Context:
If the following triangular array of numbers is continued using the pattern established, how many numbers (not how many digits) would there be in the $100^{th}$ row? As an example, the $5^{th}$ row has $11$ numbers. Use exponent notation to express your answer.
$$
\begin{align*}  &1 \\  &2 \\ 3\quad &4\quad  5\quad  \\ 6\quad   7\quad  &8\quad  9\quad  10\quad  \\ 11\quad  12\quad  13\quad  14\quad  15\quad  &16\quad  17\quad  18\quad  19\quad  20\quad  21\quad  \\ 22\quad  23\quad  24\quad  25\quad  26\quad  27\quad  28\quad  29\quad  30\quad  31\quad  &32\quad  33\quad  34 \quad 35\quad  36\quad  37\quad  38\quad  39\quad  40\quad  41\quad  42\quad  \\ \cdot \quad &\cdot\quad  \cdot\quad \\ \end{align*}
$$

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Context:
Determine the number of zeros of the polynomial \( z^9 + z^5 - 8z^3 + 2z + 1 \) that lie in the annular region \( 1 < |z| < 2 \). Use Rouche's theorem to justify your answer.
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Context:
From the same number of squares with sides 1, 2, and 3, form a square of the smallest possible size.
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Context:
John has two jobs. From job A, he makes $60 a week and receives a raise increasing his weekly pay to $78. He also gets a quarterly bonus of $50 that remains the same. From job B, he makes $100 a week and receives two raises: the first one increases his weekly pay by $10, and the second one by $5, giving him a total of $115 a week. Job B also provides a biannual bonus of $200, but after the raises, his biannual bonus is increased by 10%. Additionally, John has weekly expenses of $30 that remain constant. What is the net percentage change in John's effective quarterly income after considering his raises, bonuses, and expenses?
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Context:
Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
Determine whether the sequence \( b_n = \int_0^n \int_0^{\frac{1}{n}} \int_0^1 \int_0^{\frac{1}{n}} \log^2\left[(x-t)^2+(y-s)^2\right] \, dx \, dy \, dt \, ds \) converges or diverges as \( n \to \infty \).
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Context:
If the base of a rectangle is increased by $10\%$ and the area is unchanged, then the altitude is decreased by:

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Context:
Determine the minimum number of roots in the interval \([0, 2\pi)\) for the equation \( \sin(k_{0} x) + A_{1} \cdot \sin(k_{1} x) + A_{2} \cdot \sin(k_{2} x) = 0 \), where \( k_{0}<k_{1}<k_{2} \) are natural numbers and \( A_{1}, A_{2} \) are real numbers.
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Context:
In the figure, ABCDE is a regular pentagon. Given that $AD = AF$ and $\angle DAF = 56^\circ$, find the measure of angle $x$.
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Context:
Rosencrantz plays $n \leq 2015$ games of question, and ends up with a win rate (i.e. $\frac{\# \text { of games won }}{\# \text { of games played }}$ ) of $k$. Guildenstern has also played several games, and has a win rate less than $k$. He realizes that if, after playing some more games, his win rate becomes higher than $k$, then there must have been some point in time when Rosencrantz and Guildenstern had the exact same win-rate. Find the product of all possible values of $k$.
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Context:
20. Combine a regular hexagon with a side length of 1 and an equilateral triangle with a side length of 1, requiring no overlap, and the sides to be completely coincident, then the number of sides of the new shape is $\qquad$ .
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Context:
A cylindrical cup containing some water is tilted 45 degrees from the
 vertical. The point on the surface of the water closest to the bottom of the
 cup is 6 units away. The point on the surface of the water farthest from the
 bottom of the cup is 10 units away. Compute the volume of the water in the
 cup.
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Context:
Find the value of $k$ such that for a standardized normal variable $Z$, the probability $P(0 < Z < k)$ is equal to twice the probability $P(k < Z < 2k)$. Express your answer in terms of $k$.
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Context:
. Decompose 2008 into a sum of natural numbers whose product is maximal.
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Context:
Consider the sequence: _, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3. What is the first number in the sequence?
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Context:
A rectangle is divided by two straight lines into four smaller rectangles with perimeters of 6, 10, and 12 (the rectangle with a perimeter of 10 shares sides with the rectangles with perimeters of 6 and 12). Find the perimeter of the fourth rectangle.
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Context:
How many of the first 1000 positive integers can be expressed in the form
\[ \lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor, \]
where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?
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Context:
Suppose a 2-fair carpool system is possible but a 1-fair carpool system is not. Must the number of MOPpers, \( n \), be even? Answer 'yes' or 'no'.
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Context:
Given the operation \( \diamond \) defined as \( x\diamond 1 = x \) and \( x\diamond (n+1) = x^{x\diamond n} \) for \( n\in\mathbb{N} \), where \( x\in\mathbb{R}^+ \), let \( t \) be the unique positive real number such that \( t^t = 10^{10^{10}} \). It is known that \( 10^{10}>t \). If \( k \) and \( s \) are the unique positive real numbers satisfying \( k\diamond 50 = 10\diamond 51 \) and \( s\diamond 51 = 10\diamond 52 \), respectively, determine if \( k > s \).
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Context:
Let  $ a$  be a positive real numbers. In the coordinate plane denote by  $ S$  the area of the figure bounded by the curve  $ y=\sin x\ (0\leq x\leq \pi)$  and the  $ x$ -axis and denote  $ T$  by the area of the figure bounded by the curves  $ y=\sin x\ \left(0\leq x\leq \frac{\pi}{2}\right),\　y=a\cos x\ \left(0\leq x\leq \frac{\pi}{2}\right)$  and the  $ x$ -axis. Find the value of  $ a$  such that  $ S: T=3: 1$ .
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Context:
Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i \le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the numbers $1337$ and $123$. How many 5-digit numbers does Kelvin like?
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Context:
there are 20 cm , 21 cm , 22 cm , 23 cm , 24 cm , 25 cm rods of quantities 23 , 64 , 32 respectively . find average of this data .
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Context:
Suppose that the edges of a 10-point graph can be colored with two colors in such a way that the graph does not contain a monochromatic triangle. What is the maximum number of edges the graph can have?
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Context:
Given the power series $A(x) = \sum_{n=0}^{\infty} a_nx^n$, where $A'(x) = 2xA(x)$, find a closed-form expression for $A(x)$.
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Context:
Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \[a_{99}x^{99} + \cdots + a_1x + a_0\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?
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Context:
Consider a sequence of nested intervals defined as follows: \( L_1 = [0,1] \). For each even \( n \), let \( L_n = [a_n, b_n] \) where \( a_n \) and \( b_n \) are irrational numbers such that \( a_n < b_n \) and \( a_n, b_n \in L_{n-1} \). For each odd \( n > 1 \), let \( L_n = [x_n, y_n] \) where \( x_n \) and \( y_n \) are rational numbers such that \( x_n < y_n \) and \( x_n, y_n \in L_{n-1} \). The sequence of intervals is such that \( L_1 \supset L_2 \supset L_3 \supset L_4 \supset L_5 \dots \). Define the set \( \mathcal{S} = \bigcap_{n=1}^{\infty} L_n \). Determine the number of elements in the set \( \mathcal{S} \) and whether these elements are rational, irrational, both, or neither.
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Context:
Evaluate the limit \( \lim_{x\rightarrow 0}\frac{\int_{\cos x}^{1}{}\arctan t ^2\ \mathrm dt}{\tan^2 x} \) using L'Hopital's rule.
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Context:
Determine the maximum number of faces of an n-sided pyramid that can be perpendicular to its base.
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Context:
Expand and simplify the expression $-(4-d)(d+3(4-d))$. What is the sum of the coefficients of the expanded form?
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Context:
Is the argument that the cardinality of \( V_\omega \) is \( \aleph_0 \) valid, given that each \( V_n \) in the von Neumann Hierarchy has finitely many elements and \( V_\omega = \bigcup_{n < \omega} V_n \)?
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Context:
Eight hockey teams are competing against each other in a single round to advance to the semifinals. What is the minimum number of points that guarantees a team advances to the semifinals?
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Context:
Determine the real numbers $x$, $y$, $z > 0$ for which
$xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$
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Context:
Select numbers $a$ and $b$ between 0 and 1 independently and at random, and let $c$ be their sum. Let $A$, $B$, and $C$ be the results when $a$, $b$, and $c$, respectively, are rounded to the nearest integer.  What is the probability that $A+B=C$?
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Context:
In an equilateral triangle \(\triangle ABC\), a point \(P\) is located inside the triangle such that \(\angle PAC = x\), \(\angle PCA = 3x\), and \(\angle PBC = 2x\). Determine the value of \(x\) in degrees.
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Context:
A repunit is a positive integer, all of whose digits are 1s. Let $a_{1}<a_{2}<a_{3}<\ldots$ be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute $a_{111}$.
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Context:
Find all four-digit numbers that, after removing the middle two digits, are reduced by 120 times.
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Context:
A *strip* is the region between two parallel lines.  Let  $A$  and  $B$  be two strips in a plane.  The intersection of strips  $A$  and  $B$  is a parallelogram  $P$ .  Let  $A'$  be a rotation of  $A$  in the plane by  $60^\circ$ .  The intersection of strips  $A'$  and  $B$  is a parallelogram with the same area as  $P$ .  Let  $x^\circ$  be the measure (in degrees) of one interior angle of  $P$ .  What is the greatest possible value of the number  $x$ ?
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Context:
A rectangular piece of paper whose length is \(\sqrt{3}\) times the width has area \(A\). The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area \(B\). The ratio of the new shape's area to the original area is given in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).
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Context:
Evaluate the surface integral \(\iint_{S_1}\mathbf{F}\mathbf{\cdot} \hat{\mathbf n}\ dS\) over the sphere \(S_1\) given by \(x^2 + (y-2)^2 + z^2 = 9\) oriented outwards, where the vector field is \(\mathbf F(x,y, z) = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{[x^2+y^2+z^2]^{3/2}}\). Note that \(\mathbf{F}\) is undefined at the origin, so the divergence theorem cannot be directly applied.
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Context:
Find the limits of the following sequence, given $a_0 = 1$ and $a_n = \sum_{j=0}^{\lfloor n/2\rfloor} \frac{a_j}{(n-2j)!}$ for $n \ge 1$:

$$\lim_{n\to \infty}\frac{a_n}{n!}$$
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Context:
The train was delayed by \( t \) hours. By increasing the speed by \( m \) km/h, the driver made up for the delay over a section of \( s \) km. Determine the speed the train should have had on this section if there had been no delay.
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\boxed{\frac{\sqrt{t m (4 s + t m)} - t m}{2 t}}


Context:
设 $U=\{1,2,3, \cdots, n\}(n \geqslant 3), S$ 是 $U$ 的一个子集. 如果不属于 $S$ 的一个元素出现在 $U$ 的一个排列 $P$ 的某处,并且在该排列 $P$ 中它处在 $S$ 中的两个元素之间, 那么我们称 $S$ 是由排列 $P$ 分裂出来的(例如排列 13542 可分裂出子集 $\{1,2,3\}$, 而不能分裂出子集 $\{3,4,5\}$ ), 试求最大正整数 $m$, 使得对 $U$ 的任意 $m$ 个子集(每个子集至少有 2个元, 至多有 $n-1$ 个元), 都存在 $U$ 的一个排列分裂出所有这 $m$ 个子集.
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Context:
Piazzanos Pizzeria has a buy-1-get-1-free special on triple cheese pizzas, and a buy-2-get-1-free special on meat lovers pizzas. If the standard price for a pizza is $5, it costs $55 to purchase 10 triple cheese and a certain number of meat lovers pizzas using the special pricing. How many meat lovers pizzas were purchased?
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Context:
In an isosceles triangle, the lengths of two sides are $3cm$ and $7cm$ respectively. The perimeter of the triangle is ______ $cm$.
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Context:
Three red beads, two white beads, and one blue bead are placed in line in random order. What is the probability that no two neighboring beads are the same color? Express your answer in the form \(\frac{k}{m}\) where the fraction is in its simplest form, and find the value of \(k + m\).
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Context:
Five numbers were written on a board. Then these numbers were erased and their pairwise sums were written down:

$$
4, 8, 10, 12, 14, 18, 19, 21, 25, 29
$$

What were the five numbers originally written on the board?
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Context:
A hurricane is approaching the southern coast of Texas, and a rancher is planning to move some head of cattle 60 miles to higher ground to protect them from possible inland flooding that might occur. His animal transport truck holds 20 head of cattle. Traveling at 60 miles per hour, it took him 40 hours to transport all of his cattle to higher ground. How many head of cattle did the rancher have to move?
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Context:
Given the equations \( 2^x + 4x + 12 = 0 \) and \( \log_2((y-1)^3) + 3y + 12 = 0 \), where \( x \) and \( y \) are real numbers, find the value of \( x + y \).
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Context:
Determine the condition on the integer $k$ such that the norm of the Gaussian integer $p (1+i)^{-k}$ is between 1 and 2, where $p$ is a prime Gaussian integer with a norm greater than 2.
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Context:
Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find the value of $a+b+c$.
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Context:
In each of the $9$ small circles of the following figure, write positive integers less than $10$, without repetitions. Additionally, it is true that the sum of the $5$ numbers located around each one of the $3$ circles is always equal to $S$. Find the largest possible value of $S$.
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Context:
On the eve of the 2010 Guangzhou Asian Games, a 12-person tour group took a commemorative photo near a venue of the Asian Games. They initially stood in a formation with 4 people in the front row and 8 people in the back row. Now, the photographer plans to keep the order of the front row unchanged, and move 2 people from the back row to the front row, ensuring that the 2 moved people are not adjacent in the front row. The number of different ways to adjust their positions is _____. (Answer in numerals)
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Context:
In △ABC, a, b, and c are the sides opposite to angles A, B, and C respectively. If c = 2b and the area of △ABC is 1, find the minimum value of a.
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Context:
Quadrilateral $PQRS$ is a square. A circle with center $S$ has arc $PXC$. A circle with center $R$ has arc $PYC$. If $PQ = 3$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the nearest tenth.
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Context:
若不小于 1 的实数 $t$ ：对任意 $a, b \in[-1, t]$, 总存在 $c, d \in[-1, t]$, 使得 $(a+c)(b+d)=1$，则实数 $t$ 满足$\frac{m}{n} \leq t \leq p$，求$m+n+p$的值。
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Context:
Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio $3: 2: 1,$what is the least possible total for the number of bananas?
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Context:
The people in an infinitely long line are numbered $1,2,3,\dots$. Then, each person says either ”Karl“ or ”Lark” independently and at random. Let $S$ be the set of all positive integers $i$ such that people $i$, $i+1$, and $i+2$ all say ``Karl, and define $X = \sum\limits_{i \in S} 2^{-i}$. Then the expected value of $X^2$ is $\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$.
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Context:
Given the fractional equation $\frac{6}{x-2}-1=\frac{ax}{2-x}$, find the value of $a$ that allows the equation to have a positive root. Express your answer as a single numerical value.
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Context:
Let \[ p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3. \] Suppose that \( p(0,0) = p(1,0) = p(-1,0) = p(0,1) = p(0,-1) = p(1,1) = p(1,-1) = p(2,2) = 0. \) There is a point \( \left(\frac{a}{c},\frac{b}{c}\right) \) for which \( p\left(\frac{a}{c},\frac{b}{c}\right) = 0 \) for all such polynomials, where \( a, b, \) and \( c \) are positive integers, \( a \) and \( c \) are relatively prime, and \( c > 1 \). Find \( a + b + c \).
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Context:
Compute the partial derivative of the first derivative of \( x(t) \) with respect to \( x(t) \), represented as \[ \frac{\partial\left(\frac{d(x(t))}{dt}\right)}{\partial x(t)} \].
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Context:
Grant has worked as a freelance math worker for the last three months. The first month he made $350. The second month he made $50 more than double he made the first month. The third month he made some multiple of the sum of the first two months. Grant made a total of $5500 in his first three months. What is the relationship between the third month's earnings and the sum of the first two months' earnings?
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Context:
Starting at $12:00:00$ AM on January $1,$ $2022,$ after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365.$ Find $y + d.$
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Context:
Peter has $2022$ pieces of magnetic railroad cars, which are of two types: some have the front with north and the rear with south magnetic polarity, and some have the rear with north and the rear with south magnetic polarity (on these railroad cars the front and the rear can be distinguished). Peter wants to decide whether there is the same number of both types of cars. He can try to fit together two cars in one try. What is the least number of tries needed?
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Context:
Evaluate the integral of the function \( f(z) = \exp(1/z) \) over the unit circle in the complex plane.
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Context:
设 $a_{n}$ 是与 $\sqrt{\frac{n}{2}}$ 的差的绝对值最小的整数, $b_{n}$ 是与 $\sqrt{2 n}$ 的差的绝对值最小的整数. 记 $\left\{\frac{1}{a_{n}}\right\}$ 的前 $n$ 项和为 $S_{n},\left\{\frac{1}{b_{n}}\right\}$ 的前 $n$ 项和为 $T_{n}$ 。则 $2 T_{100}-S_{100}$ 的值为 $\qquad$.
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Context:
Vasya wrote a note on a piece of paper, folded it into quarters, and labeled the top with "MAME". Then he unfolded the note, wrote something else on it, folded the note along the creases randomly (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscription "MAME" is still on top.
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Context:
In triangle $\mathrm{V} A B C$, where $A, B, C$ represent the three interior angles, find the minimum value of $\frac{1}{\sin ^{2} A} + \frac{1}{\sin ^{2} B} + \frac{4}{1 + \sin C}$.
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Context:
Given Elmer’s new car provides a 70% better fuel efficiency and uses a type of fuel that is 35% more expensive per liter, calculate the percentage by which Elmer will save money on fuel costs if he uses his new car for his journey.
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Context:
Mason is three times younger than Sydney and Sydney is some years younger than Mason's father. If Mason is 20 years old, how old is his father?
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Context:
In convex quadrilateral ABCD, $\angle BAD = \angle BCD = 90^{\circ}$, and BC = CD. Let E be the intersection of diagonals AC and BD. Given that $\angle AED = 123^{\circ}$, find the degree measure of $\angle ABD$.
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Context:
In triangle  $ABC$ , let  $M$  be the midpoint of  $BC$ ,  $H$  be the orthocenter, and  $O$  be the circumcenter. Let  $N$  be the reflection of  $M$  over  $H$ . Suppose that  $OA = ON = 11$  and  $OH = 7.$  Compute  $BC^2$ .
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Context:
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that
$$3^r \cdot 7^s = \text{lcm}(a,b,c)= \text{lcm}(a,b,d)=\text{lcm}(a,c,d)=\text{lcm}(b,c,d),$$
depending only on $r$ and $s.$
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Context:
The average age of a group of men is increased by years when two of them, whose ages are 21 years and 23 years, are replaced by two new men with an average age of 32. How many men are in the group?
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Context:
Integers $a$, $b$, $c$, and $d$, not necessarily distinct, are chosen independently and at random from 0 to 2007, inclusive. What is the probability that $ad-bc$ is even?
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Context:
In the quadrilateral \( ABCD \), \( AB = a \), and the ratio of the measures of the four angles \( A, B, C, D \) is \( 3:7:4:10 \). Additionally, \(\angle CDB = 30^{\circ}\). Find the length of \( BC \).
Please reason step by step, and put your final answer within \boxed{}.


Context:
Given a rectangle \(A B C D\). A circle intersects the side \(A B\) at points \(K\), \(L\), and \(N\). Find the length of the segment \(M N\) if \(A K = 10\), \(K L = 17\), and \(D N = 7\).
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Context:
A couple, both phenotypically normal, have a son who suffers from both hemophilia (an X-linked recessive condition) and phenylketonuria (an autosomal recessive condition). If they have another daughter, what is the probability that she will be phenotypically normal? Express your answer as a simplified fraction.
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Context:
3 设2005条线段首尾相连，组成封闭的折线，且折线的任何两段都不在同一条直线上，那么，此折线自身相交的交点最多有多少个？
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Context:
Two circles have a radius of $9$, and one circle has a radius of $7$. Each circle is externally tangent to the other two circles, and each circle is internally tangent to two sides of an isosceles triangle, as shown in the figure. The sine of the base angle of the triangle is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
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Context:
The figure shows a shape consisting of a regular hexagon of side \( 18 \, \text{cm} \), six triangles, and six squares. The outer perimeter of the shape is \( P \, \text{cm} \). What is the value of \( P \)?
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Context:
Let  $P$  and  $Q$  be the midpoints of sides  $AB$  and  $BC,$  respectively, of  $\triangle ABC.$  Suppose  $\angle A = 30^{\circ}$  and  $\angle PQC = 110^{\circ}.$  Find  $\angle B$  in degrees.
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Context:
Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le 3$, let $D_i$, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$.

Suppose that the areas of the triangles $A_1A_2A_3$, $B_1B_2B_3$, $C_1C_2C_3$, $D_1D_2D_3$, $E_1E_2E_3$, and $F_1F_2F_3$ are $2$, $3$, $4$, $20$, $21$, and $2020$, respectively. Compute the largest possible area of $G_1G_2G_3$.
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Context:
How can we determine the number of non-isomorphic semi-direct products of a cyclic group of order 5 with a cyclic group of order 31?
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Context:
For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions:
$\bullet$ no two consecutive digits are equal, and
$\bullet$ the last digit is a prime.
Please reason step by step, and put your final answer within \boxed{}.

\boxed{\dfrac{2 \times 9^n - 2 \times (-1)^n}{5}}


Context:
Find all solutions to \( (m^2 + n)(m + n^2) = (m - n)^3 \), where \( m \) and \( n \) are non-zero integers.
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Context:
Calculate $\sin 18^{\circ}$, using only the first two terms of series (3), and estimate the resulting error.
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Context:
In Figure 1, the square \(PQRS\) is inscribed in \(\triangle ABC\). The areas of \(\triangle APQ\), \(\triangle PBS\), and \(\triangle QRC\) are 4, 4, and 12 respectively. If the area of the square is \(c\), find the value of \(c\).
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Context:
Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$ ?
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Context:
Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n=5, m=7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$?
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Context:
Given the circle equation $2x^2+2y^2+6x-4y-3=0$, find the coordinates of the center and the radius. Express the center coordinates as an ordered pair and the radius as a simplified radical.
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Context:
There are three table tennis teams, and members from different teams will play a match against each other, but members from the same team will not play against each other. There are a total of 10 players in the three teams, and 31 matches were played in total. How many players are there in each team?
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Context:
A list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list $10,34,34,22,30,22$ the sublist $22,30,22$ is good and $10,34,34,22$ is not. A list is very good if all its sublists are good. Find the minimum value of $k$ such that there exists a very good list of length $2019$ with $k$ different values on it.
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Context:
Five line segments are drawn inside a rectangle, creating a decagon. Find the sum of the six marked angles. Express your answer in degrees as a whole number.
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Context:
Consider a round table on which  $2014$  people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated  $2$  seats away from the person at the head of the table.
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Context:
The radii of the excircles of a triangle are \( r_{a}, r_{b}, \) and \( r_{c} \), and the radius of the circumcircle is \( R \). Determine the angles of the triangle if

$$
r_{a}+r_{b}=3 R \quad \text { and } \quad r_{b}+r_{c}=2 R.
$$
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Context:
If w is 40 percent less than q, q is 40 percent less than y, z is 46 percent less than y, and x is 30 percent more than w, then z is greater than x by what percent of w?
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Context:
There are three cards, each with a number from the set $\{1,2, \ldots, 10\}$. These three cards were shuffled and distributed to three people, who recorded the numbers on their respective cards. The cards were then collected and the process was repeated several times. After some repetitions, each of the three people summed their records. Knowing that the obtained sums were 13, 15, and 23, what were the numbers on the cards?
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Context:
Determine the positive integers expressible in the form  $\frac{x^2+y}{xy+1}$ , for at least  $2$  pairs  $(x,y)$  of positive integers
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Context:
A ladder  $10\text{ m}$  long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of  $1\text{ m/s}$ , how fast is the top of the ladder sliding down the wall when the bottom of the ladder is  $6\text{ m}$  from the wall?
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Context:
Three numbers, $a_1$, $a_2$, $a_3$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}$. Three other numbers, $b_1$, $b_2$, $b_3$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3$, with the sides of the brick parallel to the sides of the box. If $p$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
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Context:
In the initial draft of the floor plan for the CCTV Headquarters Building shown in the figure, $A B C D E F$ is a regular hexagon with an area of 60. Points $G, H, I, J$ are the midpoints of the sides $A B, C D, D E, F A$, respectively. Calculate the area of the shaded region.
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Context:
Let \( M \) be a set composed of a finite number of positive integers,
\[
M = \bigcup_{i=1}^{20} A_i = \bigcup_{i=1}^{20} B_i, \text{ where}
\]
\[
A_i \neq \varnothing, B_i \neq \varnothing \ (i=1,2, \cdots, 20)
\]
satisfying the following conditions:
1. For any \( 1 \leqslant i < j \leqslant 20 \),
\[
A_i \cap A_j = \varnothing, \ B_i \cap B_j = \varnothing;
\]
2. For any \( 1 \leqslant i \leqslant 20, \ 1 \leqslant j \leqslant 20 \), if \( A_i \cap B_j = \varnothing \), then \( \left|A_i \cup B_j\right| \geqslant 18 \).

Find the minimum number of elements in the set \( M \) (denoted as \( |X| \) representing the number of elements in set \( X \)).
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Context:
Tom enters a digital maze composed of 20 rooms. Each room contains a card marked with a number as shown in the figure. Tom starts in the first room and collects card 1. Subsequently, he can enter an adjacent room, but each room can only be entered once. If the number on the card in the current room is greater than any number on the cards he has already collected, he collects that card. What is the maximum number of cards Tom can collect by navigating through the maze?
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Context:
To make a very modern painting for his school, Roberto divides a square canvas into eight parts with four stripes of the same width and a diagonal, as shown in the figure. He paints the picture in blue and green, so that two adjacent parts always have different colors. In the end, he notices that he used more green than blue. What fraction of the picture was painted blue?
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Context:
Given the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, let the points $A$ and $B$ be the two endpoints of its real axis. Let $P$ be a moving point on this hyperbola, and let the slopes of lines $AP$ and $BP$ be $k_1$ and $k_2$, respectively. Find the eccentricity of the hyperbola when the expression $k_1k_2-2\left( \ln |k_1|+\ln |k_2| \right)$ reaches its minimum value.
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Context:
A wooden rectangular parallelepiped with all sides measured in whole centimeters was painted red. After that, it was cut parallel to its faces into 1 cm³ cubes. It turned out that one-third of the resulting cubes had at least one face painted red, while the remaining two-thirds had all faces unpainted. Find the length of the parallelepiped if it is 2 cm longer than its width and 4 cm longer than its height.
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Context:
设函数 f(x)=|x\log_{2023}x|，实数 a<b<c 满足 f(a)=f(b)=f(c). 若 a, b, c 构成等比数列，则其公比为__________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。
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Context:
We consider the following game for one person. The aim of the player is to reach a fixed capital  $C>2$ . The player begins with capital  $0<x_0<C$ . In each turn let  $x$  be the player’s current capital. Define  $s(x)$  as follows:  $$ s(x):=\begin{cases}x&\text{if }x<1C-x&\text{if }C-x<11&\text{otherwise.}\end{cases} $$  Then a fair coin is tossed and the player’s capital either increases or decreases by  $s(x)$ , each with probability  $\frac12$ . Find the probability that in a finite number of turns the player wins by reaching the capital  $C$ .
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Context:
Find the value of the following infinite sum, where $n$ and $z$ are fixed natural numbers with $n, z > 1$:
$$\sum_{k=1}^{\infty}\frac{(k-1)(k-2)(k-3)\cdot\ldots\cdot(k-n)}{n!z^k}$$
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Context:
Task 2. (10 points) Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$, and the points of tangency cut off segments of length 4 cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side of the square if the angle between the tangents is $60^{\circ}$.
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Context:
In the right triangular prism $ABC-A_{1}B_{1}C_{1}$ with a volume of $3\sqrt{7}$, where $AB=2$ and $AC=3$, the minimum value of the surface area of the circumscribed sphere of this prism is ______.
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Context:
If (10 ^ 4 * 3.456789) ^ some exponent is written as a single term, there would be 28 digits to the right of the decimal place. What is the exponent?
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Context:
Points $A, B, C$ in the plane satisfy $\overline{A B}=2002, \overline{A C}=9999$. The circles with diameters $A B$ and $A C$ intersect at $A$ and $D$. If $\overline{A D}=37$, what is the shortest distance from point $A$ to line $B C$?
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Context:
Given that 70% of the light bulbs are produced by Factory A with a pass rate of 95%, and 30% are produced by Factory B with a pass rate of 80%, calculate the probability of buying a qualified light bulb produced by Factory A from the market.
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Context:
Define the polynomials $P_0, P_1, P_2, \ldots$ by:
\[ P_0(x) = x^3 + 213x^2 - 67x - 2000 \]
\[ P_n(x) = P_{n-1}(x-n), \quad n \in \mathbb{N} \]
Find the coefficient of $x$ in $P_{21}(x)$.
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Context:
In a $9 × 9$ square table, 9 cells are marked: those at the intersection of the 2-nd, 5-th and 8-th rows with the 2-nd, 5-th and 8-th column. In how many ways can one get from the lower left cell to the upper right one moving only upwards and to the right without entering marked cells?
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Context:
In the AU tribe's language, there are two letters - "a" and "u". Certain sequences of these letters form words, where each word contains no fewer than one and no more than 13 letters. It is known that if any two words are written consecutively, the resulting sequence will not be a word. Find the maximum possible number of words in such a language.
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Context:
In how many ways can each of the integers $1$ through $11$ be assigned one of the letters $L$, $M$, and $T$ such that consecutive multiples of $n$, for any positive integer $n$, are not assigned the same letter?
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Context:
Evaluate the expression: $\tan \left(\arctan 2+\arctan \frac{2}{2^{2}}+\cdots+\arctan \frac{2}{12^{2}}\right)$. The original answer is in the form $-\frac{m}{n}$. Please provide the value of $m + n$.
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Context:
A $9 \times 9$ grid square is made of matches, with each side of the cells being one matchstick. Petya and Vasya take turns removing one matchstick at a time, starting with Petya. The winner is the player after whose move there are no complete $1 \times 1$ squares left. Which player can ensure their victory, regardless of how their opponent plays?
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Context:
When Jack traveled around the world, he had to go through customs and quarantine in multiple countries, as well as deal with time zone changes and layovers. In Canada, he spent 20 hours waiting in customs and 14 days in quarantine. After that, he traveled to Australia with a layover of 12 hours in a timezone 8 hours behind. In Australia, he spent 15 hours in customs and 10 days in quarantine. Finally, Jack visited Japan, passing through a layover of 5 hours in a timezone 2 hours ahead, and ended up spending 10 hours in customs and 7 days in quarantine there. Calculate the total time in hours Jack spent on his journey considering customs, quarantines, layovers, and time zone adjustments.
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Context:
Evaluate the improper integral: $$I=\int_{0}^\infty \frac{1-e^{-\frac{1}{x^2}}}{\sqrt{x}}{\rm d}x$$
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Context:
如果非负整数 m 及其各位数字之和均为 6 的倍数，则称 m 为"六合数". 求小于 2012 的非负整数中六合数的个数.
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Context:
Determine the shape of the section of a cube by a plane passing through its center perpendicular to its diagonal.
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Context:
A parallelogram $ABCD$ is inscribed in the ellipse $\frac{x^{2}}{4}+y^{2}=1$, where the slope of the line $AB$ is $k_{1}=1$. Determine the slope of the line $AD$.
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Context:
Let $A=20072009$, and $n$ is an 800-digit number formed by writing 100 $A$s side by side. Calculate the remainder when $n$ is divided by 11. Express your answer as a single integer.
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Context:
A person named Jia and their four colleagues each own a car with license plates ending in 9, 0, 2, 1, and 5, respectively. To comply with the local traffic restriction rules from the 5th to the 9th day of a certain month (allowing cars with odd-ending numbers on odd days and even-ending numbers on even days), they agreed to carpool. Each day they can pick any car that meets the restriction, but Jia’s car can be used for one day at most. The number of different carpooling arrangements is __________.
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Context:
Bernie has 2020 marbles and 2020 bags labeled \( B_{1}, \ldots, B_{2020} \) in which he randomly distributes the marbles (each marble is placed in a random bag independently). If \( E \) is the expected number of integers \( 1 \leq i \leq 2020 \) such that \( B_{i} \) has at least \( i \) marbles, compute the closest integer to \( 1000 E \).
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Context:
In triangle \(ABC\), angle \(A\) is the largest angle. Points \(M\) and \(N\) are symmetric to vertex \(A\) with respect to the angle bisectors of angles \(B\) and \(C\) respectively. If \(\angle MAN = 50^\circ\), find the measure of \(\angle A\).
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Context:
In a division operation with a remainder of 0, a student mistakenly used the divisor as 102 instead of 201 and obtained a quotient of 753. In addition, the quotient was accidentally divided by 3 due to a calculation error. What is the correct quotient of the original division operation?
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Context:
Determine the values of \( p \in [1, \infty) \) for which the sequences \( x_k \) and \( y_k \) converge with respect to the \( l^p \) norm, where \( x_k \) has the first \( k \) entries as 1 followed by 0's, and \( y_k \) has the first \( k \) entries as \( k^{-\frac{1}{3}} \) followed by 0's.
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Context:
Determine for which values of the parameter \( \beta \) the integral \( \int_{10}^{+\infty}\frac{1}{(x-5)^{\beta}\sqrt{(x-5)^2+9}}\,dx \) converges.
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Context:
An athletic facility has $5$ tennis courts. Pairs of players arrive at the courts and use a court for an exponentially distributed time with mean $40$ minutes. Suppose a pair of players arrives and finds all courts busy and $k$ other pairs waiting in queue. What is the expected waiting time to get a court?
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Context:
Let \( {ABC} \) be a triangle with \( {BC} = {20} \) and \( {CA} = {16} \) ,and let \( I \) be its incenter. If the altitude from \( A \) to \( {BC} \) ,the perpendicular
 bisector of \( {AC} \) , and the line through \( I \) perpendicular to \( {AB} \) intersect at a common point, then the length \( {AB} \) can be written as
 \( m + \sqrt{n} \) for positive integers \( m \) and \( n \) . What is \( {100m} + n \) ?
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Context:
Mr. Stevens, a businessman, owned and operated a small retail shop in Boone Mall from 1995 to 2015. During his first year in business, he sold 1,200 items. Over the next 10 years, his sales increased by 150 items each year. In his final year of business, he decided to have a closing sale and managed to sell 2,000 items. How many items in total did Mr. Stevens sell from 1995 to 2015?
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Context:
Three rods (with negligible thickness) are fixed perpendicularly to each other at a common point at one end. The lengths of the rods are 1, 2, and 3. This construction is placed on a table in such a way that the free ends of the rods lie on the plane of the tabletop. Determine the exact height of the fixed point above the table.
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Context:
Given an arithmetic-geometric sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, if $a_3 - 4a_2 + 4a_1 = 0$, find the value of $\frac{S_8}{S_4}$.
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Context:
a sum fetched total simple interest of 4016.25 at the rate of 11 p . c . p . a . in 5 years . what is the sum ?
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Context:
A factory implements a time-based wage system, where each worker is paid 6 yuan for every hour worked, for a total of 8 hours per day. However, the clock used for timing is inaccurate: it takes 69 minutes for the minute hand to coincide with the hour hand once. Calculate the amount of wages underpaid to each worker per day.
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Context:
Find the greatest value of \( \arg z \) achieved on the circle \( |z-6i|=3 \) in the complex plane.
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Context:
A high school student has two types of postcards, A and B, celebrating the school's anniversary: 2 type A postcards and 3 type B postcards. He wants to pick 4 postcards to send one to each of his 4 middle school teachers. Calculate the total number of different ways he can choose the postcards. Express your answer as a single integer.
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Context:
Determine the values of $p$ for which the $p$-norm $||x||_p = \left(\sum_{k=1}^n |x_k|^p\right)^{\frac{1}{p}}$ satisfies the Triangle Inequality.
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Context:
Find the range of values for \( \phi \) that satisfy the inequality:
\[ -1.96 \leq \frac{6.4-10\phi}{\sqrt{1-\phi^2}} \leq 1.96 \]
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Context:
Determine the minimum number of coin flips required to achieve a probability of at least 90% of obtaining three consecutive heads using a fair coin. Consider the problem as a Markov Chain with states representing the number of consecutive heads achieved.
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Context:
A square region $ABCD$ is externally tangent to the circle with equation $x^2+y^2=1$ at the point $(0,1)$ on the side $CD$. Vertices $A$ and $B$ are on the circle with equation $x^2+y^2=4$. What is the side length of this square?
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Context:
A certain set of integers is assigned to the letters of the alphabet such that $H=10$. The value of a word is the sum of its assigned letter values. Given that $THIS=50$, $HIT=35$ and $SIT=40$, find the value of $I$.
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Context:
Arwen, Elrond, Galadriel and Legolas went out to pick some flowers. They agreed to only pick a maximum of 50 flowers per person. Arwen was able to get 20 tulips, 18 roses, and 6 sunflowers. Elrond was able to get twice as many tulips as Arwen did and thrice the number of roses Arwen picked, without exceeding the 50-flower limit. Galadriel picked three times the amount of tulips that Elrond gathered and twice the amount of roses Arwen has, also following the flower limit. Legolas picked as many sunflowers as Arwen and Galadriel combined and an equal number of roses and tulips. How many flowers in total did they collect?
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Context:
2. The eight-digit number $\square 2 \square 0 \square 2 \square 2$ (the digits in $\square$ can repeat) is a multiple of 72. There are $\qquad$ such eight-digit numbers.
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Context:
If $f(x)$ is a polynomial such that $x^2-x+a$ divides $f(x)$, and $f(0)=90$ and $f(1)=92$, find the value of $a$.
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Context:
In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, it is given that $A B = B C = 4$ and $A A_{1} = 2$. Point $P$ lies on the plane $A_{1} B C$, and it holds that $\overrightarrow{D P} \cdot \overrightarrow{P B} = 0$. Find the area of the plane region formed by all such points $P$ satisfying the given condition.
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Context:
Given the function $f(x) = x^4 + ax^3 + 2x^2 + b$, where $a, b \in \mathbb{R}$. If the function $f(x)$ only has an extremum at $x=0$, find the range of values for $a$.
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Context:
Given six balls numbered 1, 2, 3, 4, 5, 6 and boxes A, B, C, D, each to be filled with one ball, with the conditions that ball 2 cannot be placed in box B and ball 4 cannot be placed in box D, determine the number of different ways to place the balls into the boxes.
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Context:
Find the limit of the following expression as x approaches 0:
$$\lim_{x \to 0} \left( \frac{\tan (\sin (x))-\sin (\tan (x))}{x^7} \right)$$
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Context:
In a $2017 \times 2017$ grid, each cell contains a light, which can be either on or off. A light is termed "faulty" if and only if it has an even number of neighboring lights that are on. Find the minimum possible number of faulty lights in this $2017 \times 2017$ grid.

Note: Two lights are considered neighbors if their cells share a common edge.
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Context:
In the interval \(1 \leqslant n \leqslant 10^{6}\), how many integers \(n\) satisfy the equation \(n = x^y\) for non-negative integer solutions \(x, y\) where \(x \neq n\)?
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Context:
Find \(\cos \frac{\alpha - \beta}{2}\), given \(\sin \alpha + \sin \beta = -\frac{27}{65}\), \(\tan \frac{\alpha + \beta}{2} = \frac{7}{9}\), \(\frac{5}{2} \pi < \alpha < 3 \pi\) and \(-\frac{\pi}{2} < \beta < 0\).
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Context:
In the Cartesian coordinate system, there are 5 points on the positive x-axis and 3 points on the positive y-axis. These points are connected to form 15 line segments. Find the maximum number of intersection points these 15 line segments can have in the first quadrant. Express your answer as a whole number.
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Context:
The side length of square \(ABCD\) is 4. Point \(E\) is the midpoint of \(AB\), and point \(F\) is a moving point on side \(BC\). Triangles \(\triangle ADE\) and \(\triangle DCF\) are folded up along \(DE\) and \(DF\) respectively, making points \(A\) and \(C\) coincide at point \(A'\). Find the maximum distance from point \(A'\) to plane \(DEF\).
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Context:
## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{1} \frac{(4 \sqrt{1-x}-\sqrt{x+1}) d x}{(\sqrt{x+1}+4 \sqrt{1-x})(x+1)^{2}}
$$
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Context:
Calculate the value of $\sin^2 36^\circ + \tan 62^\circ \tan 45^\circ \tan 28^\circ + \sin^2 54^\circ$.
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Context:
Two identical rulers are placed together. Each ruler is exactly 10 cm long and is marked in centimeters from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other. The overall length is \( L \) cm. What is the value of \( L \)?
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Context:
In a neighborhood of 200 people who own pets, the following information is given:

- 30 people own only dogs
- 25 people own only cats
- 10 people own only birds
- 7 people own only snakes
- 12 people own only fish
- 15 people own both cats and dogs
- 12 people own both birds and dogs
- 8 people own both birds and cats
- 3 people own both snakes and dogs
- 4 people own both snakes and cats
- 2 people own both snakes and birds
- 9 people own both fish and dogs
- 6 people own both fish and cats
- 14 people own both fish and birds
- 11 people own both fish and snakes
- 5 people own cats, dogs, and snakes
- 4 people own cats, dogs, and birds
- 6 people own cats, birds, and snakes
- 9 people own dogs, birds, and snakes
- 7 people own cats, fish, and dogs
- 5 people own birds, fish, and dogs
- 3 people own birds, fish, and cats
- 8 people own snakes, fish, and dogs
- 4 people own snakes, fish, and cats
- 6 people own snakes, fish, and birds
- 10 people own all five pets

How many total snakes are owned by the people in this neighborhood?
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Context:
About the pentagon $ABCDE$, we know that $AB = BC = CD = DE$, $\angle C = \angle D = 108^\circ$, $\angle B = 96^\circ$. Find the value in degrees of $\angle E$.
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Context:
For real numbers $x$, let
\[P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)\]
where $i = \sqrt{-1}$. For how many values of $x$ with $0\leq x<2\pi$ does
\[P(x)=0?\]
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Context:
Determine the number of unordered triples of distinct points in the $4 \times 4 \times 4$ lattice grid $\{0,1,2,3\}^{3}$ that are collinear in $\mathbb{R}^{3}$ (i.e. there exists a line passing through the three points).
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Context:
Leap Day, February $29$, $2004$, occurred on a Sunday. On what day of the week will Leap Day, February $29$, $2020$, occur?
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Context:
Any positive integer \( n \) can be written in the form \( n=2^{a} q \), where \( a \geq 0 \) and \( q \) is odd. We call \( q \) the odd part of \( n \). Define the sequence \( a_0, a_1, \ldots \), as follows: \( a_0 = 2^{2011} - 1 \) and for \( m \geq 0 \), \( a_{m+1} \) is the odd part of \( 3a_m + 1 \). Find \( a_{2011} \).
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Context:
In the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$, points $P, M,$ and $N$ are the midpoints of edges $AA_{1}$, $BC,$ and $CC_{1}$ respectively. What is the specific geometric shape formed by the intersection of the plane determined by these three points with the surface of the cube? Express your answer as a specific geometric shape.
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Context:
Given the function \( f(x) = x^2 + ax + b \), where \( a \) and \( b \) are real numbers, let \( M(a, b) \) be the maximum value of \( |f(x)| \) on the interval \([-1,1]\). When \( M(a, b) \leq 2 \), find the maximum value of \( |a| + |b| \).
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Context:
Let $f(x)$ be an odd function defined on $\mathbb{R}$ that satisfies $f(x+2)=-f(x)$. For $0 \leq x \leq 1$, $f(x)=\frac{1}{2}x$. Let $g(x)=f(x)+\frac{1}{2}$. Find the sum of all zeros of the function $g(x)$ in the interval $[-10,10]$.
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Context:
What is the residue of the function $\frac{1}{z-\sin z}$ at $z=0$?
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Context:
On Halloween $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.

"Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes.

"Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes.

"Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes.

How many pieces of candy in all did the principal give to the children who always tell the truth?
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Context:
Given that $\alpha \in \left( 0, \pi \right)$ and $3\cos 2\alpha = \sin \left( \frac{\pi}{4} - \alpha \right)$, find the value of $\sin 2\alpha$.
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Context:
Find the number of ordered pairs of integers $(x, y)$ such that $2167$ divides $3x^2 + 27y^2 + 2021$ with $0 \le x, y \le 2166$.
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Context:
Write all the natural numbers from 1 to 1988 on the blackboard. Perform operations \( A \) and \( B \) on these numbers repeatedly in the following sequence: first \( A \), then \( B \); repeat with \( A \), then \( B \), and so on. Operation \( A \) subtracts the same natural number from each number on the blackboard (different instances of operation \( A \) can subtract different numbers). Operation \( B \) erases two numbers from the blackboard and replaces them with their sum. Continue this process until after one application of operation \( B \), only one non-negative number remains on the blackboard. What is this number?
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Context:
Given a family of sets \(\{A_{1}, A_{2}, \ldots, A_{n}\}\) that satisfies the following conditions:
(1) Each set \(A_{i}\) contains exactly 30 elements;
(2) For any \(1 \leq i < j \leq n\), the intersection \(A_{i} \cap A_{j}\) contains exactly 1 element;
(3) The intersection \(A_{1} \cap A_{2} \cap \ldots \cap A_{n} = \varnothing\).

Find the maximum number \(n\) of such sets.
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Context:
A broken line consists of  $31$  segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.
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Context:
In the diagram, the horizontal distance between adjacent dots in the same row is 1. Also, the vertical distance between adjacent dots in the same column is 1. Find the perimeter of quadrilateral PQRS. Express your answer as a whole number.
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Context:
Let $f(x)=x^{A_0}(x+1)^{A_1}(x+2)^{A_2}\cdots(x+n)^{A_n}$. If

$$\frac{f'(x)}{f(x)}=\frac{n!}{x(x+1)(x+2)(x+3)\cdots(x+n)},$$

find the value of $A_8$.
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Context:
In a rectangular array of points, with $5$ rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered $1$ through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are selected so that each $P_i$ is in row $i.$ Let $x_i$ be the number associated with $P_i.$ Now renumber the array consecutively from top to bottom, beginning with the first column. Let $y_i$ be the number associated with $P_i$ after the renumbering. It is found that $x_1 = y_2,$ $x_2 = y_1,$ $x_3 = y_4,$ $x_4 = y_5,$ and $x_5 = y_3.$ Find the smallest possible value of $N.$
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Context:
10.5. On a plane, there are eight different points. A numbering of these points with numbers from 1 to 8 is called good if the following condition is met:

there exists a line such that all points lie on one side of it and at different distances from it, and the distances from the points to this line increase with the increase in the number. That is, the closest point is number 1, the next in distance is number 2, and so on.

What is the maximum number of different good numberings that a given set of eight points can have?
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Context:
James won a money prize at a charity raffle. He donated 25% of his winnings back to the charity. Then he spent 10% of his remaining winnings on a celebratory lunch and 15% of what was left after that on a gift for his friend. James then decided to invest 30% of his remaining money in a new business, and after a month, the investment yielded a 50% return. After all these transactions, he had $320 left. How many dollars did he initially win at the charity raffle?
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Context:
Xiaoping and Xiaoliang live in the same building, with Xiaoping residing on the fifth floor and Xiaoliang on the fourth floor. Xiaoping has to climb 80 steps to get home every day. Calculate the number of steps Xiaoliang has to climb to get home. Express your answer as a whole number.
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Context:
Tom and Jerry are running after each other on a figure-eight track. They run in the same direction at constant speeds. Initially, Jerry was exactly above Tom. After 20 minutes, Tom was exactly above Jerry, and neither of them had completed a full lap. 10 minutes later, Tom returned to the starting point. After what time from the start of running will Tom catch up with Jerry?
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Context:
对正整数 n 及实数 x(0\leq x<n)，定义f(n,x)=(1-\{x\})\cdot\text{C}_n^{[x]}+\{x\}\cdot\text{C}_n^{[x]+1},其中 [x] 表示不超过实数 x 的最大整数，\{x\}=x-[x]. 若整数 m,n\geq2 满足f\left(m,\df{1}{n}\right)+f\left(m,\df{2}{n}\right)+\cdots+f\left(m,\df{mn-1}{n}\right)=123,求 f\left(n,\df{1}{m}\right)+f\left(n,\df{2}{m}\right)+\cdots+f\left(n,\df{mn-1}{m}\right) 的值.
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Context:
In a park, pathways are laid as shown in the figure. Two workers started paving these paths simultaneously from point \(A\). They lay asphalt at constant speeds: the first worker on segment \(A-B-C\), the second worker on segment \(A-D-E-F-C\). They finished the job simultaneously, spending 9 hours on it. It is known that the second worker works 1.2 times faster than the first worker. How many minutes did the second worker spend laying asphalt on segment \(D-E\)?
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Context:
Let $n$ be the smallest positive integer such that $mn$ is a perfect $k$th power of an integer for some $k \ge 2$, where $m=2^{1980} \cdot 3^{384} \cdot 5^{1694} \cdot 7^{343}$. What is $n+k$?
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Context:
Find the volume of the solid that lies above the cone $z^2=x^2+y^2, z\geq0$, and below the sphere $x^2+y^2+z^2=4z$.
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Context:
Define \(\phi^{!}(n)\) as the product of all positive integers less than or equal to \(n\) and relatively prime to \(n\). Compute the number of integers \(2 \leq n \leq 50\) such that \(n\) divides \(\phi^{!}(n) + 1\).
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Context:
8. In the sequence $\left\{a_{n}\right\}_{n=1}^{+\infty}$, $x_{1}=1$, if for any positive integer $n$ there is $x_{n+1}=x_{n}+\left[\sqrt{x_{n}}\right]$, where the symbol $[x]$ denotes the greatest integer not exceeding $x$, then the smallest positive integer $n$ such that $x_{n} \geqslant 2019$ is $\qquad$.
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Context:
Given the parabola $y^2 = 2px$ with the axis of symmetry equation $x = -1$, find the value of $p$.
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Context:
设 $M=\{1,2,3, \cdots, 1995\}, A$ 是 $M$ 的子集且满足条件：当 $x \in A$ 时， $15 x \notin A$ ，则 $A$ 中元素的个
数最多是 $\qquad$.
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Context:
Let  $ABC$  be a triangle with  $AB=13$ ,  $BC=14$ , and  $AC=15$ .  Denote by  $\omega$  its incircle.  A line  $\ell$  tangent to  $\omega$  intersects  $\overline{AB}$  and  $\overline{AC}$  at  $X$  and  $Y$  respectively.  Suppose  $XY=5$ .  Compute the positive difference between the lengths of  $\overline{AX}$  and  $\overline{AY}$ .
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Context:
George has 10 different colors available to paint his room and he must choose exactly 2 of them. However, George prefers to include the color blue as one of his choices if possible. In how many ways can he choose the colors?
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Context:
Calculate the product of the base nine numbers $35_9$ and $47_9$, express it in base nine, and find the base nine sum of the digits of this product. Additionally, subtract $2_9$ from the sum of the digits.
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Context:
Let $z_1$, $z_2$, $z_3$, $\dots$, $z_{16}$ be the 16 zeros of the polynomial $z^{16} - 16^{4}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$.  Find the maximum possible value of the real part of
\[\sum_{j = 1}^{16} w_j.\]
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Context:
In 1860, someone deposited 100,000 florins at 5% interest with the goal of building and maintaining an orphanage for 100 orphans from the accumulated amount. When can the orphanage be built and opened if the construction and furnishing costs are 100,000 florins, the yearly personnel cost is 3,960 florins, and the maintenance cost for one orphan is 200 florins per year?
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Context:
A natural number, which does not end in zero, had one of its digits replaced with zero (if it was the leading digit, it was simply erased). As a result, the number became 9 times smaller. How many such numbers exist for which this is possible?
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Context:
Let  $ABC$  be a triangle and  $\Gamma$  the  $A$ - exscribed circle whose center is  $J$  . Let  $D$  and  $E$  be the touchpoints of  $\Gamma$   with the lines  $AB$  and  $AC$ , respectively. Let  $S$  be the area of the quadrilateral  $ADJE$ , Find the maximum value that  $\frac{S}{AJ^2}$  has and when equality holds.
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Context:
Darryl has a six-sided die with faces  $1, 2, 3, 4, 5, 6$ . He knows the die is weighted so that one face
comes up with probability  $1/2$  and the other five faces have equal probability of coming up. He
unfortunately does not know which side is weighted, but he knows each face is equally likely
to be the weighted one. He rolls the die  $5$  times and gets a  $1, 2, 3, 4$  and  $5$  in some unspecified
order. Compute the probability that his next roll is a  $6$ .
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Context:
Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$.
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Context:
Compute the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_{10} \le 1007$ such that $a_i - i$ is odd for $1 \le i \le 10$. Express your answer as ${m \choose n}$ for some $m > n$ and find the remainder when $m$ is divided by 1000.
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Context:
Given a sequence $\{a_n\}$ that satisfies: $a_1=-13$, $a_6+a_8=-2$, and $a_{n-1}=2a_n-a_{n+1}$ for $n\geqslant 2$, find the sum of the first 13 terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$.
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Context:
The ancient Chinese mathematical masterpiece "Zhou Bi Suan Jing" records the phrase "multiply the legs and the base of a right triangle by themselves, add them together, and then take the square root of the sum," which is symbolically represented as $a^{2}+b^{2}=c^{2} (a,b,c \in \mathbb{N}^{*})$. We call $a$, $b$, and $c$ Pythagorean triples. Given several sets of Pythagorean triples: $3, 4, 5$; $5, 12, 13$; $7, 24, 25$; $9, 40, 41$, and so on, we can guess that the three numbers of the fifth set of Pythagorean triples are ______.
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Context:
在凸四边形 $A B C D$ 中, $\overrightarrow{B C}=2 \overrightarrow{A D}$. 点 $P$ 是四边形所在平面上一点, 满足 $\overrightarrow{P A}+2020 \overrightarrow{P B}+\overrightarrow{P C}+2020 \overrightarrow{P D}=\mathbf{0}$. 设 $s, t$ 分别为四边形 $A B C D$ 与 $\triangle P A B$的面积，则 $\frac{t}{s}=$ $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
The angle bisector of one of the acute angles of a right triangle, which is also the altitude dropped to the hypotenuse, is divided into segments whose ratio is $1+\sqrt{2}$, starting from the vertex. Find the acute angles of the triangle.
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Context:
Two circles of radius 1 are centered at $(4,0)$ and $(-4,0).$ How many circles are tangent to both of the given circles and also pass through the point $(0,5)$?
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Context:
In chess, the winner receives 1 point and the loser 0 points. In the event of a draw (remis), each player receives $1 / 2$ point.
Fourteen chess players, no two of whom were the same age, participated in a tournament where each played against every other. After the tournament, a ranking list was created. In the case of two players with the same number of points, the younger one received a better ranking.
After the tournament, Jan noticed that the top three players had a total number of points exactly equal to the total number of points of the last nine players. Jörg added that the number of games that ended in a draw was maximized. Determine the number of drawn games.
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Context:
Find the smallest natural number $n$ such that in the two-colored complete graph $K_{n}$, there always exist two monochromatic triangles that share exactly one common vertex.
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Context:
A broken line consists of $31$ segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.
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Context:
Two irreducible fractions have their denominators equal to 600 and 700. Find the minimum value for the denominator of the sum of the fractions.
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Context:
A circle with a radius of 1 is randomly placed inside a $15 \times 36$ rectangle $ABCD$. Find the probability that this circle does not intersect with the diagonal $AC$.
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Context:
A pineapple costs a certain amount and a watermelon costs Rs 5 each. If I spend Rs 38 in total and I purchased 2 pineapples, how much does each pineapple cost?
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Context:
Let $(S_n)_{n\ge0}$ be a symmetric, simple random walk. Given $a \in \mathbb{R}$, find the following $$ \lim_{n\to \infty}\mathbb{P}\left(n^{-3/2}\sum_{i=1}^n S_i>a\right)$$
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Context:
Today, Ivan the Confessor prefers continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying $f(x)+f(y) \geq|x-y|$ for all pairs $x, y \in[0,1]$. Find the minimum of $\int_{0}^{1} f$ over all preferred functions. The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
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Context:
Let \( a, b, c, d \) be 4 distinct nonzero integers such that \( a + b + c + d = 0 \) and the number \( M = (bc - ad)(ac - bd)(ab - cd) \) lies strictly between 96100 and 98000. Determine the value of \( M \).
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Context:
Come up with a statement that is true for any number except 2 and 5.
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Context:
There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that \[\tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x}\]whenever $\tan 2023x$ is defined. What is $a_{2023}?$
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Context:
Given complex numbers \( z_{1} \) and \( z_{2} \) satisfying \( \left|z_{1}\right|=\left|z_{1}+z_{2}\right|=3 \) and \( \left|z_{1}-z_{2}\right|=3 \sqrt{3} \), find the value of \( \log_{2}\left| \left(z_{1} \overline{z_{2}}\right)^{2000} + \left(\overline{z_{1}} z_{2}\right)^{2000} \right| \).
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Context:
In triangle $ABC$, if $\sin^{2}A + \sin^{2}B = \sin^{2}C - \sqrt{2}\sin A\sin B$, find the maximum value of $\sin 2A\tan^{2}B$.
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Context:
Let $b_n$ be a sequence defined by the recurrence relation $b_{n+1} = \sum_{k=0}^n b_k$ with $b_0 = 1$. Find a closed form for $b_n$.
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Context:
## Task A-3.4.

For how many natural numbers $n$ is the value of the fraction

$$
\frac{n+2022}{n-3}
$$

an integer?
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Context:
Joe is given a permutation $p = (a_1, a_2, a_3, a_4, a_5)$ of $(1, 2, 3, 4, 5)$. A swap is an ordered pair $(i, j)$ with $1 \le i < j \le 5$, allowing Joe to swap the positions $i$ and $j$ in the permutation. For example, if Joe starts with the permutation $(1, 2, 3, 4, 5)$ and uses the swaps $(1, 2)$ and $(1, 3)$, the permutation becomes:
\[(1, 2, 3, 4, 5) \rightarrow (2, 1, 3, 4, 5) \rightarrow (3, 1, 2, 4, 5). \]
Out of all $\binom{5}{2} = 10$ swaps, Joe chooses $4$ of them to be in a set of swaps $S$. Joe notices that from $p$ he could reach any permutation of $(1, 2, 3, 4, 5)$ using only the swaps in $S$. How many different sets are possible?
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Context:
Given that \( a = (-1989)^{-1990} \), \( b = -1989^{1990} \), and \( c = -1989^{-1990} \), determine the relationship between \( a, b, \) and \( c \).
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Context:
Given $\sin 12^\circ = a$, then $\sin 66^\circ =$ ?
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Context:
In the 2011 Shanghai Spring College Entrance Examination, there were 8 colleges recruiting students. If exactly 3 students were admitted by 2 of these colleges, then the number of ways this could happen is __________.
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Context:
The square root of a certain number is $a^2 + b^2$ and $4a - 6b + 13$, then this number is.
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Context:
Given that the graph of function $g(x)$ is symmetric with respect to the line $y=x$ to the graph of function $f(x)=x^{2}$ for $x\in [0, +\infty)$, the graph of function $g(x)$ is shifted $2$ units to the right and $2$ units down to obtain the graph of function $h(x)$. If $P$ and $Q$ are two moving points on the graphs of functions $f(x)$ and $h(x)$ respectively, then the minimum distance between these two points is ______.
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Context:
\( ABCD \) is a rectangle with side lengths \( AB = CD = 1 \) and \( BC = DA = 2 \). Let \( M \) be the midpoint of \( AD \). Point \( P \) lies on the opposite side of line \( MB \) to \( A \), such that triangle \( MBP \) is equilateral. Find the value of \( \angle PCB \).
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Context:
For all positive integers $n$, let
\[ f(n) = \sum_{k=1}^n \varphi(k) \left\lfloor \frac{n}{k} \right\rfloor^2. \]
Compute $f(2019) - f(2018)$. Here, $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.
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Context:
In the following sequence of numbers, each term, starting with the third, is obtained by adding three times the previous term plus twice the term before that to the previous one:

$$a_1, a_2, 78, a_4, a_5, 3438, a_7, a_8, \ldots$$

As seen in the sequence, the third term is $78$ and the sixth term is $3438$. What is the value of the term $a_7$?
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Context:
A worker's daily wage is increased by 60%, and after the increase, they also receive a productivity bonus of $10 per day. The new total wage, including the bonus, is $45 per day. If the worker worked 8 hours per day, what was the worker's hourly wage before the increase and bonus?
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Context:
Evaluate the constant $C$ in terms of $p$ without using summation notation, given the following formula: $$C=\sum_{n_{L}=0}^{\infty}\left[\sum_{n_{G}=n_{L}}^{\infty}\left(\left(-1\right)^{n_{G}+n_{L}+1}\cdot\frac{p^{n_{G}}}{n_{L}!}\right)\right]$$
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Context:
In a business meeting, each participant shakes hands with every other participant exactly once, except for Mr. L. Mr. L arrives late and only shakes hands with those still present. With a total of 100 handshakes recorded, how many people had already left the meeting before Mr. L's arrival?

(The question is from the CMI2011 Undergraduate Entrance Examination.)
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Context:
Determine the largest integer $n$ such that $7^{2048}-1$ is divisible by $2^{n}$.
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Context:
Points \( A, B, C, D \) lie on the circumference of a circle, and \( BC = DC = 4 \), \( AE = 6 \). The lengths of segments \( BE \) and \( DE \) are both integers. Find the length of \( BD \).
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Context:
Kaylee studies in a school where the bell rings once a class starts and again when a class gets over. There is a break of 15 minutes after every class. On most days, Kaylee's classes in order are: Maths, History, Geography, Science, and Music.

However, on Monday, the schedule was altered due to a special school assembly in the morning that lasted 45 minutes, which delayed the start of the classes and consequently the ringing of the bell. The assembly added another instance of the bell ring at the start and end of it. Furthermore, after the History class, there was a surprise pop quiz which also had a start and an end bell ring. When the surprise quiz was happening, the regular class bells did not ring.

If right now she is in her Geography class, how many times would the bell have rung by now?
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Context:
A man buys 350 pens at the marked price of 270 pens from a wholesaler. If he sells these pens giving a discount of 4.3%, what is the profit percent?
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Context:
Rachel had to complete some pages of math homework, 7 pages of reading homework and 3 more pages of biology homework. She had to complete 11 pages of math and biology homework. How many pages of math homework did she have to complete?
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Context:
For how many positive integers $n$ does $1+2+...+n$ evenly divide from $6n$?
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Context:
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, find the value of $y$. Express your answer in degrees, rounded to one decimal place.
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Context:
On a lengthy, one-way, single-lane highway, cars travel at uniform speeds and maintain a safety distance determined by their speed: the separation distance from the back of one car to the front of another is one car length for each 10 kilometers per hour of speed or fraction thereof. Cars are exceptionally long, each 5 meters in this case. Assume vehicles can travel at any integer speed, and calculate $N$, the maximum total number of cars that can pass a sensor in one hour. Determine the result of $N$ divided by 100 when rounded down to the nearest integer.
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Context:
In the logging yard, there were logs 6 meters and 8 meters long of the same thickness. Which logs are more advantageous to take in order to obtain more pieces with fewer cuts to saw them into 1-meter pieces?
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\boxed{It is more advantageous to use 6-meter logs.}


Context:
Vincent has 72 inches of rope that he wants to use for a project, but he needs to cut it into smaller pieces first. He cuts it into 12 equal length pieces, but then he realizes it's too short, so he ties three pieces together. The knots then make each piece 1 inch shorter. How long are his pieces of rope after all this?
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Context:
There are  $15$  (not necessarily distinct) integers chosen uniformly at random from the range from  $0$  to  $999$ , inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is  $\frac mn$  for relatively prime positive integers  $m,n$ . Find  $100m+n$ *Proposed by Yannick Yao*
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Context:
展览馆中有 2015 幅画在展出. 每位观众选择一对画，并将它们的名字写在同一张纸上. 一个艺术家从这张纸中选择一些画对，并评判每对中哪幅画更好. 之后助手按下述规则继续评判：在每一步中，如果对某个正整数 n，在画对 (A_1,A_2) 中 A_1 比 A_2 好，在画对 (A_2,A_3) 中 A_2 比 A_3 好，\cdots\cdots，在画对 (A_{n-1},A_n) 中 A_{n-1} 比 A_n 好，则助手可以在画对 (A_1,A_n) 中评判 A_1 比 A_n 好. 求最小的正整数 k，使得无论纸上写了哪些画对，艺术家都可以对其中的 k 对评判，使得助手能在有限步内对余下所有的画对评判.
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Context:
设函数$f _ { 0 } ( x ) = \vert x \vert , f _ { 1 } ( x ) = \vert f _ { 0 } ( x ) - \vert \vert , f _ { 2 } ( x ) = \vert f _ { 1 } ( x ) - 2 \vert$ 则函数$y = f _ { 2 } ( x )$的图像与x轴所围成图形中的封闭部分的面积是.
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Context:
对正整数 $n$ 及实数 $x(0 \leqslant x<n)$, 定义
$$
f(n, x)=(1-\{x\}) \cdot C_{n}^{[x]}+\{x\} \cdot C_{n}^{[x]+1}
$$
其中 $[x]$ 表示不超过实数 $x$ 的最大整数， $\{x\}=x-[x]$ 。若整数 $m, n \geqslant 2$ 满足
$$
f\left(m, \frac{1}{n}\right)+f\left(m, \frac{2}{n}\right)+\cdots+f\left(m, \frac{m n-1}{n}\right)=123,
$$
求 $f\left(n, \frac{1}{m}\right)+f\left(n, \frac{2}{m}\right)+\cdots+f\left(n, \frac{m n-1}{m}\right)$ 的值。
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Context:
Six circles with radii equal to 1 are placed on a plane such that the distance between the centers of any two of them is greater than \( d \). What is the smallest \( d \) for which there exists a line that does not intersect any of the circles, and on each side of which lie three circles?
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Context:
In a ticket sale event, there is a tiered discount system. For the first 5 tickets purchased, 1 ticket is given for free. For every additional ticket purchased after the first 5, 2 extra tickets are given for free. If a customer purchases 10 tickets under this sale, what percentage of the original price of all the tickets received is saved?
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Context:
What is the limit of the following expression as \( n \) approaches infinity? $$\lim_{n\rightarrow\infty} \frac{(n+1)^{2n^2+2n+1}}{(n+2)^{n^2+2n+1}n^{n^2}}$$
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Context:
The product of three prime numbers is 5 times greater than their sum. Find all such triples of numbers.
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Context:
If $cos\frac{π}{5}$ is a real root of the equation $ax^{3}-bx-1=0$ (where $a$ and $b$ are positive integers), then $a+b=\_\_\_\_\_\_$.
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Context:
The set \( S \) contains 6 distinct positive integers. The average of the two smallest numbers is 5 and the average of the two largest is 22. What is the greatest possible average of all the numbers in the set \( S \)?
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Context:
Peter Pan and Crocodile are each getting hired for a job. Peter wants to get paid 6.4 dollars daily, but Crocodile demands to be paid 10 cents on day 1, 20 cents on day 2, 40 cents on day 3, 80 cents on day 4, and so on. After how many whole days will Crocodile's total earnings exceed that of Peter's?
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Context:
在锐角三角形 $A B C$ 中, $A B$ 上的高 $C E$ 与 $A C$ 上的高 $B D$ 相交于点 $H$, 以 $D E$ 为直径的圆分别交 $A B 、 A C$ 于 $F 、 G$ 两点, $F G$ 与 $A H$ 相交于点 $K$, 已知 $B C=25, B D=20, B E=7$, 求 $A K$ 的长。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
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Context:
On a circle, points \(B\) and \(D\) are located on opposite sides of the diameter \(AC\). It is known that \(AB = \sqrt{6}\), \(CD = 1\), and the area of triangle \(ABC\) is three times the area of triangle \(BCD\). Find the radius of the circle.
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Context:
Evaluate the sum $\sum_{n=1}^{\infty}\frac{(n-1)!}{\prod_{r=1}^{n}(x+r)}$ for a positive real number $x$, without using the Gamma function.
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Context:
Find the fourth derivative of the inverse function $f^{-1}(y)$ evaluated at $y=1$, where $f(x) = \sin{x} + x^2 + 1$.
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Context:
Evaluate the integral: $$\int_0^\infty \frac{1}{x^{2n+5}} \left ( \cos x - \sum_{k=0}^n \frac{(-1)^k x^{2k+2}}{(2k + 2)!} \right ) \mathrm{d}x$$
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Context:
In a group of people participating in some curriculum, 25 of them practice yoga, 15 study cooking, 8 study weaving, 2 of them study cooking only, some of them study both the cooking and yoga, 3 of them participate all curriculums. If 3 people study both cooking and weaving, how many people study both cooking and yoga?
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Context:
Let $r=H_{1}$ be the answer to this problem. Given that $r$ is a nonzero real number, what is the value of $r^{4}+4 r^{3}+6 r^{2}+4 r ?$
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Context:
Given the binomial expansion $(3x-2)^{6}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}+a_{6}x^{6}$, find the value of $a_{0}+a_{1}+2a_{2}+3a_{3}+4a_{4}+5a_{5}+6a_{6}=\_\_\_\_\_\_$.
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Context:
**
Evaluate
\[
\frac{3}{\log_8{3000^4}} + \frac{4}{\log_9{3000^4}},
\]
giving your answer as a fraction in lowest terms.

**
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Context:
Aaron has some brothers. Bennett has 6 brothers. What is the relationship between the number of Bennett's brothers and twice the number of Aaron's brothers if Aaron has 4 brothers?
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Context:
The set \( S \) consists of 9 distinct positive integers. The average of the two smallest integers in \( S \) is 5. The average of the two largest integers in \( S \) is 22. Calculate the greatest possible average of all of the integers of \( S \). Express your answer as a single number.
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Context:
A palindrome number is a positive integer that reads the same forward and backward. For example, $1221$ and $8$ are palindrome numbers, whereas $69$ and $157$ are not.

Let $A$ and $B$ be $4$-digit palindrome numbers, and $C$ be a $3$-digit palindrome number. Given that $A - B = C$, what is the value of $C$?
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Context:
14. $A, B, C$ three people went to Teacher $D$'s house to play. Teacher $D$ gave each of them a hat and wrote a four-digit number on each hat. It is known that these three four-digit numbers are all perfect squares (for example, $4=2^{2}, 100=10^{2}, 4.100$ are all squares of some number, such numbers are called perfect squares), and the tens digit of these three four-digit numbers is 0, and the units digit is not 0. Each child can only see the numbers on others' hats. These three children are very smart and honest, and the following conversation took place:
$A$ said: “The units digit of the numbers on $B$ and $C$'s hats are the same.”
$B$ and $C$ said at the same time: “After hearing what $A$ said, I know my number.”
$A$ said: “After hearing what $B$ and $C$ said, I also know my number, and the units digit of my number is an even number.”
Find: The sum of the numbers on $A, B, C$'s hats.
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Context:
In a rectangle  $ABCD$ , two segments  $EG$  and  $FH$  divide it into four smaller rectangles.  $BH$  intersects  $EG$  at  $X$ ,  $CX$  intersects  $HF$  and  $Y$ ,  $DY$  intersects  $EG$  at  $Z$ . Given that  $AH=4$ ,  $HD=6$ ,  $AE=4$ , and  $EB=5$ , find the area of quadrilateral  $HXYZ$ .
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Context:
The number of $x$-intercepts on the graph of $y=\sin(1/x)$ in the interval $(0.0001,0.001)$ is closest to
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Context:
Let \( l \) and \( m \) be two skew lines. On line \( l \), there are three points \( A \), \( B \), and \( C \) such that \( A B = B C \). From points \( A \), \( B \), and \( C \), perpendiculars \( A D \), \( B E \), and \( C F \) are drawn to line \( m \) with feet at \( D \), \( E \), and \( F \) respectively. Given \( A D = \sqrt{15} \), \( B E = \frac{7}{2} \), and \( C F = \sqrt{10} \), find the distance between \( l \) and \( m \).
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Context:
A batch of disaster relief supplies is loaded into 26 trucks. The trucks travel at a constant speed of \( v \) kilometers per hour directly to the disaster area. If the distance between the two locations is 400 kilometers and the distance between every two trucks must be at least \( \left(\frac{v}{20}\right)^{2} \) kilometers, how many hours will it take to transport all the supplies to the disaster area?
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Context:
In the country Máxico, there are two islands: the island "Mayor" and the island "Menor". The island "Mayor" has $k > 3$ states, with exactly $n > 3$ cities in each state. The island "Menor" has only one state with $31$ cities. "Aeropapantla" and "Aerocenzontle" are the airlines that offer flights in Máxico. "Aeropapantla" offers direct flights from every city in Máxico to any other city in Máxico. "Aerocenzontle" only offers direct flights from every city on the island "Mayor" to any other city on the island "Mayor".

If the percentage of flights that connect two cities in the same state is the same for the flights of each airline, what is the least number of cities that can be on the island "Mayor"?
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Context:
The function f(n) defined on the set of natural numbers N is given by:

f(n) = {
n - 3  (n >= 1000)
f[f(n + 7)]  (n < 1000)
}

Calculate the value of f(90). Express your answer as a single integer.
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Context:
3. What is the smallest radius of a sphere from which a regular quadrilateral pyramid with a base edge of 14 and an apothem of 12 can be cut?
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Context:
In the diagram, rectangle \( PRTV \) is divided into four rectangles. The area of rectangle \( PQXW \) is 9. The area of rectangle \( QRSX \) is 10. The area of rectangle \( XSTU \) is 15. Calculate the area of rectangle \( WXUV \). Express your answer as a fraction or whole number.
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Context:
The 600 students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. What is the probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group? Express your answer as a simplified fraction.
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Context:
In $\triangle ABC$, $AB = 2019$, $BC = 2020$, and $CA = 2021$. Yannick draws three regular $n$-gons in the plane of $\triangle ABC$ so that each $n$-gon shares a side with a distinct side of $\triangle ABC$ and no two of the $n$-gons overlap. What is the maximum possible value of $n$?
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Context:
Let $a=\sqrt{3}$, $b=\log_{3}2$, $c=\cos{2}$. Arrange $a$, $b$, and $c$ in increasing order.
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