4}$ . Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Evaluate the limit: \[ \lim_{n\to\infty} \frac{\sqrt[n]{(n+1)!}}{n+1} \]
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Context:
3 candidates in an election and received 1136 , 7636 and 10628 votes respectively . what % of the total votes did the winning candidate gotin that election ?
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Context:
2. Let $ABCD$ be a convex quadrilateral such that

$$
\measuredangle ABC=90^{\circ}, \measuredangle DAB=60^{\circ} \text{ and } \measuredangle BCD=120^{\circ}.
$$

Let the diagonals $AC$ and $BD$ intersect at point $M$, such that $\overline{BM}=1$ and $\overline{MD}=2$. Calculate the area of quadrilateral $ABCD$.
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Context:
Given that the first term of an arithmetic sequence is $31$, and the terms from the $16$-th term onwards are less than $1$, find the range of the common difference $d$.
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Context:
Compute the Riemann-Stieltjes integral \( \int_{-1}^{1} (x^2 + e^x) \, d(\text{Sgn}(x)) \).
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Context:
In triangle \(ABC\), it is known that \(AB = BC\) and \(AC = 4 \sqrt{3}\). The radius of the inscribed circle is 3. Line \(AE\) intersects the altitude \(BD\) at point \(E\), and the inscribed circle at points \(M\) and \(N\) (with \(M\) being between \(A\) and \(E\)). Given that \(ED = 2\), find \(EN\).
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Context:
Using the digits 1, 2, 3, 4, and 9 once each, form the smallest possible even five-digit number. What is the digit in the tens place of this number? Express your answer as a single digit.
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Context:
Petya bought a "Constructor" set, which contained 100 sticks of different lengths. The instructions for the "Constructor" state that any three sticks from the set can form a triangle. Petya decided to test this statement by forming triangles with the sticks. The sticks are arranged in the constructor in ascending order of length. What is the minimum number of checks (in the worst case) that Petya needs to perform to prove or disprove the statement in the instructions?
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Context:
How many positive integers  $a$  with  $a\le 154$  are there such that the coefficient of  $x^a$  in the expansion of \[(1+x^{7}+x^{14}+ \cdots +x^{77})(1+x^{11}+x^{22}+\cdots +x^{77})\] is zero?

*Author: Ray Li*
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Context:
If five points are given on a plane, considering all possible triples of these points, 30 angles can be formed. Let the smallest of these angles be denoted as $\alpha$. Find the maximum value of $\alpha$.
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Context:
A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$-sided die, with faces labeled $0, 1, 2, \ldots, 2018$, and with each face equally likely to come up. Every second, the mouse rolls the die. If the die lands on $k$, and the mouse has at least $k$ slices of cheese remaining, then the mouse eats $k$ slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?
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Context:
Given that $17^{-1} \equiv 31 \pmod{53}$, find $36^{-1} \pmod{53}$ as a residue modulo 53 (i.e., a value between 0 and 52 inclusive).
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Context:
设 $M=\{1,2, \cdots, 1995\}, A$ 是 $M$ 的子集, 且满足条件: 当 $x \in$ $A$ 时, $15 x \bar{\in} A$. 求 $|A|$ 的最大值.
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Context:
For the NEMO, Kevin needs to compute the product
		\[
		9 \times 99 \times 999 \times \cdots \times 999999999.
		\]
		Kevin takes exactly  $ab$  seconds to multiply an  $a$ -digit integer by a  $b$ -digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.

*Proposed by Evan Chen*
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Context:
Evaluate the limit as \(x\) approaches 0 from the left:
\[\lim_{x\to0^-}\sum_{n=1}^{\infty}\frac{\sin x}{4+n^2x^2}\]
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Context:
Find the radius of convergence for the series \( f(x) = \sum_{n=1}^\infty \left(1+\frac{(-1)^n}{n}\right)^{n^2} \cdot \frac{(2x+1)^n}{n} \).
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Context:
In the field \(\mathbb{F}_2\) of order 2, consider the ideal \(I\) in \(\mathbb{F}_2[x]\) generated by the polynomials \(x^5 + x^2 + x + 1\) and \(x^7 + x^5 + x^3 + x\). How many polynomials of degree 3 are elements of \(I\)?
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Context:
Some language has only three letters -  $A, B$  and  $C$ . A sequence of letters is called a word iff it contains exactly 100 letters such that exactly 40 of them are consonants and other 60 letters are all  $A$ . What is the maximum numbers of words one can pick such that any two picked words have at least one position where they both have consonants, but different consonants?
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Context:
Xiaohong drives to a place to attend a meeting. 30 minutes after departure, due to traffic congestion, there was a delay of 25 minutes. In order to arrive at the meeting venue on time, Xiaohong increased her speed by 25%. Eventually, she arrived at the meeting venue on time. It took Xiaohong a total of     minutes from departure to the meeting venue.
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Context:
Given a regular quadrilateral pyramid S-ABCD, points P and Q are the midpoints of SB and SD respectively. The plane passing through points A, P, and Q divides S-ABCD into two parts with volumes V1 and V2 such that V1 < V2. What is V2/V1 equal to?
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Context:
Let \( S \) be a subset of \(\{1, 2, \cdots, 1989\}\), and the difference between any two numbers in \( S \) is not equal to 4 or 7. What is the maximum number of elements in \( S \)?
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Context:
Let $P$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $P$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $AB=20$ and $CD=14$, compute the area of $ABCD$.
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Context:
Alex is a practical-minded economics student who loves to track his spending and investments. He decides to start investing in a small business project and wants to calculate his potential profits. Alex invests $500 in the project, which promises a 10% return monthly. He also incurs a monthly cost of $50 for project-related expenses. After four months, Alex wants to know his total profit. Calculate Alex's total profit after four months, considering both the returns on his investment and the monthly expenses.
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Context:
In triangle \(ABC\), \(\angle CAB = 75^\circ\) and \(\angle ABC = 45^\circ\). Point \(K\) is taken on side \(CA\), and point \(M\) on side \(CB\), such that \(CK:KA = 3:1\).

Find \(KM:AB\) if this ratio is less than \(3/4\), and if the line \(MK\) cuts off a triangle from triangle \(ABC\) that is similar to it.
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Context:
Given two permutations \(\sigma = (13624)(587)(9)\) and \(\tau=(15862)(394)(7)\) in \(S_9\), determine the number of permutations \(\alpha\) such that \(\alpha\sigma\alpha^{-1} = \tau\). How many such \(\alpha\) are there?
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Context:
Let \( n \) be a positive integer not exceeding 1996. If there exists a \( \theta \) such that \( (\sin \theta + i \cos \theta)^{n} = \sin \theta + i \cos n \theta \), find the number of possible values for \( n \).
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Context:
Evaluate the integral \( \int_{-1}^{1} \frac{x^{1/3} + 2}{\sqrt{|1 - x^{1/3}|} - \sqrt{x}} \, dx \).
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Context:
Find the minimal cardinality of a quotient ring $R/I$, where $I$ is a proper ideal of the ring $R = \mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]$. Assume that the polynomial $x^2 - x + 5$ is irreducible over both $\mathbb{F}_2$ and $\mathbb{F}_3$. Provide your answer as a single integer.
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Context:
Given that $n$ and $k$ are positive integers with $n > k$, and real numbers $a_{1}, a_{2}, \cdots, a_{n} \in (k-1, k)$. Let positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy the condition that for any $k$-element subset $I$ of $\{1,2,\cdots, n\}$, the inequality $\sum_{i \in I} x_{i} \leqslant \sum_{i \in I} a_{i}$ holds. Find the maximum value of $x_{1} x_{2} \cdots x_{n}$.
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Context:
Let \(f(z)\) be a cubic polynomial with distinct roots \(r\), \(s\), and \(t\). If \(g(z)=f(cz)\) for some complex number \(c\), and \(g(z)\) also has distinct roots, find the number of possible values of \(|c|\).
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Context:
A store has three types of boxes containing marbles in large, medium, and small sizes, respectively holding 13, 11, and 7 marbles. If someone wants to buy 20 marbles, it can be done without opening the boxes (1 large box plus 1 small box). However, if someone wants to buy 23 marbles, a box must be opened. Find the smallest number such that any purchase of marbles exceeding this number can always be done without opening any boxes. What is this smallest number?
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Context:
In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group.
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Context:
Let \( 0 < c < 1 \) and \( n \) a positive integer. Alice and Bob are playing a game. Bob writes \( n \)
 integers on the board, not all equal. On a player's turn, they erase two numbers from the
 board and write their arithmetic mean instead. Alice starts and performs at most \( {cn} \)
 moves. After her, Bob makes moves until there are only two numbers left on the board.
 Alice wins if these two numbers are different, and otherwise, Bob wins.
 For which values of \( c \) does Alice win for all large enough \( n \) ?
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Context:
7.5. Miron and Varya are playing the following game. There is a pile of 32 stones on the table. The players take turns, with Miron starting first. On their turn, a player divides any pile that has more than one stone into several equal piles. The player who cannot make a move (when there is exactly one stone in each pile before their turn) wins. Who will win if both players play optimally?
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Context:
What is the maximum number of cells in an 8x8 chessboard that can be cut by a single straight line?
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Context:
Let \( A = \{1, 2, \cdots, 2004\} \) and \( f: A \rightarrow A \) be a bijection satisfying \( f^{[2004]}(x) = f(x) \), where \( f^{[2004]}(x) \) denotes applying \( f \) 2004 times to \( x \). How many such functions \( f \) are there?
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Context:
In how many ways can 8 people be seated in a row of chairs if three of those people, Alice, Bob, and Charlie, refuse to sit next to each other in any order?
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Context:
Determine the number of Boolean functions \( F(w, x, y, z) \) that satisfy the condition \( F(\bar{w}, x, \bar{y}, z) = F(w, \bar{x}, y, \bar{z}) = F(w, x, \bar{y}, \bar{z}) \) for all Boolean variables \( w, x, y, \) and \( z \).
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Context:
Solve the equation: \[ \frac{10}{x+10}+\frac{10 \cdot 9}{(x+10)(x+9)}+\frac{10 \cdot 9 \cdot 8}{(x+10)(x+9)(x+8)}+\cdots+ \frac{10 \cdot 9 \ldots 2 \cdot 1}{(x+10)(x+9) \ldots(x+1)}=11 \]
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Context:
Given an isosceles triangle $ABC$ with base $AC$. Points $K$ and $N$ are marked on the side $BC$ (with $K$ between $B$ and $N$). It is found that $KN = AN$ and $\angle BAK = \angle NAC$. Find $\angle BAN$.
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Context:
In the game "set," consider all possible four-digit numbers that can be formed using the digits 1, 2, and 3, with each digit used exactly once in each number. A triplet of numbers forms a set if, for each digit position, either all three numbers have the same digit, or all three numbers have different digits. How many distinct sets can be formed in this game?
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Context:
The hypotenuse of a right triangle whose legs are consecutive even numbers is 34 units. What is the sum of the lengths of the two legs?
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Context:
Find the expectation of the posterior distribution given by:
\[ p_{\theta|x}(\theta) = \frac{\theta^x(1-\theta)^x\cdot6\theta(1-\theta)}{\int_0^1\theta^x\cdot6\theta(1-\theta)d\theta} \] where \( X \) is a random variable from the Bernoulli distribution with parameter \( \theta \), and the prior density \( p(\theta) = 6\theta(1 - \theta) \) for \( \theta \in (0, 1) \).
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Context:
Find the independent coefficient in the inverse of the polynomial $2x^2+2$ in the ring $\mathbb{Z}_3[x]/(x^3+2x^2+2)$. Provide your answer as an integer in $\mathbb{Z}_3$. \(\boxed{\phantom{0}}\)
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Context:
David and I each have a sheet of computer paper, mine evenly spaced by $19$ parallel lines into $20$ sections, and his evenly spaced by $29$ parallel lines into $30$ sections. If our two sheets are overlayed, how many pairs of lines are perfectly incident?
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Context:
Joseph has a refrigerator, a water heater, an electric oven, an air conditioner, and a washing machine in his house that consumes power at different rates. The total amount of money that Joseph pays for the energy used by the refrigerator is three times the amount he pays for the power used by the water heater. The electric oven uses power worth $500 in a month, twice what Joseph pays for the power the water heater uses. The air conditioner consumes power worth $300 in a month, which is half the cost of the power used by the refrigerator. The washing machine consumes power worth $100 in a month, one-fifth of the cost of the power used by the water heater. Calculate the total amount Joseph pays for the power used by all these gadgets in a month.
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Context:
Two circles are given by the equations $x^2+y^2-2mx=0$ and $(x^2+(y-2)^2=1)$. The circles are externally tangent. Find the value of the real number m. Express your answer as a numerical value, including the ± symbol if necessary to denote multiple solutions.
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Context:
Let $X$ be a nonempty set with $m$ members. How many $\sigma$-algebras can we make on this set?
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Context:
Let
\[f_1(x) = \frac{1}{x}\quad\text{and}\quad f_2(x) = 1 - x\]
Let $H$ be the set of all compositions of the form $h_1 \circ h_2 \circ \ldots \circ h_k$, where each $h_i$ is either $f_1$ or $f_2$.  For all $h$ in $H$, let $h^{(n)}$ denote $h$ composed with itself $n$ times.  Find the greatest integer $N$ such that $\pi, h(\pi), \ldots, h^{(N)}(\pi)$ are all distinct for some $h$ in $H$.
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Context:
Two brothers sold a flock of sheep, obtaining for each sheep as many rubles as there were sheep in the flock. Wishing to divide the proceeds equally, they took turns, starting with the elder brother, taking 10 rubles each from the total amount. After the elder brother took 10 rubles one more time, the younger brother was left with less than 10 rubles. To ensure an equal division, the elder brother gave the younger brother his knife. How many rubles was the knife valued at?
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Context:
对某些正整数 n>m，n 既可以表示为 2021 个加数的和，其中每个加数均为 m 的非负整数次幂，又可以表示为 2021 个加数的和，其中每个加数均为 m+1 的非负整数次幂. 求 m 的最大可能值.
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Context:
Given that \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C: \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 \), and point \( P \) lies on the hyperbola \( C \). Let \( G \) and \( I \) be the centroid and incenter of \( \triangle F_{1} P F_{2} \) respectively. If \( G I \) is parallel to the \( x \)-axis, find the circumradius \( R \) of \( \triangle F_{1} P F_{2} \).
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Context:
The inclination angle of the line $x-y+1=0$ can be calculated.
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Context:
若不等式$a{{x}^{2}}+bx+c>0$的解集为$\left( -1,2 \right)$，则不等式$c{{x2}}-\left( a+c \right)x-b<0$的解集为__________．
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Context:
Given the sets M = {x | x^2 ≤ 4} and N = {x | log2(x) ≤ 1}, determine the set M ∩ N. Express your answer in set notation.
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Context:
log base ( 5x^3 ) root ( 7x - 3 ) / log ( 7x - 3 ) is equal to
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Context:
In a Markov chain on the state space $\{0, 1, \ldots\}$, the transition probabilities are given by $p_{i, i-1}=q$ and $p_{i,i+1}=p$, where $p > q$. Determine the expected hitting time to reach state 0 from each state $k \geq 1$.
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Context:
Quadrilateral $A B C D$ satisfies $A B=8, B C=5, C D=17, D A=10$. Let $E$ be the intersection of $A C$ and $B D$. Suppose $B E: E D=1: 2$. Find the area of $A B C D$.
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Context:
Find the greatest constant \( k > 0 \) such that the inequality \( \max\left(\frac{5}{5-3c}, \frac{5b}{5-3d}\right) \geq k \cdot \frac{2+3b}{5-c-2d} \) holds for all \( 0 \leq b \leq 1 \) and \( 0 \leq c \leq d \leq 1 \).
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Context:
Identify the well-known topological space that is homeomorphic to the quotient space of \(\mathbb{R}^2\) induced by the equivalence relation \((a_0, b_0) \sim (a_1, b_1)\) if and only if \(a_0 + b_0^2 = a_1 + b_1^2\).
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Context:
Determine the values of $t > 1$ for which the set $F = \{a \in \mathbb{R} : a < 1\}$, together with the operations $a \boxplus b = a + b - ab$ and $a \boxdot b = 1 - t^{\log_t (1-a) \log_t (1-b)}$, forms a field.
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Context:
Evaluate the limit of the line integral \( \int_\mathrm{arc} \frac{\exp(iz)}{z+1} \, dz \) as the radius \( R \) of the semi-circular arc in the upper half complex plane goes to infinity. Does the integral converge to a fixed constant? If so, calculate this constant.
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Context:
Let $ABCD$ be a cyclic quadrilateral with $BC = CD = 2$. The incenter of triangle $ABD$ is $I$, and $AI = 2$. Determine the minimum value of the length of diagonal $BD$. \( \text{Express your answer as a single number.} \)
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Context:
An insurance company pays $10,000 for the first loss, $7,500 for the second loss, and $5,000 for each successive loss. If the number of losses follows a Poisson distribution with a mean of 2.5, calculate the expected payment due to losses.
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Context:
There are two types of electronic toy cars, Type I and Type II, each running on the same two circular tracks. Type I completes a lap every 5 minutes, while Type II completes a lap every 3 minutes. At a certain moment, both Type I and Type II cars start their 19th lap simultaneously. Calculate how many minutes earlier the Type I car started running compared to the Type II car. Express your answer in minutes as a whole number.
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Context:
Determine the value that \(\frac{B}{x}\) approaches, where \(B=\sqrt[3]{x\sqrt[5]{x^4\sqrt[9]{x^{24}\sqrt[17]{x^{240}\ldots}}}}\).
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Context:
Let $ ABCD$ be a parallelogram with $ AB=1$. Suppose that $ K$ is a point on the side $ AD$ such that $ KD=1, \angle ABK=90^{\circ}$ and $ \angle DBK=30^{\circ}$. Determine $ AD$.
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Context:
Determine the cardinality of the set $S$, which consists of all closed intervals on the real line with lengths being positive rational numbers.
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Context:
Simplify the matrix expression: \[ B = (Z^TZ)^{-1} + (Z^TZ)^{-1}Z^T( I - P_z)^{-1}Z(Z^TZ)^{-1} \] where \( Z \) is an \( n \times 1 \) vector, \( I \) is the identity matrix, and \( P_z = Z(Z^TZ)^{-1}Z^T \) is the projection matrix of \( Z \).
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Context:
In the triangular prism D-ABC, given that $\angle \mathrm{ACB} = \angle \mathrm{ABD} = 90^{\circ}$, $\mathrm{CA} = \mathrm{CB}$, and $\angle \mathrm{BAD} = 30^{\circ}$, if the projection of point $\mathrm{C}$ onto plane $\mathrm{ABD}$ is precisely on line $\mathrm{AD}$, find the sine of the plane angle of the dihedral angle $\mathrm{C}-\mathrm{AB}-\mathrm{D}$.
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Context:
Two trains travel in opposite directions at some speed and 45 kmph, and a man sitting in the slower train passes the faster train in 6 seconds. The length of the faster train is 135.0108 meters. What is the speed of the slower train?
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Context:
Evaluate the line integral \( \int _{C} (z^3 + 2z + \text{Re}(z))\,dz \) where \( C \) is the triangle with vertices \( z=0 \), \( z=1+2i \), and \( z=1 \).
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Context:
Given a random sample \(X_1, X_2, \dots, X_n\) from a distribution with the probability density function \(f(x, \theta) = e^{-(x-\theta)}\) for \(\theta < x < \infty\) and \(-\infty < \theta < \infty\), find the sufficient statistic for the parameter \(\theta\) using the factorization theorem.
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Context:
How many more hours are in $10$ years than seconds in $1$ day?
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Context:
What is the maximum number of points \( P \) in the plane of the convex quadrilateral \( ABCD \) for which the areas of the triangles \( PAB \), \( PBC \), \( PCD \), and \( PDA \) are equal?
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Context:
Determine the number of distinct ways to label the vertices of a cube with the integers 1 through 8, without repetition, such that the sum of the numbers on the four vertices of each face is the same. Arrangements that can be obtained through rotations of the cube are considered identical. How many different arrangements are possible?
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Context:
Calculate the mass of the plane curve defined by \( y = \frac{x^{3}}{3} \) over the segment from \( x = 0 \) to \( x = 0.1 \), given that the linear density \( \rho(x) = 1 + x^{2} \).
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Context:
Given an isosceles triangle ABC, where the base points A and B are on the right branch of the hyperbola $\frac {x^{2}}{6}- \frac {y^{2}}{3}$=1, and the vertex C is on the x-axis. If AB is not perpendicular to the x-axis, find the range of the x-coordinate t of vertex C.
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Context:
Evaluate the limit of the following expression as \( r \to 0 \):

\[ \lim_{r\to0}\left(\frac{1}{\mu_1-1}\,\mathrm{K}_0(\mathrm{j}\,k_1\,r)-\frac{1}{\mu_2-1}\,\mathrm{K}_0(-\mathrm{j}\,k_2\,r)\right) \]

Given that \( \mu_1\,k_1^2=\mu_2\,k_2^2 \), \( k_1=\sqrt{k_1^2} \), \( k_2=\sqrt{k_2^2} \), \( \mathrm{Im}[k_1]<0 \), and \( \mathrm{Im}[k_2]>0 \).
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Context:
Let x, y, and z be distinct integers such that w < x < y < z, where x is an even integer, y and z are odd integers, and w is a negative, non-integer multiple of 1/3. If the difference (y - x) > a positive integer k and (z - w) > a positive integer m, where k > m, what is the least possible value of z - w?
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Context:
Let \( A_{1}, A_{2}, \ldots, A_{11} \) be 11 points on a straight line in order, where \( A_{1} A_{11} = 56 \). Given that \( A_{i} A_{i+2} \leq 12 \) for \( i = 1, 2, \ldots, 9 \) and \( A_{j} A_{j+3} \geq 17 \) for \( j = 1, 2, \ldots, 8 \), find \( A_{2} A_{7} \).
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Context:
Cutting equilateral triangle $BFC$ out of square $ABCD$ and translating it to the left of the square creates the shaded figure $ABFCDE$. The perimeter of square $ABCD$ was 48 inches. What is the perimeter, in inches, of $ABFCDE$?

[asy]
filldraw((0,0)--(sqrt(3),1)--(sqrt(3)+2,1)--(2,0)--(sqrt(3)+2,-1)--(sqrt(3),-1)--(0,0)--cycle,gray,linewidth(1));
draw((sqrt(3),1)--(sqrt(3),-1),linewidth(1));
draw((sqrt(3)+2,-1)--(sqrt(3)+2,1)--(2,0)--(sqrt(3)+2,-1)--cycle,dashed);

label("A",(sqrt(3),1),N);
label("B",(sqrt(3)+2,1),N);
label("C",(sqrt(3)+2,-1),S);
label("D",(sqrt(3),-1),S);
label("E",(0,0),W);
label("F",(2,0),N);
[/asy]
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Context:
Given a sequence $\{a\_n\}$, where $a\_1=3$, $n(a_{n+1}-a_n)=a_n+1$, $n\in\mathbb{N}^*$. If for any $a\in[-1,1]$, $n\in\mathbb{N}^*$, the inequality $\frac{a_{n+1}}{n+1} < t^2-2at+1$ always holds, then the range of the real number $t$ is _______.
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Context:
$24, 2$ red balls and $11$ white balls are arranged in a row, satisfying the following conditions: the red balls are not adjacent, and in any consecutive $7$ balls, there is at least one red ball, then there are $\qquad$ ways to arrange them.
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Context:
设函数 $f(x)=\sqrt{2 x^{2}+2 x+41}-\sqrt{2 x^{2}+4 x+4}(x \in \mathbf{R})$, 则 $f(x)$ 的最大值是
$\qquad$.
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Context:
Find the number of non-unital ring homomorphisms from the ring \( \mathbb{Z}/20\mathbb{Z} \) to the ring \( \mathbb{Z}/12\mathbb{Z} \).
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Context:
Determine the limit \( \lim_{n\to\infty} f_n \) in the supremum norm over the interval \([-1, 1]\) for the sequence of real coefficient polynomials \(\{f_n\}\) defined recursively by \(f_0 = 0\) and \(f_{n+1}(x) = f_n(x) + \frac{x^2 - f_n^2(x)}{2}\) for \(n = 0, 1, 2, \ldots\).
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Context:
 Find the number of functions $f$ from $\{1,2,3,4,5\}$ to itself such that $f(f(f(x))) = f(f(x))$ for all $x \in \{1,2,3,4,5\}$.
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Context:
Evaluate the expression: $$ \frac{\tan(180^{\circ}/7)}{\tan(360^{\circ}/7)} - \frac{\tan(360^{\circ}/7)}{\tan(540^{\circ}/7)} - \frac{\tan(540^{\circ}/7)}{\tan(180^{\circ}/7)} $$
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Context:
Evaluate the double integral \( \mathop{\int\!\!\! \int}_{R_{1}}f(x,y)\,dA \) for the rectangle \( R_{1}=[0, 5] \times [-4, 4] \), given that \( \mathop{\int\!\!\! \int}_{R_{2}}f(x,y)\,dA=10 \) for the rectangle \( R_{2}=[0, 5] \times [0, 4] \), \( \mathop{\int\!\!\! \int}_{R_{3}}f(x,y)\,dA=24 \) for the rectangle \( R_{3}=[-5, 0] \times [-4, 0] \), and \( f(-x,y)=-f(x,y) \) for all \( (x,y) \).
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Context:
7. The cells of a $2017 \times 100$ board (2017 rows, 100 columns) are painted in a checkerboard pattern. A grasshopper standing on the board attacks all cells in its row that have the same color as the cell it stands on, as well as all cells in its column that have the opposite color. (To attack a cell, the grasshopper can jump over other pieces.) What is the maximum number of non-attacking grasshoppers that can be placed on this board?
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Context:
Given the differential equation \((1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + (\sec x)y = 0\), let \(y_1(x)\) and \(y_2(x)\) be two linearly independent solutions with a Wronskian \(W(x)\). If \(y_1(0) = 1\), \(\left(\frac{dy_1}{dx}\right)_{x=0} = 0\), and \(W\left(\frac{1}{2}\right) = \frac{1}{3}\), determine the value of \(\left(\frac{dy_2}{dx}\right)_{x=0}\) without explicitly finding \(y_1(x)\) and \(y_2(x)\).
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Context:
At a club, twenty gentlemen met. Some of them were wearing hats, and some were without hats. From time to time, one of the gentlemen took off his hat and put it on one of those who did not have a hat at that moment. In the end, ten gentlemen counted that each of them had given away a hat more times than they had received one. How many gentlemen came to the club wearing hats?
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Context:
The curvature $\varphi(M,N)$ of a curve $y=f(x)$ between two different points $M(x_{1},y_{1})$ and $N(x_{2},y_{2})$ on its graph, where the slopes of the tangents at $M$ and $N$ are $k_{M}$ and $k_{N}$ respectively, is defined as $\varphi(M,N)=\frac{|k_{M}-k_{N}|}{|MN|}$ (where $|MN|$ is the length of the segment $MN$). For the curve $f(x)=x^{3}+2$ with two different points $M(x_{1},y_{1})$ and $N(x_{2},y_{2})$ on it, and given that $x_{1}x_{2}=1$, the range of $\varphi(M,N)$ is ______.
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Context:
Evaluate the limit: $$ \lim_{n \to \infty} \left( \ \sum_{k=10}^{n+9} \frac{2^{11(k-9)/n}}{\log_2 e^{n/11}} \ - \sum_{k=0}^{n-1} \frac{58}{\pi\sqrt{(n-k)(n+k)}} \ \right) $$
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Context:
Given the planar vectors $\overrightarrow {a}$ and $\overrightarrow {b}$, with $|\overrightarrow {a}|=1$, $|\overrightarrow {b}|=2$, and $\overrightarrow {a} \cdot \overrightarrow {b} = 1$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$.
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Context:
Let \(\zeta_7 = e^{i\frac{2\pi}{7}}\) be the 7th root of unity. Determine the integer \(D\) such that the quadratic subfield of \(\mathbb{Q}(\zeta_7)\) can be expressed as \(\mathbb{Q}(\sqrt{D})\). Given that \(\omega = \zeta_7 + \zeta_7^{-1} = 2\cos(\frac{2\pi}{7})\) lies in \(\mathbb{Q}(\zeta_7)\) and generates the subfield \(K = \mathbb{Q}(\omega)\) with \([K(\zeta_7):K] = 2\), find the value of \(D\).
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Context:
Let an ordered pair of positive integers $(m, n)$ be called regimented if for all nonnegative integers $k$, the numbers $m^k$ and $n^k$ have the same number of positive integer divisors. Let $N$ be the smallest positive integer such that $(2016^{2016}, N)$ is regimented. Compute the largest positive integer $v$ such that $2^v$ divides the difference $2016^{2016} - N$.
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Context:
If real number $a > b$, then fill in the inequality for $a^2 - ab \_\_\_\_\_\_\_\_ ba - b^2$.
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Context:
In triangle ABC, the magnitudes of the sides are a, b, and c, as shown, with c ≤ b ≤ a. Through interior point P and the vertices A, B, C, lines are drawn meeting the opposite sides in A', B', C', respectively. Let s = AA' + BB' + CC'. Find the upper bound for s in terms of a, b, and c.
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Context:
In a certain kennel with 45 dogs, each of the dogs is a single color. Each of the dogs in the kennel either has long fur or does not. There are 36 dogs with long fur, and 8 dogs that are neither long-furred nor brown. How many dogs are brown?
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Context:
Find the value of \( a \) such that the line integral \( I = \int_\Gamma x(a+y)(ydx+xdy) \) over the path \( \Gamma \) equals 36. The path \( \Gamma \) consists of three segments: from \( O = (0,0) \) to \( A = (3,0) \), from \( A = (3,0) \) to \( B = (0,3) \), and from \( B = (0,3) \) back to \( O = (0,0) \). The segment \( AB \) is characterized by the quarter-circle \( \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 = 9, x \ge 0, y \ge 0\} \).
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Context:
The roots of the polynomial equation \( x^3 - 3x - 1 = 0 \) are \( a, b, \) and \( c \). Calculate the value of the expression:
\[ L = \frac{1}{a^2 - 2a + 2} + \frac{1}{b^2 - 2b + 2} + \frac{1}{c^2 - 2c + 2} \]
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Context:
Evaluate the limit: \[ \lim_{x \to 0^+} \frac{1}{\ln x} \sum_{n=1}^{\infty} \frac{x}{(1+x)^n - (1-x)^n}. \]
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Context:
17. If a square can completely cover a triangle with side lengths 3,4 and 5 , find the smallest possible side length of the square.
(2 marks)
若一個正方形可以完全覆蓋一個邊長 3、4、5 的三角形, 求正方形的最小邊長。
(2 分)
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Context:
已知 $M=\{1,2, \cdots, 8\}, A 、 B$ 是集合 $M$的两个不同子集，满足：
（1） $A$ 的元素个数比 $B$ 的元素个数少；
（2） $A$ 中的最小元素比 $B$ 中的最大元素大。
则所有满足条件的有序集合对 $(A, B)$ 的个数为$\qquad.$
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Context:
Given a right triangle $ABC$ with $AC = 3$, $BC = 4$, and $AB = 5$. Squares are constructed externally on each side of the triangle. Calculate the area of the hexagon $PQRSTU$.
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Context:
Solve the inequality \[ 8 \cdot \frac{|x+3|-|x-5|}{|2x-11|-|2x+7|} - 9 \cdot \frac{|x+3|+|x-5|}{|2x-11|+|2x+7|} \geq -8 \] and find the sum of its integer solutions that satisfy the condition \( |x| < 90 \).
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Context:
Find the domain of definition of the function \( f(x) = \int_{0}^{\infty} \frac{t^x}{1+t^4 \sin^2 t} \, dt \).
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Context:
Given the sequence \(a_n\) defined by the initial conditions \(a_1 = 1\), \(a_2 = 4\), \(a_3 = 5\), and the recurrence relation \(a_n + a_{n-1} + a_{n-2} + a_{n-3} = n^2\) for all \(n \geq 4\), find the sum of the digits of \(a_{2021}\).
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Context:
In a computer game, a player can choose to play as one of three factions: \( T \), \( Z \), or \( P \). There is an online mode where 8 players are divided into two teams of 4 players each. How many total different matches are possible, considering the sets of factions? The matches are considered different if there is a team in one match that is not in the other. The order of teams and the order of factions within a team do not matter. For example, the matches \((P Z P T ; T T Z P)\) and \((P Z T T ; T Z P P)\) are considered the same, while the matches \((P Z P Z ; T Z P Z)\) and \((P Z P T ; Z Z P Z)\) are different.
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Context:
Determine the spectrum of the infinite-dimensional matrix with elements $a_{ij} = \sqrt{j}$ if $i = j+1$, and $0$ otherwise.
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Context:
A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is 12 meters. What is the area (in square meters) of the room?
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Context:
In a right triangle, one of the acute angles $\alpha$ satisfies
\[\tan \frac{\alpha}{2} = \frac{1}{\sqrt[3]{2}}.\]Let $\theta$ be the angle between the median and the angle bisector drawn from this acute angle.  Find $\tan \theta.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Determine the value of the parameter \( a \) for which the magnitude \( |x + y| \), where \( (x, y) \) is a solution of the system of equations \( 4x^2 + 4y^2 = -a^2 + 16a - 32 \) and \( 2xy = a \), reaches its maximum value.
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Context:
A triangle has two sides of lengths 4 cm and 9 cm. The radius of the inscribed circle that touches all three sides of the triangle is 1 cm. Determine the length of the third side of the triangle.
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Context:
Jerry made 14 dollars mowing lawns and 31 dollars weed eating. He also earned an additional 20 dollars from trimming bushes. If he spends 5 dollars on snacks, 10 dollars on transportation, and saves 8 dollars every week, how long would his total earnings last him for?
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Context:
Suppose we roll a die with $n$ faces repeatedly and let $T_k$ be the number of rolls until some number appears $k$ times in a row. Find the expected value of $T_3$ for a die with $10$ faces.
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Context:
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?
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Context:
On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted.
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Context:
Let $p$ be an odd prime number less than $10^5$. Granite and Pomegranate play a game. First, Granite picks an integer $c \in \{2,3,\dots,p-1\}$. Pomegranate then picks two integers $d$ and $x$, defines $f(t) = ct + d$, and writes $x$ on a sheet of paper. Next, Granite writes $f(x)$ on the paper, Pomegranate writes $f(f(x))$, Granite writes $f(f(f(x)))$, and so on, with the players taking turns writing. The game ends when two numbers appear on the paper whose difference is a multiple of $p$, and the player who wrote the most recent number wins. Find the sum of all $p$ for which Pomegranate has a winning strategy.
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Context:
Example 3 (China Girls Mathematical Olympiad) Find all positive real numbers $a$ such that there exists a positive integer $n$ and $n$ pairwise disjoint infinite sets $A_{1}, A_{2}, \cdots, A_{n}$ satisfying $A_{1} \cup A_{2} \cup \cdots \cup A_{n}=\mathbf{Z}$, and for any two numbers $b>c$ in each $A_{i}$, we have $b-c \geqslant a^{i}$.
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Context:
Determine the error order of the second Richardson extrapolation of the trapezoidal rule in Romberg integration.
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Context:
Gavril traveled in Africa. On a sunny windy day, at noon, when the rays from the Sun were falling vertically, the boy threw a ball from behind his head with a speed of 5 m/s against the wind at an angle to the horizon. After 1 second, the ball hit his abdomen 1 meter below the point of release. Determine the maximum distance the ball's shadow moved away from Gavril's feet. The force exerted on the ball by the air is directed horizontally and does not depend on the position and speed. The acceleration due to gravity $g$ is $10 \mathrm{~m} / \mathrm{s}^{2}$.
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Context:
关于x的二次函数$f ( x ) = x ^ { 2 } - ( m + 1 ) x + 4$的图象与x轴上从0到3的长为3的一段线段只有一个交点，求m的取值范围．
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Context:
There are  $N{}$  points marked on the plane. Any three of them form a triangle, the values of the angles of which in are expressed in natural numbers (in degrees). What is the maximum  $N{}$  for which this is possible?

*Proposed by E. Bakaev*
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Context:
Given positive numbers $m$ and $n$ that satisfy $m^2 + n^2 = 100$, find the maximum or minimum value of $m + n$.
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Context:
Dylan filled his glass with ice cubes before he poured his soda into it. He counted and there were 8 ice cubes in his glass. Later when he prepares a pitcher of lemonade, he puts twice as many ice cubes per glass into the pitcher. He expects to serve 5 guests, and each guest will receive a glass of lemonade as well as his own glass. Then it is time to refill the ice cube trays, which have 14 spaces each for ice cubes. Dylan realizes that he has used 80% of the total ice cubes they initially had. How many trays does Dylan need to fill to return to the initial number of ice cubes?
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Context:
Pete chooses $1004$ monic quadratic polynomials $f_1, \cdots, f_{1004}$ such that each integer from $0$ to $2007$ is a root of at least one of them. Vasya considers all equations of the form $f_i = f_j$ for $i \neq j$ and computes their roots. For each such root, Pete has to pay Vasya $1$ ruble. Find the least possible value of Vasya's income.
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Context:
Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n \quad (n \geqslant 1), \quad \text{and} \quad \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n} = 0 \quad (n \geqslant 2).
$$

Find all possible values of $a_{2021}$.
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Context:
Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ <i>quasi-increasing</i> if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the integers $1, 2, \ldots, 7$.
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Context:
On a roulette table with 52 numbers (26 red and 26 green), you observe the selected numbers over N spins. The null hypothesis states that the table is fair, with equal probabilities for red and green (p = 0.5). The alternative hypothesis suggests the table is biased, favoring red with a probability of p = 0.5005. Given a significance level of 1% and a target power of 90%, determine the minimum number of spins (N) required for the test.
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Context:
Given the function $f(x)=ax-x^{2}-\ln x$, if the function $f(x)$ has an extreme value, and the sum of all extreme values is less than $5+\ln 2$, then the range of the real number $a$ is \_\_\_\_\_\_.
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Context:
In $\triangle ABC$, $\angle ABC = 50^\circ$, $\angle ACB = 20^\circ$, and $N$ is a point inside $\triangle ABC$. If $\angle NAB = 40^\circ$ and $\angle NBC = 30^\circ$, find the measure of $\angle NCB$.
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Context:
Let \( S = \{1, 2, \ldots, 50\} \). Determine the smallest natural number \( K \) such that in any subset of \( S \) with \( K \) elements, there exist two distinct numbers \( a \) and \( b \) such that \( (a+b) \mid a b \).
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Context:
Determine the maximum number of non-attacking $2 \times 2$ squares that can be placed on a $1000 \times 1000$ board. Two $2 \times 2$ squares are said to attack each other if they lie in the same pair of adjacent rows (or columns), and the cells between them are all empty. What is the largest number of such squares that can be placed on the board without any two attacking each other?
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Context:
Find an equivalent of the function $f(x)$ as $x \to 0^+$, where $f(x)$ is defined as:

$$f(x) = \int_{0}^{\infty} \frac{e^{-xt}}{(1 + t^3)^{1/3}} \, dt$$

This means finding a simple function $g(x)$ such that when $x \to 0^+$, we have:

$$\lim_{x \to 0^+} \frac{f(x)}{g(x)} = 1$$
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Context:
For every non-negative integer \( n \), a polynomial \( p_n(x) \) exists such that \( \int x^n e^x \,dx = p_n(x) e^x + C \). Determine \( \lfloor 1000 \cdot L \rfloor \), where \( L = \lim_{n \to \infty} \frac{|p_n(2)|}{n!} \).
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Context:
Calculate the expected length of a coin-tossing game where Player A has n coins and Player B has m coins. In each round, if two heads or two tails appear, Player A takes both coins. Otherwise, Player B takes them. What is the expected number of rounds until all coins are taken?
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Context:
For a nilpotent group $G$ of class $c$ with the upper central series $1=Z_{0}(G)<Z_1(G)<Z_2(G)<\ldots<Z_c(G)=G$, determine the nilpotency class of the quotient group $G/Z_i(G)$ for a given integer $i$. Provide your answer as a single integer.
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Context:
Given that the eccentricity of an ellipse is \( e \), with foci \( F_1 \) and \( F_2 \). A parabola \( C \) has \( F_1 \) as its vertex and \( F_2 \) as its focus. Point \( P \) is an intersection point of the parabola and the ellipse. If \( e|PF_2| = |PF_1| \), find the value of \( e \).
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Context:
Define a permutation of the set $\{1,2,3,...,n\}$ to be \textit{sortable} if, upon cancelling an appropriate term of such permutation, the remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of sortable permutations of $\{1,2,3,...,n\}$, find the remainder when \(\sum\limits_{i=1}^{2018} (-1)^i \cdot f(i)\) is divided by $1000$. Note that the empty set is considered sortable.
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Context:
Determine the computational complexity of verifying whether a matrix is orthogonal using the matrix product method and fast matrix multiplication algorithms.
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Context:
A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 63. If $m$ is also divisible by 11, what is the greatest possible value of $m$?
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Context:
Find a $3\times 3$-matrix whose null space is the $x$-axis and whose column space is the $yz$-plane.
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Context:
A train moves at a speed of 60 kilometers per hour, making stops every 48 kilometers. The duration of each stop, except for the fifth stop, is 10 minutes. The fifth stop lasts half an hour. What distance did the train travel if it departed at noon on September 29 and arrived at its destination on October 1 at 22:00?
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Context:
Evaluate the expression \(\sum_{k=1}^4 k \sum_{1 \le j_1 < j_2 < \ldots < j_k \le 4} (n_{j_1} + n_{j_2} + \ldots + n_{j_k})\) given that \(n_1 = 2\), \(n_2 = 3\), \(n_3 = 4\), and \(n_4 = 5\).
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Context:
Determine the asymptotic behavior of the function \( h(n) = \sum_{p_k < n} \frac{\log (1 + p_k)}{p_k} \), where \( p_k \) is the \( k \)th prime number.
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Context:
Two ships are moving as follows: Ship A is 1 mile due north of a pier and is traveling N $30^o$ E at 3 mph. Ship B is $\frac{3}{4}$ mile east of the pier and is traveling east at 7 mph. Determine the rate at which the angle between the two ships' paths is changing.
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Context:
A city has \( n \) schools, and the \( i \)-th school sends \( C_{i} \) students \( \left(1 \leqslant C_{i} \leqslant 39, 1 \leqslant i \leqslant n\right) \) to watch a game in an auditorium. The total number of students from all schools is \( \sum_{i=1}^{n} C_{i} = 1990 \). Each row of seats in the stand has 199 seats. It is required that all students from the same school must sit in the same row. What is the minimum number of rows needed to ensure that all students can be seated?
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Context:
Determine the structure of all continuous functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the functional equation \( f(x+y) = f(x) + f(y) + 2xy \) for all real numbers \( x \) and \( y \), given that \( f(0) = 0 \) and \( f(x+1) = f(x) + f(1) + 2x \).
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Context:
What is the remainder when the sum
\[\binom{2013}{1} + 2013\binom{2013}{3} + 2013^2\binom{2013}{5} + \dots + 2013^{1006}\binom{2013}{2013}\]
is divided by $41$?
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Context:
Al told Bob that he was thinking of  $2011$  distinct positive integers. He also told Bob the sum of those  $2011$  distinct positive integers. From this information, Bob was able to determine all  $2011$  integers. How many possible sums could Al have told Bob?

*Author: Ray Li*
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Context:
Evaluate $\int_{-1}^{3} \left[ x+\frac{1}{2} \right] dx$, where $\left[ . \right]$ denotes the greatest integer less than or equal to $x$.
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Context:
已知复数 $z$, 满足 $\frac{z}{2}$ 与 $\frac{2}{z}$ 的实部和虚部均属于 $[-1,1]$, 则 $z$ 在复平面上形成轨迹的面积为原始答案为$a - b\pi$的形式，请给出a + b的值。
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Context:
Evaluate the limit: $$\lim_{x \to \infty} \left( x - \ln\left(\frac{1+x}{x}\right)^{x^2} \right)$$
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Context:
In a game, two players take turns removing cells from a $5 \times 7$ grid. To remove a cell, a player selects an uneaten cell, places a piece on that cell, and then all cells in the "first quadrant" (the right-angled region created with the lower-left corner of that cell extending upwards and to the right) are removed. For example, placing a piece on a shaded cell in the right image will remove the 4 cells marked with $\times$ along with the shaded cell itself, where the cells represented by dotted lines were removed earlier. The goal of the game is to force the opponent to remove the last cell. How many different possible situations can appear during the entire course of the game?
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Context:
How many ways can you color the squares of a \( 2 \times 2008 \) grid in 3 colors such that no two squares of the same color share an edge?
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Context:
What is the coefficient of $a^2b^2$ in $(a+b)^4\left(c+\dfrac{1}{c}\right)^6$?
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Context:
Daniel wrote all the positive integers from 1 to $n$ inclusive on a piece of paper. After careful observation, he realized that the sum of all the digits that he wrote was exactly 10,000. Find $n$.
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Context:
Let  $ k$  be a positive constant number. Denote  $ \alpha ,\ \beta \ (0<\beta <\alpha)$  the  $ x$  coordinates of the curve  $ C: y\equal{}kx^2\ (x\geq 0)$  and two lines  $ l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}$ .　Find the minimum area of the part bounded by the curve  $ C$  and two lines  $ l,\ m$ .
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Context:
What is the minimum number of operations required to transform 2014 into 1 using only addition, subtraction, multiplication, or division by single-digit numbers (1-9)? Express your answer as a whole number.
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Context:
Compute the number of nonempty subsets $S$ of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max S + \min S}{2}$ is an element of $S$.
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Context:
Evaluate the double integral \( \int\int 2xy \, dA \) over the region bounded by \( y = 1 - x^2 \), \( y = 1 + x \), and \( y = 0 \) in the \( xy \) plane.
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Context:
Given positive integers \(n\) and \(x_1, x_2, \ldots, x_n \in (0, 1)\) such that for all \(i = 1, 2, \ldots, n\), the equation \(\sum_{i \ne j}\frac{1}{x_i - x_j} = \frac{1}{1 - x_i} - \frac{1}{x_i}\) holds, calculate the value of \(\sum_{i = 1}^n x_i\).
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Context:
Consider a random sample $X_1, X_2, \ldots, X_n$ from a uniform distribution $U(\theta, \theta + 1)$. We want to test the hypotheses $H_0: \theta = 0$ versus $H_1: \theta = \theta_1$ for some $0 < \theta_1 < 1$. Using Neyman-Pearson's lemma, find the critical value $k$ for the likelihood ratio test $\Lambda \leq k$ such that the test has size $\alpha$, i.e., $P_{\theta = 0}(\Lambda \leq k) = \alpha$. Assume $\Lambda = \frac{1}{I_{(\theta_1, \infty)}(y_1)}$, where $y_1 = \min(x_1, \ldots, x_n)$. Determine the value of $k$.
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Context:
Determine a tight asymptotic upper bound for the expected maximum load \( \mathbb{E}(X_n) \) when \( n \) balls are thrown into \( n \) bins. Each ball is placed in the least full of two bins chosen uniformly and independently at random, with the bins selected with pairwise independence.
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Context:
Find the positive integer $q \leq 50$ such that the following equation holds: $$\binom{98}{30} + 2\binom{97}{30} + 3\binom{96}{30} + \dots + 68\binom{31}{30} + 69\binom{30}{30} = \binom{100}{q}.$$
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Context:
A shipping company charges $0.30l + 0.40w + 0.50h$ dollars to process a right rectangular prism-shaped box with dimensions $l, w, h$ in inches. The customers themselves are allowed to label the three dimensions of their box with $l, w, h$ for the purpose of calculating the processing fee. A customer finds that there are two different ways to label the dimensions of their box $B$ to get a fee of $8.10, and two different ways to label $B$ to get a fee of $8.70. None of the faces of $B$ are squares. Find the surface area of $B$, in square inches.
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Context:
设 $A, B, C$ 是 $\triangle A B C$ 的三个内角, 则 $3 \cos A+2 \cos 2 B+\cos 3 C$ 的取值范围为. 请给出该范围的端点之和的整数部分。
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